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+{"text":"\\section{Introduction}\n\\IEEEPARstart{T}{he} necessity to recover digital images from their distorted and noisy observations arises in a multitude of practical applications, with some specific examples including image denoising, super-resolution, image restoration, and watermarking, just to name a few \\cite{30, 28, 29, 40}. In such cases, it is standard to assume that the observed image $v$ is formed by subjecting the original image $u$ to convolution with a point spread function\\footnote{Note that, in optical imaging, this function is also referred to as an impulse transfer function \\cite{39}.} (PSF) $i$, followed by contamination by white Gaussian noise (WGN) $\\nu$. Thus, formally,\n\\begin{equation}\\label{1}\n\\begin{split}\nv=i \\ast u + \\nu.\n\\end{split}\n\\end{equation}\nWhile $u$ and $v$ can be regarded as general members of the signal space $\\mathbb{L}_2(\\Omega)$ of real-valued functions on $\\Omega \\subseteq \\mathbb{R}^2$, the PSF $i$ is normally a much smoother function, with effectively band-limited spectrum. As a result, the convolution with $i$ has a destructive effect on the informational content of $u$, in which case $v$ typically has a substantially reduced set of features with respect to $u$. This makes the problem of reconstruction of $u$ from $v$ a problem of significant practical importance \\cite{41}.\n\nReconstruction of the original image $u$ from $v$ can be carried out within the framework of image deconvolution, which is a specific instance of a more general class of inverse problems \\cite{42}. Most of such methods are Bayesian in nature, in which case the information lost in the process of convolution with $i$ is recovered by requiring the optimal solution to reside within a predefined functional class \\cite{26, 27}. Thus, for example, in the case when $u$ is known to be an image of bounded variation, the above regularization leads to the famous Rudin-Osher-Fatemi reconstruction scheme, in which $u$ is estimated as a solution to the following problem \\cite{0, 11}\n\\begin{equation} \\label{2}\n\\hat{u} = \\underset{u}{\\arg\\min} \\left\\{ \\frac{1}{2} \\| u \\ast i - v \\|_2^2 + \\alpha \\int |\\nabla u| \\, dx dy \\right\\},\n\\end{equation}\nwhere $\\alpha>0$ is the regularization parameter. It should be noted that, if the PSF obeys $\\int i \\, dx dy \\neq 0$, the problem (\\ref{2}) is strictly convex and therefore admits a unique minimizer, which can be computed by a spectrum of available algorithms \\cite{0, 11}.\n\nA particularly non-trivial version of deconvolution is commonly referred to as blind. In this case, the original image $u$ is to be estimated without the knowledge of the PSF \\cite{42}. In this paper, however, we follow the philosophy of {\\em hybrid deconvolution} \\cite{Oleg07}, which takes advantage of any partial information on the PSF to improve the image reconstruction. Thus, in the algorithm described in this paper, the original image $u$ will be recovered from $v$ and some partial information on $i$.\n\nOptical (and, in particular, turbulent) imaging is unarguably the field of applied sciences from which the notion of deconvolution has originally emanated \\cite{Richardson72, Lucy74, 44}. In short-exposure imaging, however, computational methods of image restoration are still superseded by adaptive optics. As recently as a decade ago, the use of adaptive optics would have been considered as the only practical option. Nowadays, however, with the advent of distributed cluster computing and GPU-based image processing, it seems to be time to revisit the cost-to-performance characteristics of the existing tools of adaptive optics. Thus, in this work, our focus is on a specific tool of adaptive optics, known as the Shack-Hartmann interferometer \\cite{12, 18}. Instead of completely excluding the interferometer from our measurement system, we propose to  modify its construction through reducing the number of its local wavefront lenses. Although the advantages of such a simplification are immediate to see, its main shortcoming is obvious as well: the smaller the number of lenses is, the stronger is the effect of undersampling and aliasing. Accordingly, to overcome this problem, we propose to augment the modified Shack-Hartmann interferometer by subjecting its output to the derivative compressed sensing (DCS) algorithm of \\cite{14}. As it will be shown later in the paper, the PSF $i$ is determined by a generalized pupil function $P$, which can be expressed in a polar form as $P = A \\, e^{\\jmath \\phi}$. While the amplitude $A$ can be measured via calibration or computed as a function of the aperture geometry, the phase $\\phi$ is often influenced by environmental effects and hence it needs to be recovered from observations. It will be shown below that DCS is particularly well suited for reconstruction of $\\phi$ from incomplete measurements of its partial differences. Such an estimate can be subsequently combined with $A$ to yield an estimate of the PSF $i$, which can in turn be used by a deconvolution algorithm. Thus, the proposed method for estimation of the PSF and subsequent deconvolution of $u$ can be regarded as a hybrid deconvolution technique, which comes to simplify the design and complexity of adaptive optics on the one hand, and to make the process of reconstruction of optical images as automatic as possible, on the other hand.\n\nThe rest of the paper is organized as follows. Section II summarizes basic technical preliminaries. In Section III, we describe the SH interferometer as well as phase measurements in optical imaging. In Section IV, we explain DCS and our new approach to solve it. In Section V we describe deconvolution process to recover the original image. Experimental results are presented in Section VI, while Section VII finalizes the paper with a discussion and main conclusions.\n\n\\section{Technical Preliminaries}\nIn short exposure imaging, due to aberrations in the imaging system induced by, e.g., atmospheric turbulence, the impulse response of an optical imaging system is often unknown \\cite{15}. In order to better understand the setup under consideration, we first note that, in optical imaging, the PSF $i$ is obtained from an amplitude spread function (ASF) $h$ as $i := | h |^2$. The ASF, in turn, is defined in terms of the generalized pupil function (GPF) $P(x,y)$ as given by \\cite{17}\n\\begin{equation} \\label{3}\nh(\\xi,\\eta)=\\frac{1}{\\lambda_w z_i}\\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} P(x,y) e^{-j\\frac{2\\pi}{\\lambda z_i}( x \\, \\xi + y \\, \\eta)} \\, dxdy,\n\\end{equation}\nwhere $z_i$ is the focal distance and $\\lambda_w$ is the optical wavelength. Being a complex-valued quantity, $P(x,y)$ can be represented in terms of its amplitude $A(x,y)$ and phase $\\phi(x,y)$ as\n\\begin{equation} \\label{4}\nP(x,y) = A(x,y) \\, e^{ \\jmath \\phi(x,y)}.\n\\end{equation}\nHere, the GPF amplitude $A(x,y)$ (which is sometimes simply referred to as the aperture function) is normally a function of the aperture geometry. Thus, for instance, in the case of a circular aperture, $A(x,y)$ can be defined as \\cite{15}\n\\begin{equation} \\label{5}\nA(r)=\n \\begin{cases}\n 1, \\quad  &\\mbox{if } r \\le \\frac{D}{2} \\\\\n 0, \\quad  &\\text{otherwise}\n \\end{cases}\n\\end{equation}\nwhere $D$ denotes the pupil diameter. Thus, given $\\phi(x,y)$, one could determine $h$ and therefore $i$. Unfortunately, the phase $\\phi(x,y)$ does not have an analytic expression, and it has to be measured in practice using such tools as the Shack-Hartmann interferometer (SHI) \\cite{12}.\n\nAs will be discussed later in the paper, the SHI is capable of sensing the partial derivatives of $\\phi(x,y)$. Needless to say, in order to minimize the effect of aliasing on the estimation result, an accurate reconstruction of $\\phi(x,y)$ requires taking a fairly large number of the samples of $\\nabla \\phi (x,y)$ \\cite{Oleg2008}. In some applications, the number of sampling points (as defined by the number of local wavefront lenses) reaches as many as a few thousands. It goes without saying that reducing the number of lenses would have a positive impact on the SHI in terms of its cost and approachability. Alas, such a reduction is impossible without undersampling, which tends to have formidable effect on the overall quality of phase estimation.\n\nIn this paper, to minimize the effect of undersampling, we exploit DCS \\cite{14}. As opposed to the classical compressed sensing (CCS) \\cite{13}, in addition to the sparsifing constraints, DCS also uses constraints which are intrinsic in the definition of partial derivatives. Using these additional constraints -- which are called the cross-derivative constraints -- allows substantially improving the quality of reconstruction of $\\phi(x,y)$, as compared to the case of CCS-based estimation.\n\n\\section{Shack-Hartmann Interferometer (SHI)}\nAs it was mentioned earlier, the SHI is typically used to measure the gradient $\\nabla \\phi(x,y)$ of the GPF phase $\\phi(x,y)$, from which the values of the latter can be subsequently estimated. To this end, the unknown phase $\\phi(x,y)$ is assumed to be expandable in terms of some basis functions $\\{Z_k\\}_{k=0}^\\infty$, {\\it viz.} \\cite{18}\n\\begin{equation} \\label{6}\n\\phi(x,y)=\\sum_{k=0}^{\\infty} a_k Z_k(x,y),\n\\end{equation}\nwhere the representation coefficients $\\{a_k\\}_{k=0}^\\infty$ are assumed to be unique and stably computable. Note that, in this case, the datum of $\\{a_k\\}_{k=0}^\\infty$ uniquely identifies $\\phi(x,y)$, while the coefficients $\\{a_k\\}_{k=0}^\\infty$ can be estimated due to the linearity of (\\ref{6}) which suggests\n\\begin{equation} \\label{6.1}\n\\nabla \\phi(x,y)=\\sum_{k=0}^{\\infty} a_k \\, \\nabla Z_k(x,y),\n\\end{equation}\n\nThe most frequent choice of $\\{Z_k\\}_{k=0}^\\infty$ in adaptive optics is the Zernike polynomials (aka Zernike functions) \\cite{17}. These polynomials constitute an orthonormal basis in the space of square-integrable functions defined over the unit disk in $\\mathbb{R}^2$. Zernike polynomials can be subdivided in two subsets of the even $Z_n^m$ and odd $Z_n^{-m}$ Zernike polynomials which have very convenient analytical definitions as given by\n\\begin{align}\nZ^{m}_n(\\rho,\\varphi) &= R^m_n(\\rho)\\,\\cos(m\\,\\varphi) \\\\\nZ^{-m}_n(\\rho,\\varphi) &= R^m_n(\\rho)\\,\\sin(m\\,\\varphi)\n\\end{align}\nwhere $m$ and $n$ are nonnegative integers with $n \\ge m$, $0 \\le \\varphi < 2\\pi$ is the azimuthal angle, and $0 \\le \\rho \\le 1$ is the radial distance. The radial polynomials $R^m_n$ are defined as\n\\begin{equation}\nR^m_n(\\rho) = \\! \\sum_{k=0}^{(n-m)\/2} \\!\\!\\! \\frac{(-1)^k\\,(n-k)!}{k!\\,((n+m)\/2-k)!\\,((n-m)\/2-k)!} \\;\\rho^{n-2\\,k}.\n\\end{equation}\n\nNote that, since the Zernike polynomials above are defined using polar coordinates, it makes sense to re-express the phase $\\phi$ and its gradient in the polar coordinate system as well. (Technically, this would amount to replacing $x$ and $y$ in (\\ref{6})-(\\ref{6.1}) by $\\rho$ and $\\varphi$, respectively.) Moreover, due to the property of the Zernike polynomials to be an orthonormal basis, the representation coefficients $\\{a_k\\}_{k=0}^\\infty$ in in (\\ref{6})-(\\ref{6.1}) can be computed by orthogonal projection, namely\n\\begin{equation} \\label{71}\na_k = \\int_0^{2\\pi} \\int_0^1 \\phi(\\rho,\\varphi) \\, Z_k(\\rho,\\varphi) \\, \\rho \\, d\\rho \\, d\\varphi\n\\end{equation}\nIn practice, however, $\\phi(\\rho,\\varphi)$ is unknown and therefore the coefficients $\\{a_k\\}_{k=0}^\\infty$ need to be estimated by other means. Thus, in the case of the SHI, the coefficients can be estimated from a finite set of discrete measurements of $\\nabla \\phi(\\rho,\\varphi)$.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\hspace{-5mm}\\includegraphics[height=2.1in,width=2.1in]{fig1.pdf}\n\\caption{An example of a $10 \\times 10$ SHI array on a circular aperture. The shading indicates those blocks (i.e., lenses) which are rendered idle.} \\label{F1}\n\\end{center}\n\\end{figure}\n\nThe main function of the SHI is to acquire discrete measurements of $\\nabla \\phi$ by means of linearization. The linearization takes advantage of subdividing a (circular) aperture into rectangular blocks with their sides formed by a uniform rectangular lattice. An example of such a subdivision is shown in Fig.~\\ref{F1} for the case of a $10 \\times 10$ lattice grid. Subsequently, it is assumed that the grid is sufficiently fine to approximate a restriction of the phase $\\phi$ to each of the above blocks by a linear function. This results in a piecewise linear approximation of $\\phi$, whose accuracy improves asymptotically when the lattice size goes to infinity\\footnote{More rigorously, one can show that, as long as $\\phi$ is uniformly continuous, its piecewise linear approximation converges uniformly, as the grid size goes to infinity.}. Formally, let $\\Omega := \\{(x,y) \\in \\mathbb{R}^2 \\mid x^2+y^2 \\le D^2\\}$ be a circular aperture of radius $D$ and $\\mathcal{S} = \\{(x,y) \\in \\mathbb{R}^2 \\mid \\max\\{|x|,|y|\\} \\le D \\}$ be a square subset of $\\mathbb{R}^2$ such that $\\Omega \\subset \\mathcal{S}$. Then, for each polar coordinate $(\\rho,\\varphi) \\in \\Omega$ and an $N \\times N$ grid of square blocks of size $2 D \/ N \\times 2 D \/ N$, the phase $\\phi$ can be expressed as\n\\begin{equation} \\label{8}\n\\phi(x,y) \\approx ax + by + c,\n\\end{equation}\nfor all $(x,y)$ in a neighbourhood of $(\\rho \\, \\cos \\varphi, \\rho \\, \\sin \\varphi)$. The approximation in (\\ref{8}) suggests that\n\\begin{equation} \\label{9}\n\\begin{split}\n\\nabla\\phi(x,y) \\approx [a, b]^T\n\\end{split}\n\\end{equation}\nwhere $(\\cdot)^T$ denotes matrix transposition. While $c$ in (\\ref{8}) can be derived from boundary conditions, coefficients $a$ and $b$ should be determined via direct measurements. To this end, the SHI is endowed with an array of small focusing lenses, which are supported over each of the square blocks of the discrete grid. In the absence of phase aberrations, the focal points of the lenses are spatially identified and registered using a high-resolution CCD detector, whose imaging plane is aligned with the plane of the focal points. Then, when the wavefront gets distorted as a result of, e.g., atmospheric turbulence, the focal points are ``pushed\" towards new spatial positions, which can also be pinpointed by the same detector. The resulting displacements can therefore be measured and subsequently related to the values of $\\nabla \\phi$ at corresponding points.\n\nTo explain how the above procedure can be performed, additional notations are in order. Let $\\Omega_d$ denote a finite set of spatial coordinates defined according to\n\\begin{align}\n\\Omega_d &:= \\Big\\{ (x_d,y_d) \\in \\Omega \\,\\, \\big| \\notag\\\\\nx_d &=-D+\\frac{2D}{N} \\left(i+\\frac{1}{2}\\right), \\,\\, i=0,1,\\ldots,N-1 \\\\\ny_d &=-D+\\frac{2D}{N} \\left(j+\\frac{1}{2}\\right), \\,\\, j=0,1,\\ldots,N-1 \\notag \\\\\n&\\mbox{and } x_d^2 + y_d^2 \\le D^2 \\Big\\}. \\notag\n\\end{align}\nNote that the set $\\Omega_d$ can be thought of as a set of the spatial coordinates of the geometric centres of the wavefront lenses, restricted to the domain of aperture $\\Omega$. Under some reasonable assumptions, one can then show \\cite{19} that the focal displacement $\\Delta(x,y) = [\\Delta_x(x,y), \\Delta_y(x,y)]^T$ measured at some $(x,y) \\in \\Omega_d$ is related to the value of $\\nabla \\phi(x,y)$ as given by\n\\begin{equation}\n\\nabla \\phi(x,y) \\approx \\frac{1}{f} \\Delta \\phi(x,y), \\quad \\forall (x,y) \\in \\Omega_d,\n\\end{equation}\nwhere $f$ is the focal distance of the wavefront lenses. Such a measurement setup is depicted in Fig.~\\ref{fig222} along with an example of measured focal points.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width = 7cm, height = 5cm]{SH.pdf}\n\\label{fig:tabsubfig21}\n\\caption{Basic structure of the SHI and a resulting pattern of the focal points.}\n\\label{fig222}\n\\end{figure}\n\nNow, given a total of $M : = \\# \\Omega_d$ measurements of $\\nabla \\phi$ over $\\Omega_d$, one can try to recover a useful approximation of $\\phi$ over the whole $\\Omega$ in the form of a projection of $\\phi$ on the linear subspace spanned by all Zernike polynomials up to the order $L$ inclusive. In this case, it is possible to estimate the representation coefficients $\\{a_k\\}_{k=0}^L$ via solution of\n\\begin{align} \\label{E1}\n\\min_{\\{a_k\\}} \\sum_{(x,y) \\in \\Omega_d} \\big \\| \\sum_{k=0}^L a_k \\nabla Z_k(x,y) - f^{-1} \\Delta (x,y) \\big\\|_2^2\n\\end{align}\nsubject to appropriate boundary conditions. It is worthwhile noting that (\\ref{E1}) can be rewritten in a vector-matrix form as\n\\begin{equation} \\label{E2}\n\\min_{\\bf a} \\| {\\bf Z} \\, {\\bf a}  - {\\bf d} \\|_2^2, \\quad \\mbox{s.t.} \\,\\, {\\bf a} \\succeq 0,\n\\end{equation}\nwhere $\\bf Z$ is a $2 M \\times L+1$ matrix composed of the values of the partial derivatives of the Zernike polynomials, $\\bf d$ is a measurement (column) vector of length $2 M$, and ${\\bf a} = [a_0, a_1, \\dots, a_L]^T$ is a vector of the representation coefficients of $\\phi$. The constraint ${\\bf a} \\succeq 0$ in (\\ref{E2}) is supposed to further regularize the solution by forcing $\\bf a$ to be  a member of some convex cone as well. Thus, for example, if the mean value of $\\phi$ can be assumed to be equal to zero, the solution to (\\ref{E2}) can be computed as\n\\begin{equation}\n{\\bf a} = {\\bf Z}^{\\#} {\\bf d},\n\\end{equation}\nwhere ${\\bf Z}^{\\#}$ denotes the pseudo-inverse of $\\bf Z$, whose definition is unique and stable as long as the row-rank of $\\bf Z$ is greater or equal to $L+1$ (hence suggesting that $2 M \\ge L+1$). Having estimated $\\bf a$, the phase $\\phi$ can be approximated as\n\\begin{equation}\n\\phi(\\rho, \\varphi) \\approx \\sum_{k=0}^L a_k Z_k(\\rho, \\varphi).\n\\end{equation}\n\nA higher accuracy of phase estimation requires using higher-order Zernike polynomials, which in turn necessitates a proportional increase in the number of wavefront lenses. Moreover, as required by the linearization procedure in the SHI, the lenses have to be of a relatively small sizes (sometimes, on the order of a few microns), which may lead to the use of a few thousand lenses per one interferometer. Accordingly, to simplify the construction and to reduce the cost of SHIs, we propose to substantially reduce the number of wavefront lenses, while compensating for the induced information loss through the use of DCS, which is detailed next.\n\n\\section{Derivative Compressive Sampling}\n\\subsection{Classical Compressed Sensing}\nCentral to signal processing is the Shannon-Nyquist theorem \\cite{45}, which specifies conditions on which a band-limited signal can be stably and uniquely recovered from its discrete measurements. However, in around 2005, a different sampling theorem was formed which, in some cases, abrogates the fundamentals of its predecessor. This new theory, nowadays known as compressed sensing (aka compressive sampling), asserts that signals, which admit a sparse representation in a predefined basis\/frame, can be recovered from their discrete measurements, whose number is proportional to the $\\ell_0$-norm of the coefficients of the sparse representation. In such a case, the sparser the representation of the signal is, the smaller can be the number of  measurements required for signal reconstruction. As a result, cases are numerous in which the conditions of compressed sensing far superseded those of the Shannon-Nyquist sampling \\cite{4, 13}.\n\nDespite its widespread success in countless applications, the theory of compressed sensing is not entirely free of limitations. One of such limitations stems from the necessity to use a non-linear decoder. Indeed, while in the case of classical sampling, the reconstruction of a time-domain signal is implemented through linear interpolation (i.e., linear filtering), in the case of compressed sensing, the reconstruction involves solution of a convex optimization problem. Specifically, let us consider a typical setup of {\\em classical} compressed sensing (CCS), in which ${\\bf y} \\in \\mathbb{R}^m$ represents an observed version of ${\\bf x} \\in \\mathbb{R}^n$, related according to\n\\begin{equation} \\label{13}\n{\\bf y} = \\Psi {\\bf x},\n\\end{equation}\nwhere $\\Psi \\in \\mathbb{R}^{m\\times n}$ is an observation (sampling) matrix with $n > m$.\n\nThe recovery of ${\\bf x}$ from ${\\bf y}$ based on (\\ref{13}) is impossible to implement in a unique and stable way, unless it is known that ${\\bf x}$ is sparse and hence has a relatively low value of $\\|{\\bf x}\\|_0$. In such a case, if the sampling matrix $\\Psi$ satisfies the restricted isometry property (RIP) \\cite{4, 13} with respect to a certain class of sparse signals to which ${\\bf x}$ is assumed to belong, then CCS recovers ${\\bf x}$ as a solution to \\cite{20, 21}\n\\begin{equation}\n{\\bf x} = \\arg \\min_{{\\bf x}^\\prime} \\left\\{ \\| {\\bf x}^\\prime \\|_1 \\mid \\Psi {\\bf x}^\\prime = {\\bf y} \\right\\},\n\\end{equation}\nwhich is a convex minimization problem, which is straightforward to reformulate in terms of linear programming. Moreover, in the case when the measurements ${\\bf y}$ are error-prone, a more robust version of CCS is to recover ${\\bf x}$ as given by\n\\begin{equation}\\label{E3}\n{\\bf x} = \\arg \\min_{{\\bf x}^\\prime} \\left\\{ \\| {\\bf x}^\\prime \\|_1 \\mid \\| \\Psi {\\bf x}^\\prime - {\\bf y} \\|_2^2 \\le \\epsilon \\right\\},\n\\end{equation}\nwhere $\\epsilon > 0$ is a parameter controlling the size of the noise. Moreover, it was shown in \\cite{4, 13}, that the estimation error in the signal reconstructed according to (\\ref{E3}) can be bounded by a linear function of $\\epsilon$. This implies robustness of the CCS reconstruction towards the presence of measurement noise.\n\nIt should be finally noted that the optimization problem (\\ref{E3}) can be reformulated in its alternative Lagrangian form, in which case one can find\n\\begin{equation} \\label{17}\n{\\bf x} = \\arg \\min_{{\\bf x}^\\prime} \\left\\{ \\frac{1}{2} \\| \\Psi {\\bf x}^\\prime - {\\bf y} \\|_2^2 + \\lambda \\| {\\bf x}^\\prime \\|_1 \\right\\}\n\\end{equation}\nwhere $\\lambda > 0 $ is an optimal Lagrange multiplier \\cite{20}. In what follows, it is assumed that an optimal value of $\\lambda$ is known. (For more details on this subject, the reader is referred to \\cite{20} as well as to the later sections of this paper).\n\n\\subsection{Derivative Compressed Sensing (DCS)}\nLet the partial derivatives of $\\phi$ evaluated (by means of the SHI) at the points of set $\\Omega_d$ be column-stacked into vectors ${\\bf f}_x$ and ${\\bf f}_y$ of length $M = \\#\\Omega_d$.  In what follows, the partial derivatives ${\\bf f}_x$ and ${\\bf f}_y$ are assumed to be sparsely representable by an orthonormal basis in $\\mathbb{R}^M$. Representing such a basis by an $M \\times M$ unitary matrix $W$, the above assumption suggests the existence of two {\\em sparse} vectors ${\\bf c}_x$ and ${\\bf c}_y$ such that $W {\\bf c}_x$ and $W {\\bf c}_y$ provide accurate approximations of the partial derivatives of the original phase $\\phi$ evaluated over $\\Omega_d$.\n\nNow, the simplification of the SHI proposed in the current paper amounts to reducing the number of wavefront lenses to a minimum. Formally, such a reduction can be described by two $n \\times M$ sub-sampling matrices $\\Psi_x$ and $\\Psi_y$, where $n < M$. Specifically, let ${\\bf b}_x := \\Psi_x {\\bf f}_x$ and ${\\bf b}_y := \\Psi_y {\\bf f}_y$ be incomplete (partial) observations of ${\\bf f}_x$ and ${\\bf f}_y$, respectively. Then, the noise-free counterparts of the partial derivatives can be approximated by $W {\\bf c}_x^\\ast$ and $W {\\bf c}_y^\\ast$, respectively, where\n\\begin{equation}\n{\\bf c}_x^\\ast = \\arg \\min_{{\\bf c}_x^\\prime} \\left\\{ \\frac{1}{2} \\| \\Psi_x W {\\bf c}_x^\\prime - {\\bf b}_x \\|_2^2  + \\lambda_x \\| {\\bf c}_x^\\prime \\|_1 \\right\\}\n\\end{equation}\nand\n\\begin{equation}\n{\\bf c}_y^\\ast = \\arg \\min_{{\\bf c}_y^\\prime} \\left\\{ \\frac{1}{2} \\| \\Psi_y W {\\bf c}_y^\\prime - {\\bf b}_y \\|_2^2  + \\lambda_y \\| {\\bf c}_y^\\prime \\|_1 \\right\\}\n\\end{equation}\nfor some $\\lambda_x, \\lambda_y > 0$. Moreover, in the case when $\\lambda_x = \\lambda_y$, the above estimates can be combined together. To this end, let ${\\bf c} = [{\\bf c}_x, {\\bf c}_y]^T$, ${\\bf b} = [{\\bf b}_x, {\\bf b}_y]^T$, and $A = \\mbox{diag} \\{\\Psi_x W, \\Psi_y W\\} \\in \\mathbb{R}^{2 n \\times 2 M}$. Then,\n\\begin{equation} \\label{E4}\n{\\bf c}^\\ast = \\arg \\min_{{\\bf c}^\\prime} \\left\\{ \\frac{1}{2} \\| A {\\bf c}^\\prime - {\\bf b} \\|_2^2  + \\lambda \\| {\\bf c}^\\prime \\|_1 \\right\\},\n\\end{equation}\nwhere $\\lambda=\\lambda_x = \\lambda_y$. In this form, the problem (\\ref{E4}) is identical to (\\ref{17}) and hence it can be solved by a variety of optimization algorithms \\cite{20, 21}.\n\nThe DCS algorithm augments CCS by subjecting the minimization in (\\ref{E4}) to an additional constraint which stems from the fact that \\cite{14}\n\\begin{equation} \\label{20}\n\\frac{\\partial^2 \\phi}{\\partial x \\, \\partial y}=\\frac{\\partial^2 \\phi}{\\partial y \\, \\partial x},\n\\end{equation}\nwhich is valid for any two times continuously differentiable $\\phi(x,y)$. Thus, in particular, the constraint implies the existence of two partial differences matrices $D_x$ and $D_y$ which obey\n\\begin{equation} \\label{E5}\nD_x {\\bf f}_y = D_y {\\bf f}_x.\n\\end{equation}\nConsequently, if $T_x$ and $T_y$ are the matrices satisfying $T_x {\\bf c} = {\\bf c}_x$ and $T_y {\\bf c} = {\\bf c}_y$, respectively, then (\\ref{E5}) can be re-expressed in terms of ${\\bf c}$ as\n\\begin{equation}\nD_y T_x {\\bf c} = D_x T_y {\\bf c}\n\\end{equation}\nor\n\\begin{equation}\nB {\\bf c} = 0,\n\\end{equation}\nwhere $B := D_y T_x - D_x T_y$. Thus, DCS solves the constrained minimization problem as given by\n\\begin{align} \\label{E5}\n{\\bf c}^\\ast = \\arg \\min_{{\\bf c}^\\prime} &\\left\\{ \\frac{1}{2} \\| A {\\bf c}^\\prime - {\\bf b} \\|_2^2  + \\lambda \\| {\\bf c}^\\prime \\|_1 \\right\\}, \\\\\n&\\mbox{s.t.  } B {\\bf c}^\\prime = 0 \\notag\n\\end{align}\n\nA solution to (\\ref{E5}) can be found, for instance, by means of the Bregman algorithm \\cite{22}, in which case ${\\bf c}^\\ast$ is approximated by a stationary point of the sequence of iterations produced by\n\\begin{equation} \\label{E6}\n\\begin{cases}\n{\\bf c}^{(t+1)} = \\arg \\min_{{\\bf c}^\\prime} \\Big\\{ \\frac{1}{2} \\| A {\\bf c}^\\prime - {\\bf b} \\|_2^2  + \\\\\n\\hspace{2.5cm} + \\lambda \\| {\\bf c}^\\prime \\|_1 + \\frac{\\delta}{2} \\| B {\\bf c}^\\prime + p^{(t)} \\|_2^2 \\Big\\} \\\\\np^{(t+1)} = p^{(t)} + \\delta B {\\bf c}^{(t+1)},\n\\end{cases}\n\\end{equation}\nwhere $p^{(t)}$ is a vector of Bregman variables (or, equivalently, augmented Lagrange multipliers) and $\\delta > 0$ is a user-defined parameter\\footnote{In this work, we use $\\delta = 0.5$.}.\n\nThe ${\\bf c}$-update step in (\\ref{E6}) has the format of a standard basis pursuit de-noising (BPDN) problem \\cite{46}, which can be solved by a variety of optimization methods \\cite{23}. In the present paper, we used the FISTA algorithm of \\cite{33} due to the simplicity of its implementation as well as for its remarkable convergence properties. It should be noted that the algorithm does not require explicitly defining the matrices $A$ and $B$. Only the {\\em operations} of multiplication by these matrices and their transposes need to be known, which can be implemented in an implicit and computationally efficient manner.\n\nOnce an optimal ${\\bf c}^\\ast$ is recovered, it can be used to estimate the noise-free versions of ${\\bf f}_x$ and ${\\bf f}_y$ as $W T_x {\\bf c}^\\ast$ and $W T_y {\\bf c}^\\ast$, respectively. These estimates can be subsequently passed on to the fitting procedure of Section III to recover the values of $\\phi$, which, in combination with a known aperture function $A$, provide an estimate of the PSF $i$ as an inverse discrete Fourier transform of the autocorrelation of $P = A \\, e^{\\jmath \\phi}$. Algorithm 1 below summarizes our method of estimation of the PSF.\n\n\\begin{algorithm}\n\\setlength{\\leftmargini}{0pt}\n\\caption{PSF estimation via DCS}\n\\begin{enumerate}\n\\item {\\it Data:} ${\\bf b}_x$, ${\\bf b}_y$, and $\\lambda > 0$\n\\item {\\it Initialization:} For a given transform matrix $W$ and matrices\/operators $\\Psi_x$, $\\Psi_y$, $D_x$, $D_y$, $T_x$ and $T_y$, preset the procedures of multiplication by $A$, $A^T$, $B$ and $B^T$.\n\\item {\\it Phase recovery:} Starting with an arbitrary ${\\bf c}^{(0)}$ and $p^{(0)} = 0$, iterate (\\ref{E6}) until convergence to result in an optimal ${\\bf c}^\\ast$. Use the estimated (full) partial derivatives $W T_x {\\bf c}^\\ast$ and $W T_y {\\bf c}^\\ast$ to recover the values of $\\phi$ over $\\Omega$.\n\\item {\\it PSF estimation:} Using a known aperture function $A$, compute the inverse Fourier transform of $P = A \\, e^{\\jmath \\phi}$ to result in a corresponding ASF $h$. Estimate the PSF $i$ as $i = |h|^2$.\n\\end{enumerate}\n\\label{algo1}\n\\end{algorithm}\n\nThe estimated PSF can be used to recover the original image $u$ from $v$ through the process of deconvolution as explained in the section that follows.\n\n\\section{Deconvolution}\nThe acquisition model \\eqref{1} can be rewritten in an equivalent operator form as given by\n\\begin{equation} \\label{30}\nv = \\mathcal{H}\\{u\\}+\\nu,\n\\end{equation}\nwhere $\\mathcal{H}$ denote the operator of convolution with the estimated PSF $i$. Note that, in this case, the noise term $\\nu$ accounts for both measurement noise as well as the inaccuracies related to estimation error in $i$.\n\nThe deconvolution problem of finding a useful approximation of $u$ given its distorted measurement $v$ can be addressed in many way, using a multitude of different techniques \\cite{31, 32, 33}. In this work, we use the ROF model and recover a regularized approximation as a solution of\n\\begin{equation} \\label{E7}\nu^\\ast = \\arg \\min_u \\left\\{ \\frac{1}{2} \\| \\mathcal{H}\\{u\\} -v \\|_2^2 + \\gamma \\, \\| u \\|_{TV} \\right\\},\n\\end{equation}\nwhere $\\| u \\|_{TV} = \\int \\int | \\nabla  u | \\, dx \\, dy $ denotes the total variation (TV) semi-norm of $u$.\n\nOne computationally efficient way to solve \\eqref{E7} is to substitute a direct minimization of the cost function in \\eqref{E7} by recursively minimizing a sequence of its local quadratic majorizers \\cite{33}. In this case, the optimal solution $u^\\ast$ can be approximated by the stationary point of a sequence of intermediate solutions produced by\n\\begin{equation} \\label{E8}\n\\begin{cases}\nw^{(t)} = u^{(t)} + \\mu \\, \\mathcal{H}^\\ast \\left\\{ v - \\mathcal{H}\\{u^{(t)}\\}  \\right\\} \\\\\nu^{(t+1)} = \\arg \\min_u \\left\\{ \\frac{1}{2} \\| u - w^{(t)} \\|_2^2 + \\gamma \\, \\| u \\|_{TV} \\right\\},\n\\end{cases}\n\\end{equation}\nwhere $\\mathcal{H}^\\ast$ is the adjoint of $\\mathcal{H}$ and $\\mu$ is chosen to satisfy $\\mu > \\| \\mathcal{H}^\\ast \\mathcal{H} \\|$. In this paper, the TV denoising at the second step of \\eqref{E8} has been performed using the fixed-point algorithm of Chambolle \\cite{11}. The convergence of \\eqref{E8} can be further improved by using the same FISTA algorithm of \\cite{33}. The resulting procedure is summarized below in Algorithm 2.\n\n\\begin{algorithm}\n\\setlength{\\leftmargini}{0pt}\n\\caption{TV deconvolution using FISTA}\n\\begin{enumerate}\n\\item {\\it Initialize:} Select an initial value $u^{(0)}$; set $y^{(0)} = u^{(0)}$ and $\\tau^{(0)}=1$\\\\\n\\item {\\it Repeat until convergence:}\n\\begin{itemize}\n\\item $w^{(t)} = y^{(t)} + \\mu \\, \\mathcal{H}^\\ast \\left\\{ v - \\mathcal{H}\\{y^{(t)}\\}  \\right\\}$\n\\item $u^{(t+1)} = \\arg \\min_u \\left\\{ \\frac{1}{2} \\| u - w^{(t)} \\|_2^2 + \\gamma \\, \\| u \\|_{TV} \\right\\}$\n\\item $\\tau^{(t+1)} = 0.5  \\left( 1+\\sqrt{1+ 4 \\, (\\tau^{(t)})^2} \\right)$\n\\item $y^{(t+1)} = u^{(t+1)} + (\\tau^{(t)}\/\\tau^{(t+1)}) (u^{(t+1)} - u^{(t)})$\n\\end{itemize}\n\\end{enumerate}\n\\label{algo2}\n\\end{algorithm}\n\nIn summary, Algorithms 1 and 2 represent the essence of the proposed algorithm for hybrid deconvolution of optical images. The next section provides experimental results which further support the value and applicability of the proposed methodology.\n\n\\section{Results}\nTo demonstrate the viability of the proposed approach, its performance has been compared against reference methods. The first reference method used a dense sampling of the phase (as it would have been the case with a regular SHI), thereby eliminating the need for a CS-based phase reconstruction. The resulting method is referred below to as the dense sampling (DS) approach. Second, to assess the importance of incorporation of the cross-derivative constraints, we have used both CCS and DCS for phase recovery. In what follows, comparative results for phase estimation and subsequent deconvolution are provided for all the above methods.\n\n\\subsection{Phase recovery}\nTo assess the performance of the proposed and reference methods under controllable conditions, simulation data was used. The random nature of atmospheric turbulence necessitated the use of statistical methods to model its effect on a wavefront propagation. Specifically, in this paper, the effect of atmospheric turbulence has been described using the modified Von Karman PSF model\\cite{35}. A typical example of a GPF phase $\\phi$ is shown in subplot (a) of Fig.~\\ref{F3}. In the shown case, the size of the phase screen was set to be equal to $10 \\times 10$ cm, while the sampling was performed over a $128 \\times 128$ uniform lattice (which would have corresponded to the use of 16384 lenses of a SHI). The partial derivatives $\\partial \\phi \/ \\partial x$ and $\\partial \\phi \/ \\partial y$ are shown in subplots (b) and (c) of Fig.~\\ref{F3}, respectively.\n\nIn the present paper, the subsampling matrices $\\Psi_x$ and $\\Psi_y$ were obtained from an identity matrix $I$ through a random subsampling of its rows to result in a required compression ratio $r$ (to be specified below). To sparsely represent the partial derivatives of $\\phi$, $W$ was defined to correspond to a four-level orthogonal wavelet transform using the nearly symmetric wavelets of Daubechies with five vanishing moments \\cite{92}.\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]\n{\n\\includegraphics[width = 4cm]{orig-phase.pdf}\n\\label{fig:tabsubfig1}\n}\n\\\\\n\\subfigure[]{\n\\includegraphics[width = 4cm]{xderiv.pdf}\n\\label{fig:tabsubfig2}\n}\n\\subfigure[]{\n\\includegraphics[width = 4cm]{yderiv.pdf}\n\\label{fig1:tabsubfig3}\n}\n\\caption{An example of a simulated phase $\\phi$ (subplot (a)) along with its partial derivatives w.r.t. $x$ (subplot (b)) and $y$ (subplot (c)).}\n\\label{F3}\n\\end{figure}\n\nTo demonstrate the value of using the cross derivative constraint for phase reconstruction, the CCS and DCS algorithms have been compared in terms of the mean square errors (MSE) of their corresponding phase estimates. The results of this comparison are summarized in Fig.~\\ref{F4} for different compression ratios (or, equivalently, (sub)sampling densities) and SNR = 40 dB.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width = 9cm, height = 6cm]{density.pdf}\n\\label{fig1:tabsubfig2}\n\\caption{MSE of phase reconstruction obtained with different methods as a function of $r$. Here, the dashed and solid lines correspond to CCS and DCS, respectively, and SNR is equal to 40 dB.}\n\\label{F4}\n\\end{figure}\n\nAs expected, one can see that DCS results in lower values of MSE as compared to CCS, which implies a higher accuracy of phase reconstruction. Moreover, the difference in the performances of  CCS and DCS appears to be more significant for lower sampling rates, while both algorithms tend to perform similarly when the sampling density approaches the DS case. Specifically, when the sampling density is equal to $r=0.3$, DCS results in a ten times smaller MSE as compared to the case of CCS, whereas both algorithms have comparable performance for $r = 0.83$. This result characterizes DCS as a better performer than CCS in the case of relatively low (sub)sampling rates.\n\nSome typical phase reconstruction results are shown in Fig.~\\ref{F5}, whose left and right subplots depict the phase estimates obtained using the CCS and DCS algorithms, respectively, for the case of $r = 0.5$. To visualize the differences more clearly, error maps for both methods are shown in subplot (c) and (d)(note for better visualization error map values are multiplied by factor of 10.). A close comparison with the original phase (as shown in subplot (a) of Fig.~\\ref{F3}) reveals that DCS provides a more accurate recovery of the original $\\phi$, which further supports the value of using the cross-derivative constraints. In fact, exploiting these constraints effectively amounts to using additional measurements, which are ignored in the case of CCS.\n\n\\begin{figure}[!t]\n\\subfigure[]{\n\\includegraphics[width = 4cm]{CS-phase.pdf}\n\\label{fig:tabsubfig21}\n}\n\\subfigure[]{\n\\includegraphics[width = 4cm]{DCS-phase.pdf}\n\\label{fig:tabsubfig31}\n}\\\\\n\\subfigure[]{\n\\includegraphics[width = 4cm]{CS-diff.pdf}\n\\label{fig:tabsubfig31}\n}\n\\subfigure[]{\n\\includegraphics[width = 4cm]{DCS-diff.pdf}\n\\label{fig:tabsubfig31}\n}\n\\caption{(Subplot (a)) Phase reconstructed obtained by means of CCS for SNR = 40 dB and $r=0.5$; (Subplot (b)) Phase reconstructed obtained by means of DCS for the same values of SNR and $r$.; (Subplot (c) and (d)) Corresponding error maps for CCS and DCS.}\n\\label{F5}\n\\end{figure}\n\nTo investigate the robustness of the compared algorithms towards the influence of additive noises, their performances have been compared for a range of SNR values. The results of this comparison are summarized in Fig.~\\ref{F6}. Since the cross-derivative constraints exploited by DCS effectively restrict the feasibility region for an optimal solution, the algorithm exhibits a substantially better robustness to the additive noise as compared to the case of CCS. This fact represents another beneficial outcome of incorporating the cross-derivative constraints in the process of phase recovery.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width = 9cm, height = 6cm]{robust.pdf}\n\\caption{MSE of phase reconstruction obtained with different methods as a function of SNR. Here, the dashed and solid lines correspond to CCS and DCS, respectively, and $r=0.5$.}\n\\label{F6}\n\\end{figure}\n\nIt should be taken into account that, although the shape of $\\phi$ does not change the energy of the PSF $i$, it plays a crucial role in determining its spatial behaviour. In the section that follows, it will be shown that even small inaccuracies in reconstruction of $\\phi$ could be translated into dramatic  difference in the quality of image deconvolution.\n\n\\subsection{Image deconvolution}\nAs a next step, the phase estimates obtained using the CCS- and DCS-based methods for $r=0.5$ were combined with the aperture function $A$ to result in their respective estimates of the PSF $i$. These estimates were subsequently used to deconvolve a number of test images such as ``Satellite\", ``Saturn\", ``Moon\" and ``Galaxy\". All the test images were blurred with an original PSF, followed by their contamination with additive Gaussian noise of different levels. As an example, Fig.~\\ref{fig12} shows the ``Satellite\" image (subplot (a)) along with its blurred and noisy version (subplot (b)).\n\nUsing the PSF estimates, the deconvolution was carried out using the method detailed in \\cite{11}. For the sake of comparison, the deconvolution was also performed using the PSF recovered from dense sampling (DS) of $\\phi$. Note that this reconstruction is expected to have the best accuracy, since it neither involves undersampling nor requires a CS-based phase estimation. All the deconvolved images have been compared with their original counterparts in terms of PSNR as well as of the structural similarity index (SSIM) of \\cite{36}, , which is believed to be a better indicator of perceptual image quality \\cite{100}. The resulting values of the comparison metrics are summarized in Table 1, while Fig.~\\ref{fig11} shows the deconvolution results produced by the CCS- and DCS-based methods.\n\n\\begin{figure}[!t]\n\\subfigure[]{\n\\includegraphics[width = 4cm]{orig-image.pdf}\n\\label{fig:tabsubfig2}\n}\n\\subfigure[]{\n\\includegraphics[width = 4cm]{distorted-image.pdf}\n\\label{fig:tabsubfig3}\n}\n\\caption{Satellite image (subplot (a)) and its blurred and noisy version (subplot (b)).}\n\\label{fig12}\n\\end{figure}\n\nThe above results demonstrate the importance of accurate phase recovery, where even a relatively small phase error can have a dramatic effect on the quality of image deconvolution. Under such conditions, the proposed method produces image reconstructions of a superior quality as compared to the case of CCS. Moreover, comparing the results of Table 1, one can see that DS only slightly outperforms DCS in terms of PSNR and SSIM, while in many practical cases, the difference between the performances of these methods are hard to detect visually.\n\n\\begin{figure}[t]\n\\subfigure[]{\n\\includegraphics[width = 4cm]{CS-result.pdf}\n\\label{fig:tabsubfig2}\n}\n\\subfigure[]{\n\\includegraphics[width = 4cm]{DCS-result.pdf}\n\\label{fig:tabsubfig3}\n}\n\\caption{(Subplot (a)) Image estimate obtained with the CCS-based method for phase recovery (SSIM = 0.917); (Subplot (b)) Image estimate obtained with the DCS-based method for phase recovery (SSIM = 0.781).}\n\\label{fig11}\n\\end{figure}\n\n\\begin{table*}[t]\n\\centering \\caption{SSIM and PSNR comparisons of phase recovery\nresults}\n\\label{tab:denresults}\n\\begin{tabular}{l|cccc|cccc|cccc|cccc}\n\\hline\nImage & \\multicolumn{4}{c}{Satellite} & \\multicolumn{4}{c}{Saturn} & \\multicolumn{4}{c}{Moon} & \\multicolumn{4}{c}{Galaxy}\\\\\n\\hline\nNoise std & $10^{-5}$ & $0.001$ & $0.003$ & $0.005$ & $10^{-5}$ &  $0.001$ & $0.003$ & $0.005$  & $10^{-5}$ &  $0.001$ & $0.003$ & $0.005$  & $10^{-5}$ &  $0.001$ & $0.003$ & $0.005$ \\\\\n\\hline\n& \\multicolumn{16}{c}{PSNR comparison (in dB)}\\\\\n\\hline\nBlurred               & 14.06 & 14.06 & 14.06 & 14.05 & 17.78 & 17.78 & 17.78 &  17.78 & 19.98 & 19.97 & 19.97 &  19.97 & 18.79 & 18.79 & 18.78 &  18.78\\\\\nDS                & 27.97 & 27.75 & 25.97 & 22.43 & 31.49 & 31.08 & 28.50 &  23.89 & 25.06 & 25.04 & 24.83 &  21.76 & 23.58 & 23.60 & 23.38 &  20.93\\\\\nCS                    & 17.06 & 16.93 & 16.54 & 15.63 & 23.42 & 23.38 & 22.80 &  20.55 & 22.36 & 22.38 & 22.30 &  19.73 & 21.16 & 21.12 & 20.64 & 18.46\\\\\nDCS                   & 27.42 & 27.22 & 25.56 & 22.22 & 31.02 & 30.65 & 28.30 &  23.72 & 25.00 & 24.99 & 24.78 &  21.73 & 23.52 & 23.54 & 23.32 & 20.86\\\\\n\\hline\n& \\multicolumn{16}{c}{SSIM comparison}\\\\\n\\hline\nBlurred               & 0.200 & 0.200 & 0.199 & 0.197 & 0.226 & 0.226 & 0.226 & 0.175 & 0.512 & 0.512 & 0.509 & 0.504 & 0.257 & 0.257 & 0.257 & 0.254\\\\\nDS               & 0.730 & 0.720 & 0.554 & 0.269 & 0.688 & 0.660 & 0.506 & 0.228 & 0.645 & 0.642 & 0.607 & 0.552 & 0.493 & 0.495 & 0.501 & 0.397\\\\\nCS                    & 0.349 & 0.344 & 0.306 & 0.206 & 0.424 & 0.416 & 0.348 & 0.212 & 0.539 & 0.538 & 0.493 & 0.488 & 0.348 & 0.347 & 0.326 & 0.224\\\\\nDCS                   & 0.674 & 0.667 & 0.519 & 0.263 & 0.656 & 0.641 & 0.483 & 0.223 & 0.643 & 0.640 & 0.604 & 0.549 & 0.490 & 0.491 & 0.501 & 0.393\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\section{Discussion and conclusions}\nIn the present paper, the applicability of DCS to the problem of reconstruction of optical images has been demonstrated. It was shown that, in the presence of atmospheric turbulence, the phase $\\phi$ of the GPF $P = A \\, e^{\\jmath \\phi}$ is a random function, which needs to be measured using adaptive optics. To simplify the complexity of the latter, a CS-based approach has been proposed. As opposed to CCS, however, the proposed method performs phase reconstruction subject to an additional constraint, which stems from the property of $\\nabla \\phi$ to be a potential field. The resulting algorithm (referred to as the DCS method) has been shown to yield phase estimates of substantially better quality as compared to the case of CCS.\n\nIn this paper, our main focus has been on simplifying the structure of the SHI through reducing the number of its wavefront lenses, while compensating for the effect of undersampling by using the theory of CS augmented by the cross-derivative constraint. The solution was computed using the Bregman algorithm, which provides a computationally efficient framework to carry out the constrained phase recovery. Moreover, the resulting phase estimates were used to recover their associated PSF, which was subsequently used for image deconvolution. It was shown that the DCS-based estimation of $\\phi$ with $r=0.3$ results in image reconstructions of the quality comparable to that of DS, while substantially outperforming the results obtained with CCS.\n\nWhile the proposed method offers a practical solution to the problem of phase estimation in adaptive optics, some interesting questions about the theoretical aspects of DCS still lay open. In particular, the question of theoretical performance of CS in the presence of side information on the source signal needs to be addressed through future research.\n\n\\section*{Acknowledgment}\nThis work was supported in part by the Natural Sciences and Engineering Research Council of Canada and in part by Ontario Early Researcher Award program, which are gratefully acknowledged. The authors would also like to acknowledge Sudipto Dolui for his helpful comments as well as for providing deconvolution codes.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nBy the Nielsen-Thurston classification of surface diffeomorphisms, an automorphism $\\psi :S \\stackrel{\\sim}{\\to} S$ of a compact oriented surface $S$ is of one of three types: periodic, reducible or pseudo-Anosov \\cite{MR964685}, \\cite{MR956596}.\nA generic element of the mapping class group of $S$ is of pseudo-Anosov type.\n\nLet us assume that $\\psi$ is of pseudo-Anosov type.\nFor any closed curve $C \\subset S$, it is known that there is a sequence $\\{L_m\\}_{m \\in \\mathbb{N}}$ of closed geodesics such that $L_m$ is isotopic to $\\psi^m(C)$ for all $m \\in \\mathbb{N}$, and $\\{L_m\\}_{m \\in \\mathbb{N}}$, as a sequence of closed subsets, converges to a closed subset $\\mathcal{L}$. \nMoreover, $\\mathcal{L}$ is a geodesic lamination. \nThe definitions of a lamination, a geodesic lamination and a Lagrangian lamination are the following: \n\\begin{definition}\n\t$\\mbox{}$\n\t\\begin{enumerate}\n\t\t\\item  \n\t\tA {\\em $k$-dimensional lamination} on an $n$-dimensional manifold $M$ is a decomposition of a closed subset of $M$ into $k$-dimensional submanifolds called {\\em leaves} so that $M$ is covered by charts of the form $I^k \\times I^{n-k}$ where a leaf passing through a chart is a slice of the form $I^k \\times \\{pt\\}$.\n\t\t\\item \n\t\tA $1$-dimensional lamination $\\mathcal{L}$ on a Riemannian 2-manifold $(S,g)$ is a {\\em geodesic lamination} if every leaf of $\\mathcal{L}$ is geodesic.\n\t\t\\item\n\t\tA $n$-dimensional lamination $\\mathcal{L}$ on a symplectic manifold $(M^{2n},\\omega)$ is a {\\em Lagrangian lamination} if every leaf of $\\mathcal{L}$ is a Lagrangian submanifold.  \n\t\\end{enumerate}\n\\end{definition}\nFor more details, we refer the reader to \\cite[Chapter 15]{MR2850125}.\n\nIn \\cite{MR3289326}, Dimitrov, Haiden, Katzarkov, and Kontsevich defined the notion of a {\\em pseudo-Anosov functor} of a category.\nA pseudo-Anosov map $\\psi$ on a compact oriented surface $S$ induces a functor, also called $\\psi$, on the derived Fukaya category $D^{\\pi}Fuk(S,\\omega)$, where $\\omega$ is an area form of $S$.\nIn \\cite{MR3289326}, the authors showed that $\\psi$ is a pseudo-Anosov functor.\n\nIn \\cite[Section 4]{MR3289326}, the authors listed a number of open questions. \nOne of them is to find a symplectic automorphism $\\psi$ on a symplectic manifold $M$ of dimension greater than 2 which has invariant transversal stable\/unstable Lagrangian measured foliations. \nA slightly weaker version of the question is to define a symplectic automorphism $\\psi$ with invariant stable\/unstable Lagrangian laminations.\n\nThe goal of the present paper is to prove Theorems \\ref{branched surface thm}--\\ref{thm Lagrangian floer homology}, which answer the latter question.\n\n\\begin{theorem}\n\\label{branched surface thm}\nLet $M$ be a symplectic manifold and let $\\psi : M \\stackrel{\\sim}{\\to} M$ be a symplectic automorphism of generalized Penner type. Then, there exists a Lagrangian branched submanifold $\\mathcal{B}_{\\psi}$ such that if $L$ is a Lagrangian submanifold which is carried (resp.\\ weakly carried) by $\\mathcal{B}_{\\psi}$, then $\\psi^m(L)$ is carried (resp.\\ weakly carried) by $\\mathcal{B}_{\\psi}$ for all $m \\in \\mathbb{N}$. \n\\end{theorem} \nIn Sections 2 and 3, we will explain the terminology that appears in the statement of Theorem \\ref{branched surface thm}, i.e., a symplectic automorphism of generalized Penner type, a Lagrangian branched submanifold, and the notion of ``carried by''.\n\nWe would like to remark that Theorem \\ref{branched surface thm} is for $\\psi$ of generalized Penner type.\nHowever, there would be a generalized version of Theorem \\ref{branched surface thm}, which we do not prove in the current paper.\n\n\\begin{theorem}\n\\label{lamination thm}\nLet $M$ be a symplectic manifold and let $\\psi: M \\stackrel{\\sim}{\\to} M$ be a symplectic automorphism of generalized Penner type.\nThen, there is a Lagrangian lamination $\\mathcal{L}$ such that\nif $L$ is a Lagrangian submanifold of $M$ which is carried by $\\mathcal{B}_{\\psi}$, then there is a Lagrangian submanifold $L_m$ for all $m \\in \\mathbb{N}$, which is Hamiltonian isotopic to $\\psi^m(L)$ and converges to $\\mathcal{L}$ as closed sets as $m \\to \\infty$.\n\\end{theorem}\nWe will also prove the following generalization of Theorem \\ref{lamination thm}\n\n\\begin{theorem}\n\t\\label{generalized theorem}\n\tLet $\\psi:M \\stackrel{\\sim}{\\to} M$ be a symplectic automorphism and let $\\mathcal{B}_{\\psi}$ be a Lagrangian branched submanifold such that $\\psi(\\mathcal{B}_{\\psi})$ is carried by $\\mathcal{B}_{\\psi}$.\n\tMoreover, if the associated branched manifold $\\mathcal{B}_{\\psi}$ admits a decomposition into singular and regular disks, then there is a Lagrangian lamination $\\mathcal{L}$ such that\n\tif $L$ is a Lagrangian submanifold of $M$ which is carried by $\\mathcal{B}_{\\psi}$, then there is a Lagrangian submanifold $L_m$ for all $m \\in \\mathbb{N}$, which is Hamiltonian isotopic to $\\psi^m(L)$ and converges to $\\mathcal{L}$ as closed sets as $m \\to \\infty$.\n\\end{theorem}\nThe associated branched manifold and singular\/regular disks will be defined in Sections 3 and 4.\n\n\\begin{theorem}\n\t\\label{thm Lagrangian floer homology}\n\tLet $M$ be a plumbing space of Penner type and let $\\eta : M \\stackrel{\\sim}{\\to} M$ be the involution associated to $M$. \n\tLet assume that a transversal pair $L_1, L_2 \\subset M$ of Lagrangian submanifolds satisfies the following:\n\t\\begin{enumerate}\n\t\t\\item $\\eta(L_i) = L_i$ for $i = 0, 1$.\n\t\t\\item Let $\\tilde{L}_i = L_i \\cap M_i$. Then, $\\tilde{L}_i$ is a Lagrangian submanifold of $\\tilde{M}$ such that $\\tilde{L}_0$ and $\\tilde{L}_1$ are not isotopic to each other.\n\t\t\\item $L_0 \\cap L_1 = \\tilde{L}_0 \\cap \\tilde{L}_1$,\n\t\t\\item $L_0$ and $L_1$ are not isotopic to each other. \n\t\\end{enumerate} \n\tThen, \n\t\\begin{gather*}\n\t\\operatorname{dim} HF^0(L_1,L_2) + \\operatorname{dim}HF^1(L_1,L_2) = i(\\tilde{L}_1, \\tilde{L}_2),\n\t\\end{gather*}\n\twhere $HF^k(L_1,L_2)$ denotes $\\mathbb{Z}\/2$--graded Lagrangian Floer homology over the Novikov ring of characteristic 2 and $i(\\tilde{L}_1,\\tilde{L}_2)$ denotes the geometric intersection number of $\\tilde{L}_1$ and $\\tilde{L}_2$ in the fixed surface $\\tilde{M}$.\n\\end{theorem}\nIn Section \\ref{section pseudo-Anosov functors}, we will explain the terminology that appears in the statement of Theorem \\ref{thm Lagrangian floer homology}, i.e., a plumbing space $M$ of Penner type, the involution $\\eta$ associated to $M$, and the fixed surface $\\tilde{M}$ of $M$. \n\nThis paper consists of 5 sections.\nIn Section 2, we review plumbing spaces and generalized Dehn twists. \nWe will prove Theorem \\ref{branched surface thm} in Section 3 and Theorems \\ref{lamination thm} and \\ref{generalized theorem} in Section 4.\nIn Section \\ref{section pseudo-Anosov functors}, we will prove Theorem \\ref{thm Lagrangian floer homology}.  \n\n\n\n\n\n\n\\section{Preliminaries}\n\nIn this section, we will review plumbings of cotangent bundles and generalized Dehn twists, partly to establish notation. \n\n\n\n\\subsection{Plumbing spaces}\n\nLet $\\alpha$ and $\\beta$ be oriented spheres $S^n$. \nWe describe how to plumb $T^*\\alpha$ and $T^*\\beta$ at $p \\in \\alpha$ and $q \\in \\beta$. \nLet $U \\subset \\alpha$ and $V \\subset \\beta$ be small disk neighborhoods of $p$ and $q$.\nThen, we identify $T^*U$ and $T^*V$ so that the base $U$ (resp.\\ $V$) of $T^*U$ (resp.\\ $T^*V$) is identified with a fiber of $T^*V$ (resp.\\ $T^*U$).  \n\nTo do this rigorously, we fix coordinate charts $\\psi_1: U \\to \\mathbb{R}^n$ and $\\psi_2:V \\to \\mathbb{R}^n$.\nThen, we obtain a compositions of symplectomorphisms\n\\begin{align*}\nT^*U \\xrightarrow{({\\psi}_1^*)^{-1}} T^*\\mathbb{R}^n \\simeq \\mathbb{R}^{2n} \\xrightarrow{f} \\mathbb{R}^{2n} \\simeq T^*\\mathbb{R}^n \\xrightarrow{\\psi_2^*} T^*V,\n\\end{align*}\nwhere $f(x_1, \\cdots, x_n, y_1, \\cdots, y_n) = (y_1, \\cdots, y_n, -x_1, \\cdots, -x_n)$. \n\nA plumbing space $P(\\alpha, \\beta)$ of $T^*\\alpha$ and $T^*\\beta$ is defined by\n$T^*\\alpha \\sqcup T^*\\beta \/ \\sim$, where $x \\sim (\\psi^*_2 \\circ f \\circ \\psi^{*-1}_1)(x)$ for all $x \\in T^*U$.\nSince $\\psi^*_2 \\circ f \\circ \\psi^{*-1}_1$ is a symplectomorphism, $P(\\alpha,\\beta)$ has a natural symplectic structure induced by the standard symplectic structures of cotangent bundles.\n\nSince the plumbing procedure is a local procedure, we can plumb a finite collection of cotangent bundles of the same dimension at finitely many points. \nFor convenience, we plumb cotangent bundles of oriented manifolds.\n\nNote that we can replace $f$ by \n$$g(x_1,\\cdots,x_n,y_1,\\cdots,y_n) = (-y_1, y_2, \\cdots,y_n,x_1, -x_2, \\cdots, -x_n).$$\nIf we plumb $T^*\\alpha$ and $T^*\\beta$ at one point using $g$, this plumbing space is symplectomorphic to the previous plumbing space $P(\\alpha,\\beta)$, which is plumbed using $f$. \nHowever, if we plumb at more than one point, then by replacing $f$ with $g$ at a plumbing point, the plumbing space will change.\n\n\\begin{definition}\n\t\\label{def plumbing space}\n\tLet $\\alpha_1, \\cdots, \\alpha_m$ be oriented manifolds of dimension $n$.\n\t\\begin{enumerate}\n\t\t\\item A {\\em plumbing data} is a collection of pairs of non-negative integers $(a_{i,j}, b_{i,j})$ for all $1 \\leq i \\leq j \\leq m$ and collections of distinct points \n\t\t\\begin{gather*}\n\t\t\\{p^{i,j}_k \\in \\alpha_i \\hspace{0.2em} | \\hspace{0.2em} 1 \\leq i \\leq j \\leq m, \\hspace{0.2em} 1 \\leq k \\leq a_{i,j} + b_{i,j} \\} \\hspace{0.5em} \\text{and} \\\\\n\t\t\\{ q^{i,j}_k \\in \\alpha_j \\hspace{0.2em} | \\hspace{0.2em} 1 \\leq i \\leq j \\leq m, \\hspace{0.2em} 1 \\leq k \\leq a_{i,j} + b_{i,j}  \\}.\n\t\t\\end{gather*}\n\t\t\\item A {\\em plumbing space} $P(\\alpha_1, \\cdots, \\alpha_m)$, with the given plumbing data, is given by \n\t\t$$P(\\alpha_1, \\cdots, \\alpha_m) = T^*\\alpha_1 \\sqcup \\cdots \\sqcup T^*\\alpha_m \/ \\sim,$$\n\t\twhere the equivalence relation $\\sim$ is defined as follows:\n\t\tFirst, choose small disk neighborhoods $U^{i,j}_k \\subset \\alpha_i$ of $p^{i,j}_k$ and $V^{i,j}_k \\subset \\alpha_j$ of $q^{i,j}_k$ and orientation-preserving coordinate charts $\\psi^{i,j}_k : U^{i,j}_k \\stackrel{\\sim}{\\to} \\mathbb{R}^n$ and $\\phi^{i,j}_k:V^{i,j}_k \\stackrel{\\sim}{\\to} \\mathbb{R}^n$. \n\t\tThen for all $ x \\in T^*U^{i,j}_k$, \n\t\t\\begin{gather*}\n\t\tx \\sim (\\phi^{i,j*}_k \\circ f \\circ (\\psi^{i,j*}_k)^{-1})(x) \\hspace{0.5em}  \\text{if} \\hspace{0.5em} 1 \\leq k \\leq a_{i,j}, \\\\\n\t\tx \\sim (\\phi^{i,j*}_k \\circ g \\circ (\\psi^{i,j*}_k)^{-1})(x) \\hspace{0.5em}  \\text{if} \\hspace{0.5em} a_{i,j} +1 \\leq k \\leq a_{i,j} + b_{i,j}.\n\t\t\\end{gather*}\n\t\t\\item A {\\em plumbing point} is an identified point $p^{i,j}_k \\sim q^{i,j}_k \\in P(\\alpha_1, \\cdots, \\alpha_m).$ \n\t\\end{enumerate}\n\\end{definition}\nFigure \\ref{figure examples of plumbing space} is examples of plumbing spaces.\n\nIf $\\alpha_i$ is of dimension $n \\geq 2$, then specific choices of plumbing points do not change the symplectic topology of $P(\\alpha_1, \\cdots, \\alpha_m)$.\n\n\\begin{figure}[h]\n\t\\centering\n\t\\input{exam_plumbing_space.pdf_tex}\n\t\\caption{$P(\\alpha \\simeq S^1, \\beta \\simeq S^1)$ with plumbing data $(2,0)$ (left) and $(1,1)$ (right).}\n\t\\label{figure examples of plumbing space}\n\\end{figure}\n \n \n \n\\subsection{Generalized Dehn twist}\n\nLet\n\\begin{gather*}\nT^*S^n = \\{(u;v) \\in \\mathbb{R}^{n+1} \\times \\mathbb{R}^{n+1} \\mid \\|u\\| = 1, \\left\\langle u,v\\right\\rangle = 0 \\}, \\\\\nS^n = \\{(u;0) \\in T^*S^n \\},\n\\end{gather*}\nwhere $(u;v) \\in \\mathbb{R}^{n+1} \\times \\mathbb{R}^{n+1}$ and $<u,v>$ is the standard inner product of $u$ and $v$ in $\\mathbb{R}^{n+1}$.\nMoreover, let $0_k$ be the origin in $\\mathbb{R}^k$.\n\nWe fix a Hamiltonian function $\\mu(u;v) = \\|v\\|$ on $T^*S^n \\setminus S^n$.\nThen, $\\mu$ induces a circle action on $T^*S^n \\setminus S^n$ given by \n$$ \\sigma(e^{it})(u;v) = \\big(\\cos (t) u + \\sin (t) \\frac{v}{\\|v\\|}; \\cos (t) v - \\sin (t) \\|v\\| u  \\big).$$\nLet $r : [0,\\infty) \\to \\mathbb{R}$ be a smooth decreasing function such that $r(0) = \\pi$ and $r(t) = 0$ for all $t \\geq \\epsilon$ for a small positive number $\\epsilon$.\nIf $\\omega_0$ is the standard symplectic form of $T^*S^n$,\nwe define a symplectic automorphism $\\tau : (T^*S^n, \\omega_0) \\stackrel{\\sim}{\\to} (T^*S^n, \\omega_0) $ as follows\n\\begin{align}\n\\label{eqn generalized dehn twist definition}\n\\tau(u;v) = \\left\\{\\begin{matrix}\n\\sigma(e^{i r(\\mu{(u;v)})})(u;v) \\hspace{5pt} &\\text{if} \\hspace{5pt} v \\neq 0_{n+1},\\\\ \n(-u;0_{n+1})  \\hspace{5pt} &\\text{if} \\hspace{5pt} v = 0_{n+1}.\n\\end{matrix}\\right.\n\\end{align}  \n\nLet $(M^{2n}, \\omega)$ be a symplectic manifold and let $L \\simeq S^n$ be a Lagrangian sphere in $M$. \nBy the Lagrangian neighborhood theorem \\cite{MR0286137}, there is a neighborhood $N(L) \\supset L$ and a symplectomorphism $\\phi : T^*S^n \\stackrel{\\sim}{\\rightarrow} N(L)$.\nWe define a generalized Dehn twist $\\tau_L$ along $L$ as follows:\n\\begin{align}\n\\label{eqn definition of generalized Dehn along L}\n\\tau_L (x) = \\left\\{\\begin{matrix}\n(\\phi \\circ \\tau \\circ \\phi^{-1}) (x) \\hspace{5pt} &\\text{if} \\hspace{5pt} x \\in N(L), \\\\\nx \\hspace{5pt} &\\text{if} \\hspace{5pt} x \\notin N(L).\n\\end{matrix}\\right.\n\\end{align} \nNote that the support of $\\tau_L$ is contained in $N(L)$.\nFrom now on, a generalized Dehn twist will just be called a Dehn twist. \n\n\\begin{remark}\n\t\\label{rmk specific dehn twists}\n\tIn this paper, we will use two specific Dehn twists $\\tau, \\tilde{\\tau} : T^*S^n \\stackrel{\\sim}{\\to} T^*S^n$ which are defined by Equation \\eqref{eqn generalized dehn twist definition} and two functions $r, \\tilde{r} : [0,\\infty) \\to \\mathbb{R}$.\n\tThe function $r$ (resp.\\ $\\tilde{r}$) defining $\\tau$ (resp.\\ $\\tilde{\\tau}$) satisfies the above conditions in addition to $r(t) = \\pi$ for all $t \\leq \\tfrac{\\epsilon}{2}$ (resp.\\ $\\tilde{r}'(0) < 0)$.\n\tTwo Dehn twists $\\tau$ and $\\tilde{\\tau}$ are equivalent in the sense that $\\tau \\circ \\tilde{\\tau}^{-1}$ is a Hamiltonian isotopy.\n\\end{remark}\n\nDehn twists have been studied extensively by Seidel. \nFor example, Seidel \\cite{MR1743463} proved the following theorem.\n\\begin{theorem}\n\t\\label{lagrangian surgery theorem}\n\tLet $\\alpha$ be a Lagrangian sphere and $\\beta$ be a Lagrangian submanifold of a symplectic manifold $M$. \n\tIf $\\alpha$ and $\\beta$ intersect transversally at only one point, $\\beta \\# \\alpha $ is Lagrangian isotopic to $\\tau_{\\alpha}(\\beta)$ where $\\beta \\# \\alpha$ is a Lagrangian surgery of $\\alpha$ and $\\beta$.\n\\end{theorem}\n\nWe prove Theorem \\ref{lagrangian surgery theorem} in the special case that $\\beta$ is also a sphere and $M = P(\\alpha, \\beta)$, as an illustration of the ``spinning'' procedure.\nTo define ``spinning'', we use the following notation.\nLet $y \\in S^{n-1} \\subset \\mathbb{R}^{n}$. \nThen, \n\\begin{gather*}\n\\psi_y : T^*S^1 \\simeq S^1 \\times \\mathbb{R}  \\to T^*S^n, \\\\\n(\\theta, t) \\mapsto (\\cos \\theta (0_n,1) + \\sin \\theta (y,0); t\\cos \\theta (y,0) - t \\sin \\theta (0_n,1) )\n\\end{gather*}\nis a symplectic embedding. \nLet $W_y$ be the embedded symplectic surface $\\psi_y(T^*S^1)$. \n\\begin{definition}\n\t\\label{def spinning}\n\tGiven a curve $C$ in $T^*S^1$, its {\\em spun image} $S(C)$ is $\\cup_{y \\in S^{n-1}} \\psi_y(C)$. \n\\end{definition}\n\n\n\\begin{proof} [Proof of Theorem \\ref{lagrangian surgery theorem}]\n\tWe use $T^*\\alpha$ and $T^*\\beta$ to indicate neighborhoods of $\\alpha$ and $\\beta$ inside $M = P(\\alpha, \\beta)$. \n\tLet $p$ be the intersection point of $\\alpha$ and $\\beta$.\n\tThen, $T^*_p\\alpha = \\beta \\cap T^*\\alpha$. \n\tThe closure of $T^*_p\\alpha$ is denoted by $D_p^-$; we use $D$ to indicate that this is a disk and the subscript $p$ means that $p$ is the center of $D_p^-$. \n\tThe meaning of the negative sign in $D_p^-$ will be explained in the next section. \n\tSince $\\tau_{\\alpha}$ is supported on \n\t$T^*\\alpha$, \n\t$$\\tau_{\\alpha}(\\beta) = \\tau_{\\alpha}(\\beta \\cap T^*\\alpha) \\cup \\tau_{\\alpha}(\\beta \\setminus T^*\\alpha) = \\tau_{\\alpha}(D_p^-) \\cup (\\beta \\setminus T^*\\alpha).$$\n\n\tThere exists \n\t$\\phi : T^*S^n \\stackrel{\\sim}{\\to} T^*\\alpha$ \n\tsuch that $\\tau_{\\alpha} = \\phi \\circ \\tau \\circ \\phi^{-1}$. \n\tWithout loss of generality, $\\phi(0_n,1;0_{n+1}) = p$ and \n\t$$D_p^- = \\phi(\\{(0_n,1;ty,0) \\hspace{0.2em} | \\hspace{0.2em} t \\in \\mathbb{R}, \\hspace{0.2em} y \\in S^{n-1} \\subset \\mathbb{R}^n \\}).$$\n\tThen,\n\t\\begin{align*}\n\t(\\phi \\circ \\tau_{\\alpha} \\circ \\phi^{-1}) (D_p^-)& = (\\phi \\circ \\tau)(\\{(0_n,1;ty,0) \\hspace{0.2em} | \\hspace{0.2em} t \\in \\mathbb{R}, \\hspace{0.2em} y \\in S^{n-1} \\subset \\mathbb{R}^n \\}) \\\\\n\t &= \\cup_{y \\in S^{n-1}}\\phi( \\{\\tau(0_n,1;ty,0) \\hspace{0.2em} | \\hspace{0.2em} t \\in \\mathbb{R} \\}).\n\t\\end{align*} \n\tThus, $\\tau_{\\alpha}(D_p^+)$ is given by spinning with respect to $p$ and $\\phi$. \t\n\tSimilarly, we can construct a Lagrangian isotopy connecting $\\tau_{\\alpha}(\\beta)$ and $\\beta \\# \\alpha$ by spinning. \n\tThis completes the proof.  \n\\end{proof}\n\n\n\n\n\n\n\\section{Lagrangian branched submanifolds}\n\nIn Section 3.1, we will define Lagrangian branched submanifolds.\nIn Section 3.2, we will introduce a construction of a fibered neighborhood of a Lagrangian branched submanifolds. \nIn Section 3.3, we will defined the notion of ``carried by'' by using a fibered neighborhood.\nIn Section 3.4, we will introduce the generalized Penner construction.\nFinally, we will give a proof of Theorem \\ref{branched surface thm} in Section 3.5.\n\n\n\n\\subsection{Lagrangian branched submanifolds.}\n\nThurston \\cite{MR1435975} used train tracks, which are 1-dimensional branched submanifolds of surfaces, and defined the notion of ``carried by a train track\".\nIn this subsection, we generalize train tracks.  \n\nThe generalization of a train track is an $n$-dimensional branched submanifold of a $2n$-dimensional manifold. \nWe define the $n$-dimensional branched submanifolds with local models, as Floyd and Oertel defined a branched surface in a 3-dimensional manifold in \\cite{MR721458}, \\cite{MR746535}.\nFor our definition, we need a smooth function $s: \\mathbb{R} \\to \\mathbb{R}$ such that $s(t) = 0$ if $t \\leq 0$ and $s(t)>0$ if $t>0$. \n\n\\begin{definition}\n\t\\label{def of branched submfd}\n\tLet $M^{2n}$ be a smooth manifold.\n\t\\begin{enumerate}\n\t\t\\item A subset $\\mathcal{B} \\subset M$ is an {\\em $n$-dimensional branched submanifold} if for every $p \\in \\mathcal{B}$, there exists a chart $\\phi_p:U_p \\stackrel{\\sim}{\\to} \\mathbb{R}^{2n}$ about $p$ such that $\\phi_p(p) = 0$ and $\\phi_p(\\mathcal{B} \\cap U_p)$ is a union of submanifolds $L_0, L_1, \\cdots, L_k$ for some $k \\in \\{0, \\cdots, n\\}$, where \n\t\t\\begin{align*}\n\t\tL_i := \\{(x_1, \\cdots, x_n, s(x_1), s(x_2), \\cdots, s(x_i), 0, \\cdots, 0) \\in \\mathbb{R}^{2n} \\hspace{0.2em} | \\hspace{0.2em} x_j \\in \\mathbb{R}\\}. \n\t\t\\end{align*}\n\t\t\\item A {\\em sector} of $\\mathcal{B}$ is a connected component of the set of all points in $\\mathcal{B}$ that are locally modeled by $L_0$, i.e., $k=0$.\n\t\t\\item A {\\em branch locus $Locus(\\mathcal{B})$} of $\\mathcal{B}$ is the complement of all the sectors. \n\t\t\\item Let $(M^{2n}, \\omega)$ be a symplectic manifold.\n\t\tA subset $\\mathcal{B} \\subset M$ is a {\\em Lagrangian branched submanifold} if for every $p \\in \\mathcal{B}$, there exists a Darboux chart $\\phi_p:(U_p,\\omega|_{U_p}) \\stackrel{\\sim}{\\to} (\\mathbb{R}^{2n},\\omega_0)$ about $p$, satisfying the conditions of an $n$-dimensional branched submanifold. \n\t\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\n\t\\label{rmk some facts for lagrangian branched submanifold}\n\t$\\mbox{}$\n\t\\begin{enumerate}\n\t\t\\item At every point $p$ of a branched submanifold $\\mathcal{B}$, the tangent plane $T_p\\mathcal{B}$ is well-defined.\n\t\tMoreover, if $\\mathcal{B}$ is Lagrangian, then $T_p\\mathcal{B}$ is a Lagrangian subspace of $T_pM$.  \n\t\t\\item A point on the branch locus is (a smooth version of) an arboreal singularity in the sense of Nadler \\cite{MR3626601}. \n\t\\end{enumerate}\n\\end{remark}\n\n\\begin{exmp}\n\t\\mbox{}\n\t\\label{ex Lagrangian branched submanifold}\n\t\\begin{enumerate}\n\t\t\\item Every train track of a surface equipped with an area form is a Lagrangian branched submanifold.\n\t\t\\item Let $(M,\\omega)$ be a symplectic manifold and let $L_1$ and $L_2$ be two Lagrangian submanifold of $M$ such that \n\t\t$$L_1 \\pitchfork L_2, \\hspace{0.2em} L_1 \\cap L_2 = \\{p\\}.$$\n\t\tThe Lagrangian surgery of $L_1$ and $L_2$ at $p$ will be denoted by $L_2 \\#_p L_1$.\n\t\tThen, $L_2 \\#_p L_1 \\cup L_1$ and $L_2 \\#_p L_1 \\cup L_2$ are examples of Lagrangian branched submanifold. \n\t\\end{enumerate}\n\\end{exmp}\n\nIn Section 3.3, we will define the notion of {\\em ``carried by''} which appears in Theorems \\ref{branched surface thm} - \\ref{generalized theorem}.\nIn order to define the notion of carried by, we will construct a fibered neighborhood first in Section 3.2.\n\\vskip0.2in\n\n\\subsection{Construction of fibered neighborhoods.}\nLet $\\mathcal{B}$ be a Lagrangian branched submanifold. \nA fibered neighborhood $N(\\mathcal{B})$ of $\\mathcal{B}$ is, roughly speaking, a codimension zero compact submanifold with boundary and corners of $M$, which is foliated by Lagrangian closed disks which are called {\\em fibers}. \n\n\\begin{definition}\n\t\\label{def fibered ngbd}\n\tA {\\em fibered neighborhood of $\\mathcal{B}$} is a union $\\cup_{p \\in \\mathcal{B}} F_p$, where $\\{F_p \\hspace{0.2em} | \\hspace{0.2em} p \\in \\mathcal{B} \\}$ is a family of Lagrangian disks satisfying \n\t\\begin{enumerate}\n\t\t\\item for any $p \\in \\mathcal{B}$, $F_p \\pitchfork \\mathcal{B}$,\n\t\t\\item for any $p, q \\in \\mathcal{B}$, either $F_p = F_q$ or $F_p \\cap F_q = \\varnothing$,\n\t\t\\item there exists a closed neighborhood $U \\subset \\mathcal{B}$ of $Locus(\\mathcal{B})$, such that $\\{F_p \\hspace{0.2em} | \\hspace{0.2em} p \\in U \\}$ is a smooth family over each local sheet $L_i \\cap U$, \n\t\t\\item for each sector $S$ of $\\mathcal{B}$, $\\{F_p \\hspace{0.2em} | \\hspace{0.2em} p \\in S \\setminus U \\}$ is a smooth family,\n\t\t\\item if $p \\in S \\cap \\partial U$ where $S$ is a sector of $\\mathcal{B}$, then, for any sequence $\\{q_n \\in S \\setminus U \\}_{n \\in \\mathbb{N}}$, \n\t\t\\begin{gather*}\n\t\t\\lim_{n \\to \\infty} F_{q_n} \\text{  is a Lagrangian disk such that  } \\lim_{n \\to \\infty} F_{q_n} \\subset \\mathring{F}_p = F_p \\setminus \\partial F_p.\n\t\t\\end{gather*}\n \t\\end{enumerate} \n\\end{definition}\n\nWe will now give a specific construction of a fibered neighborhood $N(\\mathcal{B})$. \n\n\\begin{remark}\n\t\\label{rmk natural embedding}\n\tBy the Lagrangian neighborhood theorem \\cite{MR0286137}, for any Lagrangian submanifold $L$ of $M$, there exists a small neighborhood $\\mathcal{N}(L)$ of the zero section of $T^*L$ such that a symplectic embedding $i_L : \\mathcal{N}(L) \\hookrightarrow M$ is defined on $\\mathcal{N}(L)$. \n\tWithout loss of generality, we assume that $\\mathcal{N}(L)$ is a closed neighborhood. \n\tThan, $\\mathcal{N}(L)$ is foliated by closed Lagrangian disks $\\mathcal{N}(L) \\cap T^*_pL$. \n\\end{remark}\n\\vskip0.2in\n\n\\noindent{\\em Fibration over $L(\\ell)$.}\nFirst, we will construct fibers near the branch locus.\nFor each connected component $\\ell$ of $Locus(\\mathcal{B})$, we choose a small closed Lagrangian neighborhood $L(\\ell)$ of $\\ell$. \nThen, by Remark \\ref{rmk natural embedding}, there exists a symplectic embedding \n$$ i_{L(\\ell)} : \\mathcal{N}(L(\\ell)) \\hookrightarrow M.$$\nLet $U(L(\\ell)) = i_{L(\\ell)}(\\mathcal{N(L(\\ell))})$.\n\nBy choosing a sufficiently small $L(\\ell)$, without loss of generality, the following hold:\n\\begin{gather*}\ni_{L(\\ell)}(\\mathcal{N}(L(\\ell)) \\cap T^*_xL(\\ell)) \\cap \\mathcal{B} \\neq \\varnothing \\text{  for all } x \\in L(\\ell), \\\\\ni_{L(\\ell)}(\\mathcal{N}(L(\\ell)) \\cap T^*_xL(\\ell)) \\pitchfork \\mathcal{B} \\text{  for all } x \\in L(\\ell), \\\\\nU(\\ell) \\cap U(\\ell') = \\varnothing \\text{  if  }  \\ell \\neq \\ell'.\n\\end{gather*}\n\nIf $p \\in \\mathcal{B}$ is close to the branch locus, in other words, there is a connected component $\\ell$ of $Locus(\\mathcal{B})$ such that $p \\in \\mathcal{B} \\cap U(\\ell)$, then there exists $x \\in L(\\ell)$ such that $p \\in i_{L(\\ell)}(\\mathcal{N}(L(\\ell)) \\cap T^*_xL(\\ell))$.\nLet $F_p := i_{L(\\ell)}(\\mathcal{N}(L(\\ell)) \\cap T^*_xL(\\ell))$.\nThen, $F_p$ is a closed Lagrangian disk containing $p$.\n\nIf $p \\in \\ell$, then, \n\\begin{gather}\n\\label{eqn local properties}\nF_p \\pitchfork \\mathcal{B} \\text{ and } \\partial F_p \\cap \\mathcal{B} = \\varnothing.\n\\end{gather}\nMoreover, by choosing a sufficiently small $L(\\ell)$, for every $p \\in \\mathcal{B} \\cap U(\\ell)$, Equation \\eqref{eqn local properties} holds. \n\n\\begin{figure}[h]\n\t\\centering\n\t\\input{construction_of_fibered_neighborhood.pdf_tex}\n\t\\caption{Black curves are part of a Lagrangian branched submanifold and the black marked points denote a connected component $\\ell$ of $Locus(\\mathcal{B})$. \n\tin (a), $L(\\ell)$ is in red, and the fibers $F_p$, for $p \\in \\mathcal{B} \\cap U(\\ell)$, are in blue;\n\t(b) and (c) are not allowed by Equation \\eqref{eqn local properties}; \n\tand in (d), the red and green boxes are examples of $N(S)$.}\n\t\\label{figure construction of fibered neighborhood}\n\\end{figure}\n\nAfter possibly renaming $U(\\ell)$, from now we assume that\n$$U(\\ell) = \\cup_{p \\in L(\\ell)}F_p.$$\nIf $p \\in \\mathcal{B}\\cap U(\\ell)$, then there is a unique $q \\in L(\\ell)$ such that $p \\in F_q$. \nWe define $F_p := F_q$.\nThus, for $p \\in \\mathcal{B}$ which is close to $Locus(\\mathcal{B})$, i.e., $p \\in U(\\ell)$ for some connected component $\\ell$ of $Locus(\\mathcal{B})$, we can define a fiber $F_p$ at $p$.\n\\vskip0.2in\n\n\\noindent{\\em Fibration over $S \\setminus \\cup_{\\ell} U(\\ell)$.}\nIf $p \\in \\mathcal{B} \\setminus \\cup_\\ell U(\\ell)$, then there is a sector $S$ of $\\mathcal{B}$ containing $p$. \nSince $S$ is Lagrangian, there is an embedding $i_S : \\mathcal{N}(S) \\hookrightarrow M$. \nWe can assume that $\\mathcal{N}(S)$ is small enough, so that \n\\begin{gather*}\nF_q \\cap i_S\\big(\\mathcal{N}(S)\\big) \\subset \\mathring{F_q} = F_q \\setminus \\partial F_q \\text{  for any  } q \\in \\mathcal{B}\\cap U(\\ell), \\\\\n\\big( i_S(\\mathcal{N}(S)) \\setminus \\cup U(\\ell) \\big) \\cap \\big( i_{S'}(\\mathcal{N}(S')) \\setminus \\cup U(\\ell) \\big) = \\varnothing.\n\\end{gather*}\nFigure \\ref{figure construction of fibered neighborhood} (d) represents examples of $\\mathcal{N}(S)$.\nWe define $B_p$ for all $p \\in S$ by setting\n$$B_p:= i_S\\big(\\mathcal{N}(S) \\cap T^*_pS\\big).$$\n\nFor any sector $S$, let $S^\\circ:= S - \\cup_{\\ell} \\operatorname{Int} U(\\ell)$.\nThen, $S^\\circ$ is a Lagrangian submanifold with boundary. \nThe boundary of $S^\\circ$ is a union of $S(\\ell) := S \\cap \\partial\\big(U(\\ell)\\big)$.\nWe fix a tubular neighborhood of $S(\\ell)$, which is contained in $S^\\circ$, and identify the tubular neighborhood with $S(\\ell) \\times [0,1)$. \nFor convenience, we will pretend that $S(\\ell) \\times [0,1] \\subset S$ and $S(\\ell) \\times \\{0\\} = S(\\ell)$. \n\nIf $p \\in S^\\circ$ does not lie in any $S(\\ell) \\times (0,1)$, then we set $F_p:=B_p$. \n\\vskip0.2in\n\n\\noindent{\\em Interpolation on $S(\\ell)\\times[0,1]$.}\nIf there is a connected component $\\ell$ of $Locus(\\mathcal{B})$ such that $p = (p_0, t_0) \\in S(\\ell) \\times (0,1)$, we will construct $F_{p=(p_0,t_0)}$ from $F_{(p_0,0)}$ and $F_{(p_0,1)}$.\nTo do this, we need the following facts:\n\nFirst, by the definition of $F_{(p_0,0)}$, $F_{(p_0,0)} \\cap i_S\\big(\\mathcal{N}(S)\\big)$ is a Lagrangian disk which contains $(p_0,0)$, and is transversal to $\\mathcal{B}$ at $(p_0,0)$. \nAlso, $B_{(p_0,0)}$ is also a Lagrangian disk which contains $(p_0,0)$, and is transversal to $\\mathcal{B}$. \n\nBy the Lagrangian neighborhood theorem \\cite{MR0286137}, we can see $F_{(p_0,0)} \\cap i_S\\big(\\mathcal{N}(S)\\big)$ as a graph of a closed section in $T^*B_{(p_0,0)}$, i.e.,\n$$F_{(p_0,0)} \\cap i_S\\big(\\mathcal{N}(S)\\big) = i_{B_{(p_0,0})}\\big(\\text{the graph of a closed section in  }T^*B_{(p_0,0)}\\big).$$\nEvery closed section of $T^*B_{(p_0,0)}$ is an exact section because $B_{(p_0,0)}$ is a disk.\nThus, there is a function $f_{(p_0,0)} : B_{(p_0,0)} \\to \\mathbb{R}$ such that \n$$ F_{(p_0,0)} \\cap i_S\\big(\\mathcal{N}(S)\\big) = i_{B_{(p_0,0)}}\\big(\\text{the graph of } df_{(p_0,0)}\\big).$$\n\nSecond, we will fix a Riemannian metric $g$ compatible with $\\omega$ for convenience.\nBy restricting $g$ to $S$, $S$ is equipped with a Riemannian metric $g|_S$.\nThus, for any $t_0 \\in [0,1]$, there is a parallel transport induced by $g|_S$, between $T_{(p_0,t_0)}S$ and $T_{(p_0,0)}S$ along $\\gamma_{p_0}(t) = (p_0, t) \\in S$. \nAlso, $g$ induces a bijection between $T_{(p_0,0)}S$ (resp.\\ $T_{(p_0,t_0)}S$) and $T_{(p_0,0)}^*S$ (resp.\\ $T_{(p_0,t_0)}^*S$).\nThus, there is a bijective map between $B_{(p_0,t_0)}$ and $B_{(p_0,0)}$. \n\nFrom those two facts, we define a function $f_{(p_0,t)} : B_{(p_0,t)} \\to \\mathbb{R}$ as follows:\n\\begin{gather*}\nf_{(p_0,t)} : B_{(p_0,t)} \\stackrel{\\sim}{\\to} B_{(p_0,0)} \\xrightarrow{(1-t)f_{(p_0,0)}} \\mathbb{R}.\n\\end{gather*}\nThe first arrow comes from the parallel transport induced by $g$. \n\nThere is a map,\n\\begin{gather*}\nh : \\cup_{(p_0,t) \\in S(\\ell) \\times [0,1]} B_{(p_0,t)} \\to M, \\\\\nx \\in B_{(p_0,t)} \\mapsto i_{B_{(p_0,t)}} ( d f_{B_{(p_0,t)}}(x)).\n\\end{gather*}\nIt is easy to check that $h(p_0,t) = (p_0,t)$.\nMoreover, $h$ is the associated (time 1) flow of the Hamiltonian vector field of \n$$f_{(p_0,t)} : \\cup_{(p_0,t) \\in S(\\ell) \\times [0,1]} B_{(p_0,t)} \\to \\mathbb{R}.$$\nFinally, we construct $F_{(p_0,t_0)}$ by setting \n$$F_{(p_0,t_0)}:= h(B_{(p_0,t_0)}).$$\n\n\\begin{figure}[h]\n\t\\centering\n\t\\input{fibered_neighborhood.pdf_tex}\n\t\\caption{Black curves are part of a Lagrangian branched submanifold and marked points denote $\\ell$; \n\t\tin (a), $U(\\ell)$ is shaded blue, the vertical line segments are fibers;\n\t\t(b) fiber $F_p$ for $p \\notin S(\\ell) \\times (0,1]$ is in green; \n\t\tand in (c), fiber $F_p$ for $p \\in S(\\ell) \\times (0,1]$ is in red}\n\t\\label{figure fibered neighborhood}\n\\end{figure}\n\nA fibered neighborhood $N(\\mathcal{B})$ is given by the union of fibers, i.e., $ N(\\mathcal{B}) = \\cup_{p \\in \\mathcal{B}} F_p$.\nNote that the construction of $N(\\mathcal{B})$ is not unique because the construction depends on some choices, including the choices of $L(\\ell)$ and a Riemannian metric $g$.\n\\vskip0.2in\n\n\\subsection{Associated branched manifolds and the notion of ``carried by''.} \nWe constructed a fibered neighborhood $N(\\mathcal{B})$. \nFrom now on, we will define a projection map defined on $N(\\mathcal{B})$, in order to define the notion of ``carried by''.\n\nFirst, we define {\\em the associated branched manifold $\\mathcal{B}^*$} of $\\mathcal{B}$. \n\\begin{definition}\n\tLet $\\mathcal{B}$ be a Lagrangian branched submanifold of $M$ and let $N(\\mathcal{B})$ be a fibered neighborhood of $\\mathcal{B}$.\n\tThen, the {\\em associated branched submanifold} $\\mathcal{B}^*$ is defined by setting \n\t$$ \\mathcal{B}^* := N(\\mathcal{B}) \/ \\sim, \\hspace{0.2em} x \\sim y \\text{  if  } \\exists F_p \\text{  such that  } x, y \\in F_p.$$\t \n\\end{definition} \nLet  $\\pi : N(\\mathcal{B}) \\to \\mathcal{B}^*$ denote the quotient map. \n\nBefore defining the notion of ``carried by'', we note that $\\mathcal{B}^*$ is not contained in $M$.\nMoreover, since $\\mathcal{B}^*$ is a branched manifold, we can define the branch locus and sectors of $\\mathcal{B}^*$ as follows:\n\\begin{definition}\n\t\\label{def of branch locus and sector for abstract branched manifold}\n\t$\\mbox{}$\n\t\\begin{enumerate}\n\t\t\\item A {\\em sector} of $\\mathcal{B}^*$ is a connected component of \n\t\t$$ \\{ p \\in \\mathcal{B}^* \\hspace{0.2em} | \\hspace{0.2em} p \\text{  has a neighborhood which is homeomorphic to  } \\mathbb{R}^n \\}.$$\n\t\t\\item A {\\em branch locus} of $\\mathcal{B}^*$ is the complement of all the sectors. \n\t\\end{enumerate}\n\\end{definition}\n\n\\begin{figure}[h]\n\t\\centering\n\t\\input{abstract_branched_manifold.pdf_tex}\t\t\n\t\\caption{(a) represents $\\pi:N(\\mathcal{B}) \\to \\mathcal{B}^*$.\n\t\tIn $N(\\mathcal{B})$, the blue, red, and green represent $\\pi^{-1}(S_0)$, $\\pi^{-1}(S_1)$, and $\\pi^{-1}(S_2)$, where $S_i$ is the corresponding sector of $\\mathcal{B}^*$;\n\t\t(b) represents $F_x$ where $x$ is in the branch locus of $\\mathcal{B}^*$ in (a).}\n\t\\label{fig abstract branched manifold}\n\\end{figure}\n\n\\begin{remark}\n\t\\label{rmk nonuniqueness of fibered neigbhrohood}\n\t\\mbox{}\n\t\\begin{enumerate}\n\t\t\\item The construction of $N(\\mathcal{B})$ depends on the choices of a Riemannian metric, a closed neighborhood of $Locus(\\mathcal{B})$, and so on. \n\t\tThus, fibered neighborhoods $N(\\mathcal{B})$ of $\\mathcal{B}$ are not unique.\n\t\tHowever, $\\mathcal{B}^*$ is unique as a branched manifold since $\\mathcal{B}$ and $\\mathcal{B}^*$ are equivalent as branched manifolds.\n\t\t \n\t\tIn the rest of this paper, when it comes to a Lagrangian branched submanifold $\\mathcal{B}$, we will consider a triple $(\\mathcal{B}, N(\\mathcal{B}), \\mathcal{B}^*)$ with an arbitrary choice of $N(\\mathcal{B})$. \n\t\tMoreover, for any triple $(\\mathcal{B}, N(\\mathcal{B}), \\mathcal{B}^*)$, the projection map is denoted by $\\pi$ for convenience. \n\t\t\\item A fibered neighborhood$N(\\mathcal{B})$ is a union of fibers, i.e., $N(\\mathcal{B}) = \\cup_{p \\in \\mathcal{B}} F_p$.\n\t\tIn the equation, $\\mathcal{B}$ is an index set. \n\t\tHowever, there is a possibility of having two distinct points $p, q \\in \\mathcal{B}$ such that $F_p = F_q$. \n\t\tFrom now on, we will use $\\mathcal{B}^*$ as an index set.\n\t\tIn other words, we replace $F_p$ by $\\pi^{-1}(\\pi(p))$.\n\t\tBy abuse of notation, $F_x$ denotes $\\pi^{-1}(x)$ for all $x \\in \\mathcal{B}^*$. \n\t\t\\item Let $x$ be a branch point of $\\mathcal{B}^*$. \n\t\tThen, there are sectors $S_0, S_1, \\cdots, S_l$ of $\\mathcal{B}^*$ for some $l$ such that \n\t\t\\begin{gather*}\n\t\tx \\in \\bar{S}_i \\text{  for every  } i = 0, 1, \\cdots, l \\\\\n\t\tF_x \\cap \\overline{\\pi^{-1}(S_0)} = F_x  \\text{  and  } F_x \\cap \\overline{\\pi^{-1}(S_i)} \\subset \\mathring{F}_x = F_x \\setminus \\partial F_x  \\text{  for every  } i = 1, 2, \\cdots, l.\n\t\t\\end{gather*}\n\t\tFigure \\ref{fig abstract branched manifold} represents this. \n\t\\end{enumerate}\n\\end{remark}\n\t\nFrom now on, we define the notion of ``carried by''. \nIf a Lagrangian submanifold $L$ (resp.\\ a Lagrangian branched submanifold $\\mathcal{L}$) is contained in $N(\\mathcal{B})$, there is a restriction of $\\pi$ to $L$ (resp.\\ $\\mathcal{L}$).\nFor convenience, we will simply use $\\pi$ instead of $\\pi|_L : L \\to \\mathcal{B}^*$.  \n\n\\begin{definition}\n\t\\label{def singular\/regular point}\n\tLet $L$ be a Lagrangian submanifold (resp.\\ $\\mathcal{L}$ be a Lagrangian branched submanifold) of $N(\\mathcal{B})$. \n\t\\begin{enumerate}\n\t\t\\item $x \\in L$ (resp.\\ $\\mathcal{L}$) is a {\\em regular point} of $\\pi$ if $L \\pitchfork F_{\\pi(x)}$ (resp.\\ $\\mathcal{L} \\pitchfork F_{\\pi(x)}$) at $x$.\n\t\t\\item $x \\in L$ (resp.\\ $\\mathcal{L}$) is a {\\em singular point} of $\\pi$ if $x$ is not regular point of $\\pi: L' \\to \\mathcal{B}^*$.\n\t\tMoreover, values of $\\pi$ at singular points are called {\\em singular values} of $\\pi$.\n\t\t$y \\in \\mathcal{B}^*$ is a {\\em singular value} of $\\pi$ if there is a singular point $x$ of $\\pi$ such that $\\pi(x) = y$. \n\t\t\\item $L$ is {\\em minimally singular with respect to $\\mathcal{B}$} if $\\pi: L \\to \\mathcal{B}^*$ has no singular value on the branch locus of $\\mathcal{B}^*$ and $|F_x \\cap L| = |F_y \\cap L|$, for any non-singular value $x$ and $y$ which lie in the same sector of $\\mathcal{B}^*$, where $|\\cdot|$ means the cardinality of a set.\n\t\\end{enumerate}\n\\end{definition}\n\t\n\\begin{definition}\n\t\\label{def of carried by}\n\t$\\mbox{}$\n\t\\begin{enumerate}\n\t\t\\item A Lagrangian submanifold $L$ (resp.\\ a Lagrangian branched submanifold $\\mathcal{L}$) is {\\em strongly carried by} a Lagrangian branched submanifold $\\mathcal{B}$ if $L$ (resp.\\ $\\mathcal{L}$) is Hamiltonian isotopic to a Lagrangian submanifold $L'$ (resp.\\ a Lagrangian branched submanifold $\\mathcal{L}'$) such that $L'$ (resp.\\ $\\mathcal{L}'$) $\\subset N(\\mathcal{B})$ and $\\pi: L' \\to \\mathcal{B}^*$ has no singular value.\n\t\t\\item A Lagrangian submanifold $L$ (resp.\\ a Lagrangian branched submanifold $\\mathcal{L}$) is {\\em weakly carried by} a Lagrangian branched submanifold $\\mathcal{B}$ if $L$ (resp.\\ $\\mathcal{L}$) is Hamiltonian isotopic to a Lagrangian submanifold $L'$ (resp.\\ a Lagrangian branched submanifold $\\mathcal{L}'$) such that $L'$ (resp.\\ $\\mathcal{L}'$) $\\subset N(\\mathcal{B})$, $L'$ is minimally singular, and $\\pi: L' \\to \\mathcal{B}^*$ has countably many singular values.\n\t\t\\item Two Lagrangian submanifolds $L$ and $L'$ that are weakly carried by $\\mathcal{B}$ are {\\em weakly fiber isotopic} if there exists an isotopy for $L$ and $L'$ through Lagrangians that are weakly carried by $\\mathcal{B}$.\n\t\\end{enumerate}\n\\end{definition}\nIn the rest of this paper, if $L$ is weakly carried by $\\mathcal{B}$, then we will assume that $L \\subset N(\\mathcal{B})$ and $L$ is minimally singular with respect to $\\mathcal{B}$. \n \nNote that the notion of ``carried by'' used by Thurston in \\cite{MR956596} is our notion of ``strongly carried by''. \nThurston showed that for a pseudo-Anosov surface automorphism $\\psi : S \\stackrel{\\sim}{\\to} S$, there is a 1-dimensional branched submanifold $\\tau$ which is called a train track such that $\\psi(\\tau)$ is strongly carried by $\\tau$.\n\nOur higher-dimensional generalization is slightly weaker,\ni.e., for some symplectic automorphism $\\psi : (M,\\omega) \\stackrel{\\sim}{\\to} (M,\\omega)$, we construct a Lagrangian branched submanifold $\\mathcal{B}_{\\psi}$ such that $\\psi(\\mathcal{B}_{\\psi})$ is weakly carried by $\\mathcal{B}_{\\psi}$.\nIn other words, we allow non-transversality at countably many point $p \\in \\mathcal{B}_{\\psi}$. \nHowever, we allow only one type of non-transversality. \nIn the rest of the present subsection, we will describe the unique type of non-transversality.   \n\n\\begin{definition}\n\t\\label{def of singularity}\n\tLet $L$ be weakly carried by $\\mathcal{B}$. \n\tA {\\em singular component} $V$ of $\\pi : L \\to \\mathcal{B}$ is a connected component of the set of all singular points of $\\pi$.\n\\end{definition}\n\n\\begin{exmp}\n\t\\label{exmp of simplest singularity}\n\tLet $M_*$ be a symplectic manifold $T^*\\mathbb{R}^n \\simeq \\mathbb{R}^{2n}$ equipped with the canonical symplectic form.\n\tThe zero section $\\mathcal{B}_*:= \\mathbb{R}^n \\times 0 \\subset \\mathbb{R}^{2n}$ is a Lagrangian branched submanifold. \n\tWe assume that the fibered neighborhood $N(\\mathcal{B}_*)$ is $M_*$, by setting $F_p := T^*_p \\mathbb{R}^n$ for all $p \\in \\mathbb{R}^n = \\mathcal{B}_*$.\n\tThen, a Lagrangian submanifold\n\t$$L_*:= \\{(tx,x) \\in \\mathbb{R}^n \\times \\mathbb{R}^n \\hspace{0.2em} | \\hspace{0.2em} t \\in \\mathbb{R}, x \\in S^{n-1} \\subset \\mathbb{R}^n\\}$$\n\tis weakly carried by $\\mathcal{B}_*$ and $\\pi_*$ has only one singular component\n\t$$V_* := \\{ (0,x) \\hspace{0.2em} | \\hspace{0.2em} x \\in S^{n-1} \\}.$$\n\\end{exmp}\n\n\\begin{definition}\n\t\\label{def of real blow-up type}\n\tA singular component $V$ of $\\pi : L \\to \\mathcal{B}$ is of {\\em real blow-up type} if there exists an open neighborhood $U$ of $V$ and a symplectomorphism $\\phi : U \\stackrel{\\sim}{\\to} M_*$ such that $\\phi(U \\cap \\mathcal{B}) = \\mathcal{B}_*, \\phi(V) = V_*$, and $\\phi^{-1} \\circ \\pi_* \\circ \\phi = \\pi$, where $M_*, \\mathcal{B}_*$, $V_*$, and $\\pi_*$ are defined in Example \\ref{exmp of simplest singularity}.\n\\end{definition}\n\n\\begin{definition}\n\t\\label{def of fully\/weakly carried by}\n\tA Lagrangian submanifold $L$ (resp.\\ a Lagrangian branched submanifold $\\mathcal{L}$) is {\\em carried by} a Lagrangian branched submanifold $\\mathcal{B}$ if $L$ (resp.\\ $\\mathcal{L}$) is weakly carried by $\\mathcal{B}$ and every singular component of $\\pi$ (resp.\\ $\\pi$) is a singular component of real blow-up type.\n\\end{definition}\n\n\n\n\\subsection{The generalized Penner construction}  \nIn this subsection, we give a higher-dimensional generalization of Penner construction \\cite{MR930079} of pseudo-Anosov surface automorphisms. \nThe generalization replaces Dehn twists by generalized Dehn twists along Lagrangian spheres.\n\n\\underline{Generalized Penner construction} : Let $M$ be a symplectic manifold.\nA symplectic automorphism $\\psi : M \\stackrel{\\sim}{\\to} M$ is of {\\em generalized Penner type} if there are two collections $A = \\{\\alpha_1, \\cdots, \\alpha_m \\}$ and $B = \\{\\beta_1, \\cdots, \\beta_l \\}$ of Lagrangian spheres \nsuch that \n\\begin{gather*}\n\\alpha_i \\cap \\alpha_j = \\varnothing, \\hspace{0.2em} \\beta_i \\cap \\beta_j =\\varnothing, \\hspace{0.2em} \\text{for all} \\hspace{0.2em} i \\neq j, \\\\\n\\alpha_i \\pitchfork \\beta_j \\hspace{0.2em} \\text{for all} \\hspace{0.2em} i, j,\n\\end{gather*}\nso that\n$\\psi$ is a product of positive powers of Dehn twists $\\tau_i$ along $\\alpha_i$ and negative powers of Dehn twists $\\sigma_j$ along $\\beta_j$, subject to the condition that every sphere appear in the product. \n\nA Lagrangian sphere $\\alpha_i$ (resp.\\ $\\beta_j$) is called a {\\em positive} (resp.\\ {\\em negative}) sphere since only positive powers of $\\tau_i$ (resp.\\ negative powers of $\\sigma_j$) appear in $\\psi$.\n\n\\begin{remark}\n\t$\\mbox{}$\n\t\\begin{enumerate}\n\t\t\\item In Theorems \\ref{branched surface thm} and \\ref{lamination thm}, we can assume that the symplectic manifold $M$ is a plumbing space.\n\t\tEvery $\\tau_i$ (resp.\\ $\\sigma_j$) is supported on a neighborhood of $\\alpha_i$ (resp.\\ $\\beta_j$), which is denoted by $T^*\\alpha_i$ (resp.\\ $T^*\\beta_j$). \n\t\tThus, $\\psi$ is supported on the union of $T^*\\alpha_i$ and $T^*\\beta_j$. \n\t\tBy the transversality condition $\\alpha_i \\pitchfork \\beta_j$, we can identify the union with a plumbing space $P=P(\\alpha_1, \\cdots, \\alpha_m, \\beta_1, \\cdots, \\beta_l)$.\n\t\tThus, it is suffices to prove Theorems \\ref{branched surface thm} and \\ref{lamination thm} on the plumbing space $P$, which we take to be connected.\n\t\t\\item In \\cite{MR930079}, the Penner construction required that $A$ and $B$ fill the surface $S$, i.e., the complement of $A \\cup B$ is a union of disks and annuli, one of whose boundary components is a component of $\\partial S$.\n\t\tIn the current paper, we do not require the analogue of the filling condition since we only construct an invariant Lagrangian branched submanifold and an invariant Lagrangian lamination, not an invariant singular foliation on all of $M$.\n\t\\end{enumerate}\n\\end{remark}\n\nIn the rest of this subsection, we define a set of Lagrangian branched submanifolds in a plumbing space $P(\\alpha_1, \\cdots, \\alpha_m, \\beta_1, \\cdots, \\beta_l)$. \nWe start from the simplest plumbing space, having one positive and one negative sphere intersecting at only one point. \n\\begin{exmp}\n\t\\label{exmp of Lagrangian branched submanifold}\n\tLet $\\alpha$ and $\\beta$ be $n$-dimensional spheres and let $M$ be a plumbing $P(\\alpha, \\beta)$ which is plumbed at only one point $p$.\n\tLet $\\beta \\#_p \\alpha$ be the Lagrangian surgery of $\\alpha$ and $\\beta$ at $p$ such that $\\beta \\#_p \\alpha \\simeq \\tau_{\\alpha}(\\beta) \\simeq \\sigma_{\\beta}^{-1}(\\alpha)$.\n\tSee Figure \\ref{LBS example}, which represents the case $n=1$.\n\tThe cross-shape is the plumbing space $P(\\alpha,\\beta)$, where $\\alpha$ is the horizontal line and $\\beta$ is the vertical line. \n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\input{exam_lag_br_sub.pdf_tex}\n\t\t\\caption{The blue curves represent $D_p^+$ in the left hand picture and $D_p^-$ in the right hand picture, the red curves represent $N_p$ in both.}\n\t\t\\label{LBS example}\n\t\\end{figure}\n\n\tThe {\\em neck $N_p$ at $p$} connecting $\\alpha$ and $\\beta$ is the closure of $(\\beta\\#_p \\alpha) - (\\alpha \\cup \\beta)$.\n\tIn Figure \\ref{LBS example}, $N_p$ is drawn in red.\n\tThe {\\em positive disk $D_p^+$at $p$} is the closure of $\\alpha - (\\beta\\#_p \\alpha)$ and the {\\em negative disk $D_p^-$ at $p$} is the closure of $\\beta - (\\beta \\#_p \\alpha)$. \n\tThe disks $D_p^{\\pm}$ are drawn in blue in Figure \\ref{LBS example}.\n\tThen, by attaching $D_p^+$ or $D_p^-$ to $\\beta \\#_p \\alpha$, we obtain  Lagrangian branched submanifolds $(\\beta \\#_p \\alpha) \\cup \\alpha$ and $(\\beta \\#_p \\alpha) \\cup \\beta$.\n\\end{exmp}\n\nOn a general plumbing space $M = P(\\alpha_1, \\cdots, \\alpha_m, \\beta_1, \\cdots, \\beta_l)$ with positive spheres $\\alpha_i$ and negative spheres $\\beta_j$, we similarly construct Lagrangian branched submanifolds.\nMore precisely, given a plumbing point $p, N_p, D_p^+, D_p^-$ are the closures of $(\\beta_j \\#_p \\alpha_i) - (\\alpha_i \\cup \\beta_j), \\alpha_i - (\\beta_j \\#_p \\alpha_i), \\beta_j - (\\beta_j \\#_p \\alpha_i)$ respectively.\nLet $D_p$ be either $D_p^+$ or $D_p^-$.\nThen, we construct a Lagrangian branched submanifold $\\mathcal{B}$ by setting \n\\begin{align}\n\\label{eqn definition of lbs}\n\\mathcal{B} := \\cup_{i}(\\alpha_i - \\cup_{p \\in \\alpha_i}D_p^+) \\bigcup \\cup_{j}(\\beta_j - \\cup_{p \\in \\beta_j}D_p^-) \\bigcup \\cup_p N_p \\bigcup \\cup_p D_p.\n\\end{align}\nThere are $2^N$ possible choices of $\\mathcal{B}$, where $N$ is the number of plumbing points. \nLet $\\mathbb{B}$ be the set of all $2^N$ Lagrangian branched submanifolds constructed above.\n\n\n\n\\subsection{Proof of Theorem \\ref{branched surface thm}}\n\nIn this subsection, let $M = P(\\alpha_1, \\cdots, \\alpha_m, \\beta_1, \\cdots, \\beta_l)$, let $\\tau_i$ (resp.\\ $\\sigma_j$) be a Dehn twist along $\\alpha_i$ (resp.\\ $\\beta_j$), and let $\\psi$ be of generalized Penner type. \n\nIn the rest of the paper, we assume that every Dehn twist $\\tau_i$ and $\\sigma_j$ satisfies the following:\n\\begin{enumerate}\n\t\\item $\\tau_i$ (resp.\\ $\\sigma_j$) is supported on a small neighborhood $T^*\\alpha_i$ (resp.\\ $T^*\\beta_j$) of $\\alpha_i$ (resp.\\ $\\beta_j$). \n\t\\item $\\tau_i$ (resp.\\ $\\sigma_j$) agrees with the antipodal map on $\\alpha_i$ (resp.\\ $\\beta_j$).\n\\end{enumerate}\n\nWe define the following: \n\\begin{gather}\n\\label{eqn definition of disks and primes}\n\\bar{D}_p^+ := \\tau_i(D_p^+), \\hspace{0.3em} \\bar{D}_p^- := \\sigma_j^{-1}(D_p^-) \\hspace{1em} \\text{   if }  p \\in \\alpha_i \\cap \\beta_j, \\\\\n\\nonumber\n\\alpha_i' := \\alpha_i - \\cup_{p \\in \\alpha_i} (D_p^+ \\cup \\bar{D}_p^+), \\hspace{0.3em} \\beta_j' : = \\beta_j - \\cup_{p \\in \\beta_j} (D_p^- \\cup \\bar{D}_p^-).  \n\\end{gather}\nIn words, $\\bar{D}_p^+$ (resp.\\ $\\bar{D}_p^-$) is a neighborhood of an antipodal point of $p$ in $\\alpha_i$ (resp.\\ $\\beta_j$).\nWe are assuming that $D_p^{\\pm}$ and $\\bar{D}_p^{\\pm}$ are sufficiently small so that they are disjoint to each other.\n\nRecall that $\\mathbb{B}$ is the set of Lagrangian branched submanifolds defined in Section 3.2.\n\n\\begin{lemma}\n\t\\label{lem1}\n\tFor all $k$, there exists a function $F_{\\tau_k}: \\mathbb{B} \\to \\mathbb{B}$ such that $\\tau_k(\\mathcal{B})$ is carried by $F_{\\tau_k}(\\mathcal{B})$  for all $\\mathcal{B} \\in \\mathbb{B}$.\n\tSimilarly, there is a function $F_{\\sigma_j^{-1}}:\\mathbb{B} \\to \\mathbb{B}$ for all $j$ such that $\\sigma_j^{-1}(\\mathcal{B})$ is carried by $F_{\\sigma_j^{-1}}(\\mathcal{B})$. \n\\end{lemma}\n\n\\begin{proof}\n\tIn this proof, $\\tau_k$ is given by Equation \\eqref{eqn definition of generalized Dehn along L} and $\\tilde{\\tau}:T^*S^n \\stackrel{\\sim}{\\to} T^*S^n$ defined in Section 2.2, i.e., $\\tau_k = \\phi \\circ \\tilde{\\tau} \\circ \\phi^{-1}$ in a neighborhood of $\\alpha_k$, where $\\phi$ is an identification of $T^*S^n$ and a neighborhood of $\\alpha_k$.  \n\t\n\tGiven $\\mathcal{B} \\in \\mathbb{B}, \\mathcal{B}$ admits the following decomposition:\n\t\\begin{align}\n\t\\label{eqn decomposition}\n\t\\mathcal{B}= \\cup_i \\alpha_i' \\bigcup \\cup_j \\beta_j' \\bigcup \\cup_p N_p \\bigcup \\cup_p \\bar{D}_p^+ \\bigcup \\cup_p \\bar{D}_p^- \\bigcup \\cup_p D_p,\n\t\\end{align}\t\n\twhere $D_p$ is either $D_p^+$ or $D_p^-$.\n\tThis follows from Equations \\eqref{eqn definition of lbs} and \\eqref{eqn definition of disks and primes}.\n\t\n\tWe prove the first statement for $\\tau_k$; the proof for $\\sigma_j^{-1}$ is analogous.\n\tOur strategy is to apply $\\tau_k$ to $\\alpha_i', \\beta_j', N_p, \\bar{D}_p^{\\pm}$, and $D_p^{\\pm}$.\n\tWe claim the following: \n\t\\begin{itemize}\n\t\t\\item[(i)] $\\tau_k(\\alpha'_i) = \\alpha_i', \\tau_k(\\beta'_j) = \\beta_j'$ and they are strongly carried by $\\alpha_i', \\beta_j'$.\n\t\t\\item[(ii)] If $p \\notin \\alpha_k$, then\n\t\t$\\tau_k(N_p) = N_p, \\tau_k(D_{p}^{\\pm}) = D_{p}^{\\pm}, \\tau_k(\\bar{D}_{p}^{\\pm}) = \\bar{D}_{p}^{\\pm}$ and they are strongly carried by $N_p,\\hspace{2pt} D_{p}^{\\pm}, \\hspace{2pt} \\bar{D}_{p}^{\\pm}$.\n\t\t\\item[(iii)] If $ p \\in \\alpha_k$, then\n\t\t$\\tau_k(D_p^+) = \\bar{D}_p^+, \\tau_k(\\bar{D}_p^+) = D_p^+$,  $\\tau_k(\\bar{D}_p^-) = \\bar{D}_p^-$ and they are strongly carried by $\\bar{D}_p^+, D_p^+, \\bar{D}_p^-$.\n\t\t\\item[(iv)] If $ p \\in \\alpha_k$, then $\\tau_k(D_p^-)$ and $\\tau_k(N_p)$ are obtained by spinning with respect to $p$.\n\t\tMoreover, $\\tau_k(D_p^-)$ is strongly carried by $N_p \\cup (\\alpha_k - D_p^+ )$ and $\\tau_k(N_p)$ is carried by $N_p \\cup (\\alpha_k - D_p^+ )$. \n\t\\end{itemize}\n\tBy Equation \\eqref{eqn decomposition} and $(i)$--$(iv), \\tau_k(\\mathcal{B})$ is carried by $\\mathcal{B}'$ such that \n\t\\begin{align}\n\t\\label{eqn define F_}\n\t\\mathcal{B}' = \\cup_i \\alpha_i' \\bigcup \\cup_j \\beta_j' \\bigcup \\cup_p N_p \\bigcup \\cup_p \\bar{D}_p^+ \\bigcup \\cup_p \\bar{D}_p^- \\bigcup \\cup_p \\tilde{D}_p,\n\t\\end{align}\n\twhere $\\tilde{D}_p$ is $D_p$ if $p \\notin \\alpha_k$ and $D_p^+$ if $p \\in \\alpha_k$.\n\tThen, $F_{\\tau_k} : \\mathbb{B} \\to \\mathbb{B}$ is defined by $F_{\\tau_k}(\\mathcal{B}) = \\mathcal{B}'$. \t\n\n\t$(i)$ Since $\\tau_k$ agrees with the antipodal map on $\\alpha_k, \\tau_k(\\alpha_k') = \\alpha_k'$ and $\\tau_k(\\alpha_k')$ is strongly carried by $\\alpha_k'$.\n\tMoreover, since $\\tau_k$ is supported on $T^*\\alpha_k, \\alpha_i'$ does not intersect the support of $\\tau_k$ for all $i \\neq k$. \n\tThus, $\\tau_k(\\alpha_i')$ agrees with $\\alpha_i'$ and $\\tau_k(\\alpha_i')$ is strongly carried by itself.\n\tThe same proof applies to $\\tau_k(\\beta_j')$.\n\t\n\t$(ii)$ and $(iii)$ are proved in the same way.\n\t\n\t$(iv)$ We compute $\\tau_k(D_p^-)$ and $\\tau_k(N_p)$ by spinning with respect to $p$ and $\\phi$.\n\tWe assume $\\phi((1,0_{n};0_{n+1})) = p$ without loss of generality.\n\tUsing the notation from Section 2, $D_p^-$ and $N_p$ are contained in $\\cup_{y \\in S^{n-1}} \\phi(W_y)$. \n\tThus,\n\t\\begin{align}\n\t\\label{eqn spinning of disk}\n\t\\tau_k(D_p^-) &= \\cup_{y \\in S^{n-1}} (\\phi \\circ \\tilde{\\tau} \\circ \\phi^{-1}) (D_p^- \\cap \\phi(W_y))\\\\\n\t\\nonumber & = \\cup_{y \\in S^{n-1}} (\\phi(\\tilde{\\tau}|_{W_y}(\\phi^{-1}(D_p^-)\\cap W_y))) \\\\\n\t\\nonumber &= \\cup_{y \\in S^{n-1}} \\tau_k(D_p^-) \\cap \\phi(W_y), \\\\\n\t\\label{eqn spinning of neck}\n\t\\tau_k(N_p) &= \\cup_{y \\in S^{n-1}} (\\phi \\circ \\tilde{\\tau} \\circ \\phi^{-1}) (N_p \\cap \\phi(W_y)) \\\\\n\t\\nonumber &= \\cup_{y \\in S^{n-1}} \\phi(\\tilde{\\tau}|_{W_y}(\\phi^{-1}(N_p)\\cap W_y))\\\\\n\t\\nonumber &= \\cup_{y \\in S^{n-1}} \\tau_k(N_p) \\cap \\phi(W_y).\n\t\\end{align}\n\tThe restriction $\\tilde{\\tau}|_{W_y}$ is a Dehn twist on $W_y \\simeq T^*S^1$ along the zero section. \n\tThus, we obtain Figure \\ref{figure for lemma 1}, which represents intersections $\\phi(W_y) \\cap D_p^-, \\phi(W_y) \\cap N_p, \\phi(W_y) \\cap \\tau_k (D_p^-)$, and $\\phi(W_y) \\cap \\tau_k(N_p)$.\n\tEquation \\eqref{eqn spinning of neck} and Figure \\ref{figure for lemma 1} imply that $\\tau_k(N_p)$ is carried by $N_p \\cup (\\alpha_k - D_p^+)$ and $\\tau_k(D_p^-)$ is strongly carried by $N_p \\cup (\\alpha_k - D_p^+)$.  \n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\input{fig_for_lem1.pdf_tex}\t\t\n\t\t\\caption{In the left picture, the blue curve represents $D_p^-$ and the red curve represents $N_p$; in the middle picture, the red curve represents $\\tau_k(N_p)$; and in the right picture, the blue curve represents $\\tau_k(D_p^-)$.}\n\t\t\\label{figure for lemma 1}\n\t\\end{figure}\n \n\tThen, $(i)$--$(iv)$ and Equation \\eqref{eqn decomposition} prove that $\\tau_k(\\mathcal{B}) $ is carried by $F_{\\tau_k}(\\mathcal{B})$.\n\\end{proof}\n\n\\begin{lemma}\n\t\\label{lem2}\n\tIf $L$ is a Lagrangian submanifold which is carried by (resp.\\ weakly carried by) $\\mathcal{B} \\in \\mathbb{B}$, then $\\tau_k(L)$ is carried (resp.\\ weakly carried) by $F_{\\tau_k}(\\mathcal{B})$. \n\tThe case of $\\sigma_j^{-1}$ is analogous.\n\\end{lemma}\n\n\\begin{proof}\n\tWe can assume that $L$ is contained in an arbitrary small neighborhood of $\\mathcal{B}$.\n\tThen, we apply a Dehn twist $\\tau_k$ as we did in the proof of Lemma \\ref{lem1}.\n\tThe details are similar to the proof of Lemma \\ref{lem1}.\t\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{branched surface thm}]\n\tLet $\\psi: M \\stackrel{\\sim}{\\to} M$ be a symplectic automorphism of generalized Penner type.\n\tThen, we can write $\\psi = \\delta_1 \\circ \\cdots \\circ \\delta_l$ where $\\delta_k$ is a Dehn twist $\\tau_i$ or $\\sigma_j^{-1}$.\n\tWe then define $F_{\\psi} = F_{\\delta_1} \\circ \\cdots \\circ F_{\\delta_l}: \\mathbb{B} \\to \\mathbb{B}$.\n\tBy Lemma \\ref{lem1}, we have specific functions $F_{\\tau_i}$ and $F_{\\sigma_j^{-1}}$ acting on $\\mathbb{B}$. \n\n\tWe claim that $F_{\\psi}$ is a constant map, i.e., $\\operatorname{Im}(F_{\\psi})$ is a point $\\mathcal{B}_{\\psi}$, which we define as follows: in Equation \\eqref{eqn definition of lbs}, we set $D_p = D_p^+$ for $p \\in \\alpha_i \\cap \\beta_j$ if the last $\\tau_i$ in $\\psi$ appears later than the last $\\sigma_j^{-1}$, and $D_p = D_p^-$ otherwise.\n\tNote that every Dehn twist $\\tau_i$ and $\\sigma_j^{-1}$ appears in $\\psi$, thus $\\mathcal{B}_{\\psi}$ is well-defined.\n\tBy Equation \\eqref{eqn define F_}, $F_{\\psi}(\\mathcal{B}) = \\mathcal{B}_{\\psi}$ for all $\\mathcal{B} \\in \\mathbb{B}$.\n\\end{proof}\n\n\\begin{remark}\n\t\\label{rmk position of singular value}\n\t$\\mbox{}$\n\t\\begin{enumerate}\n\t\t\\item Note that a singular value of $\\pi : \\psi^m(L) \\to \\mathcal{B}^*$, which is defined in Section 3.1, can be moved by isotoping $\\psi^m(L)$.\n\t\t\\item We observe that every singular value of $\\pi:\\psi^m(\\mathcal{B}_{\\psi}) \\to \\mathcal{B}^*$ lies near $\\pi(p), \\pi\\big(\\tau_i(p)\\big),$ or $\\pi\\big(\\sigma_j^{-1}(p)\\big)$ by isotoping, where $p$ is a plumbing point. \n\t\tMore precisely, let $S_{p, \\mathcal{B}_{\\psi}}^+$ (resp.\\ $S_{p, \\mathcal{B}_{\\psi}}^-$) be the sector of $\\mathcal{B}_{\\psi}^*$ containing $\\pi(p)$ if $D_p = D_p^+$ (resp.\\ $D_p^-$), where $D_p$, $D_p^+$ and $D_p^-$ are defined in Section 3.4.\n\t\tSimilarly, let $\\bar{S}_{p, \\mathcal{B}_{\\psi}}^{\\pm}$ be $\\pi(\\bar{D}_p^{\\pm})$, where $\\bar{D}_p^{\\pm}$ is defined in Equation \\eqref{eqn definition of disks and primes}. \n\t\tThen, by isotoping $\\psi^n(\\mathcal{B}_{\\psi})$, every singular value of $\\pi:\\psi^n(\\mathcal{B}_{\\psi}) \\to \\mathcal{B}^*$ lies in the interiors of $S_{p, \\mathcal{B}_{\\psi}}^{\\pm}$ or $\\bar{S}_{p, \\mathcal{B}_{\\psi}}^{\\pm}$ for some plumbing point $p$. \n\t\t\n\t\tFor convenience, let the {\\em centers} of $S_p^\\pm$, $\\bar{S}_p^+$, $\\bar{S}_p^-$ be $p$, $\\tau(p)$, $\\sigma^{-1}(p)$ respectively. \n\t\tThen, the singular values in $S_p^\\pm$, $\\bar{S}_p^\\pm$ lie near the centers of them. \n\t\tMoreover, $S_{p, \\mathcal{B}_{\\psi}}^{\\pm}$ and $\\bar{S}_{p, \\mathcal{B}_{\\psi}}^{\\pm}$ will be simply called $S_p^{\\pm}$ and $\\bar{S}_p^{\\pm}$. \n\t\\end{enumerate}\n\\end{remark}\n\n\n\n\n\n\\section{Construction of Lagrangian laminations}\nIn this section, we will prove Theorems \\ref{lamination thm} and \\ref{generalized theorem}..\n\n\\subsection{Singular and regular disks}\nIn order to prove Theorems \\ref{lamination thm} and \\ref{generalized theorem}, we would like to construct a stable Lagrangian lamination $\\mathcal{L}$ of a symplectic automorphism $\\psi$ from a Lagrangian branched submanifold $\\mathcal{B}_{\\psi}$. \nOne of the difficulties is that singular components occur naturally.\nIn order to control the singularities, we introduce singular and regular disks.\n\nIn general, we assume that $\\mathcal{B}_{\\psi}^*$, the associated branched manifold, can be decomposed into the union of a finite number of disks $S_i \\simeq \\mathbb{D}^n$, which are called {\\em singular disks}, and $R_j \\simeq \\mathbb{D}^n$, which are called {\\em regular disks}, i.e.,\n\\begin{gather}\n\\label{eqn decomp into singular\/regular disks}\n\\mathcal{B}_{\\psi}^* = \\bigcup_i S_i \\cup \\bigcup_j R_j,\n\\end{gather} \nsuch that\n\\begin{enumerate}\n\t\\item each singular disk $S_i$ is either a closed disk contained in the interior of a sector of $\\mathcal{B}_{\\psi}^*$ or a closure of a sector,\n\t\\item $S_i \\cap S_j = \\varnothing$ for any $i \\neq j$,\n\t\\item every singular value of $\\pi: \\psi^m(\\mathcal{B}_{\\psi}) \\to \\mathcal{B}_{\\psi}$ after weakly fibered isotopy lies in $\\cup_i \\mathring{S}_i$ for all $m \\in \\mathbb{N}$, where $\\mathring{S}_i$ is the interior of $S_i$,\n\t\\item each regular disk $R_j$ is obtained by cutting up a closure of a sector minus $\\cup_i \\mathring{S_i}$,\n\t\\item $S_i$ and $R_j$ (resp.\\ $R_i$ and $R_j$ for $i \\neq j$) meet only along their boundaries.  \n\\end{enumerate}\n\n\\begin{remark}\n\t\\label{rmk singular value condition}\nFrom now on, for any compact Lagrangian submanifold $L$ which is carried by $\\mathcal{B}_{\\psi}$, we will assume that every singular value of $\\pi : L \\to \\mathcal{B}_{\\psi}$ lies in the interior of a singular disk by Remark \\ref{rmk position of singular value}.\n\\end{remark}\n\nIf $\\mathcal{B}^*$ admits Equation \\eqref{eqn decomp into singular\/regular disks}, then one obtains a decomposition of $N(\\mathcal{B})$ as follows:\n\\begin{gather*}\nN(\\mathcal{B}) = \\bigcup_i \\pi^{-1}(S_i) \\cup \\bigcup_j \\pi^{-1}(R_j).\n\\end{gather*}\n\nIn Section 4.2, we will define braids $b(L,S_i)$ for a Lagrangian $L$, which is carried by $\\mathcal{B}_{\\psi}$, and a singular disk $S_i$. \nBy Theorem \\ref{branched surface thm}, there exist sequences of braids ${b(\\psi^m(L),S_i)}_{m \\i \\mathbb{N}}$, and we will construct limits of those braid sequences as $m \\to \\infty$.\nWe then extend the limit lamination to a Lagrangian lamination of $\\pi^{-1}(S_i)$ in Section 4.3, and a Lagrangian lamination of $\\pi^{-1}(R_j)$ in Section 4.4.\n\n\\begin{remark}\n\t\\label{rmk identify with cotangent bundle of disk}\n\t\\mbox{}\n\t\\begin{enumerate}\n\t\t\\item In Section 4.3 (resp.\\ Section 4.4), we will construct a Lagrangian lamination on  $\\overline{\\pi^{-1}(\\mathring{S}_i)} \\subset \\pi^{-1}(S_i)$ (resp.\\ $\\overline{\\pi^{-1}(\\mathring{R}_j)} \\subset \\pi^{-1}(R_j)$),  the closure of $\\pi^{-1}(\\mathring{S}_i)$.\n\t\tThis is because $\\pi^{-1}(S_i)$ (resp.\\ $\\pi^{-1}(R_j)$) is not a (closed) submanifold of $M$ if $S_i$ (resp.\\ $R_j$) intersects the branch locus of $\\mathcal{B}^*$. \n\t\t\n\t\tFigure \\ref{fig abstract branched manifold} is an example. \n\t\tIf $S_1$ in Figure \\ref{fig abstract branched manifold} is a singular disk, then $\\pi^{-1}(S_1)$ is the union of the red box in Figure \\ref{fig abstract branched manifold} (a) and $F_x$.\n\n\t\\item \n\tWe note that $(\\overline{\\pi^{-1}(\\mathring{S}_i)}, \\omega)$ (resp.\\ $(\\overline{\\pi^{-1}(\\mathring{R}_j)}, \\omega)$) and $(DT^*\\mathcal{D}, \\omega_0)$ are symplectomorphic to each other, where $\\mathcal{D}$ is a closed disk, $DT^*\\mathcal{D}$ is a disk cotangent bundle of $\\mathcal{D}$, and $\\omega_0$ is the standard symplectic form of the cotangent bundle. \n\t\n\tIn order to construct a symplectomorphism, we will consider the following:\n\tLet $\\mathcal{D}$ be a largest Lagrangian disk such that \n\t$$\\mathcal{D} \\subset \\pi^{-1}(S_i) \\cap \\mathcal{B} \\hspace{0.2em} (\\text{resp. } \\pi^{-1}(R_j) \\cap \\mathcal{B}) \\text{  and  } \\pi(\\mathcal{D}) = S_i \\hspace{0.2em} (\\text{resp. } R_j).$$\n\t\n\tBy Remark \\ref{rmk natural embedding}, there exists a symplectic embedding $i_{\\mathcal{D}}:\\mathcal{N}(\\mathcal{D}) \\hookrightarrow M$. \n\tIt is easy to construct a vector field on $i_{\\mathcal{D}}(\\mathcal{N}(\\mathcal{D}))$, whose (time 1) flow moves $i_{\\mathcal{D}}(\\mathcal{N}(\\mathcal{D}) \\cap T_p^*\\mathcal{D})$ to $F_{\\pi(p)}$ for any $p \\in \\operatorname{Int}(\\mathcal{D})$. \n\tMoreover, the vector field is a symplectic vector field, i.e., the flow is a symplectomorphism, and $$\\cup_{p \\in \\operatorname{Int}(\\mathcal{D})}i_{\\mathcal{D}}(\\mathcal{N}(\\mathcal{D}) \\cap T^*_p \\mathcal{D}) \\simeq \\cup_{p \\in \\operatorname{Int}(\\mathcal{D})} F_{\\pi(p)} = \\pi^{-1}(\\mathring{S}_i) (\\text{resp.  } \\pi^{-1}(\\mathring{R}_j)).$$\n\t\n\tBy taking the closures, $i_{\\mathcal{D}}(\\mathcal{N}(\\mathcal{D})) \\simeq \\overline{\\pi^{-1}(\\mathring{S}_i)}$ (resp.\\ $\\overline{\\pi^{-1}(\\mathring{R}_j)}$). \n\tMoreover, $\\mathcal{N}(\\mathcal{D})$ is symplectomorphic to $DT^*\\mathcal{D}$.\n\tThus, $DT^*\\mathcal{D}$ and $\\overline{\\pi^{-1}(\\mathring{S}_i)}$  (resp.\\ $\\overline{\\pi^{-1}(\\mathring{R}_j)}$) are symplectomorphic.\n\t\\end{enumerate}\n\\end{remark}\n\nFrom now on, we assume that a symplectic automorphism $\\psi$ is of generalized Penner type until the end of Section 4.3. \n\\vskip0.2in\n\n\n\\noindent{\\em Decomposition of $\\mathcal{B}^*_{\\psi}$ for $\\psi$ of generalized Penner type.}\nWe will now explain how to decompose $\\mathcal{B}^*$, the associated branched manifold of $\\mathcal{B} \\in \\mathbb{B}$, into the union of specific singular and regular disks. \nNote that $\\mathbb{B}$ is defined in Section 3.4.\n\nBy Remark \\ref{rmk position of singular value}, after weakly fiber isotoping, every singular value of $\\pi:\\psi^m(\\mathcal{B}) \\to \\mathcal{B}^*$ lies in the interior of $S_p$ or $\\bar{S}_p^\\pm$, where $S_p = S_P^+$ if $D_p= D_P^+$ and $S_p = S_p^-$ if $D_p = D_p^-$.\nLet $S_p$ and $\\bar{S}_p^\\pm$ be the specific singular disks of $\\mathcal{B}^*$.\n\nWe will divide the complement of singular disks from $\\mathcal{B}^*$, i.e.,\n\\begin{gather}\n\\label{eqn regular part}\n\\mathcal{B}^* \\setminus \\big(\\cup_p S_p \\sqcup \\cup_p \\bar{S}_p^+ \\sqcup \\cup_p \\bar{S}_p^- \\big),\n\\end{gather}\ninto regular disks.\nIn order to do this, we use a symplectic submanifold $W^{2n-2} \\subset M^{2n}$, which is defined as follows: \nFor each $\\alpha_i$ (resp.\\ $\\beta_j$), there is an equator $C_{\\alpha_i}$ (resp.\\ $C_{\\beta_j}$) $\\simeq S^{n-1}$ such that\n\\begin{enumerate}\n\t\\item for any plumbing point $p \\in \\alpha_i$ (resp.\\ $\\beta_j$), $p$ lies on $C_{\\alpha_i}$ (resp.\\ $C_{\\beta_j}$),\n\t\\item if $p \\in \\alpha_i \\cap \\beta_j$, then $T^*C_{\\alpha_i} \\equiv T^*C_{\\beta_j}$ near $p$.  \n\\end{enumerate}\nNote that the equators on a Lagrangian sphere $\\alpha_i$ (resp.\\ $\\beta_j$) are defined using an identification $\\phi_{\\alpha_i} : \\alpha_i \\stackrel{\\sim}{\\to} S^n$ (resp.\\ $\\phi_{\\beta_j}: \\beta_j \\stackrel{\\sim}{\\to} S^n$).\nThus, by choosing proper identification $\\phi_{\\alpha_i}$ and $\\phi_{\\beta_j}$, we can assume the existence of $C_{\\alpha_i}$ and $C_{\\beta_j}$.\nThen,\n$$W:= \\cup_{i} T^*C_{\\alpha_i} \\bigcup \\cup_{j}T^*C_{\\beta_j}$$\nis a $(2n-2)$-dimensional symplectic submanifold of $M$.\n\nWe cut \\eqref{eqn regular part} along $\\pi(W)$.\nThese are the regular disks $R_k$.\nEach $R_k$ is a manifold with corners, where the corners are at $R_k \\cap \\pi(W) \\cap S_l$.\n\n\\subsection{Braids}\nConsider the decomposition of $\\mathcal{B}_{\\psi}^*$ into specific singular and regular disks as in the previous subsection.\nIn this subsection, for a given compact Lagrangian submanifold $L$ which is carried by $\\mathcal{B}_{\\psi}$, we define a sequence of braids $b(\\psi^m(L), S_i)$ corresponding to $\\psi^m(L)$ over the boundary of each singular disk $S_i$ of $\\mathcal{B}_{\\psi}^*$.\nLemma \\ref{lem3} gives an inductive description of the sequences $b(\\psi^m(L), S_i)$. \nWe will end this subsection by constructing limits of $b(\\psi^m(L), S_i)$ as $m \\to \\infty$. \n \nFor a singular disk $S, \\pi^{-1}(\\partial S) = \\cup_{p \\in \\partial S} F_p$ is a $\\mathbb{D}^n$-bundle over $\\partial S \\simeq S^{n-1}$. \nNote that we use $\\mathbb{D}^n$ to indicate a closed disk, and we will use $\\mathring{\\mathbb{D}}^n$ to indicate an open disk.\nLet $\\varphi : \\pi^{-1}(\\partial S) \\stackrel{\\sim}{\\to} S^{n-1} \\times \\mathbb{D}^n$ be a bundle map.\nIf $L$ is a Lagrangian submanifold which is carried by $\\mathcal{B}_{\\psi}$, then, for all $p \\in \\partial S$, $\\varphi(L \\cap F_p)$ is a finite collection of isolated points in $\\mathbb{D}^n$; recall that $\\pi : L \\to \\mathcal{B}^*$ has no singular value on $\\partial S$.\nThus, $\\varphi(L \\cap \\pi^{-1}(\\partial S))$ can be identified with a map from $\\partial S \\simeq S^{n-1}$ to the configuration space $\\operatorname{Conf}_{l}(\\mathbb{D}^n)$ of $l$ points on $\\mathbb{D}^n$ where $l = l(L,S)$, i.e., a braid.\n\nWe explained that $L \\cap \\pi^{-1}(\\partial S)$ could be identified with a braid. \nSince $L$ is a Lagrangian submanifold of $M$, the braid corresponding to $L \\cap \\pi^{-1}(\\partial S)$ satisfies a symplectic property.\nThe symplectic property is the following: \nFor the bundle map $\\varphi: \\pi^{-1}(\\partial S) \\stackrel{\\sim}{\\to} S^{n-1} \\times \\mathbb{D}^n, (\\varphi^{-1})^*(\\omega)$ is a 2-form on $S^{n-1} \\times \\mathbb{D}^n$ such that $(\\varphi^{-1})^*(\\omega)$ is zero on $\\varphi \\big(L \\cap \\pi^{-1}(\\partial S) \\big)$.\n\nFrom now on, we will define the braids on the boundary of a singular disk $S$.  \nLet $f : S^{n-1} \\to \\operatorname{Conf}_{l}(\\mathbb{D}^n)$ for some $l$. \nIn other words, there are maps \n$$f_1, \\cdots, f_{l} : S^{n-1} \\to \\mathbb{D}^n,$$ \nsuch that $f(p) = \\{f_1(p), \\cdots, f_{l}(p)\\}$ as $f_i(p) \\neq f_j(p)$ for all $i \\neq j$.\nWe define \n\\begin{align*}\nB(f) := \\{ (p,& f_i(p)) \\in S^{n-1} \\times \\mathbb{D}^n \\hspace{0.2em} | \\hspace{0.2em} i \\in \\{1, \\cdots, \\ell\\} \\},\\\\\n\\tilde{Br}_{\\partial S}:= \\{ \\varphi^{-1}\\big(B(&f)\\big) \\hspace{0.2em} | \\hspace{0.2em} f: S^{n-1} \\to \\operatorname{Conf}_{l}(\\mathbb{D}^n) \\text{ for some } l \\text{  such that},\\\\\n& (\\varphi^{-1})^*(\\omega) \\text{  is a zero on  } B(f) \\}.\n\\end{align*}\nNote that $\\tilde{Br}_{\\partial S}$ is a set of closed subsets of $\\pi^{-1}(\\partial S)$ and independent of $\\varphi$. \n\nWe define an equivalence relation on $\\tilde{Br}_{\\partial S}$ as follows:\n$b_0 \\sim b_1$ for $b_i \\in \\tilde{Br}_{\\partial S}$ if there exists a smooth 1-parameter family $b_t \\in \\tilde{Br}_{\\partial S}$ connecting $b_0$ and $b_1$.\nLet $Br_{\\partial S} := \\tilde{Br}_{\\partial S}\/\\sim$. \n\n\\begin{definition}\n\tLet $\\mathcal{B} \\in \\mathbb{B}$ and let $S$ be a singular disk of $\\mathcal{B}$.\n\tIf $L$ is a Lagrangian submanifold which is carried by $\\mathcal{B}$,\n\tthen the {\\em braid $b(L,S)$} of $L$ on $S$ is the braid isotopy class of $Br_{\\partial S}$ which is given by\n\t$$ b(L,S) = \\big[L \\cap \\pi^{-1}(\\partial S)\\big] \\in Br_{\\partial S}.$$\n\\end{definition} \nRecall that $\\mathbb{B}$ is a set of Lagrangian branched submanifold defined in Section 3.4 and for any $\\mathcal{B} \\in \\mathbb{B}$, we decompose $\\mathcal{B}$ into the union of specific singular disks and regular disks, introduced in Section 4.1.\n\n\\begin{lemma}\n\t\\label{lem3}\n\tLet $L$ be a Lagrangian submanifold of $M$ which is carried by $\\mathcal{B}$.\n\tFor a given singular disk $S$ of $F_{\\tau_i}(\\mathcal{B})$ (resp.\\ $F_{\\sigma_j^{-1}}(\\mathcal{B})$), there exist maps $f_k$ from $\\tilde{Br}_{S_{i_k}}$ to $\\tilde{Br}_{S}$, where $S_{i_k}$ is a singular disk of $\\mathcal{B}$, and there exist closed sets $\\mathring{b}_{i_k} \\in \\tilde{Br}_{S_{i_k}}$, such that $b(\\tau_i(L),S)$ (resp.\\ $b(\\sigma_j^{-1}(L),S)$) is $\\big[\\bigsqcup_k f_{k}(\\mathring{b}_{i_k})\\big] \\in Br_{\\partial S}$.\n\\end{lemma}\nRecall the functions $F_{\\tau_i}$ and $F_{\\sigma_j^{-1}}$ in Lemma \\ref{lem3} are defined in Lemma \\ref{lem1}.\n\n\\begin{proof}[Proof of Lemma \\ref{lem3}]\nIn Steps 1--3, we prove Lemma \\ref{lem3} for a particular example; this is just for notational simplicity.\nIn Step 4, we briefly describe how to prove the general case.\n\n\nThe example we consider is the Lagrangian branched submanifold $\\mathcal{B}_{\\psi}$ in $M = P(\\alpha, \\beta_1, \\beta_2)$, \nwhere $\\alpha$ and $\\beta_j$ are spheres such that $\\alpha \\cap \\beta_1 = \\{p\\}$ and $\\alpha \\cap \\beta_2 = \\{q\\}, \\tau_0$ and $\\sigma_j$ are Dehn twists along $\\alpha$ and $\\beta_j$, and $\\psi = \\tau_0 \\circ \\sigma_1^{-1} \\circ \\sigma_2^{-1}$.\nThen, $\\mathcal{B}_{\\psi}$ is given by Theorem \\ref{branched surface thm}. \n\n\\vskip0.2in\n\\noindent{\\em Step 1 (Notation).}\nFirst, we will choose $\\varphi:\\pi^{-1}(\\partial S) \\stackrel{\\sim}{\\to} S^{n-1} \\times \\mathbb{D}^n$ for $S = S_p^\\pm, S_q^\\pm, \\bar{S}_p^\\pm$, and $\\bar{S}_q^{\\pm}$.\nWe will use $\\varphi$ in the next steps. \n\nIn order to construct $\\varphi: \\pi^{-1}(\\partial S_p^+) \\stackrel{\\sim}{\\to} S^{n-1} \\times \\mathbb{D}^n$, we observe that \n$$\\pi^{-1}(S_p^+) \\cap \\mathcal{B} \\subset D_p^+,$$\nby Remark \\ref{rmk identify with cotangent bundle of disk}. \nMoreover, we can assume that $\\pi^{-1}(S_p^+) \\subset i_{D_p^+}\\big(\\mathcal{N}(D_p^+)\\big)$.\nNote that $i_{D_p^+}$ and $\\mathcal{N}(D_p^+)$ are defined in Remark \\ref{rmk natural embedding}.\nThus, by choosing coordinate charts for $D_p^+$, one obtains $\\varphi: \\pi^{-1}(S_p^+) \\stackrel{\\sim}{\\to} \\mathbb{D}^n \\times \\mathbb{D}^n$.\nBy abuse of notation, the restriction $\\varphi|_{\\pi^{-1}(\\partial S_p^+)} : \\pi^{-1}(\\partial S_p^+) \\stackrel{\\sim}{\\to} S^{n-1} \\times \\mathbb{D}^n$ is simply called $\\varphi$ again.  \nSimilarly, it is enough to choose coordinate charts for $D_p^-, D_q^\\pm, \\bar{D}_p^{\\pm}, \\bar{D}_q^{\\pm}$, in order to fix $\\varphi : \\pi^{-1}(\\partial S) \\stackrel{\\sim}{\\to} S^{n-1} \\times \\mathbb{D}^n$ for $S = S_p^-, S_q^\\pm, \\bar{S}_p^\\pm, \\bar{S}_q^\\pm$. \n\nIn order to choose specific coordinate charts for $D_p^\\pm, D_q^\\pm, \\bar{D}_p^\\pm$, and $\\bar{D}_q^\\pm$, we use the $(2n-2)$-dimensional submanifold $W \\subset M$ defined in Section 4.1. \nFor convenience, we consider the lowest nontrivial dimension, i.e., $n=2$. \nFor higher $n$, we can fix coordinate charts similarly.\n\nLet $(x_1,x_2)$ be a coordinate chart on $D_p^+ \\subset \\alpha$ such that the $x_1$-axis agrees with $W \\cap D_p^+$.\nThere are two choices for the positive $x_1$-direction corresponding to the two orientations of $W \\cap D_p^+$, or equivalently orientations of $C_{\\alpha}$.\nWe can choose either of them.\nThen, let $(y_1,y_2)$ be an oriented chart on $D_p^-$ such that the $y_1$-axis agrees with $W \\cap \\beta_1$ and $\\omega(\\partial_{x_1}, \\partial_{ y_1})>0$. \nThe positive $y_1$-direction determines an orientation of $C_{\\beta_1}$. \nOn $\\bar{D}_p^+$, there exists an oriented chart $(x_1,x_2)$ such that the positive $x_1$-direction agrees with the orientation of $C_{\\alpha}$.  \nFor the other singular disks, we obtain oriented coordinate charts from the orientations of $C_{\\alpha}, C_{\\beta_i}, \\alpha$ and $\\beta_i$ in the same way.\n\nLet $b_1 = b(L,S_p^+), b_2 = b(L,\\bar{S}_p^+), b_3 = b(L,\\bar{S}_p^-), b_4 = b(L,S_q^+ ), b_5 = b(L,\\bar{S}_q^+)$, and $b_6 = b(L,\\bar{S}_q^-)$, and let $\\mathring{b}_i$ be a representative of $b_i$.  \n\nThe boundaries of $S_p^+$ is a component of the branch locus of $\\mathcal{B}_{\\psi}^*$.\nBy Remark \\ref{rmk nonuniqueness of fibered neigbhrohood} (3), one can decompose $\\mathring{b}_1$.\nMore precisely, in this case, Remark \\ref{rmk nonuniqueness of fibered neigbhrohood} says that for any $x \\in \\partial S_p^+$, there are three sectors $S_0, S_1, S_2$ such that \n\\begin{gather*}\nx \\in S_i \\text{  for all  } i = 0, 1, 2, \\\\\nF_x \\cap \\overline{\\pi^{-1}(\\mathring{S}_0)} = F_x \\text{  and  } F_x \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)} \\subset \\mathring{F}_x \\text{  for  } i = 1, 2.\n\\end{gather*}\nMoreover, it is easy to check that $S_p^+$ is either $S_1$ or $S_2$. \nWithout loss of generality, let us label $S_1 = S_p^+$. \n\nIf $L$ is carried by $\\mathcal{B}$, we assume that $L \\subset N(\\mathcal{B})$.\nThen, one obtains \n$$L \\cap F_x \\subset \\big(F_x \\cap \\overline{\\pi^{-1}(\\mathring{S}_1)}\\big) \\cup \\big(F_x \\cap \\overline{\\pi^{-1}(\\mathring{S}_2)}\\big)$$ \nWe decompose $\\mathring{b}_1$ into $\\mathring{b}_1 = \\tilde{b}_1 \\sqcup \\bar{b}_1$, where $\\tilde{b}_1 = \\mathring{b}_1 \\cap \\overline{\\pi^{-1}(\\mathring{S}_1)}$ and $\\bar{b}_1 = \\mathring{b}_1 \\cap \\overline{\\pi^{-1}(\\mathring{S}_2)}$. \nThe decomposition $\\mathring{b}_4 = \\bar{b}_4 \\sqcup \\tilde{b}_4$ is similar.\n\t\n\tWe will explain the effects of $\\sigma_2^{-1}$ on $\\mathcal{B}_{\\psi}$ in Step 2 and $\\tau_0$ on $\\mathcal{B}_{\\psi}$ in Step 3.\n\tThe effect of $\\sigma_1^{-1}$ is similar to that of $\\sigma_2^{-1}$.\n\n\n\t\\vskip0.2in\t\n\t\\noindent{\\em Step 2 (Effect of $\\sigma_2^{-1}$ on $\\mathcal{B}_{\\psi}$).}\n\tIn the rest of this paper, we make specific choices of $\\tau_0$ and $\\sigma_j$ which are given by Equation \\eqref{eqn definition of generalized Dehn along L}, and $\\tau : T^*S^2 \\stackrel{\\sim}{\\to} T^*S^2$, which is defined in Remark \\ref{rmk specific dehn twists}.  \n\tIn other words, $\\tau_0 = \\phi_{\\alpha} \\circ \\tau \\circ \\phi_{\\alpha}^{-1}$ and $\\sigma_j = \\phi_{\\beta_j} \\circ \\tau \\circ \\phi_{\\beta_j}^{-1}$, where $\\phi_{\\alpha}$ (resp.\\ $\\phi_{\\beta_j}$) is a symplectomorphism from $T^*S^2$ to a neighborhood of $\\alpha$ (resp.\\ $\\beta_j$).\n\tThe neighborhood of $\\alpha$ (resp.\\ $\\beta_j$) will be denoted by $T^*\\alpha$ (resp.\\ $T^*\\beta_j$). \n\t\n\t\\begin{remark}\n\t\t\\label{rmk acts like antipodal}\n\t\tRecall that $\\tau$ is a Dehn twist on $T^*S^n$ which agrees with the antipodal map \n\t\t$$ T^*S^n \\stackrel{\\sim}{\\to} T^*S^n, (u;v) \\mapsto (-u;-v),$$\n\t\ton a neighborhood of the zero section $S^n$. \n\t\\end{remark}\n\t\n\tBy Lemma \\ref{lem2}, $\\sigma_2^{-1}(L)$ is carried by $\\mathcal{B}' = F_{\\sigma_2^{-1}}(\\mathcal{B}_{\\psi})$.\n\tWe label \n\t\\begin{gather*}\n\tb_1'=b(\\sigma_2^{-1}(L),S_p^+), b_2'=b(\\sigma_2^{-1}(L),\\bar{S}_p^+), b_3'=b(\\sigma_2^{-1}(L),\\bar{S}_p^-),\\\\\n\tb_4'=b(\\sigma_2^{-1}(L),S_q^-), b_5'=b(\\sigma_2^{-1}(L),\\bar{S}_q^+), b_6' = b(\\sigma_2^{-1}(L),\\bar{S}_q^-).\n\t\\end{gather*}\n\tNote that the singular disk for $b_4$ is $S_q^+$ and the singular disk for $b_4'$ is $S_q^-$, i.e., two singular disks have the same center but different sign.\n\tHowever, for $i \\neq 4$, the singular disks for $b_i$ and $b_i'$ have the same center and the same sign. \n\t\n\tFor convenience, the singular disk of $\\mathcal{B}_{\\psi}$ (resp.\\ $F_{\\sigma_2^{-1}}(\\mathcal{B}_{\\psi})$) will be called $S_i$ (resp.\\ $S_i'$), so that $b_i$ (resp.\\ $b'_i$) is a braid on $\\pi^{-1}(\\partial S_i)$ (resp.\\ $\\pi^{-1}(\\partial S_i')$).\n\tAlso, let $\\varphi_i : \\overline{\\pi^{-1}(\\mathring{S}_i)} \\stackrel{\\sim}{\\to} \\mathbb{D}^2 \\times \\mathring{\\mathbb{D}}^2$ (resp.\\ $\\varphi_i' : \\overline{\\pi^{-1}(\\mathring{S}'_i)} \\stackrel{\\sim}{\\to} \\mathbb{D}^2 \\times \\mathring{\\mathbb{D}}^2$) be the identification which is fixed in Step 1.\n\t\n\tSince $\\sigma_2^{-1}$ is supported on $T^*\\beta_2$, a small neighborhood of $\\beta_2$, $b_i$ and $b_i'$ are the same braid in $Br_{\\partial S_i}$ for $i = 1, 2, 3$, and $5$. \n\tWe will explain how $b_6'$ is constructed.\n\t\n\tWe can obtain $\\sigma_2^{-1}(\\mathcal{B}_{\\psi})$ by spinning with respect to $q$ in $T^*\\beta_2$, i.e., $\\sigma_2^{-1}(\\mathcal{B}_{\\psi})$ is the union of curves in 2-dimensional submanifold $\\phi_{\\beta_2}(W_y)$ over $y \\in S^1$.\n\tRecall that the spinning and $W_y$ are defined in Section 2.2. \n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\input{before_homotopy.pdf_tex}\n\t\t\\caption{The left picture represents $\\mathcal{B}_{\\psi} \\cap \\phi_{\\beta_2}(W_y)$ and the right picture represents $\\sigma_2^{-1}(\\mathcal{B}_{\\psi}) \\cap \\phi_{\\beta_2}(W_y)$.}\n\t\t\\label{RBG}\n\t\\end{figure}\n\n\tFigure \\ref{RBG} represents $\\mathcal{B}_{\\psi} \\cap \\phi_{\\beta_2}(W_y)$ and $\\sigma_2^{-1}(\\mathcal{B}_{\\psi}) \\cap \\phi_{\\beta_2}(W_y)$ on $\\phi_{\\beta_2}(W_y)$.\n\tWe obtain Figure \\ref{RBG} because we choose specific $\\sigma_2$.\n\t \n\tBy spinning blue, red, and green points in Figure \\ref{RBG}, we obtain $\\sigma_2^{-1}(\\mathcal{B}_{\\psi}) \\cap \\pi^{-1}(\\partial S_6')$. \n\tLet $B, R$, and $G$ be the circles obtained by spinning blue, red, and green points respectively.\n\t\n\tSince $N(\\mathcal{B}_{\\psi}) \\supset \\mathcal{B}_{\\psi}$, \n\t$\\sigma_2^{-1}\\big(N(\\mathcal{B}_{\\psi})\\big) \\cap \\pi^{-1}(\\partial S_6')$ is a neighborhood of $\\sigma_2^{-1}(\\mathcal{B}_{\\psi}) \\cap \\pi^{-1}(\\partial S_6')$.\n\tBy assuming that $N(\\mathcal{B}_{\\psi})$ is a sufficiently small neighborhood of $\\mathcal{B}_{\\psi}$, $\\sigma_2^{-1}\\big(N(\\mathcal{B}_{\\psi})\\big)  \\cap \\pi^{-1}(\\partial S_6')$ consists of three connected components, which are neighborhoods of $B, R$, and $G$. \n\tEach connected component will be called $N(B), N(R)$, and $N(G)$.  \n\t\n\tBy definition, $b_6' = \\big[\\sigma_2^{-1}(L) \\cap \\pi^{-1}(\\partial S_6') \\big]$.\n\tWithout loss of generality, we assume that $L \\subset N(\\mathcal{B}_{\\psi})$. \n\tThen, \n\t$$\\sigma_2^{-1}(L)  \\cap \\pi^{-1}(\\partial S_6') \\subset \\sigma_2^{-1}\\big(N(\\mathcal{B}_{\\psi})\\big)  \\cap \\pi^{-1}(\\partial S_6') = N(B) \\sqcup N(R) \\sqcup N(G).$$\n\tThus, strands of $\\sigma_2^{-1}(L) \\cap \\pi^{-1}(\\partial S_6')$, or equivalently $b_6'$, are divided into three groups, which are contained in $N(B), N(R)$, and $N(G)$ respectively.\n\tWe argue the group which is contained in $N(B)$ first.\n\t \n\tThe group of strands in $N(B)$ is given by $\\sigma_2^{-1}(L) \\cap N(B)$.\n\tThus, we will consider $\\sigma_2\\big( \\sigma_2^{-1}(L) \\cap N(B) \\big) = L \\cap \\sigma_2\\big(N(B)\\big)$.\n\tOne of the main difficulties is that the action of $\\sigma_2^{-1}$ on $\\sigma_2\\big(N(B)\\big)$ is not simple. \n\tTo make it simpler, we will construct a Hamiltonian isotopy $\\Phi_t$, so that there is a disk $D_B \\subset S_q^+$ such that \n\t$$(\\Phi_1 \\circ \\sigma_2^{-1})\\big(\\pi^{-1}(\\partial D_B) \\big) \\subset \\pi^{-1}(\\partial S_6').$$\n\tThen, $(\\Phi_1 \\circ \\sigma_2^{-1})\\big(\\pi^{-1}(\\partial D_B) \\cap L\\big)$ corresponds to the group of strands in $N(B)$.\n\t\n\tWe construct $\\Phi_t$ as follow:\n\tLet $H_t:\\mathbb{R}^4 \\to \\mathbb{R}^4$ be a Hamiltonian isotopy given by\n\t\\begin{align*}\n\tH_t = \\begin{pmatrix}\n\t\\cos t& 0 & -\\sin t  &0 \\\\ \n\t0& \\cos t  &0  &- \\sin t \\\\ \n\t\\sin t& 0 & \\cos t & 0\\\\ \n\t0& \\sin t & 0 & \\cos t\n\t\\end{pmatrix},\n\t\\end{align*} \n\tand let $\\delta : [0,\\infty) \\to \\mathbb{R}$ be a smooth decreasing function such that $\\delta(x) = \\tfrac{\\pi}{2}$ for all $x <1$ and $\\delta(x)= 0$ for all $x > 2$. \n\tWe choose a neighborhood $U \\subset \\beta_2$ of $\\sigma_2^{-1}(q)$ and a Darboux chart $\\phi_q: T^*U \\stackrel{\\sim}{\\to} \\mathbb{R}^4$ such that $\\phi_q(\\sigma_2^{-1}(q))$ is the origin. \n\tWe remark that $T^*\\beta_2$ denotes a neighborhood of $\\beta_2$ in $M$, which is symplectomorphic to the cotangent bundle of $\\beta_2$.\n\tThus, for a subset $U$ of $\\beta_2$, one can assume that $T^*U$ is a subset of $M$. \n\t\n\tFor convenience, let $\\phi_q(x) = (x_1;x_2)$ where $x_i \\in \\mathbb{R}^2$.\n\tThen, there is a Hamiltonian isotopy \n\t\\begin{align}\n\t\\label{eqn Hamiltonian isotopy}\n\t \\Phi_t(x) = \\left\\{\\begin{matrix}\n\t(\\phi_q^{-1} \\circ H_{t \\delta(c_1\\|x_1\\|+c_2\\|x_2\\|)} \\circ \\phi_q) (x) &\\text{  if  } x \\in T^*U, \\\\ \n\tx \\hspace{2em}  &\\text{   if  } x \\notin T^*U,\n\t\\end{matrix}\\right.\n\t\\end{align}\n\twhere $c_i$ is a positive constant and $\\|\\cdot \\|$ is the standard norm on $\\mathbb{R}^2$. \n\t\n\t\\begin{figure}[h]\n\t\t\\input{after_homotopy.pdf_tex}\n\t\t\\caption{The blue curves represent $\\tilde{D}_B \\cap \\phi_{\\beta_2}(W_y)$ in the left picture, $\\sigma_2^{-1}(\\tilde{D}_B) \\cap \\phi_{\\beta_2}(W_y)$ in the middle picture, and $\\Phi_1(\\sigma_2^{-1}(\\tilde{D}_B)) \\cap \\phi_{\\beta_2}(W_y)$ in the right picture.}\n\t\t\\label{afterhomotopy}\n\t\\end{figure}\n\tTo visualize, we use $D_q^+$ and $\\bar{D}_q^-$ instead of $S_4$ and $S'_6$ in Figure \\ref{afterhomotopy}.\n\tFigure \\ref{afterhomotopy} represents $\\phi_{\\beta_2}(W_y) \\cap D_q^+, \\phi_{\\beta_2}(W_y) \\cap \\sigma_2^{-1}(D_q^+)$ and $\\phi_{\\beta_2}(W_y) \\cap \\Phi_1(\\sigma_2^-(D_q^+))$ in the left, middle, and right pictures respectively. \n\tBy choosing proper $c_i$, we obtain a small disk $D_B \\subset S_q^+$ such that $(\\Phi_1 \\circ \\sigma_2^{-1})\\big(\\pi^{-1}(\\partial D_B) \\big) \\subset \\pi^{-1}(\\partial S_6')$. \n\tMore precisely, we obtain a disk $\\tilde{D} \\subset D_q^+$ which is in blue in the left of Figure \\ref{afterhomotopy}. \n\tBlue curves in the middle and right of Figure \\ref{afterhomotopy} represent $(\\pi \\circ \\sigma_2^{-1})\\big(\\tilde{D}_B\\big)$ and $(\\Phi_1 \\circ \\sigma_2^{-1})\\big(\\pi(\\tilde{D}_B)\\big)$.\n\tThen, $D_B$ is given by $D_B:= \\pi(\\tilde{D}_B)$. \n\n\tOn a small neighborhood of $D_B, \\sigma_2^{-1}$ agrees with the antipodal map of $\\phi_{\\beta_2}(T^*\\beta_2) \\simeq T^*S^2$, as we mentioned in Remark \\ref{rmk acts like antipodal}. \n\tThen, we obtain a map\n\t\\begin{gather*}\n\tf_1 : S^1 \\times \\mathring{\\mathbb{D}}^2 \\stackrel{\\tilde{\\varphi}_4^{-1}}{\\simeq} \\pi^{-1}(\\partial D_B) \\xrightarrow{\\Phi_1 \\circ \\sigma_2^{-1}} \\pi^{-1}(\\partial S_6') \\stackrel{\\varphi_6'}{\\simeq} S^1 \\times \\mathbb{D}^2,\\\\\n\t(\\theta, x, y) \\mapsto (\\theta + \\pi, -r_1x, -r_1y ).\n\t\\end{gather*}\n\tThe first identification $\\tilde{\\varphi}_4$ is the restriction of $\\varphi_4: \\pi^{-1}(S_4) \\stackrel{\\sim}{\\to} \\mathbb{D}^2 \\times \\mathbb{D}^2$. \n\t\\begin{remark}\n\t\t\\label{rmk radius of solid torus}\n\t\t\\mbox{}\n\t\t\\begin{enumerate}\n\t\t\t\\item Note that $\\varphi_6'\\big(\\operatorname{Im}(f_1)\\big) = (\\Phi_1 \\circ \\sigma_2^{-1})\\big(\\pi^{-1}(\\partial D_B)\\big) \\cap \\pi^{-1}(\\partial S_6')$. \n\t\t\tSimilarly, for the groups of strands in $N(R)$ and $N(G)$, one can obtain two functions $f_2$ and $f_3$ on $S^1 \\times \\mathbb{D}^2$ in the same way. \n\t\t\tThen, the images $\\operatorname{Im}(f_2)$ and $\\operatorname{Im}(f_3)$ correspond to \n\t\t\t$$(\\Phi_1 \\circ \\sigma_2^{-1})\\big(\\pi^{-1}(\\pi(N_q))\\big) \\cap \\pi^{-1}(\\partial S_6') \\subset \\pi^{-1}(\\partial S_6') \\stackrel{\\varphi_6'}{\\simeq} S^1 \\times \\mathbb{D}^2.$$ \n\t\t\tThus, $f_1$ explains the contribution of $\\tilde{b}_4$, and $f_2$ and $f_3$ explain the contribution of $\\bar{b}_4$ on the construction of $b_6'$.\t\t\t\n\t\t\t\\item The constant $r_1$ is determined by specific choices of an identification $\\phi_{\\beta_2}: T^*S^2 \\stackrel{\\sim}{\\to} T^*\\beta_2$, the fixed Dehn twist $\\tau$ in Remark \\ref{rmk specific dehn twists}, and so on. \n\t\t\tHowever, $r_1$ has to be smaller than 1. \n\t\t\tThis is because $\\operatorname{Im}(f_1), \\operatorname{Im}(f_2)$, and $\\operatorname{Im}(f_3)$ are mutually disjoint, since they corresponds to $N(B), N(R)$, and $N(G)$ respectively.\n\t\t\\end{enumerate}\n\n\t\\end{remark}\n\t\t\n\tThe strands of $b'_6$ which are contained in $N(B)$ correspond to $$\\varphi_6'^{-1}(f_1(\\tilde{\\varphi}_4(L \\cap \\pi^{-1}(\\partial D_B)))).$$ \n\tWe will prove that $L \\cap \\pi^{-1}(\\partial D_B)$ represents the same braid with $\\tilde{b}_4$. \n\tWe can assume that there is no singular value of $\\pi$ on $S_4 \\setminus D_B$. \n\tThen, $\\varphi_4(\\tilde{b}_4)$ and $\\tilde{\\varphi}_4(L \\cap \\pi^{-1}(\\partial D_B))$ represent the same braid in $S^1 \\times \\mathbb{D}^2$ because of non-singularity on $S_4 \\setminus D_B$. \n\tThus, in $S^1 \\times \\mathbb{D}^2$, $f_1(\\varphi_4(\\tilde{b}_4))$ and $f_1(\\tilde{\\varphi}_4(L \\cap \\pi^{-1}(\\partial D_B)))$ represent the same braid.\n\tIt proves that $\\varphi_6'^{-1}(f_1(\\varphi_4(\\tilde{b}_4)))$ and the group of strands in $N(B)$ represent the same braid in $Br_{\\partial S_6'}$. \n\n\\begin{remark}\n\t\\label{rmk abuse of notation}\n\tFor convenience, we simply use $f_1(\\tilde{b}_4)$, instead of $\\varphi_6'^{-1}(f_1(\\varphi_4(\\tilde{b}_4)))$.\n\tIn the rest of this paper, we will abuse notation in the same way.\n\\end{remark}\n\n\tFor the groups of strands in $N(R)$ and $N(G)$, we obtain the following maps $f_2$ and $f_3$ in the same way,\n\t\\begin{gather*}\n\tf_2: S^1 \\times \\mathbb{D}^2 \\to S^1 \\times \\mathbb{D}^2, \\\\\n\t(\\theta, x, y) \\mapsto (\\theta + \\pi, r_0\\cos \\theta+ r_2 x, r_0\\sin \\theta  + r_2y),\\\\\n\tf_3: S^1 \\times \\mathbb{D}^2 \\to S^1 \\times \\mathbb{D}^2, \\\\\n\t(\\theta, x, y) \\mapsto (\\theta + \\pi, -r_0\\cos \\theta+ r_2(x\\cos 2\\theta  - y\\sin 2\\theta ),\\\\\n\t\\hspace{7em} -r_0 \\sin \\theta+ r_2(x\\sin 2 \\theta  + y \\cos 2\\theta)),\n\t\\end{gather*}\n\twhere $r_0$ and $r_2$ are positive constants which are smaller than 1.\n\t\\begin{remark}\n\t\t\\mbox{}\n\t\t\\label{rmk radius of solid torus 2}\n\t\t\\begin{enumerate}\n\t\t\t\\item To obtain $f_1$, we used a Hamiltonian isotopy $\\Phi_t$. \n\t\t\tSimilarly, to obtain $f_2$ and $f_3$, we need a Hamiltonian isotopy.\n\t\t\tWe construct a Hamiltonian isotopy by extending a Lagrangian isotopy connecting $\\sigma_2^{-1}(N_q) \\cap \\overline{\\pi^{-1}(S_6')}$ and  \n\t\t\t$$\\varphi_6'^{-1}(\\{ (s \\cos (\\theta + \\pi), s \\sin (\\theta+\\pi), r_0 \\cos \\theta, r_0 \\sin \\theta) \\hspace{0.2em} | \\hspace{0.2em} s \\in [-1,1], \\hspace{0.5em} \\theta \\in S^1\\}),$$\n\t\t\tin $\\overline{\\pi^{-1}(\\mathring{S}_6')} \\stackrel{\\varphi_6'}{\\simeq} \\mathbb{D}^2 \\times \\mathbb{D}^2$.\n\t\t\t\\item Note that $r_0$ and $r_2$ are positive constants which are determined by specific choices.\n\t\t\tHowever, $r_0$ and $r_2$ have to satisfy $r_1 + r_2 < r_0$ since $\\operatorname{Im}(f_1), \\operatorname{Im}(f_2)$ and $\\operatorname{Im}(f_3)$ are mutually disjoint.\n\t\t\\end{enumerate}\n\t\t\t\\end{remark}\n\n\tIn the same way that we proved that $f_1(\\tilde{b}_4)$ and the group of strands in $N(B)$ represent the same braid in $Br_{\\partial S_6'}$, we can prove that $f_2(\\bar{b}_4)$ (resp.\\ $f_3(\\bar{b}_4)$) and the group of strand in $N(R)$ (resp.\\ $N(G)$) represent the same braid in $Br_{\\partial S_6'}$.  \n\tThen, $b_6'$ is represented by $f_1(\\tilde{b}_4) \\sqcup f_2(\\bar{b}_4) \\sqcup f_3(\\bar{b}_4)$.\n\tNote that we are abusing notation for convenience as we mentioned in Remark \\ref{rmk abuse of notation}.\n\t\n\tThe situation for $b_4'$ is analogous. \n\tWe obtain three maps $g_1, g_2$ and $g_3$ in the same way.\n\tAt the end, $b_4'$ is represented by $g_1(\\bar{b}_4) \\sqcup g_2(\\bar{b}_4) \\sqcup g_3(b_6)$. \n\tThis proves Lemma \\ref{lem3} for the case of $\\sigma_2^{-1}$.\n\t\n\tNote that maps $f_i$ and $g_j$ are given by specific maps acting on $S^1 \\times \\mathbb{D}^2$, but we would like to consider them as maps on $\\tilde{Br}_{\\partial S_k}$ for some $k$.\n\tThen, we summarize the effect of $\\sigma_2^{-1}$ as a matrix \n\t\\begin{align*}\n\t\\Sigma_{2,\\mathcal{B}_{\\psi}} = \\begin{pmatrix}\n\tid & 0 & 0 & 0 & 0 & 0 \\\\ \n\t0 & id  & 0  & 0 & 0 & 0 \\\\ \n\t0 & 0 & id & 0 & 0 & 0\\\\ \n\t0 & 0 & 0 & g_1 + g_2 & 0 & g_3 \\\\\n\t0 & 0 & 0 & 0 & id & 0 \\\\\n\t0 & 0 & 0 & f_1 + f_2 + f_3 & 0 & 0 \n\t\\end{pmatrix}.\n\t\\end{align*}  \n\tThus, if $\\mathring{b}_i$ is a representative of a braid $b_i$ for $L$, then $\\mathring{b}_i'$ is a representative of $b_i'$ where \n\t\\begin{gather*}\n\t\\begin{pmatrix}\n\t\\mathring{b}_1' \\\\\n\t\\mathring{b}_2' \\\\\n\t\\mathring{b}_3' \\\\\n\t\\mathring{b}_4' \\\\\n\t\\mathring{b}_5' \\\\\n\t\\mathring{b}_6' \n\t\\end{pmatrix}\n\t = \\Sigma_{2,\\mathcal{B}_{\\psi}} \n\t \\begin{pmatrix}\n\t \\mathring{b}_1 \\\\\n\t \\mathring{b}_2 \\\\\n\t \\mathring{b}_3 \\\\\n\t \\mathring{b}_4 \\\\\n\t \\mathring{b}_5 \\\\\n\t \\mathring{b}_6 \n\t \\end{pmatrix} \n\t = \\begin{pmatrix}\n\t \\mathring{b}_1 \\\\\n\t \\mathring{b}_2 \\\\\n\t \\mathring{b}_3 \\\\\n\t g_1(\\bar{b}_4) \\sqcup g_2(\\bar{b}_4) \\sqcup g_3(\\mathring{b}_6) \\\\\n\t \\mathring{b}_5 \\\\\n\t f_1(\\tilde{b}_4) \\sqcup f_2(\\bar{b}_4) \\sqcup f_3(\\bar{b}_4) \n\t \\end{pmatrix}.\n\t\\end{gather*}\n\t\n\t\\begin{remark}\n\t\t\\label{rmk non-linear algebra}\n\t\tWe remark that in surface theory, we can do linear algebra on weights, but in a higher-dimensional case, we cannot do linear algebra with the matrix $\\Sigma_{2,\\mathcal{B}_{\\psi}}$, because there is no module structure on $\\tilde{Br}_{\\partial S_i}$.\n\t\\end{remark}\t\n\\vskip0.2in\n\n\t\\noindent\n\t{\\em Step 3 (Effects of $\\tau_0$ on $\\mathcal{B}_{\\psi}$).} We use the same notation, i.e.,  $b_1, \\cdots, b_6$ denote the braids on singular disks $S-i$ of $\\mathcal{B}_{\\psi}^*$, and \n\t$$b_1' = b(\\tau_0(L),S_p^+), \\cdots, b_6' = b(\\tau_0(L),\\bar{S}_q^-),$$ \n\tso that the singular disk corresponding to $b'_i$ has the same center as the singular disk corresponding to $b_i$.\n\tWe also use $\\mathring{b}_i$ and $\\mathring{b}_i'$, $S_i$ and $S_i'$, $\\varphi_i$ and $\\varphi_i'$ to indicate representatives of braids, singular disks in $\\mathcal{B}_{\\psi}$ and $F_{\\tau_0}(\\mathcal{B}_{\\psi})$, identifications induced by fixed coordinate charts.\n\t\n\tThe situation for $\\tau_0$ is similar to that for $\\sigma_2^{-1}$.\n\tFor example, by observing how $\\tau_0$ acts on $\\overline{\\pi^{-1}(\\mathring{S}_1)}$, we obtain \n\t$$h_1: S^1 \\times \\mathbb{D}^2 \\to S^1 \\times \\mathbb{D}^2,$$ \n\texplaining the contribution of $\\tilde{b}_1$ on the construction of $b_3'$.\n\tThen, $h_1$ is given by a translation on $S^1$ and a scaling on $\\mathbb{D}^2$, as $f_1$ is.\n\tSimilarly, we obtain $h_2$ and $h_3$, which explain the contributions of $\\bar{b}_1$ on the construction of $b_3'$. \n\tThe map $h_2$ (resp.\\ $h_3$) is of the same types with $f_2$ (resp.\\ $f_3$), i.e., \n\t\\begin{gather*}\n\th_2(\\theta, x, y) = \\big(\\theta \\text{ or } \\theta + \\pi, \\pm r_1 \\cos \\theta + r_2 x, \\pm r_1 \\sin \\theta + r_2 y \\big), \\\\\n\th_3(\\theta, x, y) = \\big(\\theta \\text{ or } \\theta + \\pi, \\pm r_1 \\cos \\theta + r_2 (x \\cos 2\\theta - y \\sin 2 \\theta), \\\\\n\t\\hspace{7em} \\pm r_1 \\sin \\theta + r_2 (x \\sin 2\\theta + y \\cos 2 \\theta) \\big),\n\t\\end{gather*}\n\twhere $r_1$ and $r_2$ are constants.\n\t \n\tIf a map is of the same type to $f_1$, in other words, if the map is given by a translation on $S^1$ and a scaling on $\\mathbb{D}^2$, let the map be of {\\em scaling type}.\n\tThis is because the formula defining the map is given by a scaling on fibers.\n\tThe maps of scaling type explain how braids $b(L, S_p^\\pm)$ or $b(L,\\bar{S}_p^\\pm)$ contribute on the construction of braids $b(\\delta(L), S_{\\delta(p)}^\\pm)$ or $b(\\delta(L),\\bar{S}_{\\delta(p)}^\\pm)$ through $\\delta\\big(\\pi^{-1}(S_p^\\pm)\\big)$, where $\\delta$ is a Dehn twist.\n\t\n\tIf a map is of the same type to $f_2$ (resp.\\ $f_3$), let the map be of {\\em the first (resp.\\ second) singular type}.\n\tThis is because they are related to a creation of new singular component.\n\tThe maps of the first and second singular types explain how the braid $b(L,\\delta(S_p))$ contributes on the construction of braid $b(\\delta(L),\\bar{S}_{\\delta(p)}^\\pm)$.\n\t\n\tTo summarize, if $b_i$ contributes the construction of $b'_j$ and if the center of a singular disk corresponding to $b_i$ is either the same point or the antipodal point of the center of the singular disk corresponding to $b_j'$, maps of these three types explain the contribution of $b_i$ on the construction of $b_j'$. \n\tNote that the center of a singular disk is defined in Remark \\ref{rmk position of singular value}.\n\t\n\tThe maps of these three types explain the effects of $\\sigma_2^{-1}$ on $\\mathcal{B}$.\n\tHowever, to explain the effects of $\\tau_0$ on $\\mathcal{B}_{\\psi}$, we need maps of one more type.  \t\n\t\t\t\n\tThis is because $\\alpha$ has two plumbing points, unlike $\\beta_i$ has only one plumbing point.\n\tThus, when we apply $\\tau_0$, $b_i$ can contribute on $b_j'$ even if the centers of singular disks corresponding to $b_i$ and $b_j'$ are neither the same nor antipodals of each other. \n\tFor example, $L \\cap \\pi^{-1}\\big(\\pi(N_p)\\big)$ is stretched by $\\tau_0$.\n\tThe stretched part $\\tau_0\\big(L \\cap \\pi^{-1}(\\pi(N_p))\\big)$ has intersection with $\\pi^{-1}(S_4)$ and $\\pi^{-1}(S_5)$. \n\tThus, $b_4'$ has some strands corresponding to $\\tau_0(L \\cap \\pi^{-1}(\\pi(N_p))) \\cap \\pi^{-1}(\\partial S_4)$\n\tThese strands are the contribution of $\\bar{b}_1$ on the construction of $b_4'$.\n\tSimilarly, $\\bar{b}_1$ contributes on the construction of $b_5'$, and $\\bar{b}_4$ contributes on the constructions of $b_1'$ and $b_2'$.\n\t\n\tTo describe the contribution of $\\bar{b}_1$ on $b_4'$, without loss of generality, we assume that there is no singular value for \n\t$$\\tau_0(L \\cap \\pi^{-1}(\\pi(N_p))) \\cap \\overline{\\pi^{-1}(\\mathring{S}_4)} \\stackrel{\\pi}{\\to} S_4,$$ \n\tby Remark \\ref{rmk position of singular value}. \n\tThus, $\\tau_0(L \\cap \\pi^{-1}(\\pi(N_p))) \\cap \\overline{\\pi^{-1}(\\mathring{S}_4)}$ is a union of disjoint Lagrangian disks on $\\overline{\\pi^{-1}(\\mathring{S}_4)}$\n\tand $\\bar{b}_1$ contributes on $b_4'$ by adding strands near $\\tau_0(N_p) \\cap \\pi^{-1}(\\partial S_4)$ which are not braided to each other.\n\tThe number of the added strands is the same as the number of strands of $\\bar{b}_1$. \n\tIn the same way, $\\bar{b}_1$ contributes on the construction of $b_5'$.\n\t\n\tTo describe the contribution of $\\bar{b}_1$ on $b_4'$ as a map acting on $S^1 \\times \\mathbb{D}^2$, we define $\\bar{b}_1^\\circ \\subset \\pi^{-1}(\\partial S_1)$ such that \n\t$$\\varphi_1(\\bar{b}_1^\\circ) := \\{ (\\theta, x_0, y_0) \\hspace{0.2em} | \\hspace{0.2em} \\phi_1^{-1}(0,x_0,y_0) \\in \\bar{b}_1 \\} \\subset S^1 \\times \\mathbb{D}^2 \\stackrel{\\varphi_1}{\\simeq} \\pi^{-1}(\\partial S_1),$$\n\twhich represents a trivial braid having the same number of strands with $\\bar{b}_1$.\n\tThis is because we only need the number of the strands in $\\bar{b}_1$, not the way $\\bar{b}_1$ is braided.\n\t\n\tWe construct a Hamiltonian isotopy $\\Phi_t$ by extending a Lagrangian isotopy connecting $\\tau_0(N_p) \\cap \\pi^{-1}(\\partial S_4)$ and \n\t$$\\varphi_4'^{-1}(\\{(s \\cos \\theta, s \\sin \\theta, c_1, c_2) \\hspace{0.2em} | \\hspace{0.2em} s \\in [-1,1], \\theta \\in S^1, c_i \\hspace{0.5em} \\text{is constants} \\}) \\subset \\pi^{-1}(S_4),$$ \n\tas we did in Remark \\ref{rmk radius of solid torus 2}.\n\tThen, one obtains\n\t\\begin{gather*}\n\th_t: S^1 \\times \\mathbb{D}^2 \\stackrel{\\varphi_1}{\\simeq} \\pi^{-1}(\\partial S_1) \\xrightarrow{\\Phi_1 \\circ \\tau_0} \\pi^{-1}(\\partial S_4) \\stackrel{\\varphi_4'}{\\simeq} S^1 \\times \\mathbb{D}^2,\\\\\n\t(\\theta, x, y) \\mapsto (\\theta, r_0x + c_1, r_0y+c_2),\n\t\\end{gather*}\n\twhere $r_0$ is a positive constant number less than 1. \n\tThen, $h_t(\\bar{b}_1^{\\circ})$ represents the same braid to the strands in $b'_4$, which correspond to $\\tau_0(L \\cap \\pi^{-1}(\\pi(N_p)))$. \n\tWe recall that we are abusing notation as mentioned in Remark \\ref{rmk abuse of notation}.\n\t\n\tSimilarly, if $b_i$ contributes the construction of $b'_j$ and if the center of a singular disk corresponding to $b_i$ is neither the same point nor the antipodal point of the center of the singular disk corresponding to $b'_j$, then the contribution of $b_i$ on $b_j'$ can be described by a map like $h_t$. \n\tIf a map is of the same type with $h_t$, let the map be of {\\em trivial type}, because a map of trivial type adds strands which are not braided with each other.\n\t\n\tThen, we can describe the effect of $\\tau_0$ on $\\mathcal{B}_{\\psi}$ as a matrix\n\t\\begin{align*}\n\t\\mathrm{T}_{0,\\mathcal{B}_{\\psi}} = \\begin{pmatrix}\n\t0 & i & 0 & h_t & 0 & 0 \\\\ \n\th_1 + h_2 + h_3 & 0  & 0  & i_t & 0 & 0 \\\\ \n\t0 & 0 & id & 0 & 0 & 0\\\\ \n\th_t & 0 & 0 & 0 & i & 0 \\\\\n\ti_t & 0 & 0 & h_1 + h_2 + h_3 & 0 & 0 \\\\\n\t0 & 0 & 0 & 0 & 0 & id \n\t\\end{pmatrix}.\n\t\\end{align*} \n\tAmong the entries, $h_1, i$, and $id$ are of scaling type, $h_2$ and $h_3$ are of the first and second singular types, and $h_t$ and $i_t$ are of trivial type. \n\t\\vskip0.2in\n\t\n\t\\noindent{\\em Step 4 (General case).}\n\tA $\\psi$ of generalized Penner type is a product of Dehn twists. \n\tIn the general case, when we apply $\\psi$, each Dehn twist is followed by a Hamiltonian isotopy as $\\sigma_2^{-1}$ is followed by $\\Phi_t$ in step 2.\n\tLet $\\psi_H = (\\Phi_{1,1} \\circ \\delta_1 ) \\circ \\cdots \\circ (\\Phi_{l,1} \\circ \\delta_l)$, where $\\psi = \\delta_1 \\circ \\cdots \\circ \\delta_l$, $\\delta_i$ is a Dehn twist, and $\\Phi_{i,t}$ is a Hamiltonian isotopy which follows $\\delta_i$.\n\t\n\tAfter applying the Hamiltonian isotopy, the effect of a Dehn twist $\\tau_i$ (resp.\\ $\\sigma_j^{-1}$) on $\\mathcal{B} \\in \\mathbb{B}$ is described by a matrix $\\mathrm{T}_{i, \\mathcal{B}}$ (resp.\\ $\\Sigma_{j,\\mathcal{B}}$), whose entries are sums of maps of four types.\n\tAs we mentioned in Step 3, the maps of scaling type explain how braids $b(L, S_p^\\pm)$ or $b(L,\\bar{S}_p^\\pm)$ contribute on the construction of braids $b(\\delta(L), S_{\\delta(p)}^\\pm)$ or $b(\\delta(L),\\bar{S}_{\\delta(p)}^\\pm)$, where $\\delta$ is a Dehn twist.\n\tSimilarly, the maps of the first and second singular types explain how braids $b(L,\\delta(S_p))$ contribute on the construction of braid to $b(\\delta(L),\\bar{S}_{\\delta(p)}^\\pm)$.\n\tFinally, the maps of trivial type explain the other cases.\n\t\n\tThis completes the proof of Lemma \\ref{lem3}.\n\\end{proof}\n\n\\vskip0.2in\n\\noindent {\\em Taking the limit of a braid sequence.}\nWe have obtained braid sequences $\\{b(\\psi^m(L),S_i)\\}_{m \\in \\mathbb{N}}$, where $L$ is carried by $\\mathcal{B}_{\\psi}$, and $S_i$ is a singular disk of $\\mathcal{B}_{\\psi}^*$.\nIn the rest of this subsection, we construct a limit of $\\{b(\\psi^m(L),S_i)\\}_{m \\in \\mathbb{N}}$ as $m \\to \\infty$.\n\nWe argue with the above example, i.e., \n$$M = P(\\alpha, \\beta_1, \\beta_2), \\psi = \\tau_0 \\circ \\sigma_1^{-1} \\circ \\sigma_2^{-1}.$$ \nFor convenience, let \n$$\\mathcal{B} := \\mathcal{B}_{\\psi},\\hspace{0.2em} \\mathcal{B}' := F_{\\sigma_2^{-1}}(\\mathcal{B}),\\hspace{0.2em} \\mathcal{B}'' := F_{\\sigma_1^{-1}}(\\mathcal{B}'),$$ \nand let singular disks $S_p^+, \\bar{S}_p^+, \\bar{S}_p^-, S_q^+, \\bar{S}_q^+$, and $\\bar{S}_q^-$ of $\\mathcal{B}$ be $S_1, \\cdots, S_6$. \nUsing notation from the proof of Lemma \\ref{lem3}, we have matrices $\\mathrm{T}_{0,\\mathcal{B}''}, \\Sigma_{1,\\mathcal{B}'}$, and $\\Sigma_{2,\\mathcal{B}}$.\nThen, we obtain $\\Psi = \\mathrm{T}_{0,\\mathcal{B}''} \\cdot \\Sigma_{1,\\mathcal{B}'} \\cdot  \\Sigma_{2,\\mathcal{B}}$ by defining a multiplication of maps as a composition of them. \nNote that a product of two arbitrary matrices is not defined. \nFor example, an input of $\\Sigma_{2,\\mathcal{B}}$ and an output of $\\mathrm{T}_{0,\\mathcal{B}''}$ are tuples of braids on singular disks of $\\mathcal{B}^*$.\nThus, $\\Sigma_{2,\\mathcal{B}} \\cdot \\mathrm{T}_{0,\\mathcal{B}''}$ is defined.\nHowever, $\\mathrm{T}_{\\mathcal{B}''} \\cdot \\Sigma_{2,\\mathcal{B}}$ is not defined since an input of $\\mathrm{T}_{0,\\mathcal{B}''}$ is a tuple of braids on singular disks of $\\mathcal{B}^*$, but an output of $\\Sigma_{2,\\mathcal{B}}$ is a tuple of braids on singular disks of $\\mathcal{B}'^*$. \n\n\nLet $\\mathring{b}_i$ be a representative of $b_i = b(L,S_i)$.\nIf\n\\begin{gather*}\n\\label{eqn matrix}\n\\begin{pmatrix}\n\\mathring{b}_{1,m} \\\\\n\\mathring{b}_{2,m} \\\\\n\\mathring{b}_{3,m} \\\\\n\\mathring{b}_{4,m} \\\\\n\\mathring{b}_{5,m} \\\\\n\\mathring{b}_{6,m} \n\\end{pmatrix}\n:= \\Psi^m \n\\begin{pmatrix}\n\\mathring{b}_1 \\\\\n\\mathring{b}_2 \\\\\n\\mathring{b}_3 \\\\\n\\mathring{b}_4 \\\\\n\\mathring{b}_5 \\\\\n\\mathring{b}_6 \n\\end{pmatrix},\n\\end{gather*} \nthen $\\mathring{b}_{i,m}$ is a representative of $b_{i,m}$.\nThus, in order to keep track of braid sequences $\\{b_{i,m}\\}_{m \\in \\mathbb{N}}$, it is enough to keep track of $\\Psi^m$. \n\nEvery entry of $\\Psi^m$ is a sum of compositions of $3m$-maps.\nThe image of a composition of $3m$-maps is a solid torus.\nBy Remarks \\ref{rmk radius of solid torus} and \\ref{rmk radius of solid torus 2}, the radius of each solid torus appearing in $\\Psi^m$ decreases exponentially and converges to zero as $m \\to \\infty$. \n\nFrom another view points, we consider $\\psi_H$.\nNote that $\\psi_H$ is defined in step 4 of the proof of Lemma \\ref{lem3}. \nThe proof of Lemma \\ref{lem3} implies that \n$$\\mathring{b}_{i,m} \\subset \\psi_H^m(N(\\mathcal{B}_{\\psi})) \\cap \\pi^{-1}(\\partial S_i) \\text{  for all  } m \\in \\mathbb{N} \\text{  and for all  } i = 1, \\cdots, 6.$$ \nLet \n$$B_{i,m} := \\psi_H^{m}(N(\\mathcal{B}_{\\psi})) \\cap \\pi^{-1}(\\partial S_i).$$\nThen, $B_{i,m}$ is the disjoint union of solid tori.\nMore precisely, each solid torus in $B_{i,m}$ is the image of a composition of $3m$-maps, appearing in $\\Psi^m$.\nConversely, for each composition of $3m$-maps appearing in $\\Psi^m$, the image is a solid torus contained in $B_{i,m}$.\nThe radii of solid tori in $B_{i,m}$ are decreasing exponentially and are converging to zero as $m \\to \\infty$.\n\nSince $\\mathring{b}_{i,m} \\subset B_{i,m}$ and $B_{i+1,m} \\subset B_{i,m}$ for all $m \\in \\mathbb{N}$, there is a limit \n$$B_{i,\\infty} := \\lim_{m\\to \\infty}B_{i,m} = \\cap_{m \\in \\mathbb{N}} B_{i,m}.$$\nThus, $B_{i,\\infty}$ is the union of infinite strands as a subset of $\\pi^{-1}(\\partial S_i)$ and \n$$\\lim_{m \\to \\infty} \\mathring{b}_{i,m} = B_{i,\\infty},$$\nas a sequence of closed sets in $\\pi^{-1}(\\partial S_i)$.\n\n\\begin{remark}\n\t\\label{rmk limit of braid seq}\n\t\\mbox{}\n\t\\begin{enumerate}\n\t\t\\item We have constructed a sequence of specific representatives \n\t\t$\\{\\mathring{b}_{i,m}\\}_{m\\in\\mathbb{N}}$\n\t\tsuch that $$\\lim_{m \\to \\infty} \\mathring{b}_{i,m} = B_{i,\\infty}.$$ \n\t\tFor the purposes of extending the lamination to the singular and regular disks in Sections 4.3 and 4.4, we assume that the limit $B_{i,\\infty}$ is a specific closed subset in $\\pi^{-1}(\\partial S_i)$.\n\t\t\\item Each strand of $B_{i,\\infty}$ corresponds to an infinite sequence $\\{f_m\\}_{m\\in \\mathbb{N}}$ such that $f_1 \\circ \\cdots \\circ f_{3m}$ appears in $\\Phi^m$ for all $m \\in \\mathbb{N}$. \n\t\\end{enumerate}\n\t\\end{remark}\n\n\n\\subsection{Lagrangian lamination on a singular disk}\n\nLet $\\psi$ be of generalized Penner type and let $L$ be a Lagrangian submanifold which is carried by $\\mathcal{B}_{\\psi}$.\nIn Section 4.2, on each singular disk $S_i$, we gave an inductive description of a sequence $\\{b(\\psi^m(L),S_i)\\}_{m \\in \\mathbb{N}}$.\nThere is a limit $B_{i,\\infty}$ of the sequence. \nMoreover, the limit $B_{i,\\infty}$ depends only on $\\psi$ and $B_{i,\\infty}$ is independent to $L$.\nIn this present subsection, we will construct a Lagrangian lamination $\\mathcal{L}_i \\subset \\pi^{-1}(S_i)$ from $B_{i,\\infty}$. \n\\begin{remark}\n\tIf $\\partial S_i$ is contained in the branch locus of $\\mathcal{B}_{\\psi}^*$, $B_{i,\\infty}$ can be divided into two groups, as a braid $b$ was divided into $\\bar{b}$ and $\\tilde{b}$ in the Step 1 of the proof of Lemma \\ref{lem3}.\n\tWe will construct $\\mathcal{L}_i$ from $B_{i, \\infty} \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}$, which is one of two groups of $B_{i,\\infty}$.\n\n\tIf $\\partial S_i$ is not contained in the branch locus of $\\mathcal{B}_{\\psi}^*$, then $\\mathcal{B}_{i,\\infty} \\subset \\overline{\\pi^{-1}(\\mathring{S}_i)}$.\n\tIn this case, we will construct a Lagrangian lamination from $B_{i,\\infty} = B_{i, \\infty} \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}$. \n\tThus, we will simply say that the Lagrangian lamination is constructed from $B_{i, \\infty} \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}$.\n\\end{remark}\n\n\\begin{lemma}\n\t\\label{lem4}\n\tLet $\\psi$ be of generalized Penner type. \n\tFor each singular disk $S_i$ of $\\mathcal{B}_{\\psi}$, there is a Lagrangian lamination $\\mathcal{L}_i \\subset \\overline{\\pi^{-1}(\\mathring{S}_i)}$, such that\n\t\\begin{enumerate}\n\t\t\\item $\\mathcal{L}_i \\cap \\pi^{-1}(\\partial S_i)$ is the same braid with $B_{i,\\infty} \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}$, where $B_{i,\\infty}$ is the limit of a braid sequence, which depends only on $\\psi$. \n\t\t\\item If $L$ is a Lagrangian submanifold of $M$ which is carried by $\\mathcal{B}_{\\psi}$, then for every $m \\in \\mathbb{N}$, there is a Lagrangian submanifold $L_m$ which is Hamiltonian isotopic to $\\psi^m(L)$ and $L_m \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}$ converges to $\\mathcal{L}_i$ as a sequence of closed subsets.\n\t\\end{enumerate} \n\\end{lemma}\n\n\\begin{proof}\n\tLet $\\psi$ be of generalized Penner type, i.e., $\\psi = \\delta_1 \\circ \\cdots \\circ \\delta_l$, where $\\delta_k$ is a Dehn twist $\\tau_i$ or $\\sigma_j^{-1}$.\n\tWe will use similar notation with the previous subsection, for example, $S_i$ denotes a singular disk of $\\mathcal{B}_{\\psi}$, $\\Psi$ denotes a matrix corresponding to $\\psi$,  $\\varphi_i : \\pi^{-1}(\\partial S_i) \\stackrel{\\sim}{\\to} S^{n-1} \\times \\mathbb{D}^n$ denotes the identification induced from the fixed coordinate chart on $S_i$, and so on. \n\t\n\tIn this proof, first, we will construct $\\mathcal{L}_i \\subset \\overline{\\pi^{-1}(\\mathring{S}_i)}$ satisfying the first condition, i.e., $\\mathcal{L}_i \\cap \\pi^{-1}(\\partial S_i) = B_{i, \\infty} \\cap \\overline{\\pi(\\mathring{S}_i)}$. \n\tThen, we will show that the constructed $\\mathcal{L}_i$ satisfies the second condition.\n\t\t\n\t\\vskip.2in\n\t\\noindent{\\em Construction of $\\mathcal{L}_i$.}\t\n\tAs we mentioned in Remark \\ref{rmk limit of braid seq}, a strand of $B_{i,\\infty} \\cap \\overline{\\pi(\\mathring{S}_i)}$ is identified with an infinite sequence $\\{f_m\\}_{m \\in \\mathbb{N}}$ such that $f_1 \\circ \\cdots \\circ f_{lk}$ appears in $\\Psi^k$ for all $k \\in \\mathbb{N}$.\n\tNote that we are assuming that $\\psi= \\delta_1 \\circ \\cdots \\circ \\delta_l$ for some positive number $l$.\n\tFor each strand $\\{f_m\\}_{m \\in \\mathbb{N}}$ of $B_{i,\\infty} \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}$, we will construct a Lagrangian submanifold of $\\overline{\\pi^{-1}(\\mathring{S}_i)}$ whose boundary agrees with the strand $\\{f_m\\}_{m \\in \\mathbb{N}}$. \n\t\n\tFirst, for a given strand $\\{f_m \\}_{m \\in \\mathbb{N}}$, let us assume that $f_1$ is of trivial type. \n\tThen, the strand is identified with a straight curve\n\t$$\\{ (\\theta, x_1, \\cdots, x_n) \\hspace{0.2em} | \\hspace{0.2em} \\theta \\in S^{n-1} \\} \\subset S^{n-1} \\times \\mathbb{D}^n \\stackrel{\\varphi_i}{\\simeq} \\pi^{-1}(\\partial S_i),$$\n\twhere $x_i$ is a constant.\n\tA subsequence $\\{f_m\\}_{m \\geq 2}$ determines constants $x_i$. \n\tLet \n\t$$D:=\\{(p, x_1, \\cdots, x_n) \\hspace{0.2em} | \\hspace{0.2em} p \\in S_i \\} \\subset \\mathbb{D}^n \\times \\mathbb{D}^n \\stackrel{\\varphi_i}{\\simeq} \\overline{\\pi^{-1}(\\mathring{S}_i)}.$$\n\tThen, $\\varphi_i(D)$ is a Lagrangian disk in $\\overline{\\pi^{-1}(\\mathring{S}_i)}$, whose boundary agrees with the strands $\\{ f_m \\}_{m \\in \\mathbb{N}}$.\n\t\n\tSecond, let us assume that $f_1$ is not of trivial type, but there exists $m \\in \\mathbb{N}$ such that $f_m$ is of trivial type. \n\tLet $k>1$ be the smallest number such that $f_k$ is of trivial type appearing in $\\{ f_m \\}_{m \\in \\mathbb{N}}$. \n\tThen, $\\tilde{\\psi} = \\delta_{k_0} \\circ \\cdots \\circ \\delta_l \\circ \\delta_1 \\circ \\cdots \\circ \\delta_{k_0-1}$, where $k_0 \\cong k (\\text{mod } l)$, is of generalized Penner type such that $\\mathcal{B}_{\\tilde{\\psi}}$ has a singular disk $\\tilde{S}_j$, so that $\\tilde{B}_{j,\\infty}$, the limit of the braid sequence corresponding to $\\tilde{\\psi}$ and $\\tilde{S}_j$, has a strand identified with $\\{f_m\\}_{m \\geq k}$.\n\tThus, there is a Lagrangian disk in $\\pi^{-1}(\\tilde{S}_j)$ whose boundary agrees with $\\{f_m\\}_{m \\geq k}$.\n\tLet $D$ denote the Lagrangian disk in $\\pi^{-1}(\\tilde{S}_j)$.\n\tThen, there is a connected component of \n\t$$\\Big((\\Phi_{1,1} \\circ \\delta_1) \\circ \\cdots \\circ (\\Phi_{k_0,1} \\circ \\delta_k)\\Big)(D) \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)},$$\n\twhose boundary is $\\{ f_m \\}_{m \\in \\mathbb{N}}$, where $\\Phi_{i,t}$ is a Hamiltonian isotopy mentioned mentioned in Section 4.1.\n\t\n\tTo summarize, if there is at least one map of trivial type in $\\{f_m\\}_{m \\in \\mathbb{N}}$, then we have a Lagrangian submanifold in $\\overline{\\pi^{-1}(\\mathring{S}_i)}$, whose boundary agrees with $\\{ f_m\\}_{m \\in \\mathbb{N}}$.\n\tLet $\\mathcal{L}_{i,\\infty}$ be the union of those Lagrangian submanifolds.\n\t\n\tFinally, let us assume that for all $m \\in \\mathbb{N}, f_m$ is not of trivial type. \n\tThen, for all $k \\in \\mathbb{N}$, we will construct a sequence $\\{f_m^k\\}_{m \\in \\mathbb{N}}$ for each $k \\in \\mathbb{N}$, satisfying \n\t\\begin{enumerate}\n\t\t\\item $\\{f_m^k\\}_{m\\in\\mathbb{N}}$ is a strand of $B_{i,\\infty}$,\n\t\t\\item if $ m \\leq kl$, then $f_m^k = f_m$,\n\t\t\\item there exists a constant $N_k \\in \\mathbb{N}$ such that $f_{kl + N_k}^k$ is of trivial type.\n\t\\end{enumerate}  \n\n\tIf there is a sphere having 2 or more plumbing points, there exists a sequence $\\{f_m^k\\}_{m \\in \\mathbb{N}}$ for all $k \\in \\mathbb{N}$.\n\tThis is because of the following:\n\t\n\tWe note that the finite sequence $\\{f_t\\}_{1\\leq t \\leq kl}$ explains a contribution of the braid on a singular disk $S_{i_0}$ on the construction of the braid on a singular disk $S_{j_0}$ when one applies $\\psi^k$. \n\tIn other words, from the view point of Remark \\ref{rmk radius of solid torus}, there is a connected component of $\\psi^k(\\overline{\\pi^{-1}(\\mathring{S}_i)}) \\cap \\pi^{-1}(S_{j_0})$ or $\\psi^k(\\pi^{-1}(\\pi(N_p))) \\cap \\pi^{-1}(S_{j_0})$, where $p$ is the center of $S_{i_0}$ and $N_p$ is the neck at $p$, such that the boundary of the connected component is the image of $f_1 \\circ \\cdots \\circ f_{kl}$. \n\t\n\tIf there exists a sphere having 2 or more plumbing points, the Dehn twist along the sphere appears in $\\psi$, because of our assumption that every Dehn twist appears in $\\psi$. \n\tLet $\\delta_i$ be the Dehn twist.\n\tFor any plumbing points $p$ and $q$ of the sphere, $\\delta_i(\\pi^{-1}(\\pi(N_p)))$ intersects $\\pi^{-1}(S_q^+)$, if the sphere is positive, or $\\pi^{-1}(S_q^-)$, otherwise. \n\tThus, there is a map of trivial type in $\\Delta_i$, the matrix corresponding to $\\delta_i$.\n\t\n\tFor a sufficiently large $N$, $(\\psi^N \\circ \\delta_1 \\circ \\cdots \\circ \\delta_i) (\\pi^{-1}(\\pi(N_p)))$ intersects $\\pi^{-1}(S_{j_0})$.\n\tWe can prove this by observing that $(\\psi^N \\circ \\delta_1 \\circ \\cdots \\circ \\delta_{i-1} )(\\pi^{-1}(S_q^\\pm)) \\cap \\pi^{-1}(S_{j_0}) \\neq \\varnothing$ for some sufficiently large $N$. \n\tThus, there is a finite sequence of functions $\\{g_j\\}_{1 \\leq j \\leq Nl + i}$ such that $g_j$ is an entry of $\\Delta_{j'}$, the matrix corresponding to $\\delta_{j'}$, where $j' \\cong j (\\text{mod } l)$, and the image of $g_1 \\circ \\cdots \\circ g_{Nl+i}$ is identified to the boundary of a connected component of $(\\psi^N \\circ \\delta_1 \\circ \\cdots \\delta_i) (\\pi^{-1}(\\pi(N_p))) \\cap \\pi^{-1}(S_{j_0})$. \n\tMoreover, we can extend the finite sequence $\\{g_j\\}_{1 \\leq j \\leq Nl + i}$ to an infinite sequence $\\{g_j\\}_{j\\in \\mathbb{N}}$ such that $\\{g_j\\}_{j \\in \\mathbb{N}}$ appears in $B_{i,\\infty}$.\n\tThen, by setting $f_{kl + j}^k = g_j$, we prove the existence of $\\{f^k_m\\}_{m \\in \\mathbb{N}}$. \n\t\n\tWe obtain a strand $\\{f_m^k\\}_{k \\in \\mathbb{N}}$ for each $k \\in \\mathbb{N}$. \n\tThese strands converge to $\\{f_m\\}_{m \\in \\mathbb{N}}$ as $k \\to \\infty$. \n\tMoreover, by definition of $\\mathcal{L}_{i, \\infty}$, the boundary of $\\mathcal{L}_{i,\\infty}$ contains strands $\\{f_m^k\\}_{m \\in \\mathbb{N}}$ for all $k \\in \\mathbb{N}$. \n\tThus, the strand $\\{f_m\\}_{m \\in \\mathbb{N}}$ is contained in the boundary of $\\mathcal{L}_{i}$, where $\\mathcal{L}_{i} = \\overline{\\mathcal{L}_{i,\\infty}}$, the closure of $\\mathcal{L}_{i,\\infty}$, i.e., the closure of $\\mathcal{L}_{i,\\infty}$.\n\t\n\tIf there is no sphere with 2 or more plumbing points, then there is only one positive and one negative sphere intersecting at only one point because we are working on a connected plumbing space.\n\tFor the case, we can construct a Lagrangian lamination $\\mathcal{L}$ on $M$ by spinning. \n\tThen, $\\mathcal{L}_{i} :=\\mathcal{L} \\cap \\pi^{-1}(S_i)$ is a Lagrangian lamination which we want to construct.\n\\begin{remark}\n\tWe note that, if there is no sphere with 2 or more plumbing points, then  \n\twe can construct $\\mathcal{L}$ without using singular and regular disks. \n\\end{remark}\n\t\n\t\\vskip.2in\n\t{\\noindent {\\em Convergence to $\\mathcal{L}_i$}.} \n\tLet $L_m := \\psi_H^m(L)$. \n\tWe defined $\\psi_H$ in Step 4 of the proof of Lemma \\ref{lem3}.\n\tWe will prove that $L_m \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}$ converges to $\\mathcal{L}_i$.\n\t\n\tFirst, we will show that \n\t\\begin{gather}\n\t\\label{eqn the limit lamination in a singular disk}\n\t\\lim_{m \\to \\infty} L_m \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)} = \\lim_{m \\to \\infty} (\\psi_H^m(N(\\mathcal{B}_{\\psi})) \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}).\n\t\\end{gather}\n\tSince $\\psi_H(N(\\mathcal{B}_{\\psi})) \\subset N(\\mathcal{B}_{\\psi})$, $$\\psi_H^{m+1}(N(\\mathcal{B}_{\\psi})) \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)} \\subset \\psi_H^m(N(\\mathcal{B}_{\\psi})) \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}.$$\n\tThus, there exists the limit \n\t$$\\lim_{m \\to \\infty} (\\psi_H^m(N(\\mathcal{B}_{\\psi})) \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}) = \\cap_m (\\psi_H^m(N(\\mathcal{B}_{\\psi})) \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}).$$\n\t\n\tIf we equip a Riemannian metric $g$ on $M$, then $d_H(\\psi_H^m(\\mathcal{B}_{\\psi}), \\psi_H^m(N(\\mathcal{B}_{\\psi})))$, where $d_H$ is the Hausdorff metric induced from $g$, converges to zero as $m \\to \\infty$ because of the same reason that $B_{i,m} := \\psi_H^{m}(N(\\mathcal{B}_{\\psi})) \\cap \\pi^{-1}(\\partial S_i) $ converges to an infinite braid $B_{i,\\infty}$ in the last part of Section 4.2.\n\t\n\tSince for a large $N_0$, $L_{N_0}$ intersects $\\pi^{-1}(S_j)$ for any singular disk $S_j$, $L_{m+N_0} \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}$ intersects every connected component of $\\psi_H^m(N(\\mathcal{B}_{\\psi})) \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}$.\n\tThus,\n\t$$0 \\leq \\lim_{m \\to \\infty} d_H(L_{m+N_0} \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}, \\psi_H^m(N(\\mathcal{B}_{\\psi})) \\leq \\lim_{m \\to \\infty} 2 d_H(\\psi_H^m(\\mathcal{B}_{\\psi}), \\psi_H^m(\\mathcal{B}_{\\psi})) = 0.$$\n\tThis proves Equation \\eqref{eqn the limit lamination in a singular disk}.\n\tLet $\\mathbb{L}_i$ be the limit in Equation \\eqref{eqn the limit lamination in a singular disk}.\n\t\n\tSecond, we show that $\\mathbb{L}_i$ is $\\mathcal{L}_i$.\n\tBy the construction of $\\mathcal{L}_i$, we know that \n\t$$\\mathcal{L}_i \\subset \\psi_H^m(N(\\mathcal{B}_{\\psi})) \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)} \\text{  for every  } m \\in \\mathbb{N}.$$\n\tIt implies that $\\mathcal{L}_i \\subset \\mathbb{L}_i$.\n\tMoreover, \n\t$$\\mathcal{L}_i \\cap \\pi^{-1}(\\partial S_i) = \\mathbb{L}_i = B_{i,\\infty} \\cap \\overline{\\pi^{-1}(\\mathring{S}_i)}.$$ \n\tBecause every connected component of $\\mathbb{L}_i$ has a boundary on $\\partial S_i$, this shows $\\mathcal{L}_i = \\mathbb{L}_i$. \n\\end{proof}\t\t\t\t\n\n\n\\subsection{Lagrangian lamination on a regular disk}\n \nIn the previous subsection, we constructed Lagrangian laminations on singular disks, when boundary data for singular disks are given.\nIn the present subsection, first, we will define boundary data for a regular disk. \nThen, second, we will construct Lagrangian laminations on regular disks from the given data.\nFinally, we will prove Theorem \\ref{lamination thm} as a corollary of Lemmas \\ref{lem4} and \\ref{lem6}. \n\nBefore defining the boundary data, we remark that, by Remark \\ref{rmk identify with cotangent bundle of disk}, $\\overline{\\pi^{-1}(\\mathring{R}_i)}$ is symplectomorphic to $DT^*\\mathcal{D}$, where $\\mathcal{D}$ is a disk. \n\nWe define a data $c_{j,m}$ on the boundary of a regular disk $R_j$ for $\\psi^m(L)$, by setting  \n$$c_{j,m} := L_m \\cap \\pi^{-1}(\\partial R_j).$$\nWe defined $L_m$ in the proof of Lemma \\ref{lem4}.\nNote that $c_{j,m}$ is a closed subset, not a class of a closed subset.\n\nTo obtain a limit of $c_{j,m}$, we consider \n$$C_{j,m}:= \\psi_H^m(N(\\mathcal{B}_{\\psi})) \\cap \\pi^{-1}(\\partial R_j),$$ \nas we did in Section 4.2. \nSince $\\psi_H^(N(\\mathcal{B}_{\\psi})) \\subset N(\\mathcal{B}_{\\psi})$, $C_{j,m+1} \\subset C_{j,m}$. \nMoreover, $C_{j,m}$ is the union of solid tori (resp.\\ $S^{n-1} \\times \\mathbb{D}^n$) in $\\pi^{-1}(\\partial R_j)$ for the case $n=2$ (resp.\\ of general $n$). \nIf a symplectic manifold $M$ is equipped with a Riemannian metric $g$, we can measure the radii of solid tori in $C_{j,m}$.\nThe radii decrease exponentially and converge to zero as $m \\to \\infty$, because of the same reason that radii of solid tori comprising $B_{i,m}$ decrease exponentially and converge to zero as $m \\to \\infty$ in Section 4.2.\nThen, the limit of $c_{j,m}$ is given by \n$$C_{j,\\infty} = \\lim_{m \\to \\infty}C_{j,m} = \\cap_m C_{j,m}.$$\n\nThe next step is to smooth $R_j$. \nA regular disk $R_j$ has corners.\nWe will replace $R_j$ with a smooth disk $R_j'$.\nThis is because, at the end, a Lagrangian lamination will be given as graphs of closed sections. \nBy smoothing $R_j$, it will be easier to handle closed sections. \n\nTo smooth $R_j$, we subtract a tubular neighborhood $N(\\partial R_j) \\subset R_j$ from $R_j$. \nLet $R_j' := R_j \\setminus N(\\partial R_j)$.\nThen, $R_j'$ is a smooth disk.\nWe replace $R_j$ with $R_j'$. \nTo finish smoothing, we need to determine boundary data for $R_j'$ from $c_{j,m}$.\n \nEach connected component of $c_{j,m}$ can be identified wit a section of a bundle $\\pi^{-1}(\\partial R_j)$ over $\\partial R_j$. \nWe can extend this section to a closed section of a bundle $\\pi^{-1}(N(\\partial R_j))$ over $N(\\partial R_j)$ by computations.\nThen, the graph of the extended section is a Lagrangian submanifold of $\\pi^{-1}(N(\\partial R_j))$.\nThe boundary of the Lagrangian submanifold on $\\partial R_j'$ makes up the boundary data for $R_j'$.\n\nFrom now, we assume that a regular disk $R_j$ is a smoothed disk.\nLemma \\ref{lem5} claims that for a given data $c_{j,m}$ on a smoothed regular disk $R_j$, we can construct a Lagrangian submanifold $N_{j,m} \\subset \\overline{\\pi^{-1}(\\mathring{R}_i)}$ such that $\\partial N_{j,m} = c_{j,m} \\cap \\overline{\\pi^{-1}(\\mathring{R}_i)}$. \n\n\\begin{lemma}\n\t\\label{lem5}\n\tLet $Q$ be a closed subset of $\\partial T^*\\mathbb{D}^n$ such that there exists a Lagrangian submanifold $L \\subset T^*\\mathbb{D}^n$ so that $L \\cap \\partial T^*\\mathbb{D}^n = Q$ and $L$ is a union of Lagrangian disks transverse to fibers.\n\tThen, we can construct a Lagrangian submanifold $L$ uniquely up to Hamiltonian isotopy through Lagrangians transverse to the fibers.\n\\end{lemma}\n\nTo prove Lemma \\ref{lem5}, we will use the following:\nin Lemma \\ref{lem5}, if an identification $\\varphi: \\partial T^*\\mathbb{D}^n \\stackrel{\\sim}{\\to} S^{n-1} \\times \\mathring{\\mathbb{D}}^n$ is induced from a coordinate chart on $\\mathbb{D}^n$, $\\varphi(Q)$ represent the trivial braid because $L$ is a union of Lagrangian disks.  \n\n\\begin{proof}[Proof  of Lemma \\ref{lem5}]\n\tThe proof of Lemma \\ref{lem5} consists of two parts, the construction of $L$ and the uniqueness of $L$.\n\t\n\t\\vskip.2in\n\t\\noindent{\\em Construction.}\n\tWe start the proof with the simplest case, i.e., when $Q$ is connected.\n\tIn other words, $Q$ represents the braid with only one strand.\n\n\tBy fixing coordinate charts on $\\mathbb{D}^n$, we can write down $Q$ as a section of a disk bundle $\\partial T^*\\mathbb{D}^n$ over $\\partial \\mathbb{D}^n$, i.e.,\n\t$$ Q:= \\{ f_1(x_1,\\cdots,x_n)dx_1 + \\cdots + f_n(x_1,\\cdots,x_n)dx_n \\hspace{0.2em} | \\hspace{0.2em} x_1^2 +  \\cdots + x_n^2 =1 \\}.$$\n\tThen, the simplest case is proved by determining a function $\\phi: \\mathbb{D}^n \\to \\mathbb{R}$ such that $d \\phi = f_1dx_1 + \\cdots + f_ndx_n$ on $\\partial \\mathbb{D}^n$. \n\tThe graph of $d \\phi$ is a Lagrangian submanifold which we would like to find. \n\tNote that there are infinitely many $\\phi$ satisfying the conditions, but the Hamiltonian isotopy class of the graph of $d\\phi$ is unique through Lagrangians transverse to the fibers.\n\t\n\tIf $Q$ has 2 or more connected components $l_i$, then we can write $l_i$ as a section over $\\partial \\mathbb{D}^n$. \n\tFor each $i$, we need to determine functions $\\phi_i : \\mathbb{D}^n \\to \\mathbb{R}$ such that $d \\phi_i$ agrees with $l_i$ on $\\partial \\mathbb{D}^n$. \n\tMoreover, to avoid self-intersection, we need $d \\phi_i \\neq d\\phi_j$ for all $i \\neq j$ everywhere. \n\tThen, the union of graphs of $d \\phi_i$ on $T^*\\mathbb{D}^n$ is a Lagrangian submanifold $L$ which we want to construct.\n\t\n\tWe discuss with the simplest non-trivial case, i.e., $Q$ has two connected components $l_0$ and $l_1$, and the dimension $2n =4$. \n\tWithout loss of generality, we assume that $l_0$ is the zero section.\n\tThen, we can assume that $\\phi_0 \\equiv 0$. \t\n\tWe only need to determine $\\phi_1$ such that $d \\phi_1$ does not vanish everywhere. \n\t\n\tWe assume that there exists $\\phi_1$ satisfying the conditions.\n\tThen, we will collect combinatorial data from $\\phi_1$, and we will construct a function $\\tilde{\\phi}_1$ satisfying conditions, from the combinatorial data.\n\tThrough this, we will see what combinatorial data we need.\n\tWe will end the construction part by obtaining the combinatorial data from the given $Q$.\n\t \n\tFor convenience, we will use the polar coordinates instead of the $(x,y)$-coordinate on $\\mathbb{D}^2$.\n\tLet $r_0$ be a small positive number. \n\tWe restrict the function $\\phi_1$ on $[r_0,1] \\times S^1$.\n\tOn $\\{1\\} \\times S^1 = \\partial \\mathbb{D}^2$ agrees with $l_1$.\n\tOn $\\{r_0\\} \\times S^1, d\\phi_1$ is approximately a constant section $a dx + b dy = a( \\cos \\theta dr - r_0 \\sin \\theta d \\theta) + b(\\sin \\theta dr +r_0 \\cos \\theta d \\theta)$, where $d \\phi_1(0,0) = a dx + b dy$ and $(x,y)$ are the standard coordinate charts of $\\mathbb{D}^2$.\n\tWe remark that on $\\{r_0\\} \\times S^1$, the pair of graphs of $d\\phi_i|_{\\{r_0\\}\\times S^1}$ represents the trivial braid under the identification induced from the $(x,y)$-coordinates.  \n\tThen, the pair $(d\\phi_0 \\equiv 0, d\\phi_1)$ implies an isotopy between two representatives of the trivial braid on $[r_0,1] \\times S^1$. \n\t\n\tFor every $r_* \\in [r_0,1]$, we can find all local maxima and minima of a function \n\t$$ \\theta \\mapsto \\phi_1(r_*,\\theta).$$\n\tWe mark $(r_*, \\theta_*)$ as a red (resp.\\ blue) point if the above function has a local maxima (resp.\\ minima) at $\\theta_*$. \n\tIf $r_*=1$, there are same number of red\/blue marked points on $\\{1\\} \\times S^1$, and there are only one red\/blue marked point on $\\{r_0\\} \\times S^1$.\n\tOn $[r_0,1] \\times S^1$, we have a collection $\\mathcal{C}$ of curves shaded red and blue. \n\tIf a curve in $\\mathcal{C}$ is not a circle, then the curve has two end points on the boundary of $[r_0,1] \\times S^1$. \n\tThere are exactly two curves connecting both boundary components of $[r_0,1] \\times S^1$, and those two curves have end points of the same color.\n\t \n\tIf we write $d \\phi_1 = f d\\theta + g dr$, then $f$ is zero on curves in $\\mathcal{C}$. \n\tSince $d \\phi_1$ does not vanish, $g$ cannot be zero on the curves. \n\tThus, we can assign the sign of $g$ for each curve.\n\tFigure \\ref{local max and min} is an example of a collection $\\mathcal{C}$. \n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\input{exam_of_a_collection.pdf_tex}\n\t\t\\caption{Example of a collection $\\mathcal{C}$ on $[r_0,1] \\times S^1$.}\n\t\t\\label{local max and min}\n\t\\end{figure}\n\t\n\tConversely, if we have a collection $\\mathcal{C}$ of curves such that each curve is shaded red and blue and is equipped with a sign, then we can draw a graph of $\\tilde{\\phi}_1$ roughly. \n\tThis is because, the collection $\\mathcal{C}$ determines the sign of horizontal directional derivative of $\\tilde{\\phi}_1$, i.e., $d\\tilde{\\phi}_1(\\partial_{\\theta})$ on every point of $[r_0,1] \\times S^1$, and vertical directional derivative of $\\tilde{\\phi}_1$, i.e., $d\\tilde{\\phi}_1(\\partial_r)$ on the curves. \n\tFrom these, one obtains a (rough) graph of $\\tilde{\\phi}_1$.\n\tThus, in order to determine a function $\\phi_1$, it is enough to determine a collection $\\mathcal{C}$ of curves in $[r_0,1] \\times S^1$ from the given $Q$. \n\t\n\tFor the given $Q$, we assume that a connected component $l_0$ of $Q$ is the zero section without loss of generality. \n\tFor the other connected component $l_1$, one has $f_1, g_1:S^1 \\to \\mathbb{R}$ such that $l_1$ is the graph of $f_1 d\\theta + g_1 dr$ on $\\{1\\} \\times S^1 = \\partial \\mathbb{D}^2$. \n\tWe know that $Q$ represents the trivial braid with respect to the standard $(x,y)$-coordinate of $\\mathbb{D}^2$.\n\tThus, there is an isotopy $\\Gamma : [r_0,1] \\times S^1 \\to \\mathbb{D}^2$ such that \n\t\\begin{gather*}\n\t\\Gamma(1, \\theta) = (f(\\theta), g(\\theta)), \\hspace{0.5em} \\Gamma(r_0,\\theta) = (A r_0 \\cos \\theta, A \\sin \\theta)\\\\\n\t\\Gamma(t,\\theta) \\neq (0,0) \\text{  for all  } (t,\\theta) \\in [r_0,1] \\times S^1,\n\t\\end{gather*}\n\twhere $A$ is a constant.\n\t\n\tFor every $r \\in [r_0,1]$, let $\\gamma_r (\\theta) = \\Gamma(r, \\theta)$.\n\tThen, $\\gamma_r$ is a closed curve in $\\mathbb{D}^2$, for all $r$.\n\tMoreover, $\\Gamma$ is a path connecting $\\gamma_1$ and $\\gamma_{r_0}$ in the loop space of $\\mathring{\\mathbb{D}}^2$ without touching the origin. \n\t\n\tWe mark $(r,\\theta)$ on $[r_0,1] \\times S^1$ as a red (resp.\\ blue) point if $\\gamma_r(\\theta)$ intersects $dr$-axis from right to left (resp.\\ from left to right).\n\tThese marked points comprise curves in $[r_0,1] \\times S^1$, and we have a collection $\\mathcal{C}$ of curves, shaded red and blue, in $[r_0,1] \\times S^1$.\n\tWe know that $\\gamma_1$ has an even number of intersection points.\n\tWhen $r$ decreases, there is a series of creations\/removes of intersection points, which are given by finger moves along $dr$-axis.\n\tEach finger move does not touch the origin.\n\tThus, for a curve in $\\mathcal{C}$, every intersection point composing the curve lies on either the positive $dr$-axis or the negative $dr$-axis.\n\tThen, we can assign a sign for each curve in $\\mathcal{C}$.\n\t\n\tFigure \\ref{homotopy between loops} is an example of $\\Gamma$, corresponding to the case described by Figure \\ref{local max and min}. \n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\input{exam_of_creation_of_a_collection.pdf_tex}\n\t\t\\caption{Creation of a collection $\\mathcal{C}$.}\n\t\t\\label{homotopy between loops}\n\t\\end{figure}\t\n\tThe upper left of Figure \\ref{homotopy between loops} is $\\gamma_1$ and the upper right is $\\gamma_{r_0}$.\n\tThrough the first arrow, we observe a finger move removing two intersection points. \n\tThose two intersection points correspond to $m_2$, a local maxima shaded red, and $n_2$, a local minima shaded blue.\n\tThus, we obtain a curve connecting $m_2$ and $n_2$ in Figure \\ref{local max and min}.\n\tMoreover, the intersection points lie in the negative part of the $dr$-axis.\n\tThus, we assign a negative sign to the curve.\n\tSimilarly, we observe there are finger moves removing intersection points.\n\tWe obtain curves connecting $m_i$ and $n_i$ for $i=1, 2$, and $3$ in Figure \\ref{local max and min}. \n\tAfter the finger moves, there are only two intersection points corresponding to $m_*$ and $n_*$, and we obtain curves connecting $m_4$(resp.\\ $n_4$) and $m_*$(resp.\\ $n_*$). \n\t\n\tWe have constructed a collection $\\mathcal{C}$ of curves on $[r_0,1] \\times S^1$ from an isotopy $\\Gamma$.\n\tThus, we can obtain a function $\\phi_1 : [r_0,1] \\times S^1 \\to \\mathbb{R}$. \n\tIn order to complete the proof, we need to extend $\\phi_1$ into a small disk with radius $r_0$. \n\tWe have \n\t$$\\phi_1(x,y) = A r \\sin \\theta = Ay$$\n\ton the small disk.\n\t\n\tThe situation for the general case is analogous.\n\tIf $Q$ has more connected components $l_i$ for $i = 0, \\cdots, k$, then we have to determine $\\phi_i: \\mathbb{D}^2 \\to \\mathbb{R}$ such that $d\\phi_i = l_i$ on $\\partial \\mathbb{D}^2$, and $d\\phi_i \\neq d\\phi_j$ for all $i \\neq j$. \n\tWe fix an isotopy $\\Gamma$, and obtain a collection $\\mathcal{C}$ of curves on $[r_0,1] \\times S^1$ from $\\Gamma$.\n\tEach curve in $\\mathcal{C}$ encodes restrictions on $d\\phi_i - d\\phi_j$ for some $i$ and $j$.\n\tMore precisely,\t$(\\phi_i-\\phi_j)$ has a local maxima (resp.\\ minima) in the horizontal direction, only at a point of a curve shaded red (resp.\\ blue), and $(d\\phi_i-d\\phi_j)(\\partial_{r})$ has the sign assigned on the curve. \n\tFor the case of general dimension $2n$, we obtain combinatorial data from $Q$, i.e., a collection of curves on $[r_0, 1] \\times S^{n-1}$ assigned a sign, and construct functions on $\\mathbb{D}^n$ from the combinatorial data.\n\n\n\t\n\t\\vskip.2in\n\t\\noindent{\\em Uniqueness.}\n\tRecall that the construction consists of three steps. \n\tFirst, we choose an isotopy $\\Gamma$ connecting $Q$ and the trivial representative of the trivial braid. \n\tThen, we obtained a collection $\\mathcal{C}$ of curves from $\\Gamma$, such that each curve encodes restrictions on $d \\phi_i - d \\phi_j$.\n\tThe last step is to construct a set of functions $\\{ \\phi_i : \\mathbb{D}^n \\to \\mathbb{R}\\}$.\n\t\n\tThe construction depends on choices in the first and last steps.\n\tMore precisely, for the first step, the choice of isotopy $\\Gamma$ is not unique. \n\tIf we choose an isotopy $\\Gamma$, then there is a unique collection $\\mathcal{C}$. \n\tHowever, a set $\\{\\phi_i\\}$ of functions, which is constructed from the collection $\\mathcal{C}$, is not unique. \n\tWe will show that the Hamiltonian isotopy class of $L$, through Lagrangians transverse to the fibers, is independent to those choices. \n\t\n\tFirst, we discuss the choice in the third step. \n\tLet us assume that we have a collection $\\mathcal{C}$ of curves in $[r_0,1] \\times S^{n-1}$  and two sets of functions $\\{\\phi_i \\}_i$ and $\\{\\zeta_i \\}_i$ satisfying the restrictions encoded by $\\mathcal{C}$.\n\tThen, by setting $\\eta_{i,t} := (1-t)\\phi_i + t \\zeta_i$, we obtain a family of sets of functions such that every member of the family satisfies the restrictions encoded by $\\mathcal{C}$. \n\t\n\tLet $L_t$ be the Lagrangian submanifold corresponding to $\\{\\eta_{i,t}\\}$ for a fixed $t$.\n\tThen, $L_t$ is a Lagrangian isotopy connecting $L_0$, corresponding to $\\{\\phi_i \\}$, and $L_1$, corresponding to $\\{\\zeta_i \\}$. \n\tSince $L_t$ is a disjoint union of Lagrangian disks in $T^*\\mathbb{D}^n$,\n\t$L_0$ and $L_1$ are Hamiltonian isotopic.\n\tThus, the Hamiltonian class of $L$ through Lagrangians transverse to the fibers is independent of the choice of functions for the third step of the construction.\n\t\n\tBefore discussing the choice of the first step, note that a continuous change on a collection $\\mathcal{C}$ does not make a change on the Hamiltonian isotopy class.\n\tMore precisely, let $\\mathcal{C}_0= \\{ \\gamma_1, \\cdots, \\gamma_N \\}$ be a collection of curves and let $\\{\\phi_i\\}$ be a set of functions corresponding to $\\mathcal{C}_0$. \n\tIf $\\{\\gamma_{k,t}\\}$ is a continuous family of curves with respect to $t$ such that $\\gamma_{k,0} = \\gamma_k$ for all $k$, then we can obtain a continuous family $\\{\\phi_{1,t}, \\cdots, \\phi_{N,t}\\}$ such that $\\phi_{i,0} = \\phi_i$ and $\\{\\phi_{1,t}, \\cdots, \\phi_{N,t}\\}$ corresponds to $\\mathcal{C}_t := \\{\\gamma_{1,t}, \\cdots, \\gamma_{N,t} \\}$.\n\tThen, it is easy to check that the Hamiltonian isotopy class of the union of graphs of $d\\phi_{i,t}$ in $T^*\\mathbb{D}^n$, through Lagrangians transverse to the fibers, is independent to $t$. \n\t\n\tFinally, we will discuss the choice of $\\Gamma$.\n\tLet $\\Gamma_0$ and $\\Gamma_1$ be two isotopies obtained from the given $Q$ in the first step. \n\tThen, we can understand $\\Gamma_0$ and $\\Gamma_1$ as paths on the loop space of the configuration space of $\\mathring{\\mathbb{D}}^n$.\n\tSince the loop space is simply connected, there is a continuous family $\\{\\Gamma_t\\}_{t \\in [0,1]}$ connecting $\\gamma_0$ and $\\gamma_1$. \n\t\n\tLet $\\mathcal{C}_t$ be the collection of curves obtained from $\\Gamma_t$ and let $\\{ \\phi_i \\}$ be a set of functions constructed from $\\mathcal{C}_0$.\n\tThere is $\\{\\phi_{i,t}\\}$ corresponding to $\\mathcal{C}_t$ such that $\\phi_{i,0} = \\phi_i$. \n\tThen, if $L_t$ is the union of graphs of $d\\phi_{i,t}$, then the Hamiltonian class of $L_t$ is independent to $t$. \n\tThis shows the uniqueness of $L$, up to Hamiltonian isotopy, through Lagrangians transverse to the fibers.   \n\\end{proof}\n\nFor a smoothed regular disk $R_j$, there is a sequence of data $c_{j,m}$ for each $m \\in \\mathbb{N}$.\nThen, we can construct a sequence of Lagrangian submanifolds $N_{j,m} \\subset \\overline{\\pi^{-1}(\\mathring{R}_j)}$ such that $N_{j,m} \\cap \\partial \\overline{\\pi^{-1}(\\mathring{R}_j)} = c_{j,m}$. \nThe following lemma, Lemma \\ref{lem6}, claims that we can construct $N_{j,m}$ wisely, so that $N_{j,m}$ converges to a Lagrangian lamination $\\mathcal{N}_j$ as $m$ goes to $\\infty$.\n\n\\begin{lemma}\n\t\\label{lem6}\n\tIt is possible to construct $N_{j,m} \\subset \\overline{\\pi^{-1}(\\mathring{R}_j)}$ so that the sequence $N_{j,m}$ converges to a Lagrangian lamination $\\mathcal{N}_j \\subset \\overline{\\pi^{-1}(\\mathring{R}_j)}$ as $m \\to \\infty$. \n\\end{lemma}\n\n\\begin{proof}\n\tLet the boundary condition $c_{j,m}$ be the set $\\{ l_{1,m}, \\cdots, l_{{N_m},m} \\}$, where $l_{i,m}$ is a connected component of $c_{j,m}$, or equivalently, $l_{i,m}$ is a strand of the braid represented by $c_{j,m}$.  \n\tWe defined $C_{j,m}$ as a disjoint union of solid tori in $\\pi^{-1}(\\partial R_j)$ at the beginning of the present subsection. \n\tThen, we can divide $c_{j,m}$ into a partition, so that $l_{i,m}$ and $l_{j,m}$ are in the same subset if and only if $l_{i,m}$ and $l_{j,m}$ are in the same solid torus (resp.\\ $S^{n-1} \\times \\mathbb{D}^n$ for a higher dimensional case) in $C_{j,m}$. \n\tAfter that, we randomly choose a connected component $l_{s,m}$ from each subset of the partition. \n\t\n\tBy Lemma \\ref{lem5}, there is $\\phi_{s,m}: R_j \\to \\mathbb{R}$ such that $d\\phi_{s,m} =l_{s,m}$ on $\\partial R_j$.\n\tThen, the graph of $d\\phi_{s,m}$ is a Lagrangian disk in $\\overline{\\pi^{-1}(\\mathring{R}_i)}$. \n\tWe can choose a neighborhood $N(\\phi_{s,m})$ of the graph of $d\\phi_{s,m}$ in $\\overline{\\pi^{-1}(\\mathring{R}_i)}$, such that $N(\\phi_{s,m}) \\simeq T^*\\mathbb{D}^n$ and $N(\\phi_{s,m}) \\cap \\pi^{-1}(\\partial R_j)$ is the torus in $C_{j,m}$ containing $l_{s,m}$.\n\tMoreover, we can assume that \n\t$$d_H(N(\\phi_{s,m}), \\text{the graph of }d\\phi_{s,m}) < 2 r^m,$$ \n\twhere $d_H$ is the Hausdorff metric induced by a fixed Riemannian metric.\n\t\n\tWe apply Lemma \\ref{lem5} to $\\{ l_{t,m+1} \\in c_{j,m+1} \\hspace{0.2em} | \\hspace{0.2em} l_{t,m+1} \\subset N(\\phi_{s,m}) \\}$ in $N(\\phi_{s,m}) \\simeq T^*\\mathbb{D}^n$.\n\tThen, we can construct $\\phi_{t,m+1} : R_j \\to \\mathbb{R}$ such that $d\\phi_{i,m+1} = l_{t,m+1}$ on $\\partial R_j$ and the graph of $d\\phi_{t,m+1}$ is contained in $N(\\phi_{s,m+1})$.\n\tWe repeat this procedure inductively on $m \\in \\mathbb{N}$.\n\t\n\tLet $l$ be a strand of $C_{j,\\infty}$.\n\tThen, there is a sequence $l_{i_m,m} \\in c_{j,m}$ such that $l_{i_m,m}$ converges to $l$. \n\tIf we construct $\\phi_{i,m}$ by repeating the above procedure, we know that $$d_H(d\\phi_{i_m,m}, d\\phi_{i_n,n}) < 4 r^{\\max(m,n)}.$$\n\tThus, $d\\phi_{i_m,m}$ converges.\n\tMoreover, by assuming that $\\phi_{i,m}(p) = 0$ for every $i$ and $m$, where $p$ is a center of $R_j$, $\\phi_{i_m,m}$ converges to a function $\\phi$.\n\tThe graph of $d\\phi$ is a Lagrangian disk in $\\overline{\\pi^{-1}(\\mathring{R}_j)}$ such that whose boundary is $l$, the stand of $C_{j,\\infty}$. \n\tThe union of graphs of $d\\phi$ is the Lagrangian lamination $\\mathcal{N}_j$ which $N_{j,m}$ converges to.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{lamination thm}]\n\tBy Lemma \\ref{lem4}, there is a Lagrangian lamination $\\mathcal{L}_i$ in $\\overline{\\pi^{-1}(\\mathring{S}_i)}$ and by Lemma \\ref{lem6}, there is a Lagrangian lamination $\\mathcal{N}_j$ in $\\overline{\\pi^{-1}(\\mathring{R}_j)}$.\n\tMoreover, every Lagrangian lamination agrees with each other along boundaries, thus we can glue them.\n\tThen we obtain a Lagrangian lamination $\\mathcal{L}$ in $M$. \n\\end{proof} \n\n\n\n\\subsection{A generalization}\nIn the previous sections, we assumed that $\\psi$ is of generalized Penner type. \nIn the present subsection, we discuss a symplectic automorphism $\\psi : (M,\\omega) \\to (M,\\omega)$, not necessarily to be of generalized Penner type, with some assumptions.\n\nFirst, we assume that there is a Lagrangian branched submanifold $\\mathcal{B}_{\\psi}$ such that $\\psi(\\mathcal{B}_{\\psi})$ is (weakly) carried by $\\mathcal{B}_{\\psi}$. \nThe proof of Lemma \\ref{lem1} carries over with no change.\nThus, if a Lagrangian submanifold $L$ is (weakly) carried by $\\mathcal{B}_{\\psi}$, then $\\psi(L)$ is carried by $\\mathcal{B}_{\\psi}$. \n\nAs mentioned in Section 4.1, we assume that $\\mathcal{B}_{\\psi}^*$ admits a decomposition into a union of finite number of singular disks $S_i \\simeq \\mathbb{D}^n$ and regular disks $R_j \\simeq \\mathbb{D}^n$. \n\n\\begin{proof}[Proof of Theorem \\ref{generalized theorem}]\n\tFirst, we define data on the boundary of each singular and regular disk, in the same way we did for the case of $\\psi$ of generalized Penner type.\n\tThen, on a regular disk $R_j$, the proofs of Lemma \\ref{lem5} and Lemma \\ref{lem6} carry over with no change. \n\tThus, we can construct a Lagrangian lamination on $\\pi^{-1}(R_j)$. \n\t\n\tOn a singular disk $S_i$, we define the boundary data in the same way. \n\tIn other words, the boundary data is defined by the isotopy class of $ \\psi^m(L) \\cap \\pi^{-1}(\\partial S_i)$. \n\tWe also can obtain a matrix $\\Psi$, which explains how the sequences of braids are constructed inductively. \n\tHowever, the rest of the proof of Lemma \\ref{lem4} does not carry over.\n\tThis is because in the proof of Lemma \\ref{lem4}, functions of trivial type have a key role. \n\tTo use the same proof, we need to show that there are enough functions of trivial type. \n\tHowever, the assumptions cannot imply the existence of enough functions of trivial type. \n\t\n\tFor a singular disk $S_i$, let $\\{f_m\\}_{m \\in \\mathbb{N}}$ be a strand of the limit braid on $S_i$. \n\tWe note that each strand can be identified to an infinite sequence of functions.\n\tWe forget specific functions $f_m$, but remember their types.\n\tThen, we obtain a sequence of types. \n\tThe sequence of types determines the ``shape'' of strand, for example, how many times the strand is rotated.\n\t\n\tWe can construct a symplectomorphism $\\phi$ which is of generalized Penner type such that $\\mathcal{B}_{\\phi}$ has a singular disk $S$ so that the limit braid assigned on $S$ has a strand of the same shape. \n\tIn Section 4.3, we constructed a Lagrangian submanifold $L_0 \\subset \\overline{\\pi^{-1}(\\mathring{S})}$ such that $\\partial L_0$ is the strand. \n\tSince $\\overline{\\pi^{-1}(\\mathring{S})} \\simeq \\overline{\\pi^{-1}(\\mathring{S}_i)}$, we assume that $L_0$ is a Lagrangian submanifold in $\\overline{\\pi^{-1}(\\mathring{S}_i)}$ and $\\partial L_0$ has the same shape to the strand which we choose. \n\tBy scaling and translating $L_0$ inside $\\overline{\\pi^{-1}(\\mathring{S}_i)}$, we obtain a Lagrangian submanifold whose boundary agrees with the strand. \n\t\t\n\tThe rest of the proof is the same as the proof of Theorem \\ref{lamination thm}.\n\\end{proof}\n\n\\section{Application on the Lagrangian Floer homology}\n\\label{section pseudo-Anosov functors}\nIn this section, we will give an application of the previous sections on Lagrangian Floer homology.\nMore precisely, we will prove Theorem \\ref{thm Lagrangian floer homology} and give an example in Section \\ref{subsection pA functors - lemma}.\n\n\\subsection{Setting}\n\\label{subsection pA functors - setting}\nIn the present subsection, we will explain terminology in Theorem \\ref{thm Lagrangian floer homology}.\n\nIn Section \\ref{section pseudo-Anosov functors}, we assume that our symplectic manifold $M$ is a plumbing space $M = P(\\alpha_1, \\cdots, \\alpha_m, \\beta_1, \\cdots, \\beta_l)$ of Penner type defined as follows:\n\\begin{definition}\n\t\\label{def plumbing space of Penner type}\n\tA plumbing space $M =P(\\alpha_1, \\cdots, \\alpha_m, \\beta_1, \\cdots, \\beta_l)$ is of {\\em Penner type} if $\\alpha_i$ and $\\beta_j$ satisfy\n\t\\begin{enumerate}\n\t\t\\item $\\alpha_1, \\cdots, \\alpha_m$ and $\\beta_1, \\cdots, \\beta_l$ are $n$-dimensional spheres,\n\t\t\\item $\\alpha_i \\cap \\alpha_j = \\varnothing$, and $\\beta_i \\cap \\beta_j =\\varnothing$, for all $i \\neq j$.\n\t\\end{enumerate} \n\\end{definition}\nNote that $P(\\alpha_1, \\cdots, \\alpha_m, \\beta_1, \\cdots, \\beta_l)$ is defined in Section 2.1.\n\nFrom now on, we will define an involution $\\eta : M \\stackrel{\\sim}{\\to} M$, which is associated to $M$. \n\\vskip0.2in\n\n\\noindent{\\em Involution $\\eta_0$ on $T^*S^n$ :}\nFirst, we will define an involution $\\eta_0$ on $T^*S^n$. \nLet \n\\begin{gather*}\nS^n = \\{ x \\in \\mathbb{R}^{n+1} \\hspace{0.2em} | \\hspace{0.2em} |x|=1 \\}, \\\\\nT^*S^n = \\{ (x,y) \\in S^n \\times \\mathbb{R}^{n+1} \\hspace{0.2em} | \\hspace{0.2em} x \\in S^n, <x,y>= 0 \\}.\n\\end{gather*}\nThen, we define $\\eta_0 : T^*S^n \\stackrel{\\sim}{\\to} T^*S^n$ as follow:\n\\begin{gather*}\n\\eta_0(x_1, \\cdots, x_{n+1}, y_1, \\cdots, y_{n+1}) = (x_1, x_2, -x_3, \\cdots, -x_{n+1}, y_1, y_2, -y_3, \\cdots, -y_{n+1}).\n\\end{gather*}\n\nLet \n\\begin{gather*}\nS =\\{ (\\cos \\theta, \\sin \\theta, 0, \\cdots, 0) \\hspace{0.2em} | \\hspace{0.2em} \\theta \\in [0, 2\\pi] \\} \\subset S^n, \\\\\nT^*S = \\{ (\\cos \\theta, \\sin \\theta, 0, \\cdots, 0, -r \\sin \\theta, r \\cos \\theta, 0, \\cdots, 0) \\hspace{0.2em} | \\hspace{0.2em} \\theta \\in [0, 2\\pi], r \\in \\mathbb{R} \\} \\subset T^*S^n. \n\\end{gather*}\nThen, it is easy to check that $T^*S$ is the set of fixed points of $\\eta_0$, equivalently, $\\eta^{fixed}_0 = T^*S$.\n\\vskip0.2in\n\n\\noindent{\\em Involution $\\eta$ associated to $M$ :}\nFirst, we will construct an involution $\\eta_{\\alpha_i}$ and $\\eta_{\\beta_j}$ on $T^*\\alpha_i$ and $T^*\\beta_j$ for every $i$ and $j$.\nNote that $T^*\\alpha_i, T^*\\beta_j \\subset M$.  \n\nFor each $\\alpha_i$, we will choose an embedded circle $S_{\\alpha_i} \\subset \\alpha_i$ such that $S_{\\alpha_i}$ contains every plumbing point of $\\alpha_i$. \nThen, there is a symplectic isomorphism $\\phi_{\\alpha_i} : T^*S^n \\stackrel{\\sim}{\\to} T^*\\alpha_i$ such that $\\phi_{\\alpha_i}(S^n)= \\alpha_i$ and $\\phi_{\\alpha_i}(S) = S_{\\alpha_i}$. \nOne obtains an involution $\\eta_{\\alpha_i} : T^*\\alpha_i \\stackrel{\\sim}{\\to} T^*\\alpha_i$ by setting \n\\begin{gather*}\n\\eta_{\\alpha_i} := \\phi_{\\alpha_i} \\circ \\eta_0 \\circ (\\phi_{\\alpha_i})^{-1}.\n\\end{gather*}\nSimilarly, one obtains an involution $\\eta_{\\beta_j} : T^*\\beta_j \\stackrel{\\sim}{\\to} T^*\\beta_j$ in the same way.\n\nWithout loss of generality, one can assume that $\\eta_{\\alpha_i}(x) = \\eta_{\\beta_j}(x)$ for every $x \\in T^*\\alpha_i \\cap T^*\\beta_j$.\nFinally, the involution $\\eta : M \\stackrel{\\sim}{\\to} M$ is defined as follows:\n\\begin{gather*}\n\\eta(x) := \n\\left\\{\\begin{matrix}\n\\eta_{\\alpha_i}(x) \\text{  if  } x \\in T^*\\alpha_i,\\\\\n\\eta_{\\beta_j}(x) \\text{  if  } x \\in T^*\\beta_j. \n\\end{matrix}\\right.\n\\end{gather*}\nWe will call $\\eta$ {\\em the involution associated to $M$}.\n\n\\begin{remark}\n\t\\label{rmk properties of involution}\n\tLet $\\tilde{M}$ be the set of fixed points of $\\eta$, i.e., $\\tilde{M} = \\{ x \\in M \\hspace{0.2em} | \\hspace{0.2em} \\eta(x) = x \\}$.\n\tIt is easy to check that $\\tilde{M}$ is a $2$--dimensional symplectic submanifold of $M$. \n\tMoreover, $\\tilde{M}$ is symplectomorphic to a plumbing space $P(S_{\\alpha_1}, \\cdots, S_{\\alpha_m}, S_{\\beta_1}, \\cdots, S_{\\beta_l})$ of Penner type. \n\tNote that $S_{\\alpha_i}$ and $S_{\\beta_j}$ are embedded circles in $\\alpha_i$ and $\\beta_j$. \n\t\n\tWe call $\\tilde{M}$ {\\em the fixed surface of $M$}. \n\\end{remark}\n\n\\subsection{Proof of Theorem \\ref{thm Lagrangian floer homology}}\n\\label{subsection pA functors - proof}\n\nLet $M$ be a plumbing space of Penner type.\nLet $\\eta$ be the associated involution of $M$.\nLet $L_0$ and $L_1$ be a transversal pair of Lagrangian submanifolds such that \n\\begin{enumerate}\n\t\\item $\\eta(L_i) = L_i$.\n\t\\item Let $\\tilde{L}_i = L_i \\cap M_i$. Then, $\\tilde{L}_i$ is a Lagrangian submanifold of $\\tilde{M}$. \n\t\\item $L_0 \\cap L_1 = \\tilde{L}_0 \\cap \\tilde{L}_1$. \n\t\\item $L_0$ and $L_1$ are not isotopic to each other.\n\\end{enumerate}\nWe will compute $\\mathbb{Z}\/2$--graded Lagrangian Floer homology $HF^*(L_0,L_1)$ over the Novikov field $\\Lambda$ of characteristic 2. \nTo do this, we will prove that chain complexes $CF^*(L_0,L_1)$ and $CF^*(\\tilde{L}_0,\\tilde{L}_1)$ are the same chain complexes. \nMore precisely, we will show that those two chain complexes have the same generators and the same differential maps. \n\nFirst, it is easy to show that $CF^*(L_0,L_1)$ and $CF^*(\\tilde{L}_0,\\tilde{L}_1)$ have the same generators since $L_0$ and $L_1$ satisfy that $L_0 \\cap L_1 = \\tilde{L}_0 \\cap \\tilde{L}_1$.\nThus, $CF^*(L_0,L_1) = CF^*(\\tilde{L}_0,\\tilde{L}_1)$ as vector spaces.\n\nSecond, let $\\partial$ (resp.\\ $\\tilde{\\partial}$) denote the differential map on $CF^*(L_0,L_1)$ (resp.\\ $CF^*(\\tilde{L}_0,\\tilde{L}_1)$). \nThen, \n\\begin{gather*}\n\\partial (p) = \\sum_{\\substack{q \\in L_0 \\cap L_1 \\\\ [u]:ind([u])=1}}(\\#\\mathcal{M}(p,q;[u],J))T^{\\omega([u])}q,\n\\end{gather*}\nwhere $J$ is an almost complex structure on $M$, $u$ is a holomorphic strip connecting $p$ and $q$, and $\\mathcal{M}(p,q;[u],J)$ is the moduli space of holomorphic strips.\nWe skip the foundational details of the definition of $\\partial$.\n\nOne can easily check that $\\eta \\circ u$ is also another holomorphic strip connecting $p$ and $q$. \nLet assume that for a holomorphic strip $u$, the image of $u$ is not contained in $\\tilde{M}$. \nThen, $u$ and $\\eta \\circ u$ will be canceled together in $\\partial(p)$, since the Novikov field $\\Lambda$ is of characteristic 2.\nThus, in order to define the differential map $\\partial$, it is enough to count holomorphic strips $u$ such that the image of $u$ is contained in $\\tilde{M}$. \n\nOn the other hands, in order to define $\\tilde{\\partial} : CF^*(\\tilde{L}_0,\\tilde{L}_1) \\to CF^*(\\tilde{L}_0,\\tilde{L}_1)$, one needs to count the holomorphic strips on $\\tilde{M}$. \nThus, $\\partial(p) = \\tilde{\\partial}(p)$ for all $p \\in L_0 \\cap L_1 = \\tilde{L}_0 \\cap \\tilde{L}_1$. \n\nUnder the assumptions, $HF^*(L_0,L_1)= HF^*(\\tilde{L}_0, \\tilde{L}_1)$. \nNote that the former is defined on $M^{2n}$, but the latter is defined on a surface $\\tilde{M}$.\nThen, Lemma 2.18 of \\cite{MR3289326} completes the proof. \n\\qed\n\n\\subsection{Example \\ref{exmp of homology theorem}}\n\\label{subsection pA functors - lemma}\nIn the present subsection, we will prove Lemmas \\ref{lemma transversal intersection of Lagrangian branched submanifolds} and \\ref{lemma carried by} in order to slightly weaken the difficulty of applying Theorem \\ref{thm Lagrangian floer homology}.\nThen, we will give the Example \\ref{exmp of homology theorem}.\n\nBefore giving the statement of Lemmas \\ref{lemma transversal intersection of Lagrangian branched submanifolds} and \\ref{lemma carried by}, we will establish notation.\nIn Section \\ref{section pseudo-Anosov functors}, $M = P(\\alpha_1, \\cdots, \\alpha_m, \\beta_1, \\cdots, \\beta_l)$ is a plumbing space of Penner type. \nThen, as we did in Section 3.4, we can constructed a set $\\mathbb{B}$ of Lagrangian branched submanifolds of $M$.\n\nEvery Lagrangian branched submanifold $\\mathcal{B} \\in \\mathbb{B}$ is a union of (parts of) $\\alpha_i$ and $\\beta_j$ and Lagrangian connected sums $\\alpha_i$ and $\\beta_j$. \nHowever, there are two possible Lagrangian connect sums of $\\alpha_i$ and $\\beta_j$ at each plumbing point $p \\in \\alpha_i \\cap \\beta_j$.\nThey are $\\alpha_i \\#_p \\beta_j$ and $\\beta_j \\#_p \\alpha_i$.\nBy assuming that $\\alpha_i$ is a positive sphere and $\\beta_j$ is a negative sphere, one considers the Lagrangian connected sum $\\beta_j \\#_p \\alpha_i$, not $\\alpha_i \\#_p \\beta_j$. \nSimilarly, by assuming that $\\alpha_i$ is negative and $\\beta_j$ is positive, one can construct another set $\\mathbb{B}^{op}$ of Lagrangian branched submanifolds.\n\n\\begin{lemma}\n\t\\label{lemma transversal intersection of Lagrangian branched submanifolds}\n\tLet $\\mathcal{B}_1, \\mathcal{B}_2 \\in \\mathbb{B} \\cup \\mathbb{B}^{op}$. \n\tThen, there is a Hamiltonian isotopy $\\Phi_t : M \\to M$ such that \n\t\\begin{enumerate}\n\t\t\\item $\\Phi_t \\circ \\eta = \\eta \\circ \\Phi_t$, \n\t\t\\item $\\mathcal{B}_0 \\pitchfork \\Phi_1(\\mathcal{B}_1)$,\n\t\t\\item for every $q \\in \\mathcal{B}_0 \\cap \\Phi_1(\\mathcal{B}_1)$, $q$ is not a plumbing point or the antipodal point of a plumbing point.\n\t\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\tSince $\\mathcal{B}_1$ is a union of (parts of) compact cores and their Lagrangian connected sums, we will construct Hailtonian isotopies perturbing each compact cores $\\alpha_i$ and $\\beta_j$. \n\tThen, one obtains a perturbation of $\\mathcal{B}_1$ as a union of (parts of) perturbations of  $\\alpha_i$, $\\beta_j$ and their Lagrangian connected sums.  \n\t\n\tFirst, we choose a smooth function $f_i: \\alpha_i \\to \\mathbb{R}$ with isolated critical points such that \n\t\\begin{enumerate}\n\t\t\\item every plumbing point $p \\in \\alpha_i$, $f_i(p)=f_i(-p)=0$, where $-p$ is the antipodal point of $p$ on $\\alpha_i$, \n\t\t\\item every critical point $q$ of $f_i$ lies on $S_{\\alpha_i}$ and $q \\neq p, -p$ for any plumbing point $p \\in \\alpha_i$,\n\t\t\\item $|df_i(x)| < \\epsilon$ for all $x \\in \\alpha_i$ and for a sufficiently small fixed positive number $\\epsilon$,\n\t\t\\item $f_i \\circ \\eta_{\\alpha_i} = f_i$, where $\\eta_{\\alpha_i}$ is the involution on $T^*\\alpha_i$ defined in Section \\ref{subsection pA functors - setting}.  \n\t\\end{enumerate}\n\tWe remark that\n\t\\begin{gather*}\n\tT^*\\alpha_i \\stackrel{\\phi_{\\alpha_i}}{\\simeq} T^*S^n = \\{(x,y) \\in \\mathbb{R}^{n+1} \\times \\mathbb{R}^{n+1} \\hspace{0.2em} | \\hspace{0.2em} |x| =1, <x,y>=0 \\},\n\t\\end{gather*}\t\n\twhere $\\phi_{\\alpha_i}: T^*S^n \\stackrel{\\sim}{\\to} T^*\\alpha_i$ is the identification which we used in Section \\ref{subsection pA functors - setting}.\n\tAlso, we remark that in (3), $|df_i(x)|$ is given by the standard metric on $\\mathbb{R}^{2n+2}$.\t\n\t\n\tThen, we can extend $f_i$ to $\\tilde{f}_i : T^*\\alpha_i \\to \\mathbb{R}$ as follows.\n\tLet $\\delta: [0,\\infty) \\to \\mathbb{R}$ be a smooth decreasing function such that \n\t$$\\delta([0,\\epsilon]) = 1, \\delta([2\\epsilon, \\infty)) = 0.$$\n\tWe set \n\t$$\\tilde{f}_i : T^*\\alpha_i \\to \\mathbb{R}, \\tilde{f}_i(x,y) = \\delta(|y|) f_i(x).$$\n\tSimilarly, we can get $\\tilde{g}_j : T^*\\beta_j \\to \\mathbb{R}$ in the same way.\n\t\n\tThese Hamiltonian functions $\\tilde{f}_i$ and $\\tilde{g}_j$ induce Hamiltonian isotopies on $T^*\\alpha_i$ and $T^*\\beta_j$. \n\tMoreover, these Hamiltonian isotopies could be extended on the plumbing space $M$ since the Hamiltonian isotopies have compact supports on $T^*\\alpha_i$ and $T^*\\beta_j$.\n\t\n\tLet $\\Phi_{\\alpha_i,t}:M \\stackrel{\\sim}{\\to} M$ be the (extended) Hamiltonian isotopy associated to $\\tilde{f}_i$. \n\tThen, it is easy to check that \n\t\\begin{gather*}\n\t\\Phi_{\\alpha_i,t} \\circ \\eta = \\eta \\circ \\Phi_{\\alpha_i,t}, \\\\\n\t\\Phi_{\\alpha_i,t}(\\alpha_k) = \\alpha_k, \\text{  if  } k \\neq i, \\\\\n\t\\Phi_{\\alpha_i,t}(\\beta_j) = \\beta_j \\text{  for all  } j, \\\\\n\t\\Phi_{\\alpha_i,1}(\\alpha_i) = \\Gamma(d f_i),\n\t\\end{gather*}\n\twhere $\\Gamma(d f_i)$ is the graph of $d f_i$ in $T^*\\alpha_i \\subset M$. \n\tSimilarly, one can obtain a Hamiltonian isotopy $\\Phi_{\\beta_j,t}:M \\stackrel{\\sim}{\\to} M$ for each $\\beta_j$ in the same way. \n\t\n\tLet \n\t$$\\Phi_t = \\prod_{\\beta_j} \\Phi_{\\beta_j,t} \\circ \\prod_{\\alpha_i} \\Phi_{\\alpha_i, t}.$$\n\tThen, it is easy to check that $\\Phi_t$ satisfies the first condition of Lemma \\ref{lemma transversal intersection of Lagrangian branched submanifolds}.\n\tMoreover, one can assume that $\\Phi_1(\\mathcal{B}_1)$ is constructed from $\\Phi_1(\\alpha_i)$ and $\\Phi_1(\\beta_j)$.\n\tThus, it is easy to prove that $\\mathcal{B}_0$ and $\\Phi_1(\\mathcal{B}_1)$ satisfy the second and the last conditions of Lemma \\ref{lemma transversal intersection of Lagrangian branched submanifolds}.\n\\end{proof}\n\nFrom now on, we will explain how Lemma \\ref{lemma transversal intersection of Lagrangian branched submanifolds} weakens a difficulty of applying Theorem \\ref{thm Lagrangian floer homology}. \nThe difficulty we will consider is the last condition of Theorem \\ref{thm Lagrangian floer homology}, i.e., $L_0 \\cap L_1 = \\tilde{L}_0 \\cap \\tilde{L}_1$\n\nLet assume that $L_0$ (resp.\\ $L_1$) is a Lagrangian submanifold which is carried by $\\mathcal{B}_0$ (resp,\\ $\\mathcal{B}_1$) $\\in \\mathbb{B} \\cup \\mathbb{B}^{op}$. \nNote that $\\Phi_1(L_1)$ is carried by $\\Phi_1(\\mathcal{B}_1)$, where $\\Phi_1$ is the Hamiltonian isotopy constructed in Lemma \\ref{lemma transversal intersection of Lagrangian branched submanifolds}.\nWe will count the numbers of intersections $L_0 \\cap \\Phi_1(L_1)$ and $\\tilde{L}_0 \\cap \\Phi_1(\\tilde{L}_1)$. \nIf these numbers are the same, then $L_0 \\cap \\Phi_1(L_1) = \\tilde{L}_0 \\cap \\Phi_1(\\tilde{L}_1)$.\n\nFirst, we remark that $\\tilde{L}_0$ (resp.\\ $\\Phi_1(\\tilde{L}_1)$) is a curve which is carried by a train track $\\mathcal{B}_0 \\cap \\tilde{M}$ (resp.\\ $\\Phi_1(\\mathcal{B}_1) \\cap \\tilde{M}$). \nThen, $\\tilde{L}_0$ (resp.\\ $\\Phi_1(\\tilde{L}_1)$) has weights on the train track $\\mathcal{B}_0 \\cap \\tilde{M}$ (resp.\\ $\\Phi_1(\\mathcal{B}_1) \\cap \\tilde{M}$).\nMoreover, the number of $\\tilde{L}_0 \\cap \\Phi_1(\\tilde{L}_1)$ is the following:\n$$\\sum_{x \\in \\mathcal{B}_0 \\cap \\Phi_1(\\mathcal{B}_1)} (\\text{the weight of $\\tilde{L}_0$ at } x) \\cdot (\\text{the weight of $\\Phi_1(\\tilde{L}_1)$ at  } x).$$ \n\nTo count the number of $L_0 \\cap \\Phi_1(L_1)$, we can assume that $L_0 \\cap \\Phi_1(L_1)$ is contained in a small neighborhood of  $\\mathcal{B}_0 \\cap \\Phi_1(\\mathcal{B}_1)$.\nSince $L_0$ is carried by $\\mathcal{B}_0$, not strongly carried by, $L_0$ can have singular points. \nHowever, the singular points are lying near plumbing points or the antipodal of plumbing points.\nSince the intersection points of $\\mathcal{B}_0$ and $\\Phi_1(\\mathcal{B}_1)$ are not plumbing points or their antipodals, every $x \\in L_0 \\cap \\Phi_1(L_1)$ is a regular point of $L_0$ (resp.\\ $\\Phi_1(L_1)$). \nIt means that the number $|L_0 \\cap \\Phi_1(L_1)|$ is also give by\n$$\\sum_{x \\in \\mathcal{B}_0 \\cap \\Phi_1(\\mathcal{B}_1)} (\\text{the weight of $\\tilde{L}_0$ at } x) \\cdot (\\text{the weight of $\\Phi_1(\\tilde{L}_1)$ at  } x).$$\nThus, $|L_0 \\cap \\Phi_1(L_1)| = |\\tilde{L}_0 \\cap \\Phi_1(\\tilde{L}_1)|$.\n\n\\begin{lemma}\n\t\\label{lemma carried by}\n\tLet $L_0$ and $L_1$ be carried by $\\mathcal{B}_0, \\mathcal{B}_1 \\in \\mathbb{B} \\cup \\mathbb{B}^{op}$.\n\tThen, there is a Hamiltonian isotopy $\\Phi_t$ such that \n\t$$L_0 \\cap \\Phi_1(L_1) = \\tilde{L}_0 \\cap \\Phi_1(\\tilde{L}_1).$$\n\\end{lemma}\nThus, if $L_0$ and $L_1$ are carried by $\\mathcal{B}_0, \\mathcal{B}_1 \\in \\mathbb{B} \\cup \\mathbb{B}^{op}$, and if $L_0$ and $L_1$ satisfy conditions (1), (2), and (4) of Theorem \\ref{thm Lagrangian floer homology}, then one can apply Theorem \\ref{thm Lagrangian floer homology} for $L_0$ and $\\Phi_1(L_1)$.\n\n\\begin{exmp}\n\t\\label{exmp of homology theorem}\n\tLet $\\psi_0$ and $\\psi_1$ be symplectomorphisms of Penner type, i.e., $\\psi_0$ and $\\psi_1$ are products of positive (resp.\\ negative) powers of $\\tau_i$ and negative (resp.\\ positive) powers of $\\sigma_j$, where $\\tau_i$ and $\\sigma_j$ are Dehn twists along $\\alpha_i$ and $\\beta_j$ respectively.\n\tLet assume that $L_0$ (resp.\\ $L_1$) is a Lagrangian submanifold of $M$, which is generated from one of compact cores by applying $\\psi_0$ (resp.\\ $\\psi_1$), i.e.,\n\t$$L_0 = \\psi_0(\\alpha_k) \\text{  or  } \\psi_0(\\beta_j), \\hspace{0.2em} L_1 = \\psi_1(\\alpha_k) \\text{  or  } \\psi_1(\\beta_j).$$ \n\n\tThen, $\\eta(L_i) = L_i$ since \n\t\\begin{gather*}\n\t\\eta(\\alpha_i) = \\alpha_i \\text{ for all  } i, \\eta(\\beta_j)=\\beta_j \\text{  for all  } j, \\\\\n\t\\eta \\circ \\tau_i = \\tau_i \\circ \\eta \\text{  for all  } i, \\eta \\circ \\sigma_j = \\sigma_j \\circ \\eta \\text{  for all } j. \n\t\\end{gather*}\n\tMoreover, $\\tilde{L}_i = \\psi_i(\\tilde{\\alpha}_k)$ or $\\psi_i(\\tilde{\\beta_j})$.\n\tThus, $\\tilde{L}_i$ is a Lagrangian submanifold of $\\tilde{M}$.\n\tFinally, $L_i$ is carried by $\\mathcal{B}_{\\psi_i}$.\n\t\n\tThus, if $L_0$ and $L_1$ are not isotopic to each other, then one can apply Theorem \\ref{thm Lagrangian floer homology}. \n\\end{exmp}\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $X$ be a complex Banach space.   Let $T$ be an operator in  the Banach algebra $\\mathcal{B}(X)$ and  denote by $\\mathcal{T}$ the discrete semigroup given by $\\mathcal{T}(n):=T^{n}$ for $n\\in \\mathbb{{N}}_0$.\nThe Ces\\`{a}ro sum of order $\\alpha>0$ of $T$, $\\{\\Delta^{-\\alpha} \\mathcal{T}(n)\\}_{n\\in \\mathbb{{N}}_0}\\subset \\mathcal{B}(X)$, is  defined by\n\\begin{equation*}\n\\Delta^{-\\alpha} \\mathcal{T}(n)x = \\displaystyle\\sum_{j=0}^n k^{\\alpha}(n-j)\\mathcal{T}(j) x, \\qquad x\\in X; \\quad n\\in \\mathbb{{N}}_0,\n\\end{equation*}\nwhere\n\\begin{equation*}\\label{eq1}\nk^{\\alpha}(n) = \\frac{\\Gamma(\\alpha +n)}{\\Gamma(\\alpha) \\Gamma(n+1)}, \\qquad n\\in  \\mathbb{{N}}_0,\n\\end{equation*}\nis the Ces\\`{a}ro kernel. It is well known that Ces\\`{a}ro sums is an important concept  that appears in several contexts and ways in the literature. For instance, in Zygmund's book, it appeared in connection with summability of Fourier series \\cite[Chapter III, Section 3.11]{Zygmund} and in \\cite{Ch-Mu93} in relation with weighted norm inequalities for Jacobi polynomials and series. \nSee also \\cite{Gr31} and \\cite{Ku39}.\nThe starting point for our investigation is this definition  of fractional sum of the discrete semigroup $\\mathcal{T}$. Certain fractional sums have been used in recent years to develop a theory of fractional differences with interesting applications to boundary value problems and concrete models coming from biological issues, see for example \\cite{AtSe10} and \\cite{Go12}. Note that  this definition coincides or is connected  with other fractional sums of the discrete semigroup $\\mathcal{T}$ on the set $\\mathbb{{N}}_0,$ see \\cite[Section 1]{AtEl09} or \\cite[Theorem 2.5]{Lizama1}.\n\n\n\n\nConsider $\\phi:\\mathbb{{N}}_0\\to \\mathbb{{R}}^+$ a positive weight sequence and\n the Banach algebra  $\\ell^{1}_\\phi$ (endowed with their natural convolution product).\nSuppose $\\frac{1}{\\phi(\\cdot)}\\mathcal{T} \\in \\ell^{\\infty}(\\mathcal{B}(X)).$ It is well known and easy to show that the semigroup $\\mathcal{T}$ induces an algebra homomorphism $\\theta:\\ell^{1}_\\phi \\to \\mathcal{B}(X)$ defined by\n\\begin{equation*}\\label{homos}\n\\theta(f)x :=\\displaystyle\\sum_{n=0}^{\\infty}f(n)\\mathcal{T}(n)x,\\qquad f\\in \\ell^{1}_\\phi, \\quad x\\in X.\n\\end{equation*}\nNote that in the case that $T$ is a power bounded operator, i.e., $\\mathcal{T} \\in \\ell^{\\infty}(\\mathcal{B}(X))$, then $\\theta:\\ell^1 \\to \\mathcal{B}(X)$. Moreover, this homomorphism is a natural extension of the $Z$-transform, see for example \\cite{Elaydi} and references therein.\n\nIn general, algebra homomorphisms are  useful tools to treat different interesting aspects of operator theory: Algebra relations,  sharp norm estimations, subordination operators, or ergodic behaviour (as Katznelson-Tzafriri theorems, see \\cite{Katznelson}).\n\n\n\nAs mentioned before, it is remarkable that  Ces\\`{a}ro sums have appeared in the literature since some time ago but until now there was not noted their relationship with the theory of  fractional sums  and their algebraic structure. The first main purpose of this paper is to show how  this connection  provide   new insight on properties and characterizations of  Ces\\`{a}ro sums, notably concerning their interplay with algebra homomorphisms.\n\n\nCes\\`{a}ro sums are also a basic  tool to define $(C, \\alpha)$-bounded operators,  a natural extension of power-bounded operators.\nWe recall that a bounded operator $T \\in \\mathcal{B}(X)$ is $(C,\\alpha)$-bounded $(\\alpha >0)$ if\n\\begin{equation*}\n\\sup_{n} \\|\\frac{1}{k^{\\alpha+1}(n)}\\Delta^{-\\alpha} \\mathcal{T}(n) \\| < \\infty.\n\\end{equation*}\nSee \\cite{De00, Su-Ze13}  for  examples and  properties of  $(C,\\alpha)$-bounded operators. Note that if $T$ is power bounded,  then $T$ is a $(C, \\alpha)$-bounded operator for every $\\alpha>0$.\nHowever, there are operators that does not satisfy the power-boundedness condition,  but  $\\sup_{n\\ge 1}\n{1\\over n}\\|\\Delta^{-1} \\mathcal{T}(n) \\|<\\infty,$ as the well-known Assani example shows\n$$ T= \\left( \\begin{array}{rrr}\n-1 & 2  \\\\\n 0 & -1  \\\\\n \\end{array} \\right),\n $$\nsee \\cite[Section 4.7]{E2}; recently other examples are appeared in  \\cite{De00, Ed04,  Su-Ze13, To-Ze, Yo98}.\n\n\n\nThe following natural question then arises:  $(Q)$ Can $T$ induce an algebra homomorphism from a  proper subalgebra $\\mathcal{A} \\subset \\ell^{1}$ to $\\mathcal{B}(X)$ such that Ces\\`{a}ro sums are kernels of this homomorphism?.\n\nThe second purpose of this paper is to show that, surprisingly,  the answer to $(Q)$ is positive for every bounded operator such that their Ces\\`{a}ro sums are properly bounded (which includes $(C, \\alpha)$-bounded operators). More precisely, we  construct appropriate subalgebra  $\\tau^{\\alpha}(k^{\\alpha+1}) \\subset \\ell^{1}$ and then we  prove that the following assertions are equivalent:\n\\begin{itemize}\n\\item[(i)] $T$ is $(C, \\alpha)$-bounded operator.\n\\item[(ii)] There exists a bounded algebra homomorphism $\\theta : \\tau^{\\alpha}(k^{\\alpha+1}) \\to \\mathcal{B}(X)$ such that $\\theta(e_1)=T.$\n\\end{itemize}\nIn the limit case, the following assertions are equivalent:\n\\begin{itemize}\n\\item[(a)] $T$ is power bounded.\n\\item[(b)] There exists a bounded algebra homomorphism $\\theta : \\ell^1 \\to \\mathcal{B}(X)$ such that $\\theta(e_1)= T.$\n    \\item[(c)] For any $0<\\alpha<1$, there exist bounded algebra homomorphisms $\\theta_\\alpha : \\tau^{\\alpha}(k^{\\alpha+1}) \\to \\mathcal{B}(X)$ such that $\\theta_\\alpha(e_1)=T$ and\n        $\\displaystyle{\n        \\sup_{0<\\alpha<1}\\Vert \\theta_\\alpha\\Vert <\\infty.}\n        $\n\\end{itemize}\nThis paper is organized as follows: In order to construct a suitable Banach algebra and the corresponding homomorphism,   we introduce in Section 2 the notion of $\\alpha$-th fractional Weyl sum as follows:\n\\begin{equation*}\nW^{-\\alpha}f(n) = \\sum_{j=n}^{\\infty} k^{\\alpha}(j-n) f(j), \\qquad n\\in \\mathbb{{N}}_0.\n\\end{equation*}\nsee Definition \\ref{WeylDifference} below. We state their main algebraic properties in Proposition \\ref{WeylSumProp}. Then, we  introduce Banach algebras $\\tau^{\\alpha}(\\phi)$ as the completion of the space of sequences $c_{0,0}$ under the norm\n$ q_{\\phi}(f):=\\displaystyle\\sum_{n=0}^{\\infty}\\phi(n)|W^{\\alpha}f(n)|$  (Theorem \\ref{th3.1}). The weighted sequences $\\phi$ need to verify some summability conditions (Definition \\ref{condi}) to  prove that the space $\\tau^{\\alpha}(\\phi)$ is a Banach algebra. It is remarkable that such Banach algebras extends those defined  for $\\alpha \\in \\mathbb{N}_0$  and $\\phi=k^{\\alpha+1}$ in \\cite[Section 4]{Gale}. There they are considered to study subalgebras of analytic functions on the unit disc contained in the Koremblyum and (analytic) Wiener algebra.\n\n\n\nSection 3 contains an interesting characterization for the Ces\\`{a}ro sum of powers of a given $(C,\\alpha)$-bounded operator $T\\in \\mathcal{B}(X)$ solely in terms of certain functional equation (Theorem \\ref{TheoremEcFunc}). The obtained characterization corresponds to an extension of the well known functional equation for the corresponding discrete semigroup $\\mathcal{T},$ namely\n$$\nT^nT^m= T^{n+m}, \\quad n,m \\in \\mathbb{N}_0.\n$$\nTheorem \\ref{homomorphism} gives a complete answer to question $(Q)$ by defining\na bounded algebra homomorphism $\\theta:\\tau^{\\alpha}(\\phi)\\to \\mathcal{B}(X)$ given explicitly by $$\\theta(f)x:=\\displaystyle\\sum_{n=0}^{\\infty}W^{\\alpha}f(n)\\Delta^{-\\alpha} \\mathcal{T}(n)x,\\qquad f\\in \\tau^{\\alpha}(\\phi),\\quad x\\in X.$$\nThis homomorphism enjoys remarkable properties. The existence of  bounded homomorphisms in these new Banach algebras completely characterizes  the growth of Ces\\`{a}ro sums in Corollary \\ref{reci}; in particular   bounded homomorphisms from algebras $\\tau^{\\alpha}(k^{\\alpha+1})$ characterizes  $(C,\\alpha)$-boundedness (Corollary\n \\ref{cor5.7}). Such connection seems to be new in the current literature as well as the functional equation found in the beginning of this  section.\n\nThe $Z$-transform technique  may be traced back to De\nMoivre around the year 1730. In fact, De Moivre introduced the more\ngeneral concept of ``generating functions'' to probability theory. It is interesting compare the $Z$-transform (discrete case) versus Laplace transform (continuous case), see for example \\cite[Section 6.7]{Elaydi}. In Section 4, we use the Widder space $C^\\infty_W((\\omega,\\infty),X; \\hbox{m})$  where $\\hbox{m}$ is Borel measure on $\\mathbb{{R}}_+$, introduced in \\cite{cho}, to give a new characterization of summable vector-valued sequences in in terms of $Z$-transform  in Theorem \\ref{widder}. We complete the approach given in Section 3 involving the $Z$-transform and  resolvent operators in Theorem \\ref{th5.5}.\n\n  Finally, in Section 5  we present several applications, counterexamples and final comments on this paper. A straightforward application is to obtain the Abel means by subordination to the Ces\\`{a}ro sums, as Theorem \\ref{abels} shows. This point of view allows to improve some previous results given in \\cite{LSS}. Some results presented in this paper are inspired in similar ones obtained for $\\alpha$-times integrated semigroups, see \\cite{GM}. In Section 5.2, we show a natural connection between both operator theories. In Section 5.3, we present some counterexamples of algebra homomorphisms defined from some Banach algebras which cannot be extended to some larger algebras. A future research line, the extension of celebrated Katznelson-Tzafriri  to $(C, \\alpha)$-bounded operators, is commented in Section 5.4.\n\n\n\n\n\n\n\n\n\\bigskip\n\n\\noindent{\\bf Notation.} We denote by $\\{e_n\\}_{n\\in \\mathbb{{N}}_0}$ the set of  canonical sequences given by $e_n(j)=\\delta_{n,j}$ where $\\delta_{n,j}$ is the known Kronecker delta, i.e., $\\delta_{n,j}=1$ is $n=j$ and $0$ in other case.  Let $X$ be a Banach space and $\\ell^p(X)$  the set of vector-valued sequences \\mbox{$f:\\mathbb{{N}}_0\\to X$} such that $\\displaystyle\\sum_{n=0}^{\\infty}\\lVert f(n)\\rVert^p<\\infty,$ for $1\\le p<\\infty$;  and $c_{0,0}(X)$ the set of vector-valued sequences with finite support. When $X=\\mathbb{{C}}$ we write $\\ell^p$ and  $c_{0,0}$ respectively.  It is well known that $\\ell^1$ is a Banach algebra with the usual (commutative and associative) convolution product $$(f*g)(n)=\\displaystyle\\sum_{j=0}^n f(n-j)g(j), \\qquad n\\in \\mathbb{{N}}_0.$$\nConsider $\\phi:\\mathbb{{N}}_0\\to \\mathbb{{R}}^+$ a positive  sequence, and $\\ell^{1}_\\phi$  is the Banach spaced  formed by complex sequences $f:\\mathbb{{N}}_0\\to \\mathbb{{C}}$ such that $\\sum_{n\\in \\mathbb{{N}}_0}\\phi(n)\\vert f(n)\\vert <\\infty$.  We write $f^{\\ast n}=f\\ast f^{\\ast(n-1)}$ for $n\\ge 2$, $f^{\\ast 1}=f$ and $f^{\\ast 0}=e_0;$ in particular $e_n=e_1^{\\ast n}$ for $n\\in \\mathbb{{N}}_0$.\n\n\n\n\nThroughout the paper, we use the variable constant convention, in which $C$ denotes a constant\nwhich may not be the same from line to line. The constant is frequently written with subindexes\nto emphasize that it depends on some parameters.\n\n\n\n\\section{Weyl differences  and convolution Banach algebras}\n\\setcounter{theorem}{0}\n\\setcounter{equation}{0}\n\n\n\n\n\n\nIn this section, we  define certain spaces of sequences that corresponds to an extension in two different directions of those considered in the recent paper  \\cite[Definition 4.2]{Gale}. We consider a positive order of regularity in Weyl differences (Definition \\ref{WeylDifference}) and different order of growth of Weyl differences (Definition \\ref{condi}). These spaces correspond to Banach subalgebras of the space $\\ell^1$ and are important to  obtain a further characterization via homomorphisms for Ces\\`{a}ro sums in the next section.\n\n\n We consider the usual difference operator $\\Delta f(n)=f(n+1)-f(n),$ for $n\\in\\mathbb{{N}}_{0},$ its powers\n$\\Delta^{k+1}=\\Delta^k\\Delta=\\Delta\\Delta^k,$ for $k\\in\\mathbb{{N}},$ and we write by\n$\\Delta^0f=f $ and $\\Delta^1=\\Delta$. It is easy to see that $$\\Delta^k f(n)=\\displaystyle\\sum_{j=0}^k(-1)^{k-j}\\binom{k}{j}f(n+j),\\qquad n\\in\\mathbb{{N}}_{0},$$ see for example \\cite[(2.1.1)]{Elaydi} and then $\\Delta^m:c_{0,0}\\to c_{0,0}$ for $m\\in \\mathbb{{N}}_0$. In addition, for $\\alpha>0,$ we consider the  well-known scalar sequence $(k^{\\alpha}(n))_{n=0}^{\\infty}$ defined by $$k^{\\alpha}(n):=\\frac{\\Gamma(n+\\alpha)}{\\Gamma(\\alpha)\\Gamma(n+1)}={n+\\alpha-1\\choose \\alpha-1},\\qquad n\\in\\mathbb{{N}}_{0}.$$\nIn the classical Zygmund's monographic, the numbers $k^\\alpha(n)$ are called as Ces\\`{a}ro numbers of order $\\alpha$ (\\cite[Vol. I, p.77]{Zygmund}) and written by $k^\\alpha(n)=A^{\\alpha-1}_n$. However  the notation as function $k^\\alpha$  will facilitate the understanding of this paper. Kernels $k^{\\alpha}$ may equivalently be defined by means of the generating function:\n\\begin{equation}\\label{eq2.1}\n\\sum_{n=0}^{\\infty} k^{\\alpha}(n) z^{n} = \\frac{1}{(1-z)^{\\alpha}}, \\quad |z| < 1,\\quad \\alpha>0,\n\\end{equation}\n and satisfies the semigroup property, that is, $k^{\\alpha}*k^{\\beta}=k^{\\alpha+\\beta}$ for $\\alpha, \\beta >0$. Furthermore,  the following equality holds: for $\\alpha>0$,   \\begin{equation}\\label{double}\n k^{\\alpha}(n)=\\frac{n^{\\alpha-1}}{\\Gamma(\\alpha)}(1+O({1\\over n})), \\qquad n\\in \\mathbb{{N}}, \\end{equation}\n(\\cite[Vol. I, p.77 (1.18)]{Zygmund}) and $k^\\alpha$ is increasing (as a function of $n$) for $\\alpha >1$, decreasing for $1>\\alpha >0$ and $k^1(n)=1$ for $n\\in \\mathbb{{N}}$ (\\cite[Theorem III.1.17]{Zygmund}). It is straightforward to check that  $k^\\alpha(n)\\le k^\\beta(n)$ for $\\beta \\ge \\alpha>0$ and $n\\in \\mathbb{{N}}_0$. The Gautschi inequality states that\n\\begin{equation}\\label{gau}\nx^{1-s}<{\\Gamma(x+1)\\over \\Gamma(x+s)}<(x+1)^{1-s}, \\qquad x\\ge 1, \\quad 0<s<1,\n\\end{equation}\n(\\cite{Gautschi}), which implies that\n$$\n{(n+1)^{\\alpha-1}\\over \\Gamma(\\alpha)}<k^\\alpha(n)<{n^{\\alpha-1}\\over \\Gamma(\\alpha)}, \\qquad n\\in \\mathbb{{N}}, \\quad 0<\\alpha<1.\n$$\n\n\nNote that  when $\\alpha=0$ we have\n\\begin{equation*} \\label{rem2.2} k^{0}(n):=\\displaystyle\\lim_{\\alpha\\to 0^{+}}k^{\\alpha}(n)=e_{0}(n),\\qquad n\\in \\mathbb{{N}}_0.\n\\end{equation*}\n\n\n\\begin{lemma} \\label{duplicacion} For $\\alpha >0$, there exists $C_\\alpha>0$ such that\n$$\nk^\\alpha(2n)\\le C_\\alpha k^\\alpha(n), \\qquad n\\in \\mathbb{{N}}_0.\n$$\nIn particular for $0<\\alpha<1$, the following equality holds\n$$\nk^{\\alpha+1}(2n)< 2^\\alpha  k^{\\alpha+1}(n)\\left(1+{1-\\alpha\\over2(1+\\alpha)}\\right)^\\alpha, \\qquad n\\in \\mathbb{{N}}_0.\n$$\n\\end{lemma}\n\\begin{proof} The proof of the first inequality is  straightforward by the inequality (\\ref{double}). To show the second inequality, we use the know doubling equality for Gamma function\n$$\n\\Gamma(z)\\Gamma(z+{1\\over 2})=2^{1-2z}\\sqrt{\\pi}\\Gamma(2z), \\qquad \\Re z>0,\n$$ to obtain  that\n\\begin{eqnarray*}\nk^{\\alpha+1}(2n)&=&{\\Gamma(\\alpha +1+2n)\\over \\Gamma(\\alpha+1)\\Gamma(2n+1)}=2^\\alpha k^{\\alpha +1}(n){\\Gamma({\\alpha\\over 2}+{1\\over 2}+n)\\Gamma({\\alpha\\over 2}+{1}+n)\\over \\Gamma({\\alpha}+{1}+n)\\Gamma({1\\over 2}+n)}, \\qquad n\\ge 1.\n\\end{eqnarray*}\nWe apply the Gautschi inequality (\\ref{gau}) to get that\n\\begin{eqnarray*}\n{\\Gamma({\\alpha\\over 2}+{1\\over 2}+n)\\over \\Gamma({1\\over 2}+n)}&<&({\\alpha\\over 2}+{1\\over 2}+n)^{\\alpha \\over 2},\\cr\n{\\Gamma({\\alpha\\over 2}+{1}+n)\\over \\Gamma(\\alpha+{1}+n)}&<&(\\alpha+n)^{-\\alpha \\over 2},\n\\end{eqnarray*}\nfor $0<\\alpha<1$ and we conclude that\n$$\nk^{\\alpha+1}(2n)<2^\\alpha k^{\\alpha +1}(n)\\left(1+{1-\\alpha\\over 2(\\alpha +n)}\\right)^{\\alpha \\over 2}\\le 2^\\alpha k^{\\alpha +1}(n)\\left(1+{1-\\alpha\\over 2(1+\\alpha)}\\right)^{\\alpha},\n$$\nfor $n \\ge 1$ and $0<\\alpha<1$.\n\\end{proof}\n\n The Ces\\`{a}ro sum of order $\\alpha$ of $f$ is defined by $$\\Delta^{-\\alpha}f(n):=(k^{\\alpha}*f)(n)=\\displaystyle\\sum_{j=0}^n k^{\\alpha}(n-j)f(j),\\qquad n\\in\\mathbb{{N}}_{0}, \\alpha>0.$$\n Again we prefer to follow the notation $\\Delta^{-\\alpha}f(n)$ instead of $S_n^{\\alpha-1}$ used in \\cite{Zygmund}. Note that  $\\Delta^{-\\alpha-\\beta}f= k^\\beta\\ast (\\Delta^{-\\alpha}f)$ and then $\\Delta^{-\\alpha}\\Delta^{-\\beta}=\\Delta^{-(\\alpha+\\beta)}=\\Delta^{-\\beta}\\Delta^{-\\alpha}$ for $\\alpha, \\beta >0$,  for more details see again \\cite[Vol. I, p.76-77]{Zygmund}. Note also that $\\displaystyle\\lim_{\\alpha\\to 0}\\Delta^{-\\alpha}f(n) =f(n)$ with $\\alpha>0$ and $n\\in \\mathbb{{N}}_{0}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n We write $W=-\\Delta$, $W^{m}=(-1)^{m}\\Delta^m$ for $m\\in \\mathbb{{N}}$. The operator $W$ has inverse in $c_{0,0},$ $W^{-1}f(n)=\\displaystyle\\sum_{j=n}^{\\infty}f(j) $ and its iterations are given by the sum\n $$W^{-m}f(n)= \\displaystyle\\sum_{j=m}^{\\infty}\\frac{\\Gamma(j-n+m)}{\\Gamma(j-n+1)\\Gamma(m)}f(j)= \\sum_{j=n}^{\\infty}k^{m}(j-n)f(j),\\qquad n\\in \\mathbb{{N}}_{0}$$  for each scalar-valued sequence $f$ such that $\\displaystyle\\sum_{n=0}^{\\infty} |f(n)|n^{m} <\\infty$, see for example \\cite[p.307]{Gale}.\nThese  facts and the clear connection  with the Weyl fractional calculus motivates the following definition.\n\n\n\n\n\\begin{definition}\\label{WeylDifference}{\\rm Let $f: \\mathbb{N}_0 \\to X$  and $\\alpha>0$ be given. The  Weyl sum of order $\\alpha$ of $f$, $W^{-\\alpha}f$, is defined by $$W^{-\\alpha}f(n):=\\displaystyle\\sum_{j=n}^{\\infty} k^{\\alpha}(j-n)f(j),\\qquad n\\in\\mathbb{{N}}_0,$$\nwhenever the right hand side makes sense. The  Weyl difference of order $\\alpha$ of $f$, $W^\\alpha f$, is defined by $$W^{\\alpha}f(n):=W^m W^{-(m-\\alpha)}f(n)=(-1)^{m}\\Delta^m W^{-(m-\\alpha)}f(n),\\qquad n\\in\\mathbb{{N}}_0,$$ for $m=[\\alpha]+1,$  whenever the right hand side makes sense. In particular $W^{\\alpha}: c_{0,0}\\to c_{0,0}$ for $\\alpha \\in \\mathbb{{R}}$. }\n\\end{definition}\n\nObserve that if $\\alpha\\in\\mathbb{{N}}_0,$ the  Weyl difference of order $\\alpha$  coincides with the definition  given in \\cite[Section 4]{Gale}. Some general properties are shown in the following proposition.\n\n\n\\begin{proposition}\\label{WeylSumProp} Let $f\\in c_{0,0}(X).$ The following assertions hold:\\begin{itemize}\n\\item[(i)] For $\\alpha,\\beta>0,$ $W^{-\\alpha}W^{-\\beta}f=W^{-(\\alpha+\\beta)}f=W^{-\\beta}W^{-\\alpha}f.$\n\\item[(ii)] For $\\alpha >0$ and $n\\in\\mathbb{{N}}_0$, we have $\\displaystyle\\lim_{\\alpha\\to 0^+}W^{-\\alpha}f(n)=f(n).$\n\\item[(iii)] For $\\alpha>0,$ $W^{\\alpha}W^{-\\alpha}f=W^{-\\alpha}W^{\\alpha}f=f.$\n\\item[(iv)]For $\\alpha >0$ and $n\\in\\mathbb{{N}}_0$, we have $\\displaystyle\\lim_{\\alpha\\to 0^+}W^{\\alpha}f(n)=f(n).$\n\\item[(v)]\\label{SemPro} For all $\\alpha,\\beta\\in\\mathbb{{R}}$ we have $W^{\\alpha}W^{\\beta}f=W^{\\alpha+\\beta}f=W^{\\beta}W^{\\alpha}f.$\n\\end{itemize}\n\\end{proposition}\n\\begin{proof}\n(i) It is clear using the Fubini theorem and the semigroup property $k^{\\alpha+\\beta}=k^{\\alpha}*k^{\\beta}$ for $\\alpha, \\beta >0$. (ii) It is sufficient to apply that $f$ has finite support and $\\displaystyle\\lim_{\\alpha\\to 0^+}k^{\\alpha}(j)=e_{0}(j)$ for $j\\in \\mathbb{{N}}_0$.\n(iii) We write $m=[\\alpha]+1.$ Applying part (i), for $n\\in\\mathbb{{N}}_0,$ we have that $$W^{\\alpha}W^{-\\alpha}f(n)=W^mW^{-(m-\\alpha)}W^{-\\alpha}f(n)=W^mW^{-m}f(n)=f(n),$$ since $W^{-m}$ is the inverse of $W^m$ in $c_{0,0}(X),$ see \\cite[Section 4]{Gale}. On the other hand, \\begin{eqnarray*}\n    W^{-\\alpha}W^{\\alpha}f(n)&=&W^{-(\\alpha+1-m)}W^{-(m-1)}W^mW^{-(m-\\alpha)}f(n)=W^{-(\\alpha+1-m)}W^{1}W^{-(m-\\alpha)}f(n)\\\\\n    &=&W^{-(\\alpha+1-m)}W^{-(m-\\alpha)}f(n)-\\displaystyle\\sum_{j=n}^{\\infty}k^{\\alpha+1-m}(j-n)W^{-(m-\\alpha)}f(j+1)\\\\\n    &=&W^{-1}f(n)-\\displaystyle\\sum_{j=n+1}^{\\infty}k^{\\alpha+1-m}(j-n-1)W^{-(m-\\alpha)}f(j) \\\\\n    &=&W^{-1}f(n)-W^{-1}f(n+1)=f(n),\n    \\end{eqnarray*}\n    where we use part (i).(iv) It is sufficient to apply that $f$ has finite support and $\\displaystyle\\lim_{\\alpha\\to 0^+}k^{1-\\alpha}(j)=1$ for $j\\in\\mathbb{{N}}_0.$\n(v) It is simple to check using the previous results.\n\\end{proof}\n\n\n\\begin{example}\\label{ex2.3}\n\n\n\\begin{itemize} \\item[(i)] {\\rm Let $\\lambda \\in \\mathbb{C}\\backslash\\{0\\},$ and $p_{\\lambda}(n):=\\lambda^{-(n+1)}$ for $n\\in\\mathbb{{N}}_0.$  An easy computation shows that the sequence $p_{\\lambda}$ is a pseudo-resolvent, that is, it satisfies the Hilbert equation\n$$\n(\\mu -\\lambda ) (p_{\\lambda} * p_{\\mu})(n) = p_{\\lambda}(n) - p_{\\mu}(n), \\quad n \\in \\mathbb{N}_0.\n$$\nMoreover, the following identity holds\n\\begin{equation*}\\label{inverso}\np_\\lambda\\ast (\\lambda e_0-e_1)=e_0, \\qquad \\lambda \\in \\mathbb{C}\\backslash\\{0\\}.\n\\end{equation*}\nWe claim that the functions $p_{\\lambda}$ are eigenfunctions for the operator $W^{\\alpha}$ for $\\alpha\\in \\mathbb{{R}}$ and $\\vert \\lambda\\vert >1$ : we have, by \\eqref{eq2.1}, that\n\\begin{align*}\nW^{-\\alpha}p_{\\lambda}(n) = \\lambda^{-(n+1)}\\sum_{j=0}^{\\infty}k^{\\alpha}(j)\\lambda^{-j}=\\frac{\\lambda^{\\alpha}}{(\\lambda-1)^{\\alpha}}p_{\\lambda}(n), \\qquad n\\in \\mathbb{N}_0.\n\\end{align*}\nBy Proposition \\ref{WeylSumProp} (iii), we obtain that\n$$\nW^{\\alpha}p_{\\lambda}= \\frac{(\\lambda-1)^{\\alpha}}{\\lambda^{\\alpha}} p_{\\lambda},\\qquad |\\lambda|>1.\n$$\n}\n\\item[(ii)]{\\rm\nLet $\\alpha\\geq 0$ and $n\\in\\mathbb{{N}}_0$ be given. We define \\begin{equation*}\nh_n^{\\alpha}(j):=\\left\\{\\begin{array}{ll}\nk^{\\alpha}(n-j),&j\\leq n \\\\\n0,&j>n.\n\\end{array} \\right.\n\\end{equation*}Functions  $h_n^{\\alpha}$ are denoted by $\\Gamma^{\\alpha-1}_n$ for $\\alpha \\in \\mathbb{{N}}_0$ in \\cite[Section 4]{Gale}. Note that $h_n^\\alpha\\in c_{0,0}$ for $n\\in \\mathbb{{N}}_0$, in fact, $h_n^\\alpha\\in \\hbox{span}\\{e_j\\,\\,\\vert\\,\\, 0\\le j\\le n\\}$,\n$h_0^\\alpha=e_0$, $h_1^\\alpha= \\alpha e_0+e_1$, $h_n^{0}:=\\lim_{\\alpha \\to 0^+}h^\\alpha_n=e_{n},$ and\n\\begin{equation}\\label{cesarocano}\nh_n^{\\alpha}(j)=k^{\\alpha}(n-j)=\\sum_{l=0}^nk^\\alpha (n-l)e_l(j)=\\sum_{l=0}^nk^\\alpha (n-l)e_1^{\\ast l}(j), \\qquad 0\\le j\\le n.\n\\end{equation}\n\n\n\n\n\nThen for all $\\beta\\geq 0$ it is easy to check that $W^{-\\beta}h_n^{\\alpha}= h_{n}^{\\alpha+\\beta}$, i.e., $$W^{-\\beta}h_n^{\\alpha}(j)=\\displaystyle\\sum_{i=j}^{\\infty}k^{\\beta}(i-j)h_{n}^{\\alpha}(i)=h_{n}^{\\alpha+\\beta}(j),\\qquad j\\in\\mathbb{{N}}_0.$$\nUsing  Proposition \\ref{WeylSumProp} (iii), we obtain that\n$$\nW^{\\beta}h_{n}^{\\alpha}(j)= h_n^{\\alpha-\\beta}(j),\\qquad j\\in\\mathbb{{N}}_0,\n$$ for  $0\\leq\\beta\\leq\\alpha$ and $n\\in\\mathbb{{N}}_0.$\n}\n\\end{itemize}\n\\end{example}\n\n\n\n\n\n\n\n\n\n\nThe following remark shows an interesting duality between the operator $\\Delta^{-\\alpha}$ and $W^{-\\alpha}.$ Similar results may be found in \\cite[Section 4]{AbdeljaDual} and \\cite[Theorem 4.1 and 4.4]{Abdelja2}.\n\\begin{remark}\\label{Duality}{\\rm  Let $f,g\\in c_{0,0}$, we consider the usual duality product $\\langle\\, ,\\,\\rangle$ given by\n$$\n\\langle f,g \\rangle: = \\sum_{n=0}^{\\infty} f(n)g(n).\n$$\nBy Fubini theorem, we get that $ \\langle W^{-\\alpha} f, g \\rangle  = \\langle f, \\Delta^{-\\alpha}g \\rangle$ and consequently,\n$$ \\langle f,g \\rangle =  \\langle W^{\\alpha}f,\\Delta^{-\\alpha} g \\rangle = \\langle \\Delta^{-\\alpha} f, W^{\\alpha}g \\rangle.  $$}\n\\end{remark}\n\n\n\n\nThe next lemma includes a equality which is a important tool for further developments in this paper. The proof runs parallel to the proof of the integer case  given in \\cite[Lemma 4.4]{Gale} and we do not include here.\n\n\\begin{lemma}\\label{LemmaTech} Let $f,g\\in c_{0,0}$ and $\\alpha\\geq 0,$ then\n\\begin{eqnarray*}\\label{ConvNorm}\nW^{\\alpha}(f*g)(n)&=&\\displaystyle\\sum_{j=0}^n W^{\\alpha}g(j)\\displaystyle\\sum_{p=n-j}^n k^{\\alpha}(p-n+j)W^{\\alpha}f(p) \\\\\n&&-\\displaystyle\\sum_{j=n+1}^{\\infty} W^{\\alpha}g(j)\\displaystyle\\sum_{p=n+1}^{\\infty} k^{\\alpha}(p-n+j)W^{\\alpha}f(p).\n\\end{eqnarray*}\n\\end{lemma}\n\n\n\nFollowing definitions are inspired  in \\cite[Definition 1.3]{GM}.\n\n\\begin{definition}\\label{condi} {\\rm Let $\\alpha >0$.  We say that a positive sequence $\\phi$\nbelongs to the class $\\omega_{\\alpha, loc}$, if there is a constant $c_{\\phi}>0$ such that\n\\begin{equation}\\label{inte}\n\\left(\\sum_{n=0}^j+\\sum_{n=p+1}^{j+p}\\right)k^{\\alpha}(n)\\phi(j+p-n)\\le c_{\\phi} \\phi(j)\\phi(p), \\qquad 1\\le j\\le p.\n\\end{equation}\n\n\n\nMoreover, we denote by $\\omega_\\alpha$ the set of\nnondecreasing sequences  $\\phi\\in \\omega_{\\alpha, loc}$ which are of exponential type  and\nsatisfy $\\displaystyle{\\inf_{n\\ge 0} (k^{\\alpha+1}(n))^{-1}\\phi(n)>0\n}$}\n\\end{definition}\n\n\n\n\nExample of sequences in $\\omega_\\alpha $ are the following ones:\n\\begin{itemize}\n\\item[(i)] any nondecreasing sequence $\\phi$ satisfying $\\max(k^{\\alpha+1}(n), \\phi(2n))\\le M \\phi(n)$ for some $M>0$ and for each $n\\ge 0$ (in particular $\\phi(n)= n^\\beta(1+n^\\mu)$ with $\\beta+\\mu \\ge \\alpha$ and $\\beta,\\mu \\ge 0$ and $\\phi(n)=k^\\gamma(n)$ with $\\gamma\\ge \\alpha+1$).\n    \\item[(ii)] $\\phi(n)= k^{\\alpha+1}(n)\\rho(n)$, where $\\rho$ is a nondecreasing weight, i.e., $\\rho(n+m)\\le C\\rho(n)\\rho(m)$ for $n,m\\in \\mathbb{{N}}_0$.\n         \\item[(iii)] $\\phi(n)= k^{\\nu+1}(n)e^{\\lambda n}$ for $\\nu,\\lambda >0$.\n\\end{itemize}\nBy the equivalence $\\displaystyle{k^\\alpha(n)\\sim {n^{\\alpha-1}\\over \\Gamma(\\alpha)}}$, see formula (\\ref{double}), equivalent examples may be given in terms of $ n^{\\alpha-1}$. The particular case $\\phi(n)=k^{\\alpha +1}(n)$ will play a fundamental role in this paper, and the condition (\\ref{inte}) is improved.\n\n\\begin{lemma} For $0<\\alpha <1$, the following inequality holds\n$$\n\\left(\\sum_{n=0}^j+\\sum_{n=p+1}^{j+p}\\right)k^{\\alpha}(n)k^{\\alpha +1}(j+p-n)\\le \\left(2^{\\alpha+1}\\left(1+{1-\\alpha\\over2(1+\\alpha)}\\right)^\\alpha-1\\right) k^{\\alpha +1}(j)k^{\\alpha +1}(p), \\quad 1\\le j\\le p.\n$$\n\\end{lemma}\n\n\\begin{proof} For $1\\le j\\le p$, and $\\alpha >0$, we have that\n\\begin{eqnarray*}\n\\sum_{n=0}^jk^{\\alpha}(n)k^{\\alpha +1}(j+p-n)&\\le& k^{\\alpha +1}(j+p)\\sum_{n=0}^jk^{\\alpha}(n)=k^{\\alpha +1}(j+p)k^{\\alpha +1}(j)\\cr\n\\sum_{n=p+1}^{j+p}k^{\\alpha}(n)k^{\\alpha +1}(j+p-n)&\\le&k^{\\alpha +1}(j-1)\\sum_{n=p+1}^{j+p}k^{\\alpha}(n)\\le k^{\\alpha +1}(j)\\left(k^{\\alpha +1}(j+p)-k^{\\alpha +1}(p)\\right).\n\\end{eqnarray*}\nAs $k^{\\alpha +1}$ is an increasing sequence, we have $k^{\\alpha +1}(j+p)\\le k^{\\alpha +1}(2p)$ for $j\\le p$ and we apply the Lemma \\ref{duplicacion} to conclude the proof.\n\\end{proof}\n\n\\begin{proposition}\\label{pieces} Take $0<\\alpha \\le \\beta$ and $\\phi \\in \\omega_{\\alpha, loc}$.\n\\begin{itemize}\n\\item[(i)] Then $\\omega_{\\beta, loc}\\subset \\omega_{\\alpha, loc}$ and $\\omega_{\\beta}\\subset \\omega_{\\alpha}$.\n\\item[(ii)] $(k^\\alpha \\ast \\phi)(2n)\\le c_\\phi \\phi^2(n)$ for  $ n\\ge 1$.\n\\item[(iii)] $k^\\alpha(n)\\le c_\\phi \\phi(n)\\le a^n$ for $n \\ge 1$ and some $a>0$.\n\\item[(iv)] $k^{2\\alpha}(2n)\\le c\\phi^2(n)$ for $n\\in \\mathbb{{N}}_0$ and $c>0$.\n\\item[(v)] $\\phi(n+1)\\le C\\phi(n)$ for some $C>0$ independent of $n\\ge 1$.\n\\item[(vi)] $k^\\beta \\in \\omega_{\\alpha, loc}$ if and only if $\\beta \\ge \\alpha+1$.\n\\end{itemize}\n\n\\end{proposition}\n\n\\begin{proof} (i) Since $k^\\beta(n)\\ge k^\\alpha (n)$ for  $n \\in \\mathbb{{N}}_0$, then  $\\omega_{\\beta, loc}\\subset \\omega_{\\alpha, loc}$ and $\\omega_{\\beta}\\subset \\omega_{\\alpha}$ for $\\beta \\ge \\alpha >0$. (ii) It is enough to take $j=p$ in \\eqref{inte} to obtain the inequality. (iii) By part (ii), we have that\n$$\nk^\\alpha(n)\\phi(n)\\le  (k^\\alpha \\ast \\phi)(2n)\\le c_\\phi \\phi^2(n), \\qquad n\\ge 1,\n$$\nand we get the first inequality. For $n\\ge 1$, we apply the inequality \\eqref{inte} $n-1$ times to obtain that\n$$\nc_\\phi\\phi(n)= c_\\phi k^\\alpha(0)\\phi (n-1+1)\\le c^2_\\phi\\phi(1)\\phi(n-1)\\le \\left(c_\\phi \\phi(1)\\right)^n.\n$$\n(iv)  We combine parts (ii), (iii) and the semigroup property of kernels $k^\\alpha$ to conclude that $$ c_\\phi \\phi^2(n)\\ge(k^\\alpha \\ast \\phi) (2n)\\ge c'(k^\\alpha \\ast  k^\\alpha )(2n)=c'k^{2\\alpha}(2n) , \\qquad n\\in \\mathbb{{N}}_0,$$ for some $c'>0$. (v) Take $j=1$ and $p=n\\ge 1$  in \\eqref{inte} to get\n $$\n \\phi(n+1)=k^\\alpha(0)\\phi(n+1)\\le \\sum_{m=0}^1k^\\alpha(m)\\phi(n+1-m)\\le c_\\phi \\phi(1)\\phi(n), \\qquad n\\ge 1.\n $$\n(vi) If $k^\\beta \\in \\omega_{\\alpha, loc}$ then we apply \\eqref{double} and part (ii) to get\n$$(k^\\alpha \\ast k^\\beta)(2n)=k^{\\alpha+\\beta}(2n)\\sim 2^{\\alpha+\\beta-1}{n^{\\alpha +\\beta-1}\\over \\Gamma(\\alpha+\\beta)}\\le c {n^{2(\\beta-1)}\\over \\Gamma^2(\\beta)}, \\qquad n\\ge 1\n$$\nand we conclude  that $\\beta \\ge  \\alpha+1$. Note that $k^{\\alpha+1}\\in \\omega_{\\alpha, loc}$ and then $k^\\beta \\in \\omega_{\\alpha, loc}$ for $\\beta\\ge \\alpha +1$ for part (i) and we conclude the proof.\n\\end{proof}\n\n\n\nFor $\\alpha\\geq 0,$ and $\\phi \\in \\omega_{\\alpha, loc}$, we define the application $q_{\\phi}:c_{0,0}\\to [0,\\infty)$ given by $$q_{\\phi}(f):=\\displaystyle\\sum_{n=0}^{\\infty}\\phi(n)|W^{\\alpha}f(n)|, \\qquad f\\in c_{0,0}.$$ Note that for $\\alpha=0$ the above application corresponds to the usual norm in $\\ell^1_\\phi$. In the case $\\phi=k^{\\alpha +1}$, we write $q_\\alpha$ instead of $q_{k^{\\alpha+1}}$ and $q_0=\\Vert \\quad \\Vert_1$ for $\\alpha \\ge 0$.  By \\eqref{double}, the norm $q_{\\alpha}$ is equivalent to the norm $\\widetilde{q_{\\alpha}}$ given by $$\\widetilde{q_{\\alpha}}(f):=|f(0)|+\\displaystyle\\sum_{n=1}^{\\infty}n^{\\alpha}|W^{\\alpha}f(n)|.$$ This expression was considered for the case $\\alpha\\in\\mathbb{{N}}_0$ in \\cite[Definition 4.2]{Gale}.\n\n Part of the following result extends \\cite[Theorem 4.5]{Gale} and the proof is similar to the proof of \\cite[Proposition 1.4]{GM}.  We include the proof to give a complete view of this result.\n\n\\begin{theorem}\\label{th3.1} Let $\\alpha> 0$ and  $\\phi \\in \\omega_{\\alpha, loc}$. The application $q_{\\phi}$ defines a norm in $c_{0,0}$ and  $$q_{\\phi}(f*g)\\leq C_{\\phi}\\,q_{\\phi}(f)\\,q_{\\phi}(g), \\qquad f,g\\in c_{0,0},$$ with $C_{\\phi}>0$ independent of $f$ and $g.$ We denote by $\\tau^{\\alpha}(\\phi)$ the Banach algebra obtained as the completion of $c_{0,0}$ in the norm $q_{\\phi}.$ In the case that $\\phi \\in \\omega_{\\alpha}$ then\n\n\\begin{itemize}\n\n\\item[(i)] the operator $\\Delta$ is linear and bounded on  $\\tau^{\\alpha}(\\phi)$, $\\Delta \\in \\mathcal{B}(\\tau^{\\alpha}(\\phi))$.\n\\item[(ii)] $\\tau^{\\alpha}(\\phi)\\hookrightarrow\\tau^{\\alpha}(k^{\\alpha+1})\\hookrightarrow \\ell^1,$ and $\\lim_{\\alpha \\to 0^+}q_\\alpha(f)=\\Vert f\\Vert_1$, for $f\\in c_{0,0}$.\n\n    \\item[(iii)]  for $0<\\alpha<\\beta$, $\\tau^{\\beta}(k^{\\beta+1})\\hookrightarrow\\tau^{\\alpha}(k^{\\alpha+1})$.\n    \\item[(iv)] for $0< \\alpha <1$,\n$$\nq_{\\alpha}(f*g)\\leq \\left(2^{\\alpha+1}\\left(1+{1-\\alpha\\over2(1+\\alpha)}\\right)^\\alpha-1\\right)\\,q_{\\alpha}(f)\\,q_{\\alpha}(g), \\qquad f, g \\in \\tau^{\\alpha}(k^{\\alpha+1}).\n$$\n\\end{itemize}\n\\end{theorem}\n\\begin{proof} It is clear that $q_{\\alpha}$ is a norm in $c_{0,0}.$ Now, applying Lemma \\ref{LemmaTech} we have \\begin{eqnarray*}\nq_{\\phi}(f*g)&\\leq&\\biggl( \\displaystyle\\sum_{n=0}^{\\infty}\\sum_{j=0}^{n}\\sum_{p=n-j}^{n}+ \\displaystyle\\sum_{n=0}^{\\infty}\\sum_{j=n+1}^{\\infty}\\sum_{p=n+1}^{\\infty}\\biggr)\\phi(n)k^{\\alpha}(p-n+j)|W^{\\alpha}g(j)||W^{\\alpha}f(p)| \\\\\n&=&\\biggl( \\displaystyle\\sum_{j=0}^{\\infty}\\sum_{n=j}^{\\infty}\\sum_{p=n-j}^{n}+ \\displaystyle\\sum_{j=1}^{\\infty}\\sum_{n=0}^{j-1}\\sum_{p=n+1}^{\\infty}\\biggr)\\phi(n)k^{\\alpha}(p-n+j)|W^{\\alpha}g(j)||W^{\\alpha}f(p)| \\\\\n&=&\\biggl( \\displaystyle\\sum_{j=0}^{\\infty}\\sum_{p=0}^{\\infty}\\sum_{n=\\max(j,p)}^{p+j}+ \\displaystyle\\sum_{j=1}^{\\infty}\\sum_{p=1}^{\\infty}\\sum_{n=0}^{\\min(j,p)-1}\\biggr)\\phi(n)k^{\\alpha}(p-n+j)|W^{\\alpha}g(j)||W^{\\alpha}f(p)|\\\\\n&\\le&\\phi(0)|W^{\\alpha}g(0)||W^{\\alpha}f(0)|+c_\\phi \\displaystyle\\sum_{j=1}^{\\infty}\\sum_{p=1}^{\\infty} \\phi(j)\\phi(p)|W^{\\alpha}g(j)||W^{\\alpha}f(p)|\\le  C_{\\phi}\\,q_{\\phi}(f)\\,q_{\\phi}(g)\n\\end{eqnarray*}\nwhere we use Fubini's Theorem twice and the inequality (\\ref{inte}) to show the first inequality.\n\n\nNow take $\\phi \\in \\omega_\\alpha$. (i) It is clear that $\\Delta$ is a linear operator and\n$$\nq_\\phi (\\Delta(f))=\\sum_{n=0}^{\\infty}\\phi(n)|W^{\\alpha}f(n)-W^\\alpha f(n+1)|\\le q_\\phi(f)+\\sum_{n=1}^{\\infty}\\phi(n-1)|W^{\\alpha}f(n)\\vert\\le 2q_\\phi(f),\n$$\nfor $ f\\in \\tau^{\\alpha}(\\phi)$. (ii) It is clear that $\\tau^{\\alpha}(\\phi)\\hookrightarrow\\tau^{\\alpha}(k^{\\alpha+1})\\hookrightarrow \\ell^1.$ By the Monotone Convergence Theorem and Proposition \\ref{WeylSumProp} (ii),\n  $$\\lim_{\\alpha \\to 0^+}q_\\alpha(f)=\\lim_{\\alpha \\to 0^+}\\displaystyle\\sum_{n=0}^{\\infty}k^{\\alpha+1}(n)|W^{\\alpha}f(n)|=\\displaystyle\\sum_{n=0}^{\\infty}|f(n)| =\\Vert f\\Vert_1, \\qquad f\\in c_{0,0}.$$ (iii) Let $f\\in c_{0,0},$ and $0< \\alpha <\\beta$, then\n \\begin{eqnarray*}\nq_{\\alpha}(f)&=&\\displaystyle\\sum_{n=0}^{\\infty}k^{\\alpha+1}(n)|W^{\\alpha}f(n)|=\\displaystyle\\sum_{n=0}^{\\infty}k^{\\alpha+1}(n)|\\displaystyle\\sum_{j=n}^{\\infty}k^{\\beta-\\alpha}(j-n)W^{\\beta}f(j)| \\\\\n&\\leq&\\displaystyle\\sum_{j=0}^{\\infty}|W^{\\beta}f(j)|\\displaystyle\\sum_{n=0}^{j}k^{\\beta-\\alpha}(j-n)k^{\\alpha+1}(n)=\\displaystyle\\sum_{j=0}^{\\infty}k^{\\beta+1}(j)|W^{\\beta}f(j)|=q_{\\beta}(f),\n\\end{eqnarray*} where we have applied Proposition \\ref{WeylSumProp} (v) and the semigroup property of $k^{\\alpha}.$ (iv) This inequality follows from Lemma (2.8).\n\\end{proof}\n\n\\begin{example}\\label{NormEquiv}{\\rm Note that sequence $(h^\\alpha_n)_{n\\in \\mathbb{{N}}_0} \\subset \\tau^{\\alpha}(\\phi)$ with $\\phi \\in \\omega_{\\alpha, loc}$: By Example \\ref{ex2.3} (ii),  $q_\\phi(h^\\alpha_n)=\\phi(n)$ for $n\\in \\mathbb{{N}}_0$. Then the series $\\displaystyle\\sum_{n=0}^\\infty W^\\alpha f(n)h_n^\\alpha $ converges on $\\tau^{\\alpha}(\\phi)$ for every $f\\in \\tau^{\\alpha}(\\phi)$.  By Proposition \\ref{pieces} (iii)\n$$\n\\vert f(m)\\vert\\le  \\sum_{n=m}^\\infty k^\\alpha (n-m)\\vert W^\\alpha(f)(n)\\vert \\le c_\\phi\\sum_{n=m}^\\infty\\phi(n)\\vert W^\\alpha(f)(n)\\vert \\le c_\\phi q_\\phi(f), \\qquad m\\in \\mathbb{{N}}_0,\n$$\nwherever $k^\\alpha$  or $\\phi$ is non-decreasing functions, i.e., for $\\alpha \\ge 1$ or $\\phi \\in \\omega_{\\alpha}$ for example. And then $\nf= \\displaystyle\\sum_{n=0}^\\infty W^\\alpha f(n)h_n^\\alpha\n$ on $\\tau^{\\alpha}(\\phi)$.\n\n\n\nTake $\\phi \\in \\omega_{\\alpha}$ such that $\\phi(n) \\le Ca^n$ for $a>1$. Then $p_\\lambda \\in  \\tau^{\\alpha}(\\phi)$ for $\\vert \\lambda \\vert >a$, where sequences $p_\\lambda$ are defined in Example \\ref{ex2.3} (i), and\n$$\nq_\\phi(p_\\lambda) \\le C{\\vert \\lambda-1\\vert^\\alpha \\over \\vert \\lambda\\vert^{\\alpha}(\\vert \\lambda\\vert -a)}, \\qquad \\vert \\lambda \\vert >a.\n$$\nIn the particular case $\\phi=k^{\\gamma}$, then $p_\\lambda \\in  \\tau^{\\alpha}(k^{\\gamma})$ for $\\vert \\lambda \\vert >1$  and $\\gamma\\ge \\alpha+1,$\n \\begin{equation}\\label{normass}\nq_{k^\\gamma}(p_\\lambda)= {\\vert \\lambda-1\\vert^\\alpha \\vert \\lambda\\vert^{\\gamma-\\alpha-1}\\over (\\vert \\lambda\\vert-1)^{\\gamma}}, \\qquad \\vert \\lambda \\vert >1,\n\\end{equation}\nwhere we have applied   Example \\ref{ex2.3} (i) and the formula \\eqref{eq2.1}}.\n\n\\end{example}\n\n\n\\section{Ces\\`{a}ro sums and algebra homomorphisms}\n\\setcounter{theorem}{0}\n\\setcounter{equation}{0}\n\nIn this section we present our main results. The algebra structure of Ces\\`{a}ro sums are presented in several ways: functional equation (Theorem \\ref{TheoremEcFunc}), algebra homomorphism (Theorem \\ref{homomorphism}) and resolvent operators (Theorem \\ref{th5.5}). Note that these approach in fact  characterizes  the growth of Ces\\`{a}ro sums, as Corollary \\ref{reci} and Corollary \\ref{cor5.7} for $(C, \\alpha)$-bounded operators show.\nWe recall the following definition.\n\n\\begin{definition}\\label{cesarosum}{\\rm\nGiven a bounded operator $T \\in \\mathcal{B}(X)$, the Ces\\`{a}ro sum of order $\\alpha>0$ of $T$, $(\\Delta^{-\\alpha} \\mathcal{T}(n))_{n\\ge 0}\\subset \\mathcal{B}(X)$, is  defined by\n\\begin{equation*}\\label{cesaro}\n\\Delta^{-\\alpha} \\mathcal{T}(n)x:=(k^\\alpha \\ast  \\mathcal{T})(n)x = \\displaystyle\\sum_{j=0}^n k^{\\alpha}(n-j)T^j x, \\qquad x\\in X; \\quad n\\in \\mathbb{{N}}_0. \\end{equation*}\nNote that we keep the notation  $ \\mathcal{T}(n)=T^n$ for $n\\in \\mathbb{{N}}_0$.}\n\\end{definition}\n\n\\begin{example}\\label{canoni} {\\rm The canonical example of a family of Ces\\`{a}ro sum of order $\\alpha$ in Banach algebras $\\tau^{\\alpha}(\\phi)$ (in particular in $\\ell^1$) is the family $\\{h^\\alpha_n\\}_{n\\in \\mathbb{{N}}_0}$ given in Example \\ref{ex2.3}(ii). Note that  $(h^\\alpha_n)_{n\\in \\mathbb{{N}}_0} \\subset \\tau^{\\alpha}(\\phi)$ with $\\phi \\in \\omega_{\\alpha, loc}$, see Example \\ref{NormEquiv}.  We write $ \\mathcal{E}(n)=e^{\\ast n}_1$ to get\n$h_n^\\alpha =\\Delta^{-\\alpha} \\mathcal{E}(n)$ for $n\\in \\mathbb{{N}}_0$ by equation (\\ref{cesarocano}).}\n\\end{example}\n\n\nThe following theorem characterizes sequences of operators which are Ces\\`{a}ro sums  of some order $\\alpha>0$  and a fixed operator $T$.\n\n\\begin{theorem}\\label{TheoremEcFunc} Let $\\alpha >0$ and $T, (T_n)_{n\\in \\mathbb{{N}}_0}\\subset \\mathcal{B}(X)$. Then the following assertions are equivalent.\n\\begin{itemize}\n\\item[(i)] $T_n = \\Delta^{-\\alpha}\\mathcal{T}(n)$ for $n\\in \\mathbb{{N}}_0$.\n\\item[(ii)]\n$T_0 = I$ and the  following functional equation holds: \\begin{equation}\\label{eq4.2}\n T_nT_m=\\displaystyle\\sum_{u=m}^{n+m}k^{\\alpha}(n+m-u)T_u-\\displaystyle\\sum_{u=0}^{n-1}k^{\\alpha}(n+m-u)T_u \\qquad n\\geq 1,\\, m\\in\\mathbb{{N}}_0.\n \\end{equation}\n\\end{itemize}\n\\end{theorem}\n\\begin{proof}\nAssume (i). It is clear\n$T_0 = I$ and we claim the identity \\eqref{eq4.2}.\nTake $n\\geq 1,\\, m\\geq 0,$ then \\begin{eqnarray*}\n    T_{n}T_m&=&\\displaystyle\\sum_{j=0}^n \\displaystyle\\sum_{i=0}^mk^{\\alpha}(n-j)k^{\\alpha}(m-i)T^{j+i}\n    =\\displaystyle\\sum_{j=0}^n \\displaystyle\\sum_{u=j}^{m+j}k^{\\alpha}(n-j)k^{\\alpha}(m+j-u)T^{u} \\\\\n    &=&\\displaystyle\\sum_{j=0}^n \\displaystyle\\sum_{u=0}^{m+j}k^{\\alpha}(n-j)k^{\\alpha}(m+j-u)T^{u}\n-\\displaystyle\\sum_{j=1}^n \\displaystyle\\sum_{u=0}^{j-1}k^{\\alpha}(n-j)k^{\\alpha}(m+j-u)T^{u} \\\\\n    &=&\\displaystyle\\sum_{j=0}^n k^{\\alpha}(n-j)T_{m+j} -\\displaystyle\\sum_{j=1}^n \\displaystyle\\sum_{u=0}^{j-1}k^{\\alpha}(n-j)k^{\\alpha}(m+j-u)T^{u}.\n    \\end{eqnarray*}\n    Observe that \\begin{eqnarray*}\n    &\\quad&\\displaystyle\\sum_{j=1}^n\\displaystyle\\sum_{u=0}^{j-1}k^{\\alpha}(n-j)k^{\\alpha}(m+j-u)T^{u}=\\displaystyle\\sum_{u=0}^{n-1}\\displaystyle\\sum_{j=u+1}^n k^{\\alpha}(n-j)k^{\\alpha}(m+j-u)T^{u} \\\\\n    &\\quad&=\\displaystyle\\sum_{u=0}^{n-1}\\displaystyle\\sum_{l=u}^{n-1} k^{\\alpha}(l-u)k^{\\alpha}(m+n-l)T^{u}= \\displaystyle\\sum_{l=0}^{n-1}k^{\\alpha}(m+n-l)\\displaystyle\\sum_{u=0}^{l} k^{\\alpha}(l-u)T^{u} \\\\\n    &\\quad&=\\displaystyle\\sum_{l=0}^{n-1}k^{\\alpha}(m+n-l)T_l. \\\\\n    \\end{eqnarray*}\n    and the equality  \\eqref{eq4.2} is proven.    Conversely, assume (ii). Define $T:= T_1 -\\alpha I$ and\n\\begin{equation*}\nS_n :=\\sum_{j=0}^n k^{\\alpha}(n-j)T^j, \\quad n \\in \\mathbb{N}_0.\n\\end{equation*}\n It is clear that $S_0 = I = T_0,$ and\n$\nS_1= \\alpha I + T = T_1 .\n$\nInductively, we suppose that $S_n=T_n.$ Then using that $S_n$ satisfies \\eqref{eq4.2}, we have that $$S_{n+1}+k^{\\alpha}(1)S_n-k^{\\alpha}(n+1)I=S_nS_1=T_nT_1=T_{n+1}+k^{\\alpha}(1)S_n-k^{\\alpha}(n+1)I.$$  Then we conclude that $T_{n+1}=S_{n+1},$ and consequently $T_{n}= \\Delta^{-\\alpha} \\mathcal{T}(n)$ for all $n \\in \\mathbb{N}_0.$\n\\end{proof}\n\n\n\n\n\\begin{remark}\\label{generador}{\\rm Given  $\\{ T_n \\}_{n\\in\\mathbb{{N}}_0}\\subset  \\mathcal{B}(X)$ a sequence of bounded operators which verify the equality (\\ref{eq4.2}).  Then the operator defined by $T:=T_1-\\alpha I$ is called the generator of $\\{ T_n \\}_{n\\in\\mathbb{{N}}_0}.$  By Theorem \\ref{TheoremEcFunc}, $T_{n}= \\Delta^{-\\alpha} \\mathcal{T}(n)$ where $\\mathcal{T}(n)=T^n$ for $n\\in \\mathbb{{N}}_0$. In particular, note that $\\{h_n^{\\alpha}\\}_{n\\in\\mathbb{{N}}_0}$ satisfies \\eqref{eq4.2} in $\\tau^{\\alpha}(\\phi),$ see Example \\ref{canoni}, and the generator is the element $e_1$. }\n\\end{remark}\n\n\n\n\n\n\nThe following is one of the main results of this paper.\n\n\n\\begin{theorem}\\label{homomorphism} Let $\\alpha> 0$ and  $T \\in \\mathcal{B}(X)$ such that  $\\Vert \\Delta^{-\\alpha} \\mathcal{T}(n)\\Vert \\le C\\phi(n)$  for $n\\in \\mathbb{{N}}_0$ with $\\phi\\in \\omega_{\\alpha, loc}$ and $C>0$. Then there exists a bounded algebra homomorphism $\\theta:\\tau^{\\alpha}(\\phi)\\to \\mathcal{B}(X)$ given by $$\\theta(f)x:=\\displaystyle\\sum_{n=0}^{\\infty}W^{\\alpha}f(n)\\Delta^{-\\alpha} \\mathcal{T}(n)x,\\qquad x\\in X,\\quad f\\in \\tau^{\\alpha}(\\phi).$$\n\n Furthermore,  the following identities  hold.\n\n \\begin{itemize}\n \\item[(i)] For $n \\in \\mathbb{{N}}_0$, $\\theta(h_n^\\alpha)= \\Delta^{-\\alpha} \\mathcal{T}(n)$, in particular $\\theta (e_0)=I$ and $\\theta(e_1)=T$.\n \\item[(ii)] For  $f\\in \\tau^{\\alpha}(\\phi)$ such that $\\Delta f\\in \\tau^{\\alpha}(\\phi)$ and $x\\in X$,   $ T \\theta(\\Delta f)x= (I-T)\\theta(f)x-f(0)x.$\n \\item[(iii)] In the case that $\\displaystyle\\sup_{n\\in \\mathbb{{N}}_0}{(k^{\\beta-\\alpha}\\ast \\phi)(n)\\over  \\psi(n)}<\\infty$, for $0<\\alpha<\\beta$ and $\\psi\\in\\omega_{\\beta, loc}$, then $\\tau^{\\beta}(\\psi)\\hookrightarrow \\tau^{\\alpha}(\\phi)$ and\n $$\n \\theta(f)x= \\displaystyle\\sum_{n=0}^{\\infty}W^{\\beta}f(n)\\Delta^{-\\beta} \\mathcal{T}(n)x,\\qquad x\\in X,\\quad f\\in \\tau^{\\beta}(\\psi).\n $$\n \\item[(iv)] If $\\Vert T\\Vert \\le a$ for some $a>0$, then\n $ \\theta(f)x=\\sum_{n=0}^{\\infty}f(n)T^n (x),$ for $f\\in  \\tau^{\\alpha}(\\phi)\\cap \\ell^1_{a^n}, $\nin particular $\\theta(p_\\lambda)= (\\lambda-T)^{-1}$ for $\\vert \\lambda\\vert>a$.\n\n \\end{itemize}\n\\end{theorem}\n\\begin{proof} Note that the map $\\theta$ is well-defined, lineal and continuous, $ \\Vert \\theta(f)x\\rVert\\leq C q_{\\alpha}(f)\\lVert x\\rVert,$ for $f\\in \\tau^{\\alpha}(\\phi)$ and $x\\in X.$ To see that $\\theta$ is algebra homomorphism is sufficient to prove that $\\theta(f*g)=\\theta(f)\\theta(g)$ for $f,g\\in c_{0,0}.$ By Lemma \\ref{LemmaTech},\n we get that\\begin{eqnarray*}\n\\theta(f*g)x&=&\\displaystyle\\sum_{n=0}^{\\infty}W^{\\alpha}(f*g)(n)\\Delta^{-\\alpha}\\mathcal{T}(n)x \\\\ &= & \\displaystyle\\sum_{n=0}^{\\infty}\\displaystyle\\sum_{j=0}^n W^{\\alpha}g(j)\\displaystyle\\sum_{p=n-j}^n k^{\\alpha}(p-n+j)W^{\\alpha}f(p) \\Delta^{-\\alpha}\\mathcal{T}(n)x \\\\\n&&-\\displaystyle\\sum_{n=0}^{\\infty}\\displaystyle\\sum_{j=n+1}^{\\infty} W^{\\alpha}g(j)\\displaystyle\\sum_{p=n+1}^{\\infty} k^{\\alpha}(p-n+j)W^{\\alpha}f(p)\\Delta^{-\\alpha}\\mathcal{T}(n)x.\n\\end{eqnarray*}\nWe apply Fubini theorem to get that \\begin{eqnarray*}\n\\theta(f*g)x&=&\\displaystyle\\sum_{j=0}^{\\infty}W^{\\alpha}g(j)\\displaystyle\\sum_{p=0}^j W^{\\alpha}f(p)\\displaystyle\\sum_{n=j}^{p+j} k^{\\alpha}(p-n+j)\\Delta^{-\\alpha}\\mathcal{T}(n)x \\\\\n&&+\\displaystyle\\sum_{j=0}^{\\infty}W^{\\alpha}g(j)\\displaystyle\\sum_{p=j+1}^{\\infty} W^{\\alpha}f(p)\\displaystyle\\sum_{n=p}^{p+j} k^{\\alpha}(p-n+j)\\Delta^{-\\alpha}\\mathcal{T}(n)x \\\\\n&&-\\displaystyle\\sum_{j=1}^{\\infty}W^{\\alpha}g(j)\\displaystyle\\sum_{p=1}^j W^{\\alpha}f(p)\\displaystyle\\sum_{n=0}^{p-1} k^{\\alpha}(p-n+j)\\Delta^{-\\alpha}\\mathcal{T}(n)x \\\\\n&&-\\displaystyle\\sum_{j=1}^{\\infty}W^{\\alpha}g(j)\\displaystyle\\sum_{p=j+1}^{\\infty} W^{\\alpha}f(p)\\displaystyle\\sum_{n=0}^{j-1} k^{\\alpha}(p-n+j)\\Delta^{-\\alpha}\\mathcal{T}(n)x \\\\\n&=&\\displaystyle\\sum_{j=1}^{\\infty}W^{\\alpha}g(j)\\displaystyle\\sum_{p=1}^j W^{\\alpha}f(p)\\biggl(\\displaystyle\\sum_{n=j}^{p+j}-\\displaystyle\\sum_{n=0}^{p-1}\\biggr) k^{\\alpha}(p-n+j)\\Delta^{-\\alpha}\\mathcal{T}(n)x+W^{\\alpha}g(0)W^{\\alpha}f(0)x \\\\\n&&+\\displaystyle\\sum_{j=0}^{\\infty}W^{\\alpha}g(j)\\displaystyle\\sum_{p=j+1}^{\\infty} W^{\\alpha}f(p)\\biggl(\\displaystyle\\sum_{n=p}^{p+j}-\\displaystyle\\sum_{n=0}^{j-1}\\biggr) k^{\\alpha}(p-n+j)\\Delta^{-\\alpha}\\mathcal{T}(n)x \\\\\n&=&\\displaystyle\\sum_{j=1}^{\\infty}W^{\\alpha}g(j)\\displaystyle\\sum_{p=1}^j W^{\\alpha}f(p)\\Delta^{-\\alpha}\\mathcal{T}(p)\\Delta^{-\\alpha}\\mathcal{T}(j)x+W^{\\alpha}g(0)W^{\\alpha}f(0)x \\\\\n&&+\\displaystyle\\sum_{j=0}^{\\infty}W^{\\alpha}g(j)\\displaystyle\\sum_{p=j+1}^{\\infty} W^{\\alpha}f(p)\\Delta^{-\\alpha}\\mathcal{T}(p)\\Delta^{-\\alpha}\\mathcal{T}(j)x =\\theta(f)\\theta(g)x.\n\\end{eqnarray*}\nwhere we have used the identity (\\ref{eq4.2}).\n\n\n(i) Note that $W^\\alpha h^\\alpha_n=e_n$, see Example \\ref{ex2.3} (ii), and then $\\theta(h_n^\\alpha)= \\Delta^{-\\alpha} \\mathcal{T}(n)$ for $n\\in \\mathbb{{N}}_0$. As $e_0=h_0$ and $e_1=h_1^\\alpha-\\alpha h_0^\\alpha$,   it is clear that $\\theta(e_0)=I$ and $\\theta(e_1)=  T$. (ii) Now, for $f \\in \\tau^\\alpha(\\phi)$ such that $\\Delta f\\in \\tau^\\alpha(\\phi) $ and $x\\in X$, we have that \\begin{eqnarray*}\nT\\theta(\\Delta f)x&=& T\\biggl(\\displaystyle\\sum_{n=0}^{\\infty}W^{\\alpha}f(n+1)\\Delta^{-\\alpha}\\mathcal{T}(n)x- \\displaystyle\\sum_{n=0}^{\\infty}W^{\\alpha}f(n)\\Delta^{-\\alpha}\\mathcal{T}(n)x\\biggr)\\\\\n&=&\\displaystyle\\sum_{n=0}^{\\infty}W^{\\alpha}f(n+1)\\left(\\Delta^{-\\alpha}\\mathcal{T}(n+1)x-k^{\\alpha}(n+1)x\\right)-T\\displaystyle\\sum_{n=0}^{\\infty}W^{\\alpha}f(n)\\Delta^{-\\alpha}\\mathcal{T}(n)x \\\\\n&=&(I-T)\\theta(f)x-W^{\\alpha}f(0)\\Delta^{-\\alpha}\\mathcal{T}(0)x-\\displaystyle\\sum_{n=0}^{\\infty}W^{\\alpha}f(n+1)k^{\\alpha}(n+1)x \\\\\n&=&(I-T)\\theta(f)x-\\displaystyle\\sum_{n=0}^{\\infty}W^{\\alpha}f(n)k^{\\alpha}(n)x=(I-T)\\theta(f)x-f(0)x,\n\\end{eqnarray*}\nwhere we have applied that $\nT\\Delta^{-\\alpha}\\mathcal{T}(n)\n=\\Delta^{-\\alpha}\\mathcal{T}(n+1)-k^{\\alpha}(n+1)$ and\n $\\displaystyle\\sum_{n=0}^{\\infty}W^{\\alpha}f(n)k^{\\alpha}(n)=f(0)$ for $f\\in\\tau(\\phi).$ (iii)  Suppose that $\\displaystyle\\sup_{n\\in \\mathbb{{N}}_0}{(k^{\\beta-\\alpha}\\ast \\phi)(n)\\over  \\psi(n)}<\\infty$, with $0<\\alpha<\\beta$ and $\\psi\\in\\omega_{\\beta, loc}$, then  it is straightforward to check that $\\tau^{\\beta}(\\psi)\\hookrightarrow \\tau^{\\alpha}(\\phi)$ and $$\\displaystyle\\sum_{n=0}^{\\infty}W^{\\alpha}f(n)\\Delta^{-\\alpha}\\mathcal{T}(n)x=\\displaystyle\\sum_{n=0}^{\\infty}W^{\\beta}f(n)\\Delta^{-\\beta}\\mathcal{T}(n)x,\\qquad f\\in \\tau^{\\beta}(\\psi),\\, x\\in X,$$\nwhere we have applied Proposition \\ref{SemPro} (v) and Remark \\ref{Duality}. (iv) Now take $a>0$ such that $\\Vert T\\Vert \\le a$ and then $\\sigma(T)\\subset \\{ z\\in \\mathbb{{C}}\\,\\,\\vert\\,\\,\\vert z\\vert \\le a\\}$. For $f\\in  \\tau^{\\alpha}(\\phi)\\cap \\ell^1_{a^n}, $ we apply Remark  \\ref{Duality} to get $$ \\theta(f)x=\\sum_{n=0}^{\\infty}f(n)T^n (x), \\qquad x\\in X.\n$$\nIn particular $p_\\lambda \\in \\tau^{\\alpha}(\\phi)\\cap \\ell({a^n})$ for $\\vert \\lambda \\vert >a$ and  $\\theta(p_\\lambda)x=\\displaystyle{{1\\over \\lambda}\\sum_{n=0}^\\infty{T^n\\over \\lambda^n}x}=(\\lambda-T)^{-1}x$ for $x\\in X$.\n\\end{proof}\n\n\n\\begin{corollary}\\label{reci} Let $\\alpha>0$, $\\phi \\in \\omega_\\alpha$ and $\\theta: \\tau^{\\alpha}(\\phi)\\to {\\mathcal B}(X)$ be an algebra homomorphism. Then there exists $T\\in {\\mathcal B}(X)$ such that\n$$\n\\theta(f)x=\\sum_{n=0}^{\\infty}W^{\\alpha}f(n)\\Delta^{-\\alpha}\\mathcal{T}(n)x, \\qquad f\\in \\tau^{\\alpha}(\\phi), \\quad x\\in X;\n$$\nin particular $\\theta(h_n^\\alpha)=\\Delta^{-\\alpha}\\mathcal{T}(n)$ for $n\\in \\mathbb{{N}}_0$ and $\\theta(p_\\lambda)=(\\lambda-T)^{-1}$ for $\\vert \\lambda\\vert > \\Vert T\\Vert.$\n\\end{corollary}\n\n\n\\begin{proof} Take $T:=\\theta(e_1)$. Note that $e_1=h_1^\\alpha-\\alpha h_0^\\alpha$, see Example \\ref{ex2.3} (ii), and\n$h_n^\\alpha =\\Delta^{-\\alpha} \\mathcal{E}(n)$ for $n\\in \\mathbb{{N}}_0$ where $ \\mathcal{E}(n)=e^{\\ast n}_1$, see Example \\ref{canoni}.\nBy Example \\ref{NormEquiv}, $f=\\displaystyle\\sum_{j=0}^\\infty W^\\alpha f(n) h_n^\\alpha$ for $f\\in  \\tau^{\\alpha}(\\phi)$, we apply the continuity of $\\theta$ to get\n\\begin{eqnarray*}\n\\theta(h_n^\\alpha)x&=& \\sum_{j=0}^nk^\\alpha(n-j)\\left(\\theta(e_1)\\right)^jx=\\Delta^{-\\alpha}\\mathcal{T}(n)x;\\cr\n\\theta(f)x&=&\\sum_{n=0}^\\infty W^\\alpha f(n)\\theta( h_n^\\alpha)x=\\sum_{n=0}^{\\infty}W^{\\alpha}f(n)\\Delta^{-\\alpha}\\mathcal{T}(n)x,\n\\end{eqnarray*}\nfor $x\\in X$. By Theorem \\ref{homomorphism} (iv), we conclude the proof.\n\\end{proof}\n\n\nBy Theorem \\ref {homomorphism} and Corollary \\ref{reci}, we obtain the following characterizations of  $(C, \\alpha)$-bounded  and power-bounded operators.\n\n\\begin{corollary}\\label{cor5.7} Let $T \\in \\mathcal{B}(X)$  and $\\alpha > 0$ be given. The following assertions are equivalent:\n\\begin{itemize}\n\\item[(i)] $T$ is $(C, \\alpha)$-bounded operator.\n\\item[(ii)] There exists a bounded algebra homomorphism $\\theta : \\tau^{\\alpha}(k^{\\alpha+1}) \\to \\mathcal{B}(X)$ such that $\\theta(e_1)=T.$\n\\end{itemize}\nIn the limit case, the following assertions are equivalent:\n\\begin{itemize}\n\\item[(a)] $T$ is power bounded.\n\\item[(b)] There exists a bounded algebra homomorphism $\\theta : \\ell^1 \\to \\mathcal{B}(X)$ such that $\\theta(e_1)= T.$\n    \\item[(c)] For any $0<\\alpha<1$, there exist bounded algebra homomorphisms $\\theta_\\alpha : \\tau^{\\alpha}(k^{\\alpha+1}) \\to \\mathcal{B}(X)$ such that $\\theta_\\alpha(e_1)=T$ and\n        $\\displaystyle{\n        \\sup_{0<\\alpha<1}\\Vert \\theta_\\alpha\\Vert <\\infty.}\n        $\n\\end{itemize}\n\\end{corollary}\n\n\\begin{proof} Due to previous results, we only have to check that (c) implies (b).  Since the map $\\theta_\\alpha$ is an algebra homomorphism  then $\\theta_\\alpha(e_n)= T^n$, $\\theta_\\alpha(f)$ is well defined for $f\\in c_{0,0}$ and is independent of $\\alpha$. Take $C>0$ such that $\\sup_{0<\\alpha<1}\\Vert \\theta_\\alpha\\Vert<C$.  We define $\\theta(f):=\\theta_\\alpha(f)$ for $ f\\in c_{0,0}$ and some given $\\alpha \\in (0,1)$.\nThen $\n\\Vert \\theta(f)\\Vert=\n\\Vert \\theta_\\alpha(f)\\Vert \\le C \\, q_\\alpha(f) $ for $f\\in c_{0,0}.$\n By Theorem  \\ref{th3.1} (ii), we get that\n$\n\\Vert \\theta(f)\\Vert \\le C \\Vert f\\Vert_1,$  for $f\\in c_{0,0}$ and we conclude the result by density.\n\\end{proof}\n\n\n\n\\section{The $Z$-transform and resolvent operators}\n\\setcounter{theorem}{0}\n\\setcounter{equation}{0}\n\n\n\nLet $f:\\mathbb{{N}}_0\\to X$ be a scalar sequence on a Banach space $X$.  We also recall that the $Z$-transform of a given sequence $f:\\mathbb{N}_0 \\to X$ is defined by\n\\begin{equation} \\label{zeta}\n\\tilde f(z)= \\sum_{n=0}^{\\infty} f(n) z^{-n},\n\\end{equation}\nfor all $z$ such that this series converges. The set of numbers $z$ in the complex plane for which series \\eqref{zeta} converges\nis called the region of convergence of $\\tilde f$. The uniqueness of the inverse $Z$-transform may be established as follows: suppose that there are two sequences $f$, and $g$ with the same $Z$-transform,\nthat is,\n$$ \\sum_{n=0}^{\\infty} f(n) z^{-n}= \\sum_{n=0}^{\\infty}g(n) z^{-n}, \\qquad  \\vert z\\vert > R.$$\nIt follows from Laurent's theorem that $f(n)=g(n)$ for $n\\in \\mathbb{{N}}_0$.\n\nLet $\\phi:\\mathbb{{N}}_0\\to (0,\\infty)$ be a  sequence such that $\\phi(n)\\le Ca^n$ for some $C>0$ and $a> 0$. To follow the notation given in \\cite{cho}, we write $\\omega=\\log(a)$ and $\\omega$ is a bound for the counting measure supported on $\\mathbb{{N}}_0$, i.e., $\\epsilon_\\lambda \\in \\ell^1_\\phi$ for $\\lambda >\\omega$ where $\\epsilon_{\\lambda}(n):=e^{-\\lambda n}$ and $n\\in \\mathbb{{N}}_0$. Let $C^\\infty((\\omega,\\infty),X)$ be the space of\n$X$-valued functions on $(\\omega, \\infty)$ infinitely differentiable in the norm topology of $X$. For $r\\in C^\\infty((\\omega,\\infty),X)$, set\n$$\\Vert r\\Vert_{W, \\phi, \\omega}:=\\sup\\{{\\Vert r(\\lambda)\\Vert \\over \\Vert \\beta_{k, \\lambda}\\Vert_{1, \\phi}}\\,\\, \\vert \\,\\,k\\in \\mathbb{{N}}_0,\\lambda>\\omega\\},\n$$\nwhere $ \\beta_{k, \\lambda}(n)=n^k e^{-\\lambda n}$ for $n\\in \\mathbb{{N}}_0$ and $\\lambda >\\omega$. The Widder space $C^\\infty_W((\\omega,\\infty),X; \\phi) $ is defined by\n$$\nC^\\infty_W((\\omega,\\infty),X; \\phi)=\\{r \\in C^\\infty((\\omega,\\infty),X)\\, \\, \\vert \\, \\, \\Vert r\\Vert_{W, \\phi, \\omega}<\\infty\\}.\n$$\nEndowed with the norm  $\\Vert \\cdot \\Vert_{W, \\phi, \\omega}$, the space $C^\\infty_W((\\omega,\\infty),X;\\phi)$ is a Banach space, see more details in \\cite[Section 1]{cho}. A direct consequence of \\cite[Theorem 1.2]{cho} is the following result.\n\n\n\\begin{theorem}\\label{widder} Let $\\phi:\\mathbb{{N}}_0\\to (0,\\infty)$ be a sequence such that $\\phi(n)\\le Ca^n$ for some $C>0$ and $a> 0$. Take now $f:\\mathbb{{N}}_0\\to X$ a vector-valued sequence. Then the following assertions are equivalent.\n\n\\begin{itemize}\n\\item[(i)] $\\displaystyle{\\sup_{n\\in \\mathbb{{N}}_0}{\\Vert f(n)\\Vert \\over \\phi(n)}}<\\infty.$\n\\item[(ii)] There exists $\\theta: \\ell^1_\\phi\\to X$ such that $\\theta(\\lambda p_\\lambda)= \\tilde f(\\lambda)$ for $ \\lambda > a$.\n\\item[(iii)] $\\tilde f\\circ \\exp\\in C^\\infty_W((\\log(a),\\infty),X; \\phi).$\n\n\\end{itemize}\n\\begin{proof} To show that (i) implies (ii), we define $\\theta( g):=\\sum_{n=0}^\\infty g(n)f(n)$ for $g=(g(n))_{n\\ge 0}\\in \\ell^1_\\phi$. Now consider the part (ii). We define $h(n):=\\theta(e_n)$ for $n\\in \\mathbb{{N}}_0$.\nIt is clear that $\\displaystyle{\\sup_{n\\in \\mathbb{{N}}_0}{\\Vert h(n)\\Vert \\over \\phi(n)}}<\\infty$ and\n$$\n\\tilde f(\\lambda)=\\theta(\\lambda p_\\lambda)=\\sum_{n\\in \\mathbb{{N}}_0}\\theta(e_n)\\lambda^n=\\tilde h(\\lambda), \\qquad \\vert \\lambda\\vert > a,\n$$\nwhere we conclude that $h(n)=f(n)$ for $n\\in \\mathbb{{N}}_0$ and part (i) is proved. Now take again part (ii).\nDue to  \\cite[Theorem 1.2]{cho},\n$$\n\\theta(\\epsilon_\\mu)=\\theta(\\exp(\\mu) p_{\\exp(\\mu)})= (\\tilde f\\circ \\exp)(\\mu), \\qquad \\mu >\\log(a),\n$$\nand we conclude the part (iii). Finally  suppose that $\\tilde f\\circ \\exp\\in C^\\infty_W((\\log(a),\\infty),X; \\phi).$ Again by \\cite[Theorem 1.2]{cho}, there exists a bounded homomorphism  $\\theta: \\ell^1_\\phi\\to X$ such that $\\theta(\\epsilon_\\mu)= (\\tilde f\\circ \\exp)(\\mu)$ for $ \\mu > \\log(a)$. Since $\\epsilon_\\mu(n)=e^{-\\mu n}=e^{\\mu}p_{e^\\mu}(n)$, we conclude that  $\\theta(\\lambda p_\\lambda)= \\tilde f(\\lambda)$ for $ \\lambda > a$.\n\\end{proof}\n\\end{theorem}\n\n\\begin{remark}{\\rm  Note that  Theorem \\ref{widder} is closely connected to \\cite[Theorem 4.2]{cho}, where the Banach space $X$ has the Radon-Nikodym property, RNP, to may identity  the Widder space $C^\\infty_W((\\omega,\\infty),X; \\hbox{m})$ and $L^\\infty(\\mathbb{{R}}_+, X; \\hbox{m})$. The RNP is a  well-known property in the theory of function spaces. This property passes to closed subspaces (hereditary property) and is enjoyed by any reflexive space, any separable dual space, and any $\\ell^1(\\Gamma)$ space, where $\\Gamma$ is a set, see definitions and more details in \\cite[Section 1.2]{ABHN}.}\n\\end{remark}\n\n\nIn the well-known scalar version, $X=\\mathbb{{C}}$,   the following $Z$-transforms are obtained directly:\n\\begin{eqnarray*}\n\\widetilde{e_n}(z)&=& z^{-n}, \\qquad z\\not=0, \\quad n\\in \\mathbb{{N}}_0;\\\\\n\\widetilde{ k^{\\alpha}}(z)&=& \\displaystyle{z^\\alpha \\over (z-1)^{\\alpha}}, \\qquad \\vert z\\vert >1;\\\\\n\\widetilde{p_\\lambda}(z)&=&\\displaystyle{z\\over z\\lambda-1}, \\qquad \\vert z\\vert >{1\\over \\vert \\lambda\\vert}, \\,\\, \\lambda\\in \\mathbb{{C}}\\backslash\\{0\\}, \\\\\n\\widetilde{h_n^{\\alpha}}(z)&=&\\displaystyle\\sum_{j=0}^{n}k^{\\alpha}(n-j)z^{-j},\\qquad z\\neq 0.\n\\end{eqnarray*}\nIt is also well-known that \\begin{equation}\\label{convo}\n\\widetilde{(f\\ast g)}(z)= \\widetilde{f}(z)\\widetilde {g}(z), \\end{equation} wherever these $Z$-transforms converge on $z\\in \\mathbb{{C}}$,\nsee these results and many other properties of the $Z$-transform in, for example \\cite[Chapter 6]{Elaydi}. In particular, given $\\alpha>0$ and $f:\\mathbb{{N}}_0\\to X$ such that $\\tilde f(z)$ converges for $\\vert z\\vert >R$, then\n$$\n\\widetilde{(\\Delta^{-\\alpha} f)}(z)=  \\displaystyle{z^\\alpha \\over (z-1)^{\\alpha}}\\widetilde{f}(z), \\qquad \\vert z\\vert >\\max\\{R, 1\\}.\n$$\n\n\n\n\n\\bigskip\n\nWe denote by $\\,_nf(m):=f(n+m)$ for all $m,n\\in\\mathbb{{N}}_0.$ Next technical lemma for the $Z$-transform is applied in  Theorem \\ref{th5.5}. Similar results hold for the Laplace transform, see for example \\cite[Proposition 4.1]{Ke-Li-Mi}.\n\n\n\n\\begin{lemma}\\label{rt} Let $X$ be a Banach space,  $f:\\mathbb{{N}}_0\\to \\mathbb{{C}}$ a scalar sequence and $S:\\mathbb{{N}}_0\\to \\mathcal{B}(X)$ a vector-operator valued sequence. Then \\begin{eqnarray*}\n\\frac{1}{\\mu-\\lambda}\\widetilde{f}(\\mu)\\biggl( \\mu\\widetilde{S}(\\lambda)x-\\lambda\\widetilde{S}(\\mu)x \\biggr)&=&\\displaystyle\\sum_{n=0}^{\\infty}\\lambda^{-n}\\sum_{m=0}^{\\infty}\\mu^{-m}(f*\\,_nS)(m)x,\\\\\n\\frac{1}{\\mu-\\lambda}\\biggl( \\mu\\widetilde{f}(\\lambda)-\\lambda\\widetilde{f}(\\mu) \\biggr)\\widetilde{S}(\\mu)x&=&\\displaystyle\\sum_{n=0}^{\\infty}\\lambda^{-n}\\sum_{m=0}^{\\infty}\\mu^{-m}(\\,_nf*S)(m)x,\n\\end{eqnarray*}\nfor $|\\lambda|>|\\mu|$ sufficiently large where the double $Z$-transform converge and $x\\in X$.\n\\end{lemma}\n\\begin{proof}\nTo show the first equality, note that, $$\n\\displaystyle\\widetilde{\\,_nS}(\\mu)x=\\sum_{m=0}^{\\infty}\\mu^{-m}S(m+n)x=\\mu^{n}\\sum_{j=n}^{\\infty}\\mu^{-j}S(j)x=\\mu^{n}\\biggl(\\widetilde{S}(\\mu)x-\\sum_{j=0}^{n-1}\\mu^{-j}S(j)x\\biggr),$$ for $x\\in X$ and $n\\geq 1$. Then we get that\n\\begin{displaymath}\\begin{array}{l}\n\\displaystyle\\sum_{n=0}^{\\infty}\\lambda^{-n}\\sum_{m=0}^{\\infty}\\mu^{-m}(f*\\,_nS)(m)x=\\widetilde{f}(\\mu)\\displaystyle\\sum_{n=0}^{\\infty}\\lambda^{-n}\\widetilde{\\,_nS}(\\mu)x=\\widetilde{f}(\\mu)\\biggl( \\widetilde{S}(\\mu)x+\\displaystyle\\sum_{n=1}^{\\infty}\\lambda^{-n}\\widetilde{\\,_nS}(\\mu)x\\biggr)\\\\\n=\\displaystyle\\widetilde{f}(\\mu)\\widetilde{S}(\\mu)x\\sum_{n=0}^{\\infty}\\biggl(\\frac{\\mu}{\\lambda}\\biggr)^n-\\widetilde{f}(\\mu)\\displaystyle\\sum_{n=1}^{\\infty}\\biggl(\\frac{\\mu}{\\lambda}\\biggr)^n\\sum_{j=0}^{n-1}\\mu^{-j}S(j)x.\n\\end{array}\\end{displaymath} where we have applied the equality (\\ref{convo}). Finally, as $$\\displaystyle\\sum_{n=1}^{\\infty}\\biggl(\\frac{\\mu}{\\lambda}\\biggr)^n\\sum_{j=0}^{n-1}\\mu^{-j}S(j)x=\\sum_{j=0}^{\\infty}\\mu^{-j}S(j)x\\displaystyle\\sum_{n=j+1}^{\\infty}\\biggl(\\frac{\\mu}{\\lambda}\\biggr)^n=\\frac{\\mu}{\\lambda-\\mu}\\widetilde{S}(\\lambda)x,$$ we conclude that $$\\displaystyle\\sum_{n=0}^{\\infty}\\lambda^{-n}\\sum_{m=0}^{\\infty}\\mu^{-m}(f*\\,_n S)(m)x=\\frac{1}{\\lambda-\\mu}\\widetilde{f}(\\mu)\\biggl( \\lambda\\widetilde{S}(\\mu)x-\\mu\\widetilde{S}(\\lambda)x \\biggr),$$\nfor $|\\lambda|>|\\mu|$ sufficiently large  and $x\\in X$. Following these ideas, the second equality  is also shown.\n\\end{proof}\n\n\n\n\\begin{theorem}\\label{th5.5} Let $\\alpha\\geq 0$, $X$ a Banach space,  $\\{T_n\\}_{n\\in\\mathbb{{N}}_0}\\subset \\mathcal{B}(X)$ such that $T_0=I$, $\\Vert T_n\\Vert \\le C \\phi(n) \\le C 'a^n$ ($\\phi\\in \\omega_{\\alpha} $ and\n$a>1$) for all $n\\in\\mathbb{{N}}_0$ with $C,C'>0$.\n The following statements are equivalent: \\begin{itemize}\n\\item[(i)] The operator-valued sequence $\\{T_n\\}_{n\\in\\mathbb{{N}}_0}$ satisfies the equation \\eqref{eq4.2}.\n\\item[(ii)] There exists  a bounded algebra homomorphism $\\theta: \\tau^{\\alpha}(\\phi)\\to  \\mathcal{B}(X)$ such that $\\theta(h_n^\\alpha)=T_n$ for $n\\in \\mathbb{{N}}_0$.\n\\item[(iii)] The family $\\{ R(\\lambda) \\}_{|\\lambda|>a}$ defined by\n$$\\displaystyle R(\\lambda)x:=\\frac{(\\lambda-1)^{\\alpha}}{\\lambda^{\\alpha+1}}\\displaystyle\\sum_{n=0}^{\\infty}\\lambda^{-n}T_n(x),\\qquad |\\lambda|>a,\\, x\\in X,$$  is a pseudo-resolvent.\n\\end{itemize}\nIn these cases the generator of  $\\{T_n\\}_{n\\in\\mathbb{{N}}_0}$, defined by $T:=T_1-\\alpha I$ in  Remark \\ref{generador}, satisfies that $T_n = \\Delta^{-\\alpha}\\mathcal{T}(n)$ for $n\\in \\mathbb{{N}}_0$,  $\\theta(e_1)=T$, $\\{\\lambda \\in \\mathbb{{C}}\\,\\,\\vert \\, \\vert \\lambda\\vert>a\\}\\subset \\rho(T)$ and\n$$R(\\lambda)=(\\lambda -T)^{-1}, \\qquad \\vert\\lambda\\vert >a.\n$$\n\n\\end{theorem}\n\\begin{proof}\nThe proof (i)$\\Rightarrow$(ii) is a direct consequence of Theorem \\ref{TheoremEcFunc} and Theorem \\ref{homomorphism}. To show that (ii)$\\Rightarrow$(iii), we use that Corollary \\ref{reci}. Finally we prove (iii)$\\Rightarrow$(i). It is clear that $$R(\\lambda)=\\frac{\\widetilde{{\\frak T}}(\\lambda)}{\\lambda\\widetilde{k^{\\alpha}}(\\lambda)}, \\qquad  \\vert\\lambda\\vert >a,$$ where ${\\frak T}=\\{T_n\\}_{n\\in\\mathbb{{N}}_0}$ and $\\widetilde{{\\frak T}}$  is given by (\\ref{zeta}).  Since $\\{ R(\\lambda) \\}_{|\\lambda|>a}$ is a pseudo-resolvent, then $$(\\mu-\\lambda)\\frac{\\widetilde{{\\frak T}}(\\lambda)\\widetilde{{\\frak T}}(\\mu)}{\\lambda\\widetilde{k^{\\alpha}}(\\lambda)\\mu\\widetilde{k^{\\alpha}}(\\mu)}=\\frac{\\widetilde{{\\frak T}}(\\lambda)}{\\lambda\\widetilde{k^{\\alpha}}(\\lambda)}-\\frac{\\widetilde{{\\frak T}}(\\mu)}{\\mu\\widetilde{k^{\\alpha}}(\\mu)},\\qquad \\vert\\lambda\\vert, \\vert \\mu\\vert >a, \\quad \\mu\\not=\\lambda,$$ so $$\\widetilde{{\\frak T}}(\\lambda)\\widetilde{{\\frak T}}(\\mu)=\\frac{1}{\\mu-\\lambda}\\biggl( \\mu\\widetilde{k^{\\alpha}}(\\mu)\\widetilde{{\\frak T}}(\\lambda)-\\lambda\\widetilde{k^{\\alpha}}(\\lambda)\\widetilde{{\\frak T}}(\\mu) \\biggr), \\qquad \\vert\\lambda\\vert, \\vert \\mu\\vert >a, \\quad \\mu\\not=\\lambda.$$ On the other hand, note that the condition (\\ref{eq4.2}) is expressed by \\begin{displaymath}\\begin{array}{l}\n\\displaystyle(k^{\\alpha}*\\,_n{\\frak T})(m)-(\\,_nk^{\\alpha}*{\\frak T})(m)+k^{\\alpha}(n)T_m =\\sum_{u=n}^{n+m}k^{\\alpha}(n+m-u)T_u-\\sum_{u=0}^{m-1}k^{\\alpha}(n+m-u)T_u,\\end{array}\\end{displaymath} for $m\\geq 1$ and $n\\geq 0.$  We apply Lemma \\ref{rt}  and do some simple operations to get that\n$$\n\\sum_{n=0}^{\\infty}\\lambda^{-n}\\sum_{m=0}^{\\infty}\\mu^{-m}\\left(\\displaystyle(k^{\\alpha}*\\,_n{\\frak T})(m)-(\\,_nk^{\\alpha}*{\\frak T})(m)+k^{\\alpha}(n)T_m \\right)={ \\mu\\widetilde{k^{\\alpha}}(\\mu)\\widetilde{{\\frak T}}(\\lambda)-\\lambda\\widetilde{k^{\\alpha}}(\\lambda)\\widetilde{{\\frak T}}(\\mu)\\over\\mu-\\lambda},\n $$ for $\\vert\\lambda\\vert, \\vert \\mu\\vert >a, $ and $ \\mu\\not=\\lambda$. Then we conclude that $\\{T_n\\}_{n\\in\\mathbb{{N}}_0}$ satisfies \\eqref{eq4.2}, as consequence of the injectivity of the double $Z$-transform.\n Finally, by Corollary \\ref{reci}\n $$\n R(\\lambda)=\\theta(p_\\lambda)=(\\lambda -T)^{-1}, \\qquad \\vert\\lambda\\vert >a,\n $$\nand we finish the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Applications, examples and final comments}\n\\setcounter{theorem}{0}\n\\setcounter{equation}{0}\n\nIn this last section, we present some applications, comments, examples and counterexamples of some results presented in this paper.\n\n\n\n\\subsection{Bounds for Abel means} Given $T\\in {\\mathcal B}(X)$ and $0\\leq r<1$ the Abel mean of order $r$ of operator $T$, $A_r(T),$ is defined by\n$$\nA_r(T)x:= (1-r)\\sum_{n=0}^\\infty r^nT^n(x), \\qquad x\\in X,\n$$\nwhen this series converges for some $r\\in[0,1)$, see for example \\cite{LSS}. Note that for $0< r<\\frac{1}{r(T)}$ then ${1\\over r}\\in\\rho(T) $ and\n$$\nA_r(T)=\\frac{(1-r)}{r}(\\frac{1}{r}-T)^{-1}, \\qquad  0< r <\\min\\{1, {1\\over r(T)}\\},\n$$\nwhere $r(T)=\\lim_{n\\to\\infty}\\lVert T^{n}\\rVert^{\\frac{1}{n}}$ denotes the spectral radius of $T.$\n\nThe next theorem improves  \\cite[Proposition 2.1 (i)]{LSS} given for $\\alpha \\in \\{0,1\\}$.\n\\begin{theorem}\\label{abels} Take $\\alpha\\ge 0 $ and $T\\in\\mathcal{B}(X)$. Then\n$$\nA_r(T)x=(1-r)^{\\alpha+1}\\sum_{n=0}^\\infty r^{n}\\Delta^{-\\alpha}\\mathcal{T}(n)x, \\qquad 0\\leq r <\\min\\{1, {1\\over r(T)}\\}.\n$$\nIn the case that $\\Vert\\Delta^{-\\alpha}\\mathcal{T}(n)\\Vert\\le Ck^{\\gamma+1}(n)$ for $n\\ge 1$ and $\\gamma\\ge \\alpha$ then\n$$\n\\Vert A_r(T)\\Vert\\le  C(1-r)^{-(\\gamma-\\alpha)}, \\qquad 0\\leq r<1.\n$$\nIn particular if $T$ is a $(C, \\alpha)$-bounded operator, then $ \\sup_{0\\leq r<1}\\Vert  A_r(T)\\Vert <\\infty$.\n\\end{theorem}\n\n\\begin{proof} Let $\\alpha\\ge 0$, and $p_{1\\over r}(n)=r^{n+1}$ for $0<r<1$. By Remark \\ref{Duality}, we have that\n\n\\begin{eqnarray*}\nA_r(T)x &=& (1-r)\\sum_{n=0}^\\infty r^nT^n(x)= \\frac{1-r}{r}\\sum_{n=0}^\\infty W^\\alpha p_{1\\over r}(n)\\Delta^{-\\alpha}\\mathcal{T}(n)x\\\\\n&=& \\frac{(1-r)^{\\alpha+1}}{r}\\sum_{n=0}^\\infty p_{1\\over r}(n)\\Delta^{-\\alpha}\\mathcal{T}(n)x\n= (1-r)^{\\alpha+1}\\sum_{n=0}^\\infty r^{n}\\Delta^{-\\alpha}\\mathcal{T}(n)x,\n\\end{eqnarray*}\n where we have used Example \\ref{ex2.3} (i) for $0<r<\\min\\{1, {1\\over r(T)}\\}$. For $r=0$ is obvious.\n\nIn the case that $\\Vert\\Delta^{-\\alpha}\\mathcal{T}(n)\\Vert\\le Ck^{\\gamma+1}(n)$ for $n\\ge 1$ and $\\gamma\\ge \\alpha$, there exists a bounded algebra homomorphism $\\theta:\\tau^{\\alpha}(k^{\\gamma+1})\\to \\mathcal{B}(X)$ by Theorem \\ref{homomorphism}. Note that $p_{1\\over r}\\in  \\tau^{\\alpha}(k^{\\gamma+1})$ and\n${A_r(T)= \\frac{1-r}{r}\\theta( p_{1\\over r}),}$ for  $0<r<1.$\nBy formula (\\ref{normass}), we obtain that\n$$\n\\Vert A_r(T)\\Vert \\le C \\frac{1-r}{r}q_{k^{\\gamma+1}}(p_{1\\over r})=  C\\frac{1-r}{r}{r\\over (1-r)^{\\gamma+1-\\alpha}}={C\\over(1-r)^{\\gamma-\\alpha}}, \\qquad 0<r<1,\n$$\nand we conclude the proof.\n\\end{proof}\n\n\\begin{remark}{\\rm  If we consider $\\lVert T^n \\rVert\\leq C n^{\\gamma},$ with $\\gamma\\geq 0,$ using that $n^{\\gamma}\\leq \\Gamma(\\gamma+1)k^{\\gamma+1}(n)$ which follows easily from \\eqref{gau}, we get that  $$\\lVert A_r(T)\\rVert\\leq  C\\Gamma(\\gamma+1)(1-r)^{-\\gamma},$$ which improves the bound of \\cite[Proposition 2.1 (i) (2.3)]{LSS}. Use similar arguments to improve the bound of \\cite[Proposition 2.1 (i) (2.4)]{LSS}.}\n\\end{remark}\n\n\n\\begin{remark}{\\rm An inverse result exists on Banach lattices, see \\cite[Corollary 3.2]{LSS}, which proves that for any $\\alpha>-1$ and a positive bounded operator $T,$  $\\{ (1-r)^{\\alpha}A_r(T),\\ 0\\leq r<1\\}$ is bounded if only if $\\lVert \\Delta^{-1}\\mathcal{T}(n)\\rVert \\leq C (n+1)^{\\alpha},$ $n\\in\\mathbb{{N}}_0.$ In particular,  $T$ is Abel-mean bounded if only if  is $(C,1)$-bounded. Note that there are examples of positive $(C,1)$-bounded operators in Banach lattices which are not power bounded, see remarks following \\cite[Corollary 3.2]{LSS}.}\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\alpha$-Times integrated semigroups and Ces\\`{a}ro sums}\n\n\n\n\nNow, let $A$ be a closed linear operator on $X,$ $\\alpha> 0$ and $\\{S_{\\alpha}(t)\\}_{t\\geq 0}\\subset {\\mathcal B}(X)$ an $\\alpha$-times integrated semigroup generated by $A$, that is,  $S_\\alpha(0)=0$, the map $[0,\\infty)\\to X$, $r\\mapsto S_\\alpha(r)x$  is strongly continuous and\n$$\nS_\\alpha(t)S_\\alpha(s)x={1\\over \\Gamma(\\alpha)}\\left(\\int_t^{t+s}(t+s-r)^{\\alpha-1}S_\\alpha(r)xdr- \\int_0^{s}(t+s-r)^{\\alpha-1} S_\\alpha(r)xdr\\right), \\qquad x\\in X,\n$$\n for $t,s >0$; for $\\alpha=0,$ $\\{S_{0}(t)\\}_{t\\geq 0}$ is an  usual $C_0$-semigroup, $S_0(0)=I$  and $S_0(t+s)=S_0(t)S_0(s)$ for $t,s>0$. In the case that  $\\{S_{\\alpha}(t)\\}_{t\\geq 0}$ is a non-degenerate family and $\\Vert S_\\alpha(t)\\Vert \\le C e^{\\omega t}$ for $C>0$, $\\omega \\in \\mathbb{{R}}$, then there exists a closed operator, $(A, D(A))$, called the generator of $\\{S_{\\alpha}(t)\\}_{t\\geq 0}$, such that\n \\begin{equation}\\label{resolvent}\n (\\lambda-A)^{-1}x=\\lambda^\\alpha \\int_0^\\infty e^{-\\lambda t} S_\\alpha (t)xdt, \\qquad \\Re \\lambda>\\omega, \\qquad x\\in X.\n \\end{equation}\n Moreover the following integral equality holds\n\\begin{equation}\\label{integral}\nA\\int_0^tS_\\alpha(s)xds=S_\\alpha(t)x-{t^\\alpha\\over \\Gamma(\\alpha+1)}x, \\qquad t > 0, \\quad x\\in X.\n\\end{equation}\n\n\n\n \\begin{theorem} \\label{resolventinte}Suppose that $\\{S_{\\alpha}(t)\\}_{t\\geq 0}$ is an $\\alpha$-times integrated semigroup generated by $(A, D(A))$ such that $\\Vert S_\\alpha(t)\\Vert \\le C e^{\\omega t}$ with $0\\le\\omega<1$. Then $1\\in \\rho(A)$, $R:=(1-A)^{-1}$,  ${\\mathcal R}(n)=R^n$ and\n\\begin{eqnarray*}\\label{convoresol}\n\\Delta^{-\\alpha}{\\mathcal R}(n)x&=&(I-A)\\int_0^\\infty {e^{-t}t^{n}\\over n!}S_\\alpha(t)xdt, \\qquad n\\in \\mathbb{{N}}_0,\\cr\n&=&\\int_0^\\infty {e^{-t}t^{n-1}\\over (n-1)!}S_\\alpha(t)xdt+k^{\\alpha+1}(n)x-k^{\\alpha+1}(n-1)x, \\qquad n\\ge 1, \\quad x\\in X,\n\\end{eqnarray*}\nIn particular if $\\{S_{\\alpha}(t)\\}_{t\\geq 0}$  has temperated growth, i.e.  $\\Vert S_\\alpha(t)\\Vert \\le C t^\\alpha$ for $t>0$, then $(I-A)^{-1}$ is a $(C, \\alpha)$-bounded operator.\n \\end{theorem}\n\\begin{proof} Take $\\lambda$ such that $\\lambda\\in\\rho(A) $ and then\n$$\n\\frac{(-1)^n}{n!}\\frac{d^n}{d\\lambda^n}(\\lambda^{-\\alpha}(\\lambda-A)^{-1})= \\sum_{j=0}^n{k^\\alpha(n-j)\\over \\lambda^{\\alpha+n-j}}(\\lambda-A)^{-j-1}.\n$$\nIn other hand, for $\\lambda $ such that $\\Re \\lambda >\\omega$, we apply formula (\\ref{resolvent}) to get that\n$$\n\\frac{(-1)^n}{n!}\\frac{d^n}{d\\lambda^n}(\\lambda^{-\\alpha}(\\lambda-A)^{-1})x= \\int_0^{\\infty} {t^n\\over n!}e^{-\\lambda t}S_{\\alpha}(t)x\\,dt, \\qquad x\\in X.\n$$\nFinally we take $\\lambda=1$ and write $R:=(1-A)^{-1}$,  ${\\mathcal R}(n)=R^n$ to conclude the first equality for $n\\in \\mathbb{{N}}_0$.  Now for $n\\ge 1$, we have that\n\\begin{eqnarray*}\n\\Delta^{-\\alpha}{\\mathcal R}(n)x&=& \\int_0^\\infty {e^{-t}t^{n}\\over n!}S_\\alpha(t)xdt+A\\int_0^\\infty {e^{-t}t^{n-1}\\over (n-1)!}\\left(1-{t\\over n}\\right)\\int_0^tS_\\alpha(s)xdsdt\\\\\n&=&\\int_0^\\infty {e^{-t}t^{n-1}\\over (n-1)!}S_\\alpha(t)xdt+k^{\\alpha+1}(n)x-k^{\\alpha+1}(n-1)x, \\qquad x\\in X,\n\\end{eqnarray*}\nwhere we have apply the equality (\\ref{integral}).\n\nIn the case that  $\\Vert S_\\alpha(t)\\Vert \\le C t^\\alpha$, we use the second equality and that the sequence $k^{\\alpha+1}$ is increasing to conclude that\n$\\displaystyle{\\sup_{n\\in \\mathbb{{N}}_0}}{\\lVert \\Delta^{-\\alpha}{\\mathcal R}(n)\\rVert \\over k^{\\alpha+1}(n)}<\\infty$ and $(I-A)^{-1}$ is a $(C, \\alpha)$-bounded operator.\n\\end{proof}\n\nClassical examples of   generators of  temperated  $\\alpha$-times integrated semigroup are differential operators $A$ such that their symbol $\\hat{A}$ is of the form $\\hat{A}=ia$ where $a$ is a real elliptic homogeneous polynomial on $\\mathbb{{R}}^n$ or $a\\in C^{\\infty}(\\mathbb{{R}}^n\\setminus\\{0\\})$ is a real homogeneous function on $\\mathbb{{R}}^n$ such that if $a(t)=0$ then $t=0,$ see \\cite[Theorem 4.2]{Hieber}, and other different examples in \\cite[Section 6]{Hieber}.\n\n\\begin{remark}{\\rm  In the case of uniformly bounded $C_0$-semigroups, i.e. $\\{T(t)\\}_{t\\ge 0}\\subset {\\mathcal B}(X)$ such that $\\sup_{t>0}\\Vert T(t)\\Vert <\\infty$, the resolvent $(1-A)^{-1}$ is power-bounded due to\n$$\n(1-A)^{-n}x=\\int_0^\\infty {t^{n-1}\\over (n-1)!} e^{-t}T(t)xdt, \\qquad x\\in X.\n$$\nNote that Theorem \\ref{resolventinte} includes a natural extension of this fact: the resolvent $(1-A)^{-1}$ is a $(C, \\alpha)$-bounded operator when $A$ generates a   temperated  $\\alpha$-times integrated semigroup.\n\nWe may also consider the homomorphism $\\theta$ defined in Theorem \\ref{homomorphism}, and in this case\n$$\n\\theta(\\Delta f)x = -A\\theta (f)x - (I-A)f(0)x, \\qquad f\\in\\tau^{\\alpha}(k^{\\alpha+1}), \\quad x\\in D(A),\n$$ when $A$ generates a  temperated  $\\alpha$-times integrated semigroup.}\n\\end{remark}\n\n\\subsection{Counterexamples of bounded homomorphisms}\n\n\\begin{example}{\\rm In \\cite[Section 2]{DL} there is an example of a positive, Ces\\`{a}ro bounded  but not\npower bounded operator $T$ on the space $\\ell^1$. As the author comments in \\cite[Section 4. Examples]{De00}, $\\Vert  T^n\\Vert_1\\le K{n\/ \\ln (n)}$  where $K$ is the uniform bound of the Ces\\`{a}ro averages of $T$. In this example $T$ is also a contraction in $\\ell^\\infty$. In \\cite[Section (VI)]{E2}, it is proven that $\\sup_{n\\ge 0}\\Vert T^n\\Vert_p\\ge (2^k)^{1\\over p}$ for any $k\\ge 1$ and $1\\le p< \\infty$. We conclude that $T$ is not power bounded in $\\ell^p$ $(1 \\le p < \\infty)$ and $T$ is a Ces\\`{a}ro bounded in $\\ell^p$ $(1 \\le p \\le \\infty)$ . By Corollary \\ref{cor5.7}, there exists a bounded homomorphism $\\theta: \\tau^1(k^2)\\to {\\mathcal B}(\\ell^p)$ such that $\\theta(e_1)=T$ and extends to $\\theta: \\ell^1\\to {\\mathcal B}(\\ell^p)$ if and only $p=\\infty$. }\n\\end{example}\n\n\n\n\n\n\\begin{example}{\\rm In \\cite{To-Ze}, a simple\nmatrix construction,  which  unifies  different approaches to the\nRitt condition and  ergodicity of matrix semigroups, is studied in detail. Consider the Banach space ${\\frak X}:= X\\oplus X$ with norm\n$$\n\\Vert x_1\\oplus x_2\\Vert_{X\\oplus X}:= \\sqrt{\\Vert x_1\\Vert^2+\\Vert x_2\\Vert^2}, \\qquad x_1\\oplus x_2\\in {\\frak X}.\n$$\n Let the bounded linear operator ${\\frak T}$ on ${\\frak X}$ be defined by the operator matrix\n$$\n{\\frak T}:=\\left(\\begin{matrix}T &T-I\\\\ 0&T \\end{matrix}\\right)\n$$\nwhere $T\\in {\\mathcal B}(X)$. In \\cite[Lemma 2.1]{To-Ze}, some connected properties between $T$ and ${\\frak T}$ are given. Now we consider as $X= \\ell^2$ and the backward shift operator $T\\in {\\mathcal L}(\\ell^2)$ defined by\n$$T((x_n)_{n\\ge 0}):=(x_{n+1})_{n\\ge 0}, \\qquad (x_n)_{n\\ge 0}\\in \\ell^2. $$\nBy \\cite[Example 3.1]{To-Ze}, $\\Vert {\\frak T}^n\\Vert\\ge 2n$ and $  {\\frak T}\n $ is a $(C, 1)$-bounded operator. We apply Corollary \\ref{cor5.7} to conclude that there exists an algebra homomorphisms $\\theta: \\tau^1(k^2)\\to {\\mathcal B}({\\frak X})$ such $\\theta(e_1)={\\frak T}$ and it is not extended continuously to $\\ell^1$.  In \\cite[Remark 3.2]{To-Ze}, the growth    $\\Vert {\\frak T}^n\\Vert\\ge 2n$ is pointed at as the fastest possible for a Ces\\`{a}ro bounded operator. }\n\\end{example}\n\n\n\n\n\\begin{example}{\\rm In \\cite[Proposition 4.3]{LSS}, the following example is given.\nFor any $\\gamma$ with $0<\\gamma<1$, there exists a positive linear operator $T$ on an $L_1$-space such that\n$$\\sup_{n\\ge0}\\Vert\\frac{\\Delta^{-\\gamma}\\mathcal{T}(n)}{k^{\\gamma+1}(n)}\\Vert=\\infty, \\quad\\textrm{but}\\quad \\sup_{n\\ge0}\\Vert\\frac{\\Delta^{-\\beta}\\mathcal{T}(n)}{k^{\\beta+1}(n)}\\Vert<\\infty \\quad\\textrm{for all }\\beta>\\gamma.$$\nBy  Corollary \\ref{cor5.7},  we conclude that there exists a bounded algebra homomorphism $\\theta$ such that $\\theta:\\tau^{\\beta}(k^{\\beta+1})\\to \\mathcal{B}(X)$ for all $\\beta >\\gamma$, $\\theta(e_1)=T$, and the homomorphism $\\theta$ is not extended continuously to the algebra $\\tau^{\\gamma}(k^{\\gamma+1})$ with $0<\\gamma<1$.}\n\\end{example}\n\n\n\n\n\n\n\\begin{example}{\\rm In \\cite[Proposition 4.4 (i)]{LSS}, the following operator is constructed.\nLet $dim X=\\infty$. For any integer $j\\ge0$, there exists a bounded linear operator $T$ on $X$ such that\n$$\\sup_{n\\ge0}\\Vert\\frac{\\Delta^{-(j+1)}\\mathcal{T}(n)}{k^{j+2}(n)}\\Vert<\\infty,  \\quad\\textrm{but}\\quad \\sup_{n\\ge0}\\Vert\\frac{\\Delta^{-\\gamma}\\mathcal{T}(n)}{k^{\\gamma+1}(n)}\\Vert=\\infty \\quad\\textrm{for }0\\le\\gamma<j+1.$$\nBy  Corollary \\ref{cor5.7},  we conclude that there exists a bounded algebra homomorphism $\\theta$ such that $\\theta:\\tau^{j+1}(k^{j+2})\\to \\mathcal{B}(X)$, $\\theta(e_1)=T$, and the homomorphism $\\theta$ is not continuously extended to the algebra $\\tau^{\\gamma}(k^{\\gamma+1})$ with $0\\le\\gamma<j+1$.}\n\\end{example}\n\n \\begin{example}{\\rm In \\cite[Proposition 4.4 (ii)]{LSS}, the following operator is constructed.\nLet $dim X=\\infty$. There exists a bounded linear operator $T$ on $X$ with $r(T)=1$, $\\Vert T\\Vert=2,$  and\n$$\\Vert A_r(T)\\Vert\\le 1-r,  \\quad0<r<1;\\quad\\textrm{and}\\quad \\sup_{n\\ge0}\\Vert\\frac{\\Delta^{-j}\\mathcal{T}(n)}{k^{j+1}(n)}\\Vert=\\infty, \\quad\\textrm{for }j\\ge 1.$$\nSince $k^j(n)\\le k^{j+1}(n)$ for $n\\ge 0$, we also conclude that $\\displaystyle{\\Vert\\frac{\\Delta^{-j}\\mathcal{T}(n)}{k^{j}(n)}\\Vert=\\infty}$ for $j\\ge 1$ and the converse of Theorem \\ref{abels} does not hold for $\\gamma <\\alpha$.}\n\\end{example}\n\n\n\n\n\n\\subsection{Application to Katznelson-Tzafriri theorem}\n\n\nLet $A(\\mathbb{T})$ be the regular convolution Wiener algebra formed by all continuous periodic functions $f(t)=\\sum_{n=-\\infty}^{\\infty}\\hat{f}(n) e^{int}, \\quad t\\in [-\\pi,\\pi],$ where $(\\hat{f}(n))_{n\\in\\mathbb{{Z}}}$ are the Fourier coefficients of $f,$ that is $$\\hat{f}(n)=\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi}f(t)e^{-int}\\,dt,\\qquad n\\in \\mathbb{{Z}},$$ with the norm $\\lVert f \\rVert_{A(\\mathbb{T})}:=\\sum_{n=-\\infty}^{\\infty}|\\hat{f}(n)|,$ and $A_+(\\mathbb{T})$ be the convolution closed subalgebra of $A(\\mathbb{T})$ where the functions satisfies that $\\hat{f}(n)=0$ for $n<0.$ Note that both $A(\\mathbb{T})$ and $\\ell^1_{\\mathbb{{Z}}},$ and $A_+(\\mathbb{T})$ and $\\ell^1$ are isometrically isomorphic, where $\\ell^1_{\\mathbb{{Z}}}$ denotes the complex summable sequences indexed in $\\mathbb{{Z}}.$\n\nKatznelson and Tzafriri proved in 1986 the following well known  theorem: if $T\\in\\mathcal{B}(X)$ is power-bounded and $f\\in A_+(\\mathbb{T})$ is of spectral synthesis in $A(\\mathbb{T})$ with respect to $\\sigma(T)\\cap \\mathbb{T},$ then $$\\displaystyle\\lim_{n\\to\\infty}\\lVert T^n\\theta(\\hat{f}) \\rVert=0,$$ see \\cite[Theorem 5]{Katznelson}. Moreover, for $T\\in\\mathcal{B}(X)$ a power-bounded operator, $\\displaystyle\\lim_{n\\to\\infty}\\lVert T^n-T^{n+1}\\rVert=0$ if and only if $\\sigma(T)\\cap \\mathbb{T}\\subseteq\\{1\\},$ see \\cite[Theorem 1]{Katznelson}.\n\nThe authors have got some similar results for $(C,\\alpha)$-bounded operators, which will appear in a forthcoming paper. We define $A^{\\alpha}(\\mathbb{T})$ a new regular Wiener algebra contained in $A(\\mathbb{T}),$ and $A_+^{\\alpha}(\\mathbb{T})$ a convolution closed subalgebra of $A^{\\alpha}(\\mathbb{T}),$ which is isometrically isomorphic to $\\tau^{\\alpha}(k^{\\alpha+1}).$ The result prove that if $\\alpha> 0,$ $T\\in\\mathcal{B}(X)$ is a $(C,\\alpha)$-bounded operator and $f\\in A_+^{\\alpha}(\\mathbb{T})$ is of spectral synthesis in $A^{\\alpha}(\\mathbb{T})$ with respect to $\\sigma(T)\\cap \\mathbb{T},$ then $$\\displaystyle\\lim_{n\\to\\infty}\\frac{1}{k^{\\alpha+1}(n)}\\lVert \\Delta^{-\\alpha} \\mathcal{T}(n)\\theta(\\hat{f}) \\rVert=0.$$\n\nOn the continuous case, Katznelson-Tzafriri theorems have been proved for $C_0$-semigroups and extended later for $\\alpha$-times integrated semigroups, see \\cite{Esterle} and \\cite{GMM} respectively.\n\n\n\n\n\n\n\n\n\\subsection*{\\it Acknowledgments.} This work was done while the second author was on  sabbatical leave, visiting  the University of Zaragoza. He is grateful to the members of the Analysis Group for their kind hospitality.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nThe heat kernel method has become a ubiquitous tool in both mathematics and \nphysics (see \\cite{Fu1} for a recent overview). In mathematics it appears e.g.\nin the study of the spectral geometry of a Laplace-type differential operator \non a Riemannian space and in the proof of index theorems \\cite{ABP,Gi1}. In \nphysics, the euclidean one-loop effective action for a given quantum field \ntheory can be expressed in terms of the determinant of such a differential \noperator \\cite{DeW}, which in turn can be written in terms of the associated \nheat kernel. The heat kernel therefore appears in many places, from quantum \ngravity \\cite{DeW} to chiral perturbation theory \\cite{Bal}. Anomalies can \nalso be studied with the heat kernel method (see \\cite{Bal,Ber}). Physicists \nfrequently refer to it as the Schwinger-DeWitt \\cite{Sch,DeW} or proper-time \nmethod. Exact expressions for the heat kernel exist only for special spaces. \nIn the general case one may use its asymptotic expansion in the \nproper-time\\footnote{An alternative is the so-called covariant perturbation \ntheory \\cite{BarV}. It provides a (partial) summation of the Schwinger-DeWitt \nseries and can account for nonlocal effects.}.\nThe coefficients in this expansion are the so-called heat kernel \ncoefficients (see sect 2 for a precise definition).\n\nSeveral methods have been developed to find the heat kernel coefficients (see\n e.g. \\cite{AvS} for a review). DeWitt \\cite{DeW} determined the first two \ncoefficients\\footnote{We frequently abbreviate `diagonal value of a heat \nkernel coefficient' to `coefficient'.} with a covariant recursive method.\nSakai \\cite{Sak} relied on Riemannian coordinates to find the third coefficient\nin the scalar case (i.e. a single scalar field on a curved space).\nFor the general case, this coefficient was found by Gilkey \\cite{Gi2} using a \nnoncovariant pseudo-differential-operator technique. The integrated and traced\nfourth and fifth coefficients for an arbitrary field theory in flat space were \nfound in \\cite{vdV} through the evaluation of a noncovariant Feynman graph. \nAvramidi \\cite{Avr} presented a new covariant nonrecursive procedure and found\nthe fourth coefficient for the general case (for the scalar case see also\n \\cite{ABC}). More recently, string-inspired world line path integral methods \nhave been used \\cite{FHSS1} to determine the functional trace of the first \neight heat kernel coefficients for the case of a matrix potential in flat \nspace without gauge connection (see also \\cite{BELS}). \n\nIn this paper the explicit diagonal value of the fifth heat kernel coefficient\nin the general case is presented for the first time. In physics this \ncoefficient is of importance e.g. in analysing the short distance behavior and \nanomalies of ten-dimensional quantum field theories (see \\cite{FrT}). \nHowever, the number of terms in the higher heat kernel coefficients grows \nrapidly, leading one to expect more than a thousand terms for the fifth \ncoefficient\\footnote{Even restricting to the a single scalar field on a curved\nspace, the $j$-th coefficient already contains $1,4,17,92,668$ $R^j$-terms for \n$j=1,2,3,4,5$ (see appendix A of \\cite{FKW}).}. This would seem to preclude \nwriting down this coefficient in an intelligable form. Indeed, a computer \nwould appear to be an essential piece of equipment in determining and storing \nthe fifth coefficient. Contrary to these expectations, we will show here that \nwith a suitable index-free notation one obtains a compact expression for this \ncoefficient, containing only 26 terms. \n\nUsing standard matrix notation for the field indices, the heat kernel \ncoefficients are scalars and this suggests that it may be possible to write \nthem in a form which is free of spacetime indices as well. \nSince the heat kernel coefficients are covariant, we may determine them in a \nspecial gauge and adapted coordinates. Following earlier authors, we select\nthe Fock-Schwinger gauge and Riemann normal coordinates. \nExamination of the relevant recursion relations then shows that only certain \nmaximally symmetrized multiple (covariant) derivatives of the matrix potential\nand Yang-Mills or Riemann curvature tensors appear. The heat kernel \ncoefficients being formal scalars, it\nturns out that we need to keep track of the rank of these tensors only (note  \nthe similarity with totally antisymmetric tensors, i.e. differential forms). \nThis provides the basis for our index-free notation. Using instead a fully \ncovariant method, Avramidi \\cite{Avr} has arrived at similar conclusions by \nexpanding in so-called covariant Taylor series, obtaining in this way the \nfirst four coefficients. \nOur non-covariant procedure seems to be no less efficient and, by maintaining \nindex-free notation and manifest hermiticity, yields even compacter answers for\nthe heat kernel coefficients. Thus, in the special case of a flat space with a\ngauge connection, we can also present the answer for the sixth coefficient. It\ncontains only 75 terms.\n\nUsing index-free notation also permits us to investigate the general structure \nof the heat kernel coefficients. Without actually solving the recursion \nrelations, we can show that certain Lorentz scalars are absent from all \ncoefficients. In flat space, this happens for the first time in the fifth \ncoefficient. This may therefore be relevant for the anomalies of quantum field\ntheories in ten or more dimensions.\n\nAs we already mentioned above, a brute force approach with a computer algebra\nprogram would have produced an unwieldy result.\nOur results were obtained without the aid of a computer. However, it is \nrelatively easy to program our index-free method and we used FORM \\cite{Ver} \nand Mathematica \\cite{Wol} to run some checks.\n\nAn outline of this paper is as follows. In sect 2 we recall the main features\nof the heat kernel method {\\it alias} the Schwinger-DeWitt formalism. In sect \n3 we introduce our index-free notation and use it to determine first the heat \nkernel coefficients in flat space without gauge connection, expanding in powers\nof either derivatives (sect 3.1) or of the matrix potential (sect 3.2). \nIn sect 3.1 we also use our index-free notation to prove that certain scalars \nare absent from all heat kernel coefficients. In sect 4 we show that the \ncorresponding heat kernel coefficients with a gauge connection can be obtained\nfrom a simple covariantization process. This involves not only replacing \npartial derivatives by covariant ones but also adding new field strength \ndependent terms. The latter kind of terms are shown to arise only as shifts in\nthe potential and in its covariant derivatives. These shifts being understood,\nthe heat kernel coefficients do not change their form upon `turning on the \ngauge field'. In sect 5 we employ Riemann normal coordinates to generalize to \na curved space. In particular, we present in subsect 5.1 an explicit expansion\nof the vielbein to all orders in these coordinates (such an expansion, usually\ngiven for the metric, but only to some finite order in the normal coordinates \ncan be found in many places, e.g. \\cite{Sak,McL}). We give a similar result in\nsubsect 5.2 for the gauge connection to all orders in normal coordinates. \nBased on this, we find in subsect 5.3 the explicit form for any heat kernel \ncoefficient up to and including terms of fourth order in the Yang-Mills and \nRiemann curvatures. The curved coefficients can be obtained from the \ncorresponding gauged but flat coefficients via further simple covariant \nsubstitutions. To complete the fifth coefficient, we need to find the few \nterms of fifth order in the curvatures. This we do in subsect 5.4 by \nspecializing to a locally symmetric space. \nIn sect 6 we present the explicit answers for the first five coefficients. \nWe indicate how to return to more conventional notation and compare with \nearlier results. Our conclusions are given in sect 7. Several appendices \nfollow (in particular, the sixth coefficient in flat space is given in \nappendix C).\n\\section{Schwinger-DeWitt formalism}\nConsider a set of fields $\\phi_i(x)$, $i=1\\dots n$, defined over a compact\n$d$-dimensional Riemannian manifold\\footnote{The fields can be considered \nto be sections of a smooth vector bundle. The heat kernel coefficients do \nnot explicitly depend on the dimension or signature of space(time). We assume\nspace to have no boundary (for the case with boundary, see e.g. \\cite{BGV}).}\nwith coordinates $x^\\mu$, $\\mu=1\\dots d$ and metric $g_{\\mu\\nu}$ (see appendix A \nfor our notation and conventions). The fields are acted upon by a Laplace-type\nwave operator $\\Delta$\n\\bee\\label{wave}\n\\Delta \\= -\\, \\de^2 - X \\quad ,\\quad \\de^2 \\= g^{\\mu\\nu}\\de_\\mu\\de_\\nu\n\\ene\nHere the covariant derivative $\\de$ includes connection terms as needed for \nthe fields $\\phi_i$. $X$ is a hermitian $n\\times n$ matrix potential (we suppress\nthe field or `bundle' indices). The wave operator $\\Delta$ is hermitian with \nrespect to the inner product\n\\bee\n(\\phi ,\\psi) \\= \\int d^dx \\sqrt{g}\\ \\phi^\\ast \\psi\n\\ene\nFor most bosonic gauge field theories of interest one can achieve a wave \noperator of Laplace-type as in (\\ref{wave}) by a suitable gauge choice. For \nfermionic (gauge) fields one squares the wave operator to obtain again \n(\\ref{wave}) (see \\cite{BaV} for wave operators not of this form).\n\nFollowing Schwinger and DeWitt, we introduce the proper-time parameter $\\tau$\nand define the heat kernel $K$ associated with $\\Delta$ by\n\\bee\\label{hkd}\n({\\pa\\ov\\pa\\tau} +\\Delta)\\, K(x,x\\pr;\\tau) \\= 0 \\quad ,\\quad\nK(x,x\\pr;0) \\= I\\delta(x,x')\n\\ene\nwhere $I$ is the $n\\times n$ unit matrix and the bi-scalar $\\delta$ function is\ndefined by\n\\bee\n\\int d^dx\\, {\\sqrt g}\\,\\delta(x,x')\\,\\phi(x) \\= \\phi(x\\pr) \n\\ene\nfor any scalar field $\\phi$. As we mentioned in the introduction, an exact \nsolution for the kernel $K$ exists only for special spaces. We instead make\nDeWitt's ansatz\n\\bee\\label{hkc}\nK(x,x\\pr;\\tau) \\= (4\\pi\\tau)^{-d\/2} {\\cal D}(x,x\\pr)^{1\/2} e^{-\\sigma(x,x\\pr)\/2\\tau}\n\\su_{j=0}^\\infty  a_j(x,x\\pr)\\,{\\tau^j\\ov j!} \n\\ene\nwhich is known to be an asymptotic expansion in $\\tau$ \\cite{Gi2}. Note our\nunconventional normalization for the heat kernel coefficients $a_j$, which \nhowever agrees with \\cite{Avr}. They transform as scalar densities of weight \n$-1\/2$ at both $x$ and $x\\pr$. The bi-scalar $\\sigma$ is the geodetic interval \n(one half of the distance squared between $x$ and $x\\pr$) and satisfies\n\\bee\\label{sig}\n\\sigma_;\\sp\\mu\\sigma_{;\\mu} \\= 2\\sigma\\quad ,\\quad [\\sigma] \\equiv\\sigma(x,x) \\= 0\n\\ene\nwhere we use Synge's bracket notation to indicate evaluation on the diagonal. \nThe bi-scalar ${\\cal D}$ is the Van Vleck-Morette determinant defined by\n\\bee\n{\\cal D}(x,x\\pr) \\= \ng^{-1\/2}\\,\\det(-\\sigma_{;\\mu\\nu\\pr}) \\, {g\\pr}{\\phantom g\\!\\!}^{-1\/2}\n\\ene\nwhere a prime refers to the point $x\\pr$. It satisfies\n\\bee\\label{vvm}\n(2\\sigma_;\\sp\\mu\\de_\\mu + \\sigma_;\\sp\\mu\\sb\\mu - d)\\, {\\cal D}^{1\/2} \\= 0\n\\quad ,\\quad [{\\cal D}] \\= 1\n\\ene\nInserting (\\ref{hkc}) into (\\ref{hkd}) and using (\\ref{sig}) and (\\ref{vvm}), \none finds that the heat kernel coefficients must satisfy the following \nrecursion relations for $j\\geq 0$\n\\bee\\label{ajd}\n(\\sigma^\\mu\\de_\\mu + j) a_j \\= -\\, j {\\cal D}^{-1\/2} \\Delta\\, {\\cal D}^{1\/2} a_{j-1}\n\\quad ,\\quad [a_0] \\= I\n\\ene\nwhere it is to be understood that $a_{-1}$ vanishes. Note that whereas $\\sigma$ \nand ${\\cal D}$ depend only on the metric, the $a_j$ are matrix valued and \ndepend in addition on the detailed form of the wave operator. The hermiticity \nof the wave operator implies\n\\bee\na_j(x,x\\pr)^\\dagger \\= a_j(x\\pr,x) \\quad\\Rightarrow\\quad [a_j]^\\dagger\\= [a_j]\n\\ene\nTo keep this property manifest we introduce the following notation: for any \nmatrix valued function $F$ we define\n\\bee\\label{brac}\n\\{F\\}\\equiv F\\, +\\, F^\\dagger\\quad {\\rm if}\\quad F\\neq F^\\dagger\\quad \n{\\rm else}\\quad F\n\\ene\nThus $\\{F\\}=F$ when $F$ is selfadjoint. Frequently one is interested in the \nfunctional trace of the heat kernel coefficients\n\\bee\nb_j \\,\\equiv\\, {\\rm Tr} \\, a_j\\equiv {\\rm tr}\\,\\int d^dx\\,\\sqrt{g}\\ a_j(x,x)\n\\ene\nwhere ${\\rm tr}$ denotes the matrix trace over field indices only. To determine e.g.\nchiral anomalies one would need the non-traced version. In this paper we will \ndetermine the non-traced diagonal heat kernel coefficients.\n\\section{Flat space without gauge connection}\n\\setcounter{equation}{0}\nIn a flat space without gauge connection the recursion relations (\\ref{ajd}) \nfor the heat kernel coefficients become\n\\bee\\label{rec}\n(x^\\mu\\pa_\\mu + j)\\, a_j\\= j\\,\\big(\\pa^2+X) a_{j-1}\n\\quad ,\\quad [a_0] \\= I\n\\ene\nHere we have set $x\\pr$ to zero (we will not differentiate at $x\\pr$).\nWe are mostly interested in the diagonal values of the heat kernel \ncoefficients. However, it is easy to see that this in turn requires knowledge \nof some derivatives of preceding heat kernel coefficients on the diagonal. \nIndeed, taking the diagonal value of (\\ref{rec}) yields\n\\bee\\label{[aj]}\n[a_j] \\= [\\pa^2 a_{j-1}]\\,  + X\\,[a_{j-1}]\n\\ene\nThus in particular \n\\bee\n[a_1] \\= X\n\\ene\nbut for $j>1$ we must first find $a_{j-1}$ and $\\pa^2 a_{j-1}$ on the diagonal.\nApplying $\\pa^2$ to (\\ref{rec}) and then going on the diagonal gives\n\\bee\n[\\pa^2 a_{j-1}]\\={j-1\\ov j+1}\\,\\Big( [\\pa^2\\pa^2 a_{j-2}] + X\\,[\\pa^2 a_{j-2}]\n +2X_,\\sp\\mu\\,[\\pa_\\mu a_{j-2}] + X_,\\sp\\mu\\sb\\mu\\,[a_{j-2}] \\Big)\n\\ene\nSetting $j=2$ and substituting the result in (\\ref{[aj]}), we find\n\\bee\n[a_2] \\= \\sfrac13\\,\\pa^2 X \\,+\\, X^2 \n\\ene\nbut for $j>2$ we require some new diagonal values of derivatives of $a_{j-2}$.\nThis recursive procedure ends after $j$ steps since the diagonal value of any \nderivative of $a_0$ vanishes. Differentiating (\\ref{rec}) $n$ times and \ntaking the diagonal value yields\n\\bee\\label{panaj}\n[\\pa_{\\mu_1}\\dots\\pa_{\\mu_n} a_j] \\=\n -\\,{j\\ov j+n} [\\pa_{\\mu_1}\\dots\\pa_{\\mu_n}\\Delta a_{j-1}]\n\\ene\n\nTo solve these recursion relations in an effective way, we introduce a short \nhand notation for them. Using comma notation for partial derivatives, but \nwriting only the {\\it number} $n$ of uncontracted derivatives taken, we can \nabbreviate (\\ref{panaj}) as\n\\bee\\label{ajn}\n\\sa_{j,n}\\={j\\ov j+n}\\,\\Big(\\,\\sa_{j-1,(2),n}\n                            \\,+\\,(\\Xi\\,\\sa_{j-1})_{,n} \\,\\Big)\n\\ene\nHere, the index $n$ stands for all partial derivatives taken on the left hand \nside of (\\ref{panaj}). The {\\it sans serif} symbols serve to emphasize that we\nare using this short hand notation and at the same time imply evaluation on the\ndiagonal. The $\\pa^2$ on the right hand side has been abbreviated to a 2 in \nparenthesis. Of course one must first distribute the $n$ derivatives over the \nfactors of the second term in (\\ref{ajn}) before taken the diagonal value.\nDoing so yields\n\\bee\\label{ajndis}\n\\sa_{j,n}\\={j\\ov j+n}\\,\\Big(\\,\\sa_{j-1,(2),n}\n                   \\,+\\,\\su_{p=0}^n {n{\\cal H} p} \\Xi_{(p}\\,\\sa_{j-1,n-p)}\\,\\Big)\n\\ene\nwhere the parenthesis around the $p$ plus $n-p$ indices imply total \nsymmetrization. We write the result of replacing $n$ by $2n$ and contracting \nall derivatives as\n\\bee\n\\sa_{j,(2n)}\\={j\\ov j+2n}\\,\\Big(\\,\\sa_{j-1,(2n+2)}\\,+\\,\n\\su_{p=0}^{2n} {2n{\\cal H} p} \\Xi_{(p}\\,\\sa_{j-1,2n-p)}\\,\\Big)\n\\ene\nwhere the parenthesis now imply not only total symmetrization, but also full \ncontraction. Here we introduced the following notation: for any functions \n$F(x)$, $G(x)$ and $H(x)$ we define\n\\bea\\label{distr}\n\\Phi_{(2n)} &=& F_{,(\\mu_1\\mu_1\\dots\\mu_n\\mu_n)}(0) \\non\\\\\n\\Phi_{(k}\\Gamma_{2n-k)} &=& F_{,(\\mu_1\\dots\\mu_k}(0)\\,G_{,\\mu_{k+1}\\dots\\mu_{2n})}(0)\\,\n                      \\delta^{\\mu_1\\mu_2}\\dots\\delta^{\\mu_{2n-1}\\mu_{2n}} \\\\\n\\Phi_{(k}\\Gamma_\\el\\H_{2n-k-\\el)} &=& F_{,(\\mu_1\\dots\\mu_k}(0)\\,\n           G_{,\\mu_{k+1}\\dots\\mu_{k+\\el}}(0)\\,H_{,\\mu_{k+\\el+1}\\dots\\mu_{2n})}(0)\\,\n               \\delta^{\\mu_1\\mu_2}\\dots\\delta^{\\mu_{2n-1}\\mu_{2n}} \\non\n\\ena\nwith similar expressions in case the $2n$ derivatives are distributed over yet\nmore factors. \nUnless otherwise noted, the use of parenthesis as on the left \nhand side of (\\ref{distr}) indicates that the $2n$ derivatives are to be \ntotally symmetrized {\\it and} fully contracted. Note that\n\\bee\\label{F2n}\n\\Phi_{(2n)}\\= (\\pa^2)^n F(0) \n\\ene\nso in this case the symmetrization in (\\ref{distr}) is superfluous (it is\nrelevant when a gauge connection is present: see (\\ref{Xj})). The notation of \n(\\ref{F2n}) is due to Avramidi \\cite{Avr} (evaluation at the origin is not \nimplied there). We generalized it here by allowing distribution of the \ncontracted and symmetrized derivatives over any number of factors. \nNote that e.g.\n\\bea\\label{FGex}\n\\Phi_{(1}\\Gamma_{1)} &=& F_{,\\mu}(0)\\, G_{,\\mu}(0) \\non\\\\ \n\\Phi_{(1}\\Gamma_{3)} &=& F_{,\\mu}(0)\\, G_{,\\mu\\nu\\n}(0) \\= \\Phi_{(1}\\Gamma_{1)(2)} \\\\ \n\\Phi_{(2}\\Gamma_{2)} &=& \\sfrac13 F_{,\\mu\\m}(0)\\, G_{,\\nu\\n}(0) \n                 + \\sfrac23 F_{,\\mu\\nu}(0)\\, G_{,\\mu\\nu}(0) \\non \n\\ena\nThe general rule for such a reduction is given in (\\ref{combi}).\n\\subsection{Expansion in derivatives}\nWe first discuss some generic features of the derivative expansion, which can \nbe understood without solving the recursion relations. Using only dimensional \nanalysis, it follows that through second order in derivatives the diagonal \nvalues of the heat kernel coefficients must have the structure\n\\bee\\label{dima}\n[a_j] = \\alpha_j X^j\\ +\\su_{k=1}^{j-1}\\alpha_{j\\,k}\\,X^{k-1} X_{,\\mu\\m} X^{j-k-1} \n+\\su_{k=1}^{j-2}\\su_{\\el=1}^{j-k-1} \\alpha_{j\\,k\\el}\\,\n  X^{k-1} X_{,\\mu} X^{j-k-\\el-1} X_{,\\mu} X^{\\el-1} +\\dots\n\\ene\nThe $\\alpha$'s are numerical coefficients which, in order for (\\ref{dima}) to be\nhermitian, must satisfy\n\\bee\n\\alpha_{j\\ j-k}\\= \\alpha_{j\\,k}\\quad ,\\quad\\alpha_{j\\,\\el\\,k}\\= \\alpha_{j\\,k\\,\\el}\n\\ene\nThese expectations are borne out by our results. In particular, all terms in \n(\\ref{dima}) will turn out to have nonvanishing coefficients. At fourth order \nin derivatives, dimensional analysis allows 11 ways of distributing the \nderivatives, namely\n\\bea\\label{fourd}\n& X_{,\\mu\\m\\nu\\n} &\\non\\\\\n&X_{,\\mu\\m\\nu} X_{,\\nu}\\quad ,\\quad \n X_{,\\mu\\m} X_{,\\nu\\n} \\quad ,\\quad X_{,\\mu\\nu} X_{,\\mu\\nu}& \\\\\n&X_{,\\mu\\m} X_{,\\nu}X_{,\\nu}\\quad ,\\quad X_{,\\nu} X_{,\\mu\\m} X_{,\\nu}\\quad ,\\quad\n X_{,\\mu\\nu} X_{,\\mu}X_{,\\nu}\\quad ,\\quad X_{,\\mu} X_{,\\mu\\nu} X_{,\\nu}& \\non\\\\ \n&X_{,\\mu} X_{,\\mu} X_{,\\nu} X_{,\\nu}\\quad ,\\quad \n X_{,\\mu} X_{,\\nu} X_{,\\mu} X_{,\\nu}\\quad ,\\quad\n X_{,\\mu} X_{,\\nu} X_{,\\nu} X_{,\\mu}& \\non\n\\ena\nHere and through (\\ref{overlap}) below, we list only equivalence classes of\nLorentz scalars, where two scalars are considered equivalent if they become\nequal upon omission of all undifferentiated factors $X$ and\/or reversing the \norder of the factors\\footnote{By working with real fields only, we can always \narrange $X$ to be a real symmetric matrix. The same is true for its derivatives\nand $\\{F\\}$ as in (\\ref{brac}) then means that, unless $F$ is a palindrome, we \nmust add to $F$ its transpose.}. We therefore do not write the curly \nbrackets as in (\\ref{brac}).\n\nWe now claim that two of the Lorentz scalars in (\\ref{fourd}) can not appear \nin the heat kernel coefficients. To prove our assertion, we need not solve the\nrecursion relations explicitly. It suffices to keep track of the way the \nderivatives get distributed over the various factors as the recursion proceeds.\nHence, we omit numerical factors as well as any undifferentiated matrix $X$. \nWe may even drop the ordering labels $j$ respectively $j-1$ in (\\ref{ajndis}) \nand thus abbreviate $\\sa_{j,n}$ to $\\sa_n$, where $n$ is the number of \nderivatives. Writing only Lorentz scalars, we thus have the following chain of\nsubstitutions\n\\bea\\label{chain0}\n\\sa &\\ra & \\sa_{(2)} \\non\\\\\n\\sa_{(2n)} &\\ra & \\sa_{(2n+2)} +\\su_{p=1}^{2n}\\Xi_{(p}\\sa_{2n-p)} \n                      \\quad ,\\quad n\\geq 1\\\\\n\\Xi_{(p}\\sa_{2n-p)} &\\ra & \\Xi_{(p}\\sa_{2n-p)(2)} \n                    +\\su_{q=1}^{2n-p}\\Xi_{(p}\\Xi_q\\sa_{2n-p-q)}\n                      \\quad ,\\quad p=1\\dots 2n \\non\n\\ena\netc. We truncate this hierarchy at level $2N$ in derivatives, i.e. we drop any\nterm with more than $2N$ derivatives (note that for the $j$-th coefficient, \n$N\\leq j-1$). To be definite, consider the case $N=2$ where the first few \nsteps are\n\\bea\\label{ex0}\n\\sa &\\ra & \\sa_{(2)} \\non\\\\\n    &\\ra & \\sa_{(4)} +\\Xi_{(1}\\sa_{1)} +\\Xi_{(2)}\\sa \\\\\n    &\\ra &  \\Xi_{(4)} +\\Xi_{(3}\\sa_{1)} +\\Xi_{(2}\\sa_{2)} +\\Xi_{(1}\\sa_{3)}\n                     +\\Xi_{(1}\\Xi_{1)}\\sa +\\Xi_{(2)}\\sa \\non\n\\ena\nWe used\\footnote{Note that in general we have \n$\\Xi_{(p}\\sa_{2n-p)(2)}\\neq\\Xi_{(p}\\sa_{2n-p+2)}$. Thus, before being able to \niterate again, we must `remove the box'. See (\\ref{mobo}) for an example and \napp. B for the general solution. Since we shall not go beyond $N=2$ this \ncomplication is irrelevant here.} that $\\Xi_{(1}\\sa_{1)(2)} = \\Xi_{(1}\\sa_{3)}$, \nsee (\\ref{FGex})\nIterating once more, we see that at fourth order in derivatives a generic heat\nkernel coefficient can contain only the following 9 equivalence classes of\nLorentz scalars\n\\bea\\label{nine}\n&\\Xi_{(4)}& \\non\\\\\n&\\Xi_{(3}\\Xi_{1)}\\quad ,\\quad \\Xi_{(2}\\Xi_{2)}\\quad,\\quad\\Xi_{(2)}\\,\\Xi_{(2)}&\n\\non\\\\\n&\\Xi_{(1}\\Xi_2\\Xi_{1)}\\quad,\\quad \\Xi_{(2}\\Xi_1\\Xi_{1)}\\quad ,\\quad \n                                          \\Xi_{(2)}\\Xi_{(1}\\Xi_{1)} & \\\\\n&\\Xi_{(1}\\Xi_1\\Xi_1\\Xi_{1)}\\quad ,\\quad \\Xi_{(1}\\Xi_{1)}\\,\\Xi_{(1}\\Xi_{1)}& \\non \n\\ena\nComparison with (\\ref{fourd}) shows that expressions with overlapping sets of \nsymmetrized and contracted derivatives, namely\n\\bea\\label{overlap}\n&\\Xi_{(1}\\Xi^{(2)}\\Xi_{1)} \\equiv X_{,\\nu} X_{,\\mu\\m} X_{,\\nu}&    \\\\\n&\\Xi_{(1}\\Xi^{(1}\\Xi^{1)}\\Xi_{1)} \\equiv X_{,\\mu} X_{,\\nu} X_{,\\nu} X_{,\\mu}\n\\ \\quad {\\rm or} \\quad\\ \n\\Xi_{(1}\\Xi^{(1}\\Xi_{1)}\\Xi^{1)} \\equiv X_{,\\mu} X_{,\\nu} X_{,\\mu} X_{,\\nu}& \\non\n\\ena\nare absent. This holds for all heat kernel coefficients (recall that we dropped\nany undifferentiated matrix $\\Xi$). The absence of the first (second and third)\nentry in (\\ref{overlap}) can be first observed in $\\sa_5$ (respectively \n$\\sa_6$). This may hence be relevant to the short distance behavior and \nanomalies of quantum field theories in ten or more dimensions. We conclude \nthat at fourth order in derivatives only 9 of the 11 {\\it a priori} allowed \nscalars appear. This is only a small reduction in the number of terms, but at \nhigher orders the relative number of such absent Lorentz scalars increases \nrapidly. At sixth order in derivatives we find that dimensional analysis and \nhermiticity would permit 85 different Lorentz scalars similar to those in \n(\\ref{fourd}). However, continuing (\\ref{ex0}), we discover that only 53 of \nthese scalars can actually appear in the heat kernel coefficients.\n\nWe now give our explicit solution for $\\sa_j$ through fourth order in \nderivatives. To have manifest hermiticity, we use here the convention that in \nan $N$-fold sum with $k_1,\\dots,k_N$ as summation variables (below \n$k_1,\\,k_2,\\dots =k,\\,\\el,\\dots$ etc, but skip $o$), the label $k_{N+1}$ has \nby definition the value $(k_N)_{\\rm max}-k_N +1$ (so e.g. in the second line\nbelow, $\\el\\equiv j-k$).\n\\bea\\label{deriv}\n&&\\sa_j \\= \\Xi^j \\non\\\\\n&&\n+\\su_{k=1}^{j-1} {k\\el\\ov j+1}\\,\\Xi^{k-1}\\Xi_{(2)}\\Xi^{\\el-1} \\non\\\\\n&&\n+\\su_{k=1}^{j-2}\\su_{\\el=1}^{j-k-1} {2k\\el\\ov j+1}\\,\n\\Xi^{k-1}\\Xi_{(1}\\Xi^{m-1}\\Xi_{1)}\\Xi^{\\el-1} \\non\\\\\n&&\n+\\su_{k=1}^{j-2}{k(k+1)\\el(\\el+1)\\ov 2(j+1)(j+2)}\\,\n\\Xi^{k-1}\\Xi_{(4)}\\Xi^{\\el-1} \\non\\\\\n&&\n+\\su_{k=1}^{j-3}\\su_{\\el=1}^{j-k-2} \\Big( {k\\el m\\ov j+1}\\,\n \\Xi^{k-1}\\Xi_{(2)}\\Xi^{m-1}\\Xi_{(2)}\\Xi^{\\el-1} \\non\\\\\n&&\\qquad\\qquad\n+ {3k(k+1)\\el(\\el+1)\\ov (j+1)(j+2)}\\,\n   \\Xi^{k-1}\\Xi_{(2}\\Xi^{m-1}\\Xi_{2)}\\Xi^{\\el-1} \\non\\\\\n&&\\qquad\\qquad\n+ {2k(k+1)\\el(\\el+m+1)\\ov (j+1)(j+2)}\\,\n \\{\\Xi^{k-1}\\Xi_{(3}\\Xi^{m-1}\\Xi_{1)}\\Xi^{\\el-1}\\} \\Big) \\non\\\\\n&&\n+\\su_{k=1}^{j-4}\\su_{\\el=1}^{j-k-3}\\su_{m=1}^{j-k-\\el-2} \\Big( \n{2k\\el m\\ov j+1}\\,\n \\{\\Xi^{k-1}\\Xi_{(2)}\\Xi^{m-1}\\Xi_{(1}\\Xi^{n-1}\\Xi_{1)}\\Xi^{\\el-1}\\} \\non\\\\\n&&\\qquad\\qquad\n+{6k(k+1)\\el(\\el+n+1)\\ov (j+1)(j+2)}\\,\n \\{\\Xi^{k-1}\\Xi_{(2} \\Xi^{m-1}\\Xi_1\\Xi^{n-1}\\Xi_{1)}\\Xi^{\\el-1}\\} \\non\\\\\n&&\\qquad\\qquad\n+ {6 k(k+m+1)\\el(\\el+n+1)\\ov (j+1)(j+2)}\\,\n \\Xi^{k-1}\\Xi_{(1}\\Xi^{m-1}\\Xi_2\\Xi^{n-1}\\Xi_{1)}\\Xi^{\\el-1} \\Big)\\non\\\\\n&&\n+\\su_{k=1}^{j-5}\\su_{\\el=1}^{j-k-4}\\su_{m=1}^{j-k-\\el-3}\n \\su_{n=1}^{j\\! -\\! k\\! -\\!\\el\\! -\\! m\\! -\\! 2}\n\\Big( {4k\\el p\\ov j+1}\\,\n \\Xi^{k-1}\\Xi_{(1}\\Xi^{m-1}\\Xi_{1)}\\Xi^{p-1}\\Xi_{(1}\\Xi^{n-1}\\Xi_{1)}\\Xi^{\\el-1} \\non\\\\\n&&\\qquad\\qquad\n+ {12 k(k+m+1)\\el(\\el+n+1)\\ov (j+1)(j+2)}\\,\n \\Xi^{k-1}\\Xi_{(1}\\Xi^{m-1}\\Xi_1\\Xi^{p-1}\\Xi_1\\Xi^{n-1}\\Xi_{1)}\\Xi^{\\el-1} \\Big)\\non\\\\\n&&\n+\\ O(\\pa^6) \n\\ena\nWith the restriction of at most four derivatives, this result constitutes an \nexplicit solution for {\\it all} heat kernel coefficients in flat space without\na gauge connection. It is complete for $j\\leq 3$ and yields all terms but one \nin $\\sa_4$. To find the coefficient of the `missing' $\\Xi_{(6)}$ term in $\\sa_4$\nit is best to expand in powers of the matrix potential. This will be presented \nin the next subsection. Also note that the last two terms in (\\ref{deriv})\ndo not appear until $\\sa_6$. Taken together, (\\ref{deriv}) and (\\ref{Xres})\nyield the complete answer for the first six heat kernel coefficients. We refer\nto sect 6 for the explicit answers for the first five coefficients (to obtain \nthe flat space results, replace each $\\Z$ in (\\ref{a1-a5}) by an $\\Xi$ and omit\nall hats and daggers). \nThe result for $\\sa_6$ is presented in appendix C.\n\\subsection{Expansion in the potential}\nWe return to (\\ref{ajn}), now paying attention to the order in the matrix \npotential rather than in derivatives. Thus we are looking for an expansion \nthat starts as\n\\bee\n[a_j] \\= \\alpha\\pr_j (\\pa^2)^{j-1} X\\ +\\ O(X^2)\n\\ene\n$\\alpha\\pr_j$ being some $j$-dependent numerical factor.\nWe will show that the recursion relations yield an expression of the form\n\\bea\\label{Xexp}\n&&\\sa_j = \\alpha\\pr_j \\Xi_{(2j-2)}\\ \n+\\su_k\\su_\\el \\alpha\\pr_{jk\\el} \\Xi_{(2k-\\el}\\Xi_{\\el)(2j-2k-4)} \\non\\\\\n&&\n\\quad + \\su_k\\su_\\el\\su_m\\su_n\\su_p \\alpha\\pr_{jk\\el mnp}\n\\Xi^{\\phantom{p}}_{(2k-m}\\Xi_{m-n\\phantom{)}}^{(2\\el-p}\\Xi^{p)}_{n)(2j-2k-2\\el-6)}\n\\ +\\, O(\\Xi^4)\n\\ena\nwhere we used our shorthand notation (the exponents $2\\el -p$ and $p$ on the\nsecond and third factor of the last term also count derivatives).\nThis is not manifestly hermitian and verifying hermiticity \nthus provides a strong check on our results for the $\\alpha\\pr$ coefficients. \nFurthermore, at third order in $\\Xi$, (\\ref{Xexp}) shows overlapping sets of \nderivatives. The results of the previous subsection imply that it must\nbe possible to remove these overlapping terms. Below we show how this can be \ndone.\n\nFor convenience, we repeat our starting point, eq (\\ref{ajn}) \n\\bee\n\\sa_{j,n}\\={j\\ov j+n}\\,\\Big(\\,\\sa_{j-1,n,(2)}\n                            \\,+\\,(\\Xi\\,\\sa_{j-1})_{,n} \\,\\Big)\n\\ene\nThe first term on the right hand side, which is of zeroth order in $\\Xi$, can be\neliminated by substituting for it from the left hand side, i.e. replace \n$j\\ra j-1,\\,n\\ra n+2$ and contract one pair of derivatives\n\\bee\n\\sa_{j-1,n,(2)}\\= {j-1\\ov j+n+1}\\,\\Big(\\,\\sa_{j-2,n,(4)}\\,\n+\\,(\\Xi\\,\\sa_{j-2})_{,n,(2)} \\,\\Big)\n\\ene\nIterating this yields\n\\bee\n\\sa_{j,n}\\=\\su_{k=0}^{j-1}\\ (\\Xi\\sa_{j-k-1})_{,n,(2k)} \\,\n\\prod_{q=0}^k {j-q\\ov j+n+q}\\=\\su_{k=0}^{j-1}\\ {{j{\\cal H} k+1}\\ov {j+k+n{\\cal H} k+1}}\n\\ (\\Xi\\,\\sa_{j-k-1})_{,n,(2k)} \n\\ene\nClearly, each term is now at least of first order in $\\Xi$. We prefer to reverse\nthe order of summation and write this as\n\\bee\n\\sa_{j,n}\\=\\su_{k=0}^{j-1} C_{j\\,k}^n\\,(\\Xi\\sa_k)_{,n,(2j-2k-2)} \n\\ene\nwhere we defined the combinatorical coefficients $C$ by\n\\bee\\label{C}\nC_{j\\,k}^n \\= {{j{\\cal H} k}\\ov {2j-k+n-1{\\cal H} j-k}}\n\\ene\nNow separate off the $k=0$ term and distribute the derivatives over $\\Xi\\sa_k$ \nto obtain\n\\bee\n\\sa_{j,n}\\= C_{j\\,0}^n\\,\\Xi_{n,(2j-2)}\n+\\su_{k=1}^{j-1}\\su_{p=0}^{2j'+n} {2j'+n{\\cal H} p} C_{j\\,k}^n\\,\n\\Xi_{(2j'+{\\hat n}-p} \\sa_{k,p)} \n\\ene\nwith $j'\\equiv j-k-1$. The set of indices $\\mu_1\\dots\\mu_n$ is labeled ${\\hat n}$\nhere to indicate that the elements of this set are to be included in the \nindicated symmetrization, but they remain uncontracted. By substituting for \n$\\sa_{k,p}$ from the left hand side we find\n\\bea\n&&\\sa_{j,n} \\= C_{j\\,0}^n\\,\\Xi_{n,(2j-2)}\n      +\\su_{k=1}^{j-1}\\su_{p=0}^{2j'+n} {2j'+n{\\cal H} p}\n C_{j\\,k}^n C_{k\\,0}^p\\,\\Xi_{(2j'+{\\hat n}-p}\\Xi_{p)(2k-2)} \\non\\\\\n&&\\quad\n+\\su_{k=2}^{j-1}\\su_{\\el=1}^{k-1}\\su_{p=0}^{2j'+n} {2j'+n{\\cal H} p}\n C_{j\\,k}^n C_{k\\,\\el}^p\\,\\Xi_{(2j'+{\\hat n}-p} (\\Xi\\sa_\\el )_{p)(2k')}\n\\ena\nwith $k'\\equiv k-\\el-1$.\nThis shows explicitly the terms of second order in $\\Xi$. To proceed to third \norder, we take $n=0$ and distribute the derivatives over $\\Xi\\sa_\\el$. In \ngeneral\nrequires two binomial sums, one for each of the sets of derivatives marked $p$ \nand $2k'$, and leads to overlapping sets of derivatives as in (\\ref{Xexp}).\nTo avoid this and to allow us to lump the two sets of derivatives together, we\nuse that for any functions $F(x)$ and $G(x)$\n\\bea\\label{mobo}\n& F_{(1}\\,G_{1)(2)} &=  F_{(1}\\,G_{3)} \\non\\\\\n& F_{(2}\\,G_{2)(2)} &= \\sfrac56 F_{(2}\\,G_{4)} +\\sfrac16 F_{(2)}\\,G_{(4)}\n\\ena\netc. Choosing $F=X$, $G=Xa_1$ and taking $x=0$ , the first identity shows that\nwe can trivially lump the derivatives together in computing $\\sa_5$ and the \nsecond identity shows how to achieve the same for $\\sa_6$. In appendix B, we \nshow how to do this for arbitrary values of $j$. Restricting for simplicity \nhere to $j\\leq5$, we obtain as our final result\n\\bea\\label{Xres}\n&&\\sa_j \\= {1\\ov {2j-1{\\cal H} j}}\\,\\Xi_{(2j-2)} \\ \n  +\\su_{k=1}^{[j\/2]}\\su_{p=0}^{j-2k} {2j'{\\cal H} p}\n   C_{j\\,k}^0 C_{k\\,0}^p\\,\\{\\Xi_{(2j'-p}\\Xi_{p)(2k-2)}\\} \\non\\\\\n&&\\quad\n+\\su_{k=2}^{j-1} \\su_{\\el=1}^{k-1} \\su_{p=0}^{2j'}\n \\su_{q=0}^{2k'+p} {2j'{\\cal H} p} {2k'+p{\\cal H} q}\n C_{j\\,k}^0 C_{k\\,\\el}^p C_{\\el\\,0}^q\\,\n\\Xi_{(2j'-p}\\Xi_{2k'+p-q}\\Xi_{q)(2\\el-2)} \\non\\\\\n&&\\quad\n+ O(\\Xi^4) \n\\ena\nwith $j'\\equiv j-k-1$, $k'\\equiv k-\\el-1$ and the $C$-symbols were defined in \n(\\ref{C}). Note that we rewrote the second order terms in manifestly hermitian\nform, the range of the double sum having been correspondingly restricted (we \nuse $[n]$ to denote the integer part of $n$).\nExpression (\\ref{Xres}) is one of the main results of this paper.\n\\section{Flat space with gauge connection}\n\\setcounter{equation}{0}\nWe will show that the heat kernel coefficients in the presence of a gauge \nconnection can be obtained from their `trivial' counterparts without such a \nconnection by making simple covariant substitutions of the kind\n\\bee\n\\pa_{\\mu_1}\\dots\\pa_{\\mu_j} X\\ \\ra\\ \\de_{(\\mu_1}\\dots\\de_{\\mu_j)} X\\, +\\, \nF{\\rm -dependent\\ terms}\n\\ene\nHere partial derivatives are turned into totally symmetrized covariant \nderivatives and in general there are additional field strength dependent terms.\n\nDenote the nonabelian vector connection by $A$. The associated covariant \nderivative and field strength are defined by\n\\bee\n\\de_\\mu \\=\\pa_\\mu +A_\\mu \\quad ,\\quad F_{\\mu\\nu} \\= [\\de_\\mu,\\de_\\nu]\n\\ene\nWe take $\\de$ and thus $F$ to be antihermitian. The heat kernel coefficients \nsatisfy\n\\bee\n(x^\\mu\\de_\\mu + j)\\, a_j \\= j\\,(\\de^2 +X)\\, a_{j-1}\\quad ,\\quad [a_0] \\= I\n\\ene\nTo solve these recursion relations, we find it convenient to work in \nFock-Schwinger gauge\n\\bee\\label{FSgauge}\nx^\\mu A_\\mu (x)\\=0\n\\ene\nThis is equivalent to the requirement that all partial derivatives of the \ngauge connection vanish upon total symmetrization at the origin, i.e. for\n$j\\geq 1$\n\\bee\\label{symA}\nA_{(\\mu_1\\, ,\\ \\dots\\,\\mu_j)}(0) \\= 0 \n\\ene\nIn particular, the gauge field vanishes at the origin and for any $n\\geq 0$\n\\bee\\label{BOXndA}\n(\\pa^2)^n \\pa{\\cal D} A(0) \\= 0\n\\ene\nIn Fock-Schwinger gauge the recursion relations simplify to ({\\it cf} eq \n(\\ref{rec}))\n\\bee\n(x^\\mu\\pa_\\mu + j)\\, a_j \\= j\\,(\\pa^2 +\\hX)\\,a_{j-1}\\quad ,\\quad [a_0] \\= I\n\\ene\nwhere we defined the differential operator\n\\bee\n\\hX \\= X + A^\\mu\\sb{,\\mu} +2A^\\mu\\pa_\\mu + A^\\mu A_\\mu \n\\ene\nLet $Z$ be its non-differential-operator part, i.e.\n\\bee\\label{Zdef}\nZ \\= X + A^\\mu\\sb{,\\mu} + A^\\mu A_\\mu\n\\ene\nWe will soon need a covariant expression for the partial derivatives of $Z$\nat the origin. It is well known and easily verified that in Fock-Schwinger \ngauge the partial derivatives of the gauge field at this point have covariant\nvalues given for $j\\geq 1$ by\n\\bee\\label{Aj}\nA^\\nu\\sb{,\\mu_1\\dots\\mu_j}(0) \\=\n{j\\over j+1}\\,F_{(\\mu_1}\\sp\\nu\\sb{;\\ \\dots\\,\\mu_j)}(0)\n\\ene\nHere we use semicolon notation for Yang-Mills covariant derivatives.\nEq (\\ref{Aj}) implies\n\\bea\\label{dAjAAj}\nA^\\nu\\sb{,\\nu\\mu_1\\dots\\mu_j}(0)\n&=& {j+1\\over j+2}\\,F_{(\\nu}\\sp\\nu\\sb{;\\mu_1\\dots\\mu_j)}(0)\\quad ,\\quad j\\geq 1 \\\\\nA^2\\sb{,\\mu_1\\dots\\mu_j}(0)\n&=& \\su_{k=1}^{j-1}\\,{j{\\cal H} k}\\,{k(j-k)\\ov (k+1)(j-k+1)} \nF_{(\\mu_1}\\sp\\nu\\sb{;\\,\\dots\\mu_k}(0) F_{\\mu_{k+1}}\\sp\\nu\\sb{;\\,\\dots\\mu_j)}(0)\n\\ \\ \\ ,\\ \\ \\ j\\geq 2 \\non\n\\ena\nIf we now define new covariant {\\it sans serif\\\/} symbols $\\Y_j$\n\\bee\\label{defY}\n\\Y_j\\sp\\nu \\equiv {j\\over j+1}\\,F_{(\\mu_1}\\sp\\nu\\sb{;\\ \\dots\\mu_j)}(0)\n\\ene\nwhich are to be treated formally as vectors, then we can abbreviate \n(\\ref{Aj}, \\ref{dAjAAj}) as\n\\bea\nA^\\nu\\sb{,\\mu_1\\dots\\mu_j}(0) \\=\\Y_j\\sp\\nu \\quad &,&\\quad\nA^\\nu\\sb{,\\nu\\mu_1\\dots\\mu_j}(0)\\=\\Y_{(\\nu}\\sp\\nu\\sb{;\\, j)}\n\\quad ,\\quad j\\geq 1 \\non\\\\\nA^2\\sb{,\\mu_1\\dots\\mu_j}(0) &=&\\su_{k=1}^{j-1}\\,{j{\\cal H} k}\\,\\Y_{(k}\\Y_{j-k)}\n\\ \\ \\ ,\\ \\ \\ j\\geq 2 \n\\ena\nWe further note that at the origin, due to (\\ref{symA}), we can immediately \ncovariantize the partial derivatives of the matrix potential as follows\n\\bee\\label{Xj}\n\\Xi_j\\,\\equiv\\,X_{,\\mu_1\\dots\\mu_j}(0) \\= X_{;(\\mu_1\\dots\\mu_j)}(0)\n\\ene\nHere we used the same symbol $\\Xi_j$ as in sect 3 to mean now a totally\nsymmetrized $j$-fold covariant derivative of the matrix potential at the \norigin. Thus, the desired covariant expression for {\\it any} partial derivative\nof $Z$ at the origin is\n\\bee\\label{Zj}\n\\Z_j\\,\\equiv\\, Z_{,\\mu_1\\dots\\mu_j}(0) \\= \n\\Xi_j +\\Y_{(\\nu}\\sp\\nu\\sb{;\\, j)}\\, +\\su_{k=1}^{j-1} {j{\\cal H} k}\\,\\Y_{(k}\\Y_{j-k)}\n\\ene\nNo implicit contractions occur here except for the scalar product between\n$\\Y_k$ and $\\Y_{j-k}$. Replacing $j$ by $2j$ and contracting all indices yields\n\\bee\\label{Z2j}\n\\Z_{(2j)}\\=\\Xi_{(2j)}\\,+\\su_{k=1}^{2j-1} {2j{\\cal H} k}\\,\\Y_{(k}\\Y_{\\el)}\n         \\=\\Xi_{(2j)}\\,+\\su_{k=1}^j {2j{\\cal H} k}\\,\\{\\Y_{(k}\\Y_{\\el)}\\}\n\\ene\nwhere $\\el=2j-k$ is to be understood. In the second expression we used the \nnotation of (\\ref{brac}) to obtain a manifestly hermitian result. In sect. 5 \nwe determine the generalization of (\\ref{Zj}, \\ref{Z2j}) to a curved space.\nIn that case we do not find a closed expression which holds for all values \nof $j$.\n\\subsection{Expansion in derivatives}\nReturning to our short hand notation, we have \n\\bee\\label{ajng}\n\\sa_{j,n}\\={j\\ov j+n}\\,\\Big(\\,\\sa_{j-1,(2),n}\\,+\\,(\\HX\\,\\sa_{j-1})_{,n} \\,\\Big)\n\\ene\nSimilar to (\\ref{chain0}), we now have the following chain of substitutions\n(we again omit numerical factors and ordering labels for the heat kernel \ncoefficients, but this time we keep undifferentiated matrices $\\Xi$)\n\\bea\\label{chain1}\n\\sa         &\\ra& \\sa_{(2)}\\, +\\, \\Xi\\,\\sa \\non\\\\\n\\sa_{(2n)}  &\\ra& \\sa_{(2n+2)}\\, +\\, (\\HX\\,\\sa)_{(2n)}\\quad ,\\quad n\\geq 1\n\\ena\netc. In the first line we used that $\\hX$ at the origin equals $\\Xi$. To remove\n$\\hX$ from the second line of (\\ref{chain1}) as well, we use that\nat the origin\n\\bee\\label{lem0}\n(\\HX\\,\\sa)_{(2n)} \\=(\\Z\\dg\\sa)_{(2n)}\\quad ,\\quad n\\geq 0\n\\ene\nNote that the right hand side no longer contains a differential operator. To \nprove this, we use $\\hX = Z+2A^\\mu\\pa_\\mu$ and $Z\\dg = Z - 2A^\\mu\\sb{,\\mu}$ and \nnote that for any function $F$ one has (see appendix D for the proof)\n\\bee\\label{AF}\n(\\pa^2)^n \\pa_\\mu (A^\\mu F)(0)\\= 0 \\quad ,\\quad  n\\geq 0\n\\ene\nTaking for $F$ a heat kernel coefficient, (\\ref{lem0}) follows. Adding one \nmore step to the hierarchy (\\ref{chain1}), we obtain\n\\bea\\label{chain1n}\n\\sa        &\\ra& \\sa_{(2)}\\,+\\, \\Xi\\,\\sa \\non\\\\\n\\sa_{(2n)} &\\ra& \\sa_{(2n+2)}\\, +\\su_{p=0}^{2n} \\Z_{(p}\\do\\sa_{2n-p)}\n                      \\quad ,\\quad n\\geq 1  \\\\\n\\Z_{(p}\\do\\sa_{2n-p)} &\\ra& \\Z_{(p}\\do\\sa_{2n-p)(2)} \\,\n+\\,\\Z_{(p}\\do (\\HX\\,\\sa)_{2n-p)}  \\quad ,\\quad p=0\\dots 2n \\non\n\\ena\nExcept for $p=0$ and $p=2n$, (\\ref{lem0}) can not be used to eliminate $\\hX$ \nfrom the last term. Instead we proceed as follows (write $r$ for $2n-p$ and \nkeep binomial coefficients here)\n\\bea\\label{Zhi}\n\\Z_{(p}\\do(\\HX\\,\\sa)_{r)}\n&=&\\su_{q=0}^r {r{\\cal H} q} \\Z_{(p}\\do\\Big(\\Z_q\\sa_{r-q)}\\,+\\,\n        2\\A^\\nu\\sb{q}\\sa_{r-q)\\nu}\\Big) \\non\\\\\n&=&\\su_{q=0}^r {r{\\cal H} q} \\Z_{(p}\\do\\Z_q\\sa_{r-q)}\\ +\\su_{q\\pr=0}^{r-1} \n           {r{\\cal H} q\\pr+1} \\Z_{(p}\\do 2\\A^\\nu\\sb{q\\pr+1}\\sa_{r-q\\pr-1)\\nu} \\non\\\\\n&=&\\su_{q=0}^r {r{\\cal H} q} \\Z_{(p}\\do \\Big(\\Z_q\\sa_{r-q)} \\,+\\,\n                    {r-q\\ov q+1}\\,2\\A^\\nu\\sb{q+1}\\sa_{r-q-1)\\nu}\\Big) \\non\\\\\n&\\equiv &\\su_{q=0}^r {r{\\cal H} q} \\Z_{(p}\\do \\HZ_q\\sa_{r-q)} \n\\ena\nIn the second line we shifted the summation index\\footnote{\\mbox{The \n$q\\pr=-1$ term is absent because the gauge connection vanishes at the origin, \nsee (\\ref{symA})}.} $q$ so as to \ncollect terms with the same number of derivatives on the heat kernel \ncoefficient. In the third line we can use (\\ref{Aj}) to replace $\\A_{q+1}$ by \n$\\Y_{q+1}$ and thus obtain a covariant answer. The last line defines $\\HZ_q$ \nin terms of the previous line. The hatted $\\Z$ is designed to absorb the new \nfield strength dependent term (this term \nexists even for $q=0$ and we will write $\\HZ$ for $\\HZ_0$. In case $r=0$ too, \n$\\HZ_0=\\Xi$). With this definition of $\\HZ$ we can replace the last line of the\nhierarchy (\\ref{chain1n}) by\n\\bee\n\\Z_{(p}\\do\\sa_{2n-p)}\\ \\ra\\ \\Z_{(p}\\do\\sa_{2n-p)(2)} \\,\n+\\su_{q=0}^{2n-p} \\Z_{(p}\\do\\HZ_q\\sa_{2n-p-q)} \\quad ,\\quad p=0\\dots 2n\n\\ene\nIterating once more does not bring new features\\footnote{Other than those \nremarked upon in footnote 6.} and we arrive at the following conclusion.\\\\\n\\\\\n\\noindent\n{\\it The diagonal values of the heat kernel coefficients in the presence of a \ngauge connection are obtained from those without this gauge connection by\nmaking the following covariant substitutions in (\\ref{deriv})}\n\\bea\\label{Subs}\n\\Xi_{(2j)}              &\\ra& \\Z_{(2j)} \\non\\\\\n\\Xi_{(j}\\Xi_{k)}         &\\ra& \\Z_{(j}\\do\\Z_{k)} \\non\\\\\n\\Xi_{(j}\\Xi_n\\Xi_{k)}     &\\ra& \\Z_{(j}\\do\\HZ_n\\Z_{k)} \\\\\n\\Xi_{(j}\\Xi_p\\Xi_n\\Xi_{k)} &\\ra& \\Z_{(j}\\do\\HZ_p\\HZ_n\\Z_{k)} \\non\n\\ena\n{\\it etc, where $j,k\\geq 1$, $n,p,\\dots\\geq 0$, $\\Z_j$ is defined in (\\ref{Zj})\nand the $\\HZ$ act to their right as follows}\n\\bea\\label{Zhatdef}\n\\HZ_n\\Z_{k)} &=&\\Z_n\\Z_{k)}\\, +\\,{2\\ov n+1}\\, k\\,\\Y_{n+1}\\sp\\nu\\Z_{k-1)\\nu}\\non\\\\\n\\HZ_p\\HZ_n\\Z_{k)} &=& \\Z_p\\HZ_n\\Z_{k)}\\, +\\,{2\\ov p+1}\\,\\Y_{p+1}\\sp\\nu\n            (n\\HZ_{n-1}\\sp\\nu \\Z_{k)} + k\\HZ_n\\Z_{k-1)\\nu})\n\\ena\n\\\\\nSince they are unaffected, we omitted $\\Z_{(j}\\do\\dots$ from the left of each \nterm in (\\ref{Zhatdef}), the dots representing possible further $\\HZ$. Note \nthat $\\HZ_n$ acts on each one of the $k$ indices on $\\Z_k$ in the same way, \nhence the factor $k$. Similarly $\\HZ_p$ acts on $n$ plus $k$ indices, etc. \nThus, upon `turning on the gauge field', (\\ref{deriv}) becomes\n\\bee\\label{gauged}\n\\sa_j \\= \\Xi^j\\,\n+\\su_{k=1}^{j-1} {k\\el\\ov j+1}\\,\\Xi^{k-1}\\Z_{(2)}\\Xi^{\\el-1} \n+\\su_{k=1}^{j-2}\\su_{\\el=1}^{j-k-1} {2k\\el\\ov j+1}\\,\n\\Xi^{k-1}\\Z_{(1}\\do\\HZ^{m-1}\\Z_{1)}\\Xi^{\\el-1}\n+ \\dots\n\\ene\nNote that only an $\\Xi$ which is `sandwiched' between $\\Xi_{(j}$ and $\\Xi_{k)}$ \ngets replaced by a $\\HZ$. Formally, the substitutions in (\\ref{Subs}) do not \nchange the number of terms or their numerical coefficients. In this sense the \nheat kernel coefficients retain their original appearance in the gauging \nprocess. If we use the curly bracket notation of (\\ref{brac}), we should take \nnote that for $j\\neq k$\n\\bee\n\\{\\Z_{(j}\\do\\HZ_n\\Z_{k)}\\} \\= \\Z_{(j}\\do\\HZ_n\\Z_{k)} +\\Z_{(k}\\do\\HZ_n\\Z_{j)}\n\\ene\nwith no dagger on $\\HZ_n$ in the second term (this can be shown to be \nhermitian). Finally, repeating the steps of (\\ref{ex0}), we find that the \ncovariant analogues of (\\ref{overlap}) are absent.\n\\subsection{Expansion in the potential}\nUpon replacing $X$ in sect. 3.2 by the differential operator $\\hX$ and\nretracing our steps, we find that the heat kernel coefficients for the case \nwith a gauge connection are obtained from those in (\\ref{Xres}) through the \nsame substitutions as in (\\ref{Subs}). Thus\n\\bea\\label{Zres}\n&&\\sa_j \\= {1\\ov {2j-1{\\cal H} j}}\\,\\Z_{(2j-2)} \\ \n  +\\su_{k=1}^{[j\/2]}\\su_{p=0}^{j-2k} {2j'{\\cal H} p}\n   C_{j\\,k}^0 C_{k\\,0}^p\\,\\{\\Z_{(2j'-p}\\do\\Z_{p)(2k-2)}\\} \\non\\\\\n&&\\quad\n+\\su_{k=2}^{j-1} \\su_{\\el=1}^{k-1} \\su_{p=0}^{2j'}\n \\su_{q=0}^{2k'+p} {2j'{\\cal H} p} {2k'+p{\\cal H} q}\n C_{j\\,k}^0 C_{k\\,\\el}^p C_{\\el\\,0}^q\\,\n\\Z_{(2j'-p}\\do\\HZ_{2k'+p-q}\\Z_{q)(2\\el-2)} \\non\\\\\n&&\\quad\n+ O(\\Z^4) \n\\ena\nwith $j'\\equiv j-k-1$, $k'\\equiv k-\\el-1$ and the $C$-symbols were defined in \n(\\ref{C}). The action of $\\HZ_n$ was defined in (\\ref{Zhatdef}) (it does not \nact on the set of indices labeled $(2\\el-2)$). \n\\section{Curved space}\n\\setcounter{equation}{0}\nIn this section we shall generalize our results to a curved space with metric \n$g_{\\mu\\nu}$. Riemann normal coordinates $x^\\mu$ can be defined by\\footnote{\nWe use the same symbol for normal coordinates as for general coordinates.\nThis should not cause confusion. In these coordinates and at the origin there \nis no need to distinguish co- from contravariant indices. We exploit this to\nposition indices in such a way as to cause minimal clutter. Normal coordinates\n$x^\\mu$ may alternatively be defined in terms of the affine connection by \n$x^\\mu x^\\nu \\Gamma_{\\mu\\nu}^\\lambda (x) = 0$ (which is equivalent to \n$\\Gamma_{(\\mu_1\\mu_2 ,\\,\\dots\\mu_j)}^{\\ \\lambda}(0) = 0$ for $j\\geq 2$).\nNote the similarity with the Fock-Schwinger gauge (\\ref{FSgauge}).}\n\\bee\\label{Rnc}\ng_{\\mu\\nu}(0) \\= \\delta_{\\mu\\nu}\\quad ,\\quad x^\\mu g_{\\mu\\nu}(x) \\= x^\\mu g_{\\mu\\nu}(0) \n\\ene\nwhich is equivalent to\n\\bee\\label{gsym0}\n\\quad g_{\\mu\\nu}(0) \\= \\delta_{\\mu\\nu}\\quad ,\\quad\ng_{\\mu (\\mu_1 ,\\mu_2\\dots\\mu_j)}(0) \\= 0 \\quad ,\\quad j\\geq 2\n\\ene\nThese properties also hold for the inverse metric $g^{\\mu\\nu}$.\nTaking the origin of our normal coordinate system to coincide with the point\n$x\\pr$, we have\n\\bee\\label{signc}\n\\sigma(x,0)\\= \\sfrac12\\, x^2 \\= \\sfrac12\\, x^\\mu x^\\nu\\delta_{\\mu\\nu}\\quad ,\\quad\n{\\cal D}(x,0) \\= g(x)^{-1\/2}\n\\ene\nand DeWitt's ansatz (\\ref{hkc}) for the heat kernel becomes\n\\bee\\label{hkn}\nK(x,0;\\tau) \\= (4\\pi\\tau)^{-d\/2} g(x)^{-1\/4}\\, e^{-x^2\/4\\tau}\n\\su_{j=0}^\\infty  a_j(x,0) \\,{\\tau^j\\ov j!}\n\\ene\nUsing (\\ref{signc}), the recursion relations (\\ref{ajd}) become\n\\bee\\label{curvrec}\n(x^\\mu\\pa_\\mu + j)\\, a_j \\= j\\,(\\pa_\\mu g^{\\mu\\nu}\\pa_\\nu + 2A^\\mu\\pa_\\mu + Z)\\,a_{j-1}\n\\quad ,\\quad [a_0]\\= I\n\\ene\nHere\n\\bee\nZ\\,\\equiv\\, \\ZM + \\ZS \\quad ,\\quad \n\\ZM\\,\\equiv\\, X + A^\\mu\\sb{,\\mu} + A^\\mu A_\\mu\\quad ,\\quad\n\\ZS\\,\\equiv\\,\\sfrac12 B_,\\sp\\mu\\sb\\mu -\\sfrac14 B_,\\sp\\mu B_{,\\mu}\n\\ene\nwhere we defined\n\\bee\nB \\,\\equiv\\, \\ln\\,{\\cal D} \\= -\\,\\ln\\,\\sqrt{g}\n\\ene\nNote the position of the explicit inverse metric in (\\ref{curvrec}). The \nquantities $\\ZM$ and $\\ZS$ are the matrix and scalar parts of $Z$ respectively.\nAlso note the similarity between $\\ZM-X$ and $\\ZS$. In the flat space limit \n$\\ZS$ vanishes and $\\ZM$ reduces to the quantity earlier defined as $Z$ in \n(\\ref{Zdef}). We now reserve the symbol $Z$ to mean the sum of $\\ZM$ and $\\ZS$.\n\\subsection{Scalar part}\nWe write $\\ZS$ with explicit inverse metric as\n\\bee\\label{Zscalar}\n\\ZS\\=\\sfrac12\\,(g^{\\mu\\nu} B_{,\\nu} )_{,\\mu}\\,-\\,\\sfrac14\\,B_{,\\mu} g^{\\mu\\nu} B_{,\\nu}\n\\ene\nOur task is to find a covariant expression for the partial derivatives of $\\ZS$\nat the origin of the normal coordinate system. This in turn requires the\nexpansion for the (inverse) metric in normal coordinates, which is a \nwell-known problem. We find it easier to work with the vielbein, the metric \nbeing defined as usual by\n\\bee\ng_{\\mu\\nu} \\= e_\\mu\\sp{a} e_\\nu\\sp{b}\\,\\delta_{ab}\n\\ene\nwhere $a$ and $b$ are tangent-space indices.\nA recursive formula for the vielbein in normal coordinates was given in \n\\cite{ABC}. Here we shall give its solution to all orders in the curvature.\nUsing matrix notation for the vielbein, i.e.\n\\bee\n(\\E_j)_\\mu\\sp{a} \\,\\equiv\\, e_\\mu\\sp{a}\\sb{,\\mu_1\\dots\\mu_j}(0)\n\\ene\nand defining the following {\\it sans serif} curvature symbols\\footnote{Except\nfor the normalization factor, this agrees with the definition in \\cite{Avr}.\nSee (\\ref{Rcon}) for our curvature conventions.}\n(compare with (\\ref{defY}))\n\\bee\\label{defK}\n\\K_j\\sp{\\mu\\nu}\\={j-1\\over j+1}\\,\nR_{(\\mu_1}\\sp\\mu\\sb{\\mu_2}\\sp\\nu\\sb{;\\dots\\mu_j)}(0)\\quad ,\\quad j\\geq 2\\quad . \n\\ene\nwe can write the recursive formula of \\cite{ABC} for the vielbein as follows\n\\bee\\label{recE}\n\\E_0\\=\\I\\quad,\\quad \\E_1\\= 0 \\quad ,\\quad\n\\E_j\\= -\\,\\K_j\\, -\\,\\su_{k=2}^{j-2} {j{\\cal H} k}\\,{k(k+1)\\ov j(j+1)}\\,\n\\K_k\\E_\\el\\ \\ , \\ \\ j\\geq 2\n\\ene\nwhere $\\I$ is the $d\\times d$ unit matrix and total symmetrization on the \n$k+\\el\\equiv j$ indices is implied. We treat $\\K_j$ as a symmetric matrix in \nthe index pair $\\mu\\nu$. Taking the trace of such a matrix yields\n\\bee\\label{trKj}\n{\\rm tr}[\\K_j] \\= {j-1\\over j+1}\\,\nR_{(\\mu_1\\mu_2;\\dots\\mu_j)}(0)\\quad ,\\quad j\\geq 2 \n\\ene\nwhere the Ricci tensor and its covariant derivatives appear. Note that we \ntraced over the world indices. This is not to be confused with the trace over \nfield or `bundle' indices which does not occur in this paper. In taking the \ntrace over a product of $\\K$-matrices, say ${\\rm tr}[\\K_j\\K_k\\K_\\el]$, total\nsymmetrization on the $j+k+\\el$ indices is implied. Such traces appear in our \nfinal expression for the heat kernel coefficients in curved space, e.g.\n\\bee\\quad{\\rm tr}[\\K_{(2}\\K_{2)}] \\= \n\\sfrac19\\,R_{(\\kappa}\\sp\\mu\\sb\\kappa\\sp\\nu(0)\\, R_\\lambda\\sp\\nu\\sb{\\lambda)}\\sp\\mu(0)\\=\n\\sfrac1{27} R^{\\kappa\\lambda} R_{\\kappa\\lambda} +\\sfrac1{18} R^{\\kappa\\lambda\\mu\\nu} R_{\\kappa\\lambda\\mu\\nu}\n\\ene\nwhere the parenthesis on the left hand side imply symmetrization and\ncontraction (see (\\ref{combi}) for the general case). In this way we \nmaintain an index-free notation. Returning now to the recursion formula \n(\\ref{recE}), the first few cases are\n\\bee\n\\E_2 = -\\K_2\\quad,\\quad\n\\E_3 = -\\K_3\\quad,\\quad\n\\E_4 = -\\K_4 +\\sfrac95 \\K_{(2}\\K_{2)} \\quad ,\\quad\n\\E_5 = -\\K_5 + 2\\K_{(2}\\K_{3)} + 4\\K_{(3}\\K_{2)} \n\\ene\nwhere the parenthesis mean symmetrization only.\nNote that for $j\\geq 5$ the $\\E_j$ are not symmetric matrices.\nBy iteration of (\\ref{recE}) we obtain as solution to all orders\n\\bea\\label{solE}\n\\E_j &=& -\\,\\K_j\\, -\\su_{n=1}^{[j\/2]-1} (-1)^n\n\\su_{k_1=2}^{j-2n}\\,\\su_{k_2=2}^{j_2-2n+2}\\!\\!...\\!\n\\su_{k_n=2}^{j_n-2} {j{\\cal H} k_1,..,k_n} \\Big(\n \\prod_{i=1}^n {k_i(k_i+1)\\ov j_i(j_i+1)} \\K_{k_i}\\Big) \\K_{j_{n+1}}\\non\\\\\nj_1\\! &\\equiv &\\! j\\quad ,\\quad \nj_i\\equiv j -\\su_{\\el=1}^{i-1} k_\\el \\quad ,\\quad i\\geq 2\n\\ena\nwhere total symmetrization of the $j$ indices is to be understood. \nTo the best of our knowledge, such an explicit expression for the vielbein in \nnormal coordinates was not available in the literature up to now\\footnote{We \nhave been informed by C. Schubert and U. M\\\"uller that they have obtained \nresults equivalent to (\\ref{solE}). Details will be published elsewhere \n\\cite{MSV}.}.\nFor the inverse metric it follows that\n\\bee\\label{invmet}\n\\sg\\sp{\\mu\\nu}\\sb{j} \\= 2\\,\\K_j\\sp{\\mu\\nu}\n+\\,2\\su_{k=2}^{[j\/2]} {j{\\cal H} k}\\,\n\\alpha_{k\\el}\\,\\{\\K_k\\K_\\el\\}\\sp{\\mu\\nu}\\ +\\, O(\\K^3) \\quad ,\\quad\n\\alpha_{k\\el} \\,\\equiv\\, 1\\,+\\,{k\\el\\ov j(j+1)}\n\\ene\nwhere $\\el\\equiv j-k$ (see (\\ref{alphaord}) for our summation conventions). \nFinding the heat kernel coefficients through fourth order in $\\K$ turns out \nto require knowledge of the inverse metric through second order only. See \nappendix E.\nExplicit expansions for the metric and its inverse to some finite order in\nnormal coordinates are well known, see e.g. \\cite{Sak,McL} and our eq \n(\\ref{invmetexpl}). We next use\n\\bee\nB(x) =\\, -{\\rm tr}\\ln E\\= -\\su_{k=1}^\\infty {1\\ov k}{\\rm tr}[(I-E)^k]\n\\ene\nThus $B$ and its first derivative vanish at the origin and for $j\\geq 2$, after\ncollecting terms of the same degree in $\\K$, we find ({\\it cf} (\\ref{Bexpl}))\n\\bea\\label{Bj}\n\\B_j &=& {\\rm tr}[\\K_j]\\,\n+\\su_{k=2}^{[j\/2]} {j{\\cal H} k} \\beta_{k\\el} \\tt[\\K_k\\K_\\el]\\ \n+\\su_{k=2}^{[j\/3]}\\su_{\\el=k}^{[(j-k)\/2]} {j{\\cal H} k,\\el}\n \\beta_{k\\el m}\\tt[\\K_k\\K_\\el\\K_m] \\non\\\\\n&&\\quad\n+\\su_{k=2}^{[j\/4]}\\su_{\\el=k\\pr\\pr}^{[j\/2]-k}\\su_{m=k}^{j-k-2\\el} \n{j{\\cal H} k,\\el,m} \\beta_{k\\el mn}\\tt[\\K_k\\K_\\el\\K_m\\K_n]\\ \n+\\ O(\\K^5) \n\\ena\nwhere $\\tt$ is a weighted trace defined below and the lower limit $k\\pr\\pr$ \nmeans: when $\\el=k$, reduce the upper limit for the sum over $m$ to $[j\/2]-k$.\nThe coefficients are given by\n\\bea\\label{betas}\n\\beta_{k\\el}   &=& 1\\,-\\,P_2\\Big[{k(k+1)\\ov j(j+1)}\\Big] \\={2k\\el\\ov j(j+1)}\\non\\\\\n\\beta_{k\\el m} &=& 2\\,-\\,P_3\\Big[{k(k+1)(j+\\el+1)\\ov j(j+1)(j-\\el+1)}\\Big] \\=\n{4k\\el m (j^2+4j+3+k\\el+\\el m+mk)\\ov j(j+1) D_{k\\el} D_{\\el m} D_{mk}} \\non\\\\\n\\beta_{k\\el mn}&=& 2\\,-\\,P_4\\Big[{k(k+1)\\ov (k+\\el) D_{k\\el}}\n\\Big(1-{n(n+1)(j+m+1)\\ov j(j+1)D_{k\\el n}}-{m(m+1)\\ov 2(m+n)D_{mn}}\\Big)\\Big] \n\\non\\\\\n&&\\quad D_{k\\el} \\equiv k+\\el+1 \\quad ,\\quad D_{k\\el m} \\equiv k+\\el+m+1\n\\ena\nHere $P_N$ denotes the group consisting of all cyclic and anticyclic \npermutations on $N$ objects (so $\\abs{P_2} = 2,\\,\\abs{P_N} =2N$ for \n$N>2$) and its action is defined by\n\\bee\nP_N[f(k_1,\\dots,k_N)]\\=\\su_\\pi  f(k_{\\pi(1)},\\dots,k_{\\pi(N)})\n\\quad ,\\quad j\\equiv\\su_{i=1}^N k_i\n\\ene\nwhere the sum runs over all cyclic and anticyclic permutations. We used here \nthat the trace of a product of symmetric matrices is invariant under such \npermutations in order to simplify the sums. Thus, in ${\\rm tr}[\\K_k\\K_\\el\\K_m]$ the\norder of the matrices is irrelevant and we can assume without loss of \ngenerality that $k\\leq\\el\\leq m$. Similarly, in ${\\rm tr}[\\K_k\\K_\\el\\K_m\\K_n]$ we \ncan assume $k$ to be the smallest label and $\\el\\leq n$ (when $\\el=k$ we \narrange $m\\leq n$). Ordering subtleties do not occur before fourth order, \nwhere we must distinguish\\footnote{Thus in obtaining the logarithm of the Van \nVleck-Morette determinant through ninth order in the normal coordinates, see \n(\\ref{Bexpl}), we may treat the $\\K_j$ as commuting objects !}\nbetween ${\\rm tr}[\\K_2\\K_2\\K_3\\K_3]$ and ${\\rm tr}[\\K_2\\K_3\\K_2\\K_3]$. \nThe weighted trace $\\tt$ is defined by\n\\bee\\label{symmfac}\n\\tt[\\K_{k_1}\\K_{k_2}\\dots\\K_{k_N}]\\= \n    {S\\ov \\abs{P_N}}\\,{\\rm tr}[\\K_{k_1}\\K_{k_2}\\dots\\K_{k_N}]\n\\ene\nwith $S$ the total number of distinguishable cyclic and anticyclic\npermutations (including the identity) of the product \n$\\K_{k_1}\\K_{k_2}\\dots\\K_{k_N}$. We refer to appendix F for a table\ncontaining all symmetry factors for $N\\leq 4$.\nSubstituting (\\ref{Bj}) and (\\ref{invmet}) into (\\ref{Zscalar}) we find\nfor any $j\\geq 0$ but with a cutoff at third order in $\\K$\n(see appendix E for the details and (\\ref{Zs1-5}) for examples)\n\\bea\\label{ZSj}\n\\Zs_j &=& \\sfrac12{\\rm tr}[\\K_{j\\,\\mu\\m}]\\ \n+\\su_{k=2}^j {j+1{\\cal H} k} \\K_{(k}\\sp{\\mu\\nu}{\\rm tr}[\\K_{j-k\\,\\mu)\\nu}]  \\non\\\\\n&& \n+\\su_{k=1}^{[j\/2]\\pr} {j{\\cal H} k}\n\\Big( {j+1\\ov j+3}{\\rm tr}[\\K_{\\mu k} \\K_{\\mu\\el}]\n\\, -\\,\\sfrac12{\\rm tr}[\\K_{\\mu(k}] {\\rm tr}[\\K_{\\el)\\mu}] \\Big) \\non\\\\\n&&\n+\\su_{k=2}^{[{j+2\\ov 3}]}\\su_{\\el=k}^{[{j-k+2\\ov 2}]} {j+2{\\cal H} k,\\el}\\,\n\\sfrac12\\beta_{k\\el m}\\tt[\\K_{\\mu k-1}\\K_m\\K_{\\mu\\,\\el-1}] \\non\\\\\n&&\n-\\su_{k=1}^{j-3}\\su_{\\el=2}^{[{j-k+1\\ov 2}]} {j{\\cal H} k,\\el} {\\el\\ov j-k+2}\n{\\rm tr}[\\K_{\\mu(k}] \\tt[\\K_\\el \\K_{m)\\mu}] \\non\\\\\n&&\n-\\su_{k=1}^{[j\/2]-1}\\su_{\\el=k\\pr}^{j-k-2} {j{\\cal H} k,\\el} \n{\\rm tr}[\\K_{\\mu(k}]\\,\\K_m\\sp{\\mu\\nu}{\\rm tr}[\\K_{\\el)\\nu}]  \\non\\\\\n&&\n+\\su_{k=4}^j {j+1{\\cal H} k} \\su_{\\el=2}^{[k\/2]} {k{\\cal H}\\el}\n\\Big( 1 + {\\el (k-\\el)\\ov k(k+1)}\\Big) \\{ \\K_{(\\el}\\K_{k-\\el}\\}^{\\mu\\nu}\n{\\rm tr}[\\K_{j-k\\,\\mu)\\nu}] \\non\\\\\n&&\n+\\su_{k=2}^{j-2} \\su_{\\el=2}^{[(j-k+2)\/2]} {j+1{\\cal H} k,\\el}\n{2\\el\\ov j-k+3} \\K_{(k}\\sp{\\mu\\nu}\\tt[\\K_\\el\\K_{j-k-\\el\\,\\mu)\\nu}] \\non\\\\\n&&\n+\\ O(\\K^4) \n\\ena\nThis expression is complete for $j\\leq 5$. The coefficient $\\beta_{k\\el m}$ which\nappears here is obtained by replacing $j$ by $j+2$ in (\\ref{betas}) and \n$\\{\\K_k\\K_\\el\\}$ is defined as in (\\ref{brac}). The prime on $[j\/2]$ in the \nsecond sum implies division by 2 when $j$ is even and this limit is reached. \nSimilarly, the lower limit $\\el=k\\pr$ means division by 2 when $\\el=k$. \nWe recall that total symmetrization on {\\it all} indices within a given trace \n(except for those being traced over) is to be understood and thus one has e.g.\n\\bee\n{\\rm tr}[\\K_{\\mu(1}] {\\rm tr}[\\K_3\\K_{1)\\mu}] \\= {\\rm tr}[\\K_{\\mu(1}] {\\rm tr}[\\K_2\\K_{2)\\mu}] \n\\ene\nReplacing $j$ by $2j$ in (\\ref{ZSj}) and contracting fully, the second and last\ntwo terms vanish. Keeping also terms of fourth order in $\\K$ we find ({\\it cf}\n(\\ref{Zs(2)-(8)}))\n\\bea\\label{Zs2j}\n\\Zs_{(2j)} &=& \\sfrac12{\\rm tr}[\\K_{(2j+2)}] \\non\\\\\n&&\n+\\su_{k=1}^{j\\pr} {2j{\\cal H} k}\\,\\Big( {2j+1\\ov 2j+3}{\\rm tr}[\\K_{(k+1}\\K_{\\el+1)}]\n                   \\, -\\,\\sfrac12{\\rm tr}[\\K_{\\mu(k}] {\\rm tr}[\\K_{\\el)\\mu}] \\Big)\\non\\\\\n&&\n+\\su_{k=2}^{[{2j+2\\ov 3}]}\\su_{\\el=k}^{[{2j-k+2\\ov 2}]} {2j+2{\\cal H} k,\\el}\\,\n\\sfrac12\\,\\beta_{k\\el m}\\tt[\\K_{(k}\\K_\\el\\K_{m)}] \\non\\\\\n&&\n-\\su_{k=1}^{2j-3}\\su_{\\el=1}^{[{2j-k-1\\ov 2}]} {2j{\\cal H} k,\\el}\n\\gamma_{\\el m} {\\rm tr}[\\K_{\\mu(k}]\\tt[\\K_m\\K_{\\el)\\mu}]  \\non\\\\\n&&\n-\\,\\su_{k=1}^{j-1}\\,\\su_{\\el=k\\pr}^{2j-k-2}\\, {2j{\\cal H} k,\\el} \n{\\rm tr}[\\K_{\\mu(k}]\\,\\K_m\\sp{\\mu\\nu}{\\rm tr}[\\K_{\\el)\\nu}] \\non\\\\\n&&\n+\\su_{k=2}^{[{j+1\\ov 2}]} \\su_{\\el=k\\pr\\pr}^{j-k+1} \\su_{m=k}^{2j-k-2\\el+2} \n {2j+2{\\cal H} k,\\el,m}\\,\\sfrac12\\,\\beta_{k\\el mn} \\tt[\\K_{(k}\\K_\\el\\K_m\\K_{n)}]\\non\\\\\n&&\n-\\su_{k=1}^{2j-5} \\su_{\\el=1}^{[{2j-k-2\\ov 3}]} \n \\su_{m=\\el+1}^{[{2j-k-\\el\\ov 2}]} {2j{\\cal H} k,\\el,m} \\gamma_{\\el mn}\n{\\rm tr}[\\K_{\\mu(k}] \\tt[\\K_n\\K_m\\K_{\\el)\\mu}] \\non\\\\\n&&\n-\\su_{k=1}^{[{j-1\\ov 2}]} \\su_{\\el=k+1}^{(j-k)\\pr}\n \\su_{m=1}^{[{2j-k-\\el-1\\ov 2}]}  {2j{\\cal H} k,\\el,m} 2\\gamma_{k\\el}\\gamma_{mn}\n\\tt[\\K_{\\mu(k}\\K_\\el] \\tt[\\K_n\\K_{m)\\mu}] \\non\\\\\n&&\n-\\su_{k=1}^{j-2}\\,\\su_{\\el=k\\pr}^{2j-k-4} \\su_{m=2}^{[{2j-k-\\el\\ov 2}]}\\ \\,\n{2j{\\cal H} k,\\el,m}\\alpha_{mn}{\\rm tr}[\\K_{\\mu(k}]\\,\\{\\K_m\\K_n\\}^{\\mu\\nu}{\\rm tr}[\\K_{\\el)\\nu}]\\non\\\\\n&&\n-\\su_{k=1}^{2j-5} \\su_{\\el=1}^{[{2j-k-3\\ov 2}]} \\su_{m=\\el+1}^{2j-k-\\el-2}\n {2j{\\cal H} k,\\el,m} 2\\gamma_{\\el m}\n{\\rm tr}[\\K_{\\mu(k}]\\,\\K_n\\sp{\\mu\\nu} \\tt[\\K_m\\K_{\\el)\\nu}] \\non\\\\\n&&\n+ \\ O(\\K^5)\n\\ena\nThis expression is complete for $j\\leq 3$. The $\\alpha_{mn}$ are defined in \n(\\ref{invmet}) (with $j\\ra m+n\\equiv 2j-k-\\el$) and the $\\beta$-coefficients are \nthose of (\\ref{betas}) with $j\\ra 2j+2$. The $\\gamma$-coefficients are defined by\n\\bee\n\\gamma_{k\\el} \\,\\equiv\\,  {\\el\\ov k+\\el+2} \\quad ,\\quad\n\\gamma_{k\\el m}\\,\\equiv\\, {k+\\el+m+1\\ov 2(k+1)}\\,\\beta_{k+1\\,\\el\\, m}\n\\ene\nThe upper limit $j\\pr$ means: divide by 2 when $k=j$. A lower limit $k\\pr$ \nmeans: divide by 2 when $\\el=k$. The upper limit $(j-k)\\pr$ means: divide by 2\nwhen $\\el=j-k$ and $m=k$. Finally, the lower limit $k\\pr\\pr$ means: when\n$\\el=k$, reduce the upper limit for the sum over $m$ to $j-k+1$.\n\\subsection{Matrix part}\nAbbreviate the $j$-th partial derivative of the gauge connection at the origin\nby\n\\bee\n\\A_j \\= A_{\\mu,\\mu_1\\dots\\mu_j}(0)\n\\ene\nThen the recursion for the gauge connection is given by (until further notice, \nparenthesis imply symmetrization only)\n\\bee\n\\A_0 \\= 0\\quad ,\\quad \n\\A_j \\=\\Y_j +\\su_{k=2}^{j-1} {j{\\cal H} k} \\,{\\el+1\\ov j+1}\\,\\E_{(k}\\Y_{\\el)}\n\\ \\ ,\\ \\ j\\geq 1\n\\ene\nWe iterate this and use (\\ref{solE}) to obtain as solution to all orders\n\\bea\\label{solA}\n\\A_j\\!\\! &=&\\!\\!\\Y_j +\\su_{n=1}^{[{j-1\\ov 2}]} (-1)^n \\su_{k=1}^{j-2n}\n\\su_{k_1=2}^{j_1-2n+2}\\!...\\!\\!\n\\su_{k_{n-1}=2}^{j_{n-1}-2} {j{\\cal H} k,..,k_{n-1}} {k+1\\ov j+1} \\Y_k \n\\Big(\\prod_{i=1}^{n-1} {k_i(k_i+1)\\ov j_i(j_i+1)} \\K_{k_i}\\Big)\\K_{j_n} \\non\\\\\nj_i &\\equiv & j-\\su_{\\el=0}^{i-1} k_\\el\\quad ,\\quad k_0\\,\\equiv\\, k \n\\ena\nwhere symmetrization on the $j$ indices is to be understood. We used that, \nsince the $\\K_j$ are symmetric matrices, we may write them to the right of the\nvector $\\Y_{k_0}$ in the reverse order. Through second order in $\\K$, \n(\\ref{solA}) reads ({\\it cf} (\\ref{Aexpl}))\n\\bea\\label{expA}\n\\A_j &=&\\Y_j\\ -\\su_{k=1}^{j-2}{j{\\cal H} k} {k+1\\ov j+1}\\,\\Y_{(k}\\K_{\\el)} \\\\\n&&\\quad\n+\\su_{k=1}^{j-4}\\su_{\\el=2}^{j-k-2} {j{\\cal H} k,\\el} \n{(k+1)\\el(\\el+1)\\ov (j+1)(j-k)(j-k+1)}\\,\\Y_{(k}\\K_\\el\\K_{m)}\\ +\\,O(\\K^3)\\non\n\\ena\nWe thus find (this is exact for $j\\leq 5$; see (\\ref{Zm1-5}) for examples)\n\\bea\\label{Zmj}\n&&\n\\Zm_j = \\Xi_j\\, +\\Y_{(\\nu}\\sp\\nu\\sb{;j)}\\, +\\su_{k=1}^{j-1} \\Y_{(k}\\Y_{\\el)}\\ \n+\\su_{k=2}^j{j+1{\\cal H} k}{j+k+2\\ov j+2}\\,\\K_{(k}\\sp{\\mu\\nu}\\Y_\\mu\\sp\\nu\\sb{;j-k)}\n\\non\\\\\n&&\n+ \\su_{k=1}^{[j\/2]-1}\\su_{\\el=k}^{j-k-2} {j{\\cal H} k,\\el}\\,\n\\big({m\\ov k+m+1} + {m\\ov \\el+m+1}\\big)\\, \\{\\Y_{(k}\\K_m\\Y_{\\el)}\\}\\non \\\\\n&&\n+\\su_{k=2}^{j-2}\\su_{\\el=2}^{j-k} {j+1{\\cal H} k,\\el}\\,\\Big({3\\el+m+1\\ov\\el+m+1}\n\\,-\\,{\\el(\\el+1)\\ov (j+2)(k+\\el+1)}\\,-\\,{k(k+1)\\ov (k+\\el)(k+\\el+1)}\\Big)\\non\\\\\n&&\\qquad\\qquad\\qquad\\qquad \n\\times(\\K_{(k}\\K_\\el)\\sp{\\mu\\nu} \\Y_\\mu\\sp\\nu\\sb{;j-k-\\el)}\\ +\\ O(\\K^3\\Y,\\K^2\\Y^2)\n\\ena\nIf we now replace $j$ by $2j$ in (\\ref{Zmj}) and contract all indices, the\nsecond, fourth and last term vanish. Including terms of fourth order in\ncurvatures, we find ({\\it cf} (\\ref{Zm(2)-(8)}))\n\\bea\\label{Zm2j}\n&&\n\\Zm_{(2j)} =\\Xi_{(2j)}\\,+\\,\\su_{k=1}^j {2j{\\cal H} k}\\{\\Y_{(k}\\Y_{\\el)}\\}\\non\\\\\n&&\n+ \\su_{k=1}^{j-1}\\su_{\\el=k}^{2j-k-2} {2j{\\cal H} k,\\el}\n\\big({m\\ov k+m+1} + {m\\ov \\el+m+1}\\big) \\{\\Y_{(k}\\K_m\\Y_{\\el)}\\} \\\\\n&&\n+\\su_{k=1}^{j-2}\\su_{\\el=k}^{2j-k-4}\\su_{m=2}^{2j-k-\\el-2}{2j{\\cal H} k,\\el,m}\n{mn\\,N_{k\\el m}\\ov D_{km} D_{\\el n} D_{mn} D_{kmn} D_{\\el mn}}\n\\{\\Y_{(k}\\K_m\\K_n\\Y_{\\el)}\\}\\ +\\,O(\\K^3) \\non\n\\ena\nwhere the $D$-symbols were defined in (\\ref{betas}) and \n\\bea\nN_{k\\el m} &=& 5 + 5k\\el + 12j + 16jmn\\ +\\{6m +(k+m)(9m+10n) \\non\\\\\n           & &\\qquad\\quad +k^2(\\el+m+2n+2) +(3k+4\\el+2m)m^2 +4k\\el m\\}_S \\non\\\\\n&&\\{f(k,\\el,m,n)\\}_S \\equiv  f(k,\\el,m,n) + f(\\el,k,n,m) \n\\ena\nWe recognize the first line of (\\ref{Zm2j}) as the flat spacetime result \n(\\ref{Z2j}). As we already mentioned there, we are not able to give a closed \nexpression to all orders in $\\K$ here. However, the terms of $N$-th order in \n$\\K$ in $\\Zm_{(2j)}$ will be of the form \n\\bee\n\\{\\Y_{(k}\\K_{m_1}\\dots\\K_{m_N}\\!\\Y_{\\el)}\\}\\quad ,\\quad\n2j \\= k+\\el +\\su_{i=1}^N m_i\\quad ,\\quad k\\leq\\el\n\\ene\nand this shows that $\\ZM$ has a much simpler structure than $\\ZS$.\n\\subsection{Expansion in derivatives}\nWe retrace the steps of sect 4.1, starting with (compare with (\\ref{ajng}))\n\\bee\\label{ajnc}\n\\sa_{j,n}\\={j\\ov j+n}\\,\\Big(\\,\\sa_{j-1,}\\sp\\mu\\sb{\\mu,n}\\,\n           +\\,(\\HX\\,\\sa_{j-1})_{,n} \\,\\Big)\n\\ene\nwhere now\n\\bee\n\\hX \\= Z + 2 A^\\mu\\pa_\\mu \\= \\ZM +\\ZS+ 2 A^\\mu\\pa_\\mu\n\\ene\nExcept for $n=0$ or 1, we can {\\it not} replace the explicit pair of indices\nin (\\ref{ajnc}) by our shorthand notation as in (\\ref{ajng}). However,\nfor even values of $n$ and after full contraction we can use the following \nlemma: for any scalar $F(x)$\n\\bee\\label{lem2}\n\\Phi\\sp\\mu\\sb{\\mu\\,(2n)} \\= \\Phi_{(2n+2)} \\quad ,\\quad n\\geq 0\n\\ene\nThis follows immediately from the defining properties of the inverse metric in\nnormal coordinates, see (\\ref{gsym0}). Using this lemma, we obtain the \nfollowing chain of substitutions (omit numerical factors and ordering labels, \nbut keep undifferentiated $\\Z$'s)\n\\bea\\label{chain2}\n\\sa        &\\ra& \\sa_{(2)}\\, +\\, \\Z\\,\\sa\\non\\\\\n\\sa_{(2n)} &\\ra& \\sa_{(2n+2)}\\, +\\, (\\Z\\dg\\sa)_{(2n)}\\quad ,\\quad n\\geq 1\\\\\n\\Z_{(p}\\do\\sa_{2n-p)} &\\ra& \\Z_{(p}\\do\\sa^\\mu\\sb{2n-p)\\mu} \\,\n+\\,\\Z_{(p}\\do (\\HX\\,\\sa)_{2n-p)}  \\quad ,\\quad p=0\\dots 2n \\non\n\\ena\netc, where\n\\bee\n\\Z\\=\\Zm +\\Zs \\=\\Xi +\\sfrac12{\\rm tr}[\\K_{(2)}] \\=X(0)+\\sfrac16 R(0)\n\\ene\nIn the second step we used that (\\ref{AF}), with the understanding that \n$\\pa^2= \\pa_\\mu\\pa_\\mu$, remains true in normal coordinates. We can remove $\\hX$\nfrom the last line exactly as in the first three lines of (\\ref{Zhi}). However,\nthe first term on the right hand side in the last line of (\\ref{chain2}) yields\na new term as compared to flat space ($r\\equiv 2n-p$)\n\\bea\\label{rewr}\n\\Z_{(p}\\do\\sa^\\mu\\sb{r)\\mu} \\,-\\,\\Z_{(p}\\do\\sa_{r)(2)} &\\equiv& \n              \\Z_{(p}\\do({\\sf h}^{\\mu\\nu}\\sa_{,\\vert\\nu\\vert})_{r)\\mu} \\non\\\\\n&=&\n-\\,{p\\ov r+1}\\,\\Z_{\\mu(p-1}\\do({\\sf h}^{\\mu\\nu}\\sa_{,\\vert\\nu\\vert})_{r+1)} \\non\\\\\n&=&\n-\\,{p\\ov r+1}\\su_{q=0}^{r+1} {r+1{\\cal H} q}\n         \\Z_{\\mu(p-1}\\do {\\sf h}^{\\mu\\nu}\\sb{q}\\sa_{r-q+1)\\nu}  \\\\\n&=&\n-\\su_{q=0}^r {r{\\cal H} q} {p(r-q)\\ov (q+1)(q+2)}\\,\n         \\Z_{\\mu(p-1}\\do \\sg^{\\mu\\nu}\\sb{q+2}\\sa_{r-q-1)\\nu} \\non\n\\ena\nThe first step merely defines ${\\sf h}^{\\mu\\nu}\\equiv\\sg^{\\mu\\nu}-\\delta^{\\mu\\nu}$.\nThe vertical bars indicate that the index $\\nu$ is not to be included in the\nindicated symmetrization. In the second step we use that, due to (\\ref{gsym0})\n\\bee\n\\Z_{(p}\\do({\\sf h}^{\\mu\\nu}\\sa_{,\\vert\\nu\\vert})_{r\\,\\mu)}\\= 0 \n\\ene\nWriting this out with respect to the position of the index $\\mu$ shows the\nequality of the first two lines of (\\ref{rewr}). Next we use the binomial \ntheorem and in the last step we shift $q$ and use that $h$ and its first \nderivative vanish at the origin. We absorb this new term through a redefinition\nof $\\HZ_q$ in (\\ref{Zhi}) and arrive at the following conclusion.\\\\\n\\\\\n\\noindent\n{\\it The diagonal values of the heat kernel coefficients in curved space are \nobtained from those in flat space by formally making the same substitutions as\nin (\\ref{Subs}), where $\\Z_j$ now stands for $\\Zm_j+\\Zs_j$ (see (\\ref{ZSj}, \n\\ref{Zmj}) and the action of $\\HZ$ is defined by}\n\\bee\\label{TheoII}\n\\Z_{(j}\\do\\HZ_n\\Z_{k)}\\= \\Z_{(j}\\do\\Z_n\\Z_{k)}\\, \n                          +\\,{2k\\ov n+1}\\,\\Z_{(j}\\do\\A^\\nu\\sb{n+1}\\Z_{k-1)\\nu}\n\\, -\\,{jk\\ov(n+1)(n+2)}\\,\\Z_{\\mu(j-1}\\do\\sg^{\\mu\\nu}\\sb{n+2}\\Z_{k-1)\\nu}\n\\ene\n{\\it etc, where the (contravariant) gauge connection and inverse metric \nfollow from (\\ref{invmet}) and (\\ref{expA}).}\\\\\n\\\\\n\\noindent\nThese redefinitions being understood, the heat kernel coefficients formally \nremain unchanged upon going to curved spacetime.\n\\subsection{Locally symmetric space}\nThe results of the previous sections nearly suffice to find $\\sa_5$. The few \nmissing terms are of fifth order in the Yang-Mills and Riemann curvature.\nThey are most easily found by considering a locally symmetric space, i.e. a\nspace with covariantly constant curvatures. In that case one can find closed \nexpressions which hold through all orders in these curvatures. This situation \nhas been considered in detail by Avramidi \\cite{ls}. Here, we merely want to\nfind those terms in a given heat kernel coefficient for a {\\it general} \ncurved space which do not contain explicit covariant derivatives. We therefore\ndo not take account of the consequences of the requirement that the Riemann \ncurvature is covariantly constant (i.e. $\\de R=0$ would imply $[\\de,\\de]R=0$).\n\nThus, consider the case where only $\\Xi_0$, $\\Y_1$ and $\\K_2$ are \nnonvanishing. Define\n\\bee\nY_\\nu(x) \\= {1\\ov 2} \\, F_{\\mu\\nu}(0)\\,x^\\mu\\quad ,\\quad\nK_{\\mu\\nu}(x) \\= {1\\ov 3} \\,R_{\\mu\\rho\\nu\\sigma}(0)\\, x^\\rho x^\\sigma\n\\ene\nThe vielbein is then given to all orders in $K$ by (see (\\ref{solE}) \n\\bee\nE[K] \\=\\su_{k=0}^\\infty {1\\ov (2k+1)!} (-3K)^k \\= {\\sinh\\,S\\ov S}\\quad ,\\quad\nS\\,\\equiv\\,\\sqrt{-3 K}\n\\ene\nNote that the vielbein is an even function of $S$ and hence it depends only on\n$K$. Since $K$ is a symmetric matrix, so is $E$. It follows that the inverse \nmetric simply equals $E^{-2}$. Differentiating $B= -{\\rm tr}\\ln E$ we find\n\\bee\\label{dB}\nB_{,\\mu}\\= {1\\ov 2} {\\rm tr}[ K_{,\\mu} L] \\quad ,\\quad\nL[K] \\equiv {3\\ov S}\\,\\big(\\coth S \\,-\\,{1\\ov S}\\,\\big) \n\\ene\nwhere $L$ is an even function\\footnote{Except for the extra factor $3\/S$,\nthis is the well known Langevin function.} of $S$. In general, the commutator \nof $K$ and $K_{,\\mu}$ does not vanish, but the trace insures that (\\ref{dB})\nholds. Differentiating $L$ we find\n\\bee\nL_{,\\mu} \\= \\langle K_{,\\mu} L\\pr \\rangle \\quad ,\\quad\n\\langle K_{,\\mu}K^n\\rangle \\equiv {1\\over n}\\su_{k=0}^n K^k K_{,\\mu} K^{n-k}\n\\ene\nWith this notation, the result for $\\ZS$ in a locally symmetric space reads\n\\bee\\label{ZSloc}\n\\ZS[K] = \\sfrac14 (E^{-2})^{\\mu\\nu} \\Big({\\rm tr}[ K_{,\\mu\\nu} L + K_{,\\mu} L_{,\\nu} ]\n-\\sfrac14 {\\rm tr}[K_{,\\mu} L]{\\rm tr}[K_{,\\nu} L] \\Big) \n+\\sfrac14 \\langle E^{-2} K_{,\\mu} L\\rangle^{\\mu\\nu} {\\rm tr}[ K_{,\\nu} L ]\n\\ene\nTo obtain a similar result for $\\ZM$ we note that the covariant \ngauge connection is given by\n\\bee\nA[Y,K] \\=\\Big({\\sinh\\,S\/2\\ov S\/2}\\Big)^2\\,Y\n\\ene\nThe contravariant gauge connection is therefore\n\\bee\nA[Y,K] \\= ({\\rm sech}\\,S\/2)^2\\,Y\n\\ene\nand we obtain the following result for $\\ZM$ in a locally symmetric space\n\\bea\\label{ZMloc}\n&&\\ZM[X,Y,K]\\= X-\\pa{\\cal D}(\\tanh\\,S\/2)^2\\,Y +Y\\Big({\\tanh\\,S\/2\\ov S\/2}\\Big)^2 Y \\\\\n&&\\= X +\\pa\\cdot (\\sfrac34 K +\\sfrac38 K^2+\\,\\dots )Y\n + Y (I +\\sfrac12 K +\\sfrac{17}{80} K^2 +\\sfrac{93}{1120} K^3+\\,\\dots)Y\\non\n\\ena\n\\section{Explicit results for $\\sa_1$ through $\\sa_5$}\n\\setcounter{equation}{0}\nAs an application, we give here the explicit results for the diagonal values \nof the first five heat kernel coefficients, obtained from (\\ref{Subs}), \n(\\ref{Zres}) and (\\ref{TheoII}). The number of terms equals 1,2,4,10 and 26 \nrespectively. The sixth coefficient in flat space can be found in appendix C.\n\\bea\\label{a1-a5}\n\\sa_1 &=& \\Z \\non\\\\\n\\sa_2 &=& \\Z^2 \n+\\sfrac13\\Z_{(2)}  \\non\\\\\n\\sa_3 &=& \\Z^3 \n+\\sfrac12\\{\\Z\\Z_{(2)}\\} \n+\\sfrac12\\Z_{(1}\\do\\Z_{1)}\n+\\sfrac1{10}\\Z_{(4)} \\non\\\\\n\\sa_4 &=& \\Z^4 \n+\\sfrac35\\{\\Z^2\\Z_{(2)}\\} \n+\\sfrac45\\Z\\Z_{(2)}\\Z\n+\\sfrac25\\Z_{(1}\\do\\HZ\\Z_{1)} \n+\\sfrac45\\{\\Z\\Z_{(1}\\do\\Z_{1)}\\} \\non\\\\\n&&\n+\\sfrac15\\{\\Z\\Z_{(4)}\\}\n+\\sfrac25\\{\\Z_{(1}\\do\\Z_{3)}\\}\n+\\sfrac25\\Z_{(2}\\do\\Z_{2)} \n+\\sfrac15\\Z_{(2)}\\sp{2} \n+\\sfrac1{35}\\Z_{(6)} \\non\\\\\n\\sa_5 &=& \\Z^5 \n+\\sfrac23\\{\\Z^3\\Z_{(2)}\\}\n+\\{\\Z^2\\Z_{(2)}\\Z\\}\n+\\sfrac13\\Z_{(1}\\do\\HZ^2\\Z_{1)}\n+\\sfrac23\\{\\Z\\Z_{(1}\\do\\HZ\\Z_{1)}\\}\n+\\{\\Z^2\\Z_{(1}\\do\\Z_{1)}\\}\\non\\\\\n&&\n+\\sfrac43\\Z\\Z_{(1}\\do\\Z_{1)}\\Z \n+\\sfrac27\\{\\Z^2\\Z_{(4)}\\}\n+\\sfrac37\\Z\\Z_{(4)}\\Z \n+\\sfrac67\\{\\Z\\Z_{(3}\\do\\Z_{1)}\\}\n+\\sfrac8{21}\\{\\Z_{(1}\\do\\HZ\\Z_{3)}\\} \\non\\\\\n&&\n+\\sfrac{16}{21}\\{\\Z\\Z_{(1}\\do\\Z_{3)}\\}\n+\\sfrac27  \\Z_{(2}\\do\\HZ\\Z_{2)}\n+\\sfrac67\\{\\Z\\Z_{(2}\\do\\Z_{2)}\\}\n+\\sfrac13\\Z_{(2)}\\Z\\Z_{(2)}\n+\\sfrac13\\{\\Z\\Z_{(2)}\\sp{2}\\} \\non\\\\\n&&\n+\\sfrac97\\Z_{(1}\\do\\HZ_2\\Z_{1)}\n+\\sfrac67\\{\\Z_{(1}\\do\\HZ_1\\Z_{2)}\\}\n+\\sfrac13\\{\\Z_{(1}\\do\\Z_{1)}\\Z_{(2)}\\}\n+\\sfrac1{14}\\{\\Z\\Z_{(6)}\\}\n+\\sfrac3{14}\\{\\Z_{(1}\\do\\Z_{5)}\\}\\non\\\\\n&&\n+\\sfrac5{14}\\{\\Z_{(2}\\do\\Z_{4)}\\}\n+\\sfrac5{14}  \\Z_{(3}\\do\\Z_{3)} \n+\\sfrac2{21}\\{\\Z_{(2)}\\Z_{(4)}\\}\n+\\sfrac4{21} \\Z_{(3)}\\do\\Z_{(3)}\n+\\sfrac1{126}\\Z_{(8)}\n\\ena\nThe $\\Z$'s appearing here were defined in the previous sections, but to be \nquite explicit and to avoid misunderstanding of our conventions, we give them \nbelow. We have $\\Z_j\\=\\Zm_j + \\Zs_j$ with the matrix quantities, see \n(\\ref{Zmj}), given by\n\\bea\\label{Zm1-5}\n\\Zm_1 &=&\\Xi_1 +\\Y_{(\\mu}\\sp\\mu\\sb{;1)} \\non\\\\\n\\Zm_2 &=&\\Xi_2 +\\Y_{(\\mu}\\sp\\mu\\sb{;2)} +2\\Y_{(1}\\Y_{1)}\n              +\\sfrac92 \\K_{(2}\\sp{\\mu\\nu}\\Y_{\\mu)\\nu} \\non\\\\\n\\Zm_3 &=&\\Xi_3 +\\Y_{(\\mu}\\sp\\mu\\sb{;3)} +3\\{\\Y_{(1}\\Y_{2)}\\} \n+\\sfrac25\\Big( 21 \\K_{(2}\\sp{\\mu\\nu}\\Y_\\mu\\sp\\nu\\sb{;1)}\n              +16 \\K_{(3}\\sp{\\mu\\nu}\\Y_{\\mu)\\nu} \\Big)\\non\\\\\n\\Zm_4 &=&\\Xi_4 +\\Y_{(\\mu}\\sp\\mu\\sb{;4)} +4\\{\\Y_{(1}\\Y_{3)}\\} \n                           +6\\Y_{(2}\\Y_{2)}+12\\Y_{(1}\\K_2\\Y_{1)} \\non\\\\\n&&\\quad\n+\\sfrac53\\Big( 8 \\K_{(2}\\sp{\\mu\\nu}\\Y_\\mu\\sp\\nu\\sb{;2)}\n             + 9 \\K_{(3}\\sp{\\mu\\nu}\\Y_\\mu\\sp\\nu\\sb{;1)} \n            + (5 \\K_{(4} +9\\K_{(2}\\K_2)\\sp{\\mu\\nu}\\Y_{\\mu)\\nu} \\Big) \\non\\\\\n\\Zm_5 &=& \\Xi_5 +\\Y_{(\\mu}\\sp\\mu\\sb{;5)} +5\\{\\Y_{(1}\\Y_{4)}\\}\n   +10\\{\\Y_{(2}\\Y_{3)}\\} + 24\\Y_{(1}\\K_3\\Y_{1)} +9\\{\\Y_{(1}\\K_2\\Y_{2)}\\} \\non\\\\\n&&\\quad\n+\\sfrac37\\Big( 45 \\K_{(2}\\sp{\\mu\\nu}\\Y_\\mu\\sp\\nu\\sb{;3)}\n             +\\sfrac{200}3 \\K_{(3}\\sp{\\mu\\nu}\\Y_\\mu\\sp\\nu\\sb{;2)}\n             + 9\\,(5\\K_{(4} +31\\K_{(2}\\K_2)\\sp{\\mu\\nu}\\Y_\\mu\\sp\\nu\\sb{;1)}\\non\\\\\n&&\\qquad\\quad\n             + 4\\,(6\\K_{(5} +51\\K_{(3}\\K_2 +60\\K_{(2}\\K_3)\\sp{\\mu\\nu}\\Y_{\\mu)\\nu}  \n\\Big)\n\\ena\nIn (\\ref{Zm1-5}) the parenthesis around the indices imply symmetrization \nonly (but note that $\\Zm_4$ and $\\Zm_5$ appear in (\\ref{a1-a5}) with at least\none respectively two contracted pair(s) of indices). All terms in (\\ref{Zm1-5})\nwith explicit indices vanish upon full contraction. From (\\ref{Zm2j}) we find\n\\bea\\label{Zm(2)-(8)}\n\\Zm_{(2)}&=&\\Xi_{(2)}+2  \\Y_{(1}\\Y_{1)}  \\non\\\\\n\\Zm_{(4)}&=&\\Xi_{(4)}+4\\{\\Y_{(1}\\Y_{3)}\\}+6\\Y_{(2}\\Y_{2)}+12\\Y_{(1}\\K_2\\Y_{1)}\n\\non\\\\\n\\Zm_{(6)}&=&\\Xi_{(6)}  \n+ 6\\{\\Y_{(1}\\Y_{5)}\\}\n+15\\{\\Y_{(2}\\Y_{4)}\\}\n+24  \\Y_{(3}\\Y_{3)} \n+40  \\Y_{(1}\\K_4\\Y_{1)} \\non\\\\\n&&\n+66\\{\\Y_{(1}\\K_3\\Y_{2)}\\}\n+50\\{\\Y_{(1}\\K_2\\Y_{3)}\\} \n+72  \\Y_{(2}\\K_2\\Y_{2)} \n+153 \\Y_{(1}\\K_2\\K_2\\Y_{1)} \\non\\\\\n\\Zm_{(8)}&=&\\Xi_{(8)} \n+ 8\\{\\Y_{(1}\\Y_{7)}\\}\n+28\\{\\Y_{(2}\\Y_{6)}\\} \n+56\\{\\Y_{(3}\\Y_{5)}\\}\n+70  \\Y_{(4}\\Y_{4)} \n+84  \\Y_{(1}\\K_6\\Y_{1)} \\non\\\\\n&&\n+225  \\{\\Y_{(1}\\K_5\\Y_{2)}\\}\n+\\sfrac{980}3 \\{\\Y_{(1}\\K_4\\Y_{3)}\\} \n+273  \\{\\Y_{(1}\\K_3\\Y_{4)}\\}\n+126  \\{\\Y_{(1}\\K_2\\Y_{5)}\\}  \\non\\\\\n&&\n+480   \\Y_{(2}\\K_4\\Y_{2)}\n+520 \\{\\Y_{(2}\\K_3\\Y_{3)}\\}\n+288 \\{\\Y_{(2}\\K_2\\Y_{4)}\\}\n+\\sfrac{1120}3\\Y_{(3}\\K_2\\Y_{3)}  \\non\\\\\n&&\n+\\sfrac{6336}5 \\Y_{(1}\\K_3\\K_3\\Y_{1)}\n+ 870 \\{\\Y_{(1}\\K_2\\K_4\\Y_{1)}\\}\n+1440 \\{\\Y_{(1}\\K_2\\K_3\\Y_{2)}\\}  \\non\\\\\n&&\n+\\sfrac{6966}5 \\{\\Y_{(1}\\K_3\\K_2\\Y_{2)}\\}\n+1092 \\{\\Y_{(1}\\K_2\\K_2\\Y_{3)}\\}\n+\\sfrac{7776}5 \\Y_{(2}\\K_2\\K_2\\Y_{2)} \\non\\\\\n&&\n+ 3348 \\Y_{(1}\\K_2\\K_2\\K_2\\Y_{1)}\n\\ena\nThe last term in $\\Zm_{(8)}$ was obtained from (\\ref{ZMloc}). From (\\ref{ZSj})\nwe find that the scalar quantities are given explicitly by\n\\bea\\label{Zs1-5}\n\\Zs_1 &=&\\sfrac12{\\rm tr}[\\K_{1\\mu\\m}] \\non\\\\\n\\Zs_2 &=&\\sfrac12{\\rm tr}[\\K_{2\\mu\\m} +\\sfrac65\\K_2\\K_{\\mu\\m}]\\,\n-\\sfrac12{\\rm tr}[\\K_{\\mu(1}]{\\rm tr}[\\K_{1)\\mu}]\\, +3\\K_{(2}\\sp{\\mu\\nu}{\\rm tr}[\\K_{\\mu)\\nu}]\\non\\\\\n\\Zs_3 &=&\\sfrac12{\\rm tr}[\\K_{3\\mu\\m} + 4\\K_2\\K_{1\\mu\\m}]\n-\\sfrac32{\\rm tr}[\\K_{\\mu(1}] {\\rm tr}[\\K_{2)\\mu}]\n+ 4\\K_{(3}\\sp{\\mu\\nu}{\\rm tr}[\\K_{\\mu)\\nu}] + 6\\K_{(2}\\sp{\\mu\\nu}{\\rm tr}[\\K_{1\\mu)\\nu}]   \\non\\\\\n\\Zs_4 &=&\\sfrac1{14}{\\rm tr}\\Big[7\\K_{4\\mu\\m} +40\\K_2\\K_{2\\mu\\m} \n           +30\\K_3\\K_{1\\mu\\m} +48\\K_2\\K_2\\K_{\\mu\\m}\\Big]\\ \n-\\sfrac32{\\rm tr}[\\K_{\\mu(2}]{\\rm tr}[\\K_{2)\\mu}] \\non\\\\ \n&&\n-\\sfrac25{\\rm tr}[\\K_{\\mu(1}]{\\rm tr}[5\\K_{3)\\mu} +6\\K_2\\K_{1)\\mu}]\\ \n-  6 {\\rm tr}[\\K_{\\mu(1}]\\,\\K_2\\sp{\\mu\\nu}{\\rm tr}[\\K_{1)\\nu}]\\ \n+10\\K_{(3}\\sp{\\mu\\nu}{\\rm tr}[\\K_{1\\mu)\\nu}] \\non\\\\\n&&\n+ (5\\K_{(4} +36\\K_{(2}\\K_2)\\sp{\\mu\\nu}{\\rm tr}[\\K_{\\mu)\\nu}] \\ \n+ 2\\K_{(2}\\sp{\\mu\\nu}{\\rm tr}[5\\K_{2\\mu)\\nu}+6\\K_2\\K_{\\mu)\\nu}] \\non\\\\\n\\Zs_5 &=&\\sfrac12{\\rm tr}\\Big[\\K_{5\\mu\\m} +\\sfrac{15}2\\K_2\\K_{3\\mu\\m} \n      + 15\\K_3\\K_{2\\mu\\m} + 48\\K_2\\K_2\\K_{1\\mu\\m} \\Big] \\non \\\\\n&&\n-\\sfrac52{\\rm tr}[\\K_{\\mu(1}]{\\rm tr}[\\K_{4)\\mu} + 4\\K_2\\K_{2)\\mu}] \\ \n-        {\\rm tr}[\\K_{\\mu(2}]{\\rm tr}[5\\K_{3)\\mu}+ 6\\K_2\\K_{1)\\mu}] \\non\\\\\n&&\n- 30 {\\rm tr}[\\K_{\\mu(1}]\\,\\K_2\\sp{\\mu\\nu}{\\rm tr}[\\K_{2)\\nu}]\\ \n- 10 {\\rm tr}[\\K_{\\mu(1}]\\,\\K_3\\sp{\\mu\\nu}{\\rm tr}[\\K_{1)\\nu}] \\non\\\\\n&&\n+ 15 \\K_{(2}\\sp{\\mu\\nu}{\\rm tr}[\\K_{3\\mu)\\nu} + 4\\K_{ 2}\\K_{1\\mu)\\nu}]\\ \n+  4 \\K_{(3}\\sp{\\mu\\nu}{\\rm tr}[5\\K_{2\\mu)\\nu} +6\\K_{ 2}\\K_{\\mu)\\nu}] \\non\\\\\n&&\n+ 3\\,(5\\K_{(4} +36\\K_{(2}\\K_2)\\sp{\\mu\\nu}{\\rm tr}[\\K_{1\\mu)\\nu}]\\non\\\\\n&&\n+ 6\\,( \\K_{(5} +12\\K_{(2}\\K_3 +12\\K_{(3}\\K_2 )\\sp{\\mu\\nu}{\\rm tr}[\\K_{\\mu)\\nu}]\n\\ena\nwhere the parenthesis mean symmetrization only. Upon full contraction we find\nfrom (\\ref{Zs2j})\n\\bea\\label{Zs(2)-(8)}\n\\Zs_{(2)} &=&\\sfrac12{\\rm tr}[\\K_{(4)} +\\sfrac65\\K_{(2}\\K_{2)}] \\,\n            -\\sfrac12{\\rm tr}[\\K_{\\mu(1}]{\\rm tr}[\\K_{1)\\mu}] \\non\\\\\n\\Zs_{(4)} &=&\\sfrac1{14}{\\rm tr}\\Big[7\\K_{(6)} +40\\K_{(2}\\K_{4)}\n                        +30\\K_{(3}\\K_{3)} +48\\K_{(2}\\K_2\\K_{2)} \\Big] \\,\n-\\sfrac32{\\rm tr}[\\K_{\\mu(2}]{\\rm tr}[\\K_{2)\\mu}]  \\non\\\\\n&&\n- 2 {\\rm tr}[\\K_{\\mu(1}]{\\rm tr}[\\K_{3)\\mu} +\\sfrac65\\K_2\\K_{1)\\mu}] \\,\n- 6 {\\rm tr}[\\K_{\\mu(1}]\\,\\K_2\\sp{\\mu\\nu} {\\rm tr}[\\K_{1)\\nu}] \\non\\\\\n\\Zs_{(6)} &=&\\sfrac12{\\rm tr}\\Big[\\K_{(8)} +\\sfrac{28}3\\K_{(2}\\K_{6)} \n + \\sfrac{70}3\\K_{(3}\\K_{5)} +\\sfrac{140}9\\K_{(4}\\K_{4)} \n +\\sfrac{272}3\\K_{(2}\\K_2\\K_{4)} +\\sfrac{400}3\\K_{(2}\\K_3\\K_{3)} \\non\\\\\n&&\n +\\sfrac{432}5 \\K_{(2}\\K_2\\sp{2}\\K_{2)}\\Big]\n-\\sfrac37{\\rm tr}[\\K_{\\mu(1}]{\\rm tr}\\Big[7\\K_{5)\\mu} +40\\K_2\\K_{3)\\mu}\n   +30\\K_3\\K_{2)\\mu} +48\\K_{2}\\K_2\\K_{1)\\mu}\\Big]  \\non\\\\\n&&\n-\\sfrac{15}2{\\rm tr}[\\K_{\\mu(2}]{\\rm tr}[\\K_{4)\\mu} +4\\K_2\\K_{2)\\mu}]  \\,\n- \\sfrac15{\\rm tr}[5\\K_{\\mu(3}+6\\K_{\\mu(1}\\K_2]{\\rm tr}[5\\K_{3)\\mu}+6\\K_2\\K_{1)\\mu}] \\non\\\\\n&&\n- 3{\\rm tr}[\\K_{\\mu(1}]\\, (5\\K_4 +36\\K_2\\K_2)\\sp{\\mu\\nu} {\\rm tr}[\\K_{1)\\nu}]\\,\n-12{\\rm tr}[\\K_{\\mu(1}]\\,\\K_2\\sp{\\mu\\nu} {\\rm tr}[5\\K_{3)\\nu}+6\\K_2\\K_{1)\\nu}] \\non\\\\\n&&\n-60{\\rm tr}[\\K_{\\mu(1}]\\,\\K_3\\sp{\\mu\\nu} {\\rm tr}[\\K_{2)\\nu}] \\,\n-45{\\rm tr}[\\K_{\\mu(2}]\\,\\K_2\\sp{\\mu\\nu} {\\rm tr}[\\K_{2)\\nu}]  \\non\\\\\n\\Zs_{(8)} &=&\\sfrac12{\\rm tr}[\\K_{(10)}] \n+\\sfrac8{11}{\\rm tr}\\Big[9\\K_{(2}\\K_{8)} +500\\K_{(2}\\K_4\\K_{4)} \n+1680\\K_{(2}\\K_2\\sp{2}\\K_{4)} +1296\\K_{(2}\\K_2\\sp{3}\\K_{2)} \\Big] \\non\\\\\n&&\n+\\sfrac{21}{11}{\\rm tr}\\Big[12\\K_{(3}\\K_{7)} +24\\K_{(4}\\K_{6)} +15\\K_{(5}\\K_{5)}\n               +\\sfrac{304}5\\K_{(2}\\K_2\\K_{6)} +290\\K_{(2}\\K_3\\K_{5)} \\non\\\\\n&&\\qquad\\qquad\n+ 460\\K_{(2}\\K_3\\K_2\\K_{3)} \\Big]\\ \n+ 540{\\rm tr}[\\K_{(3}\\K_3\\K_{4)}]\n+1788{\\rm tr}[\\K_{(2}\\K_2\\K_3\\K_{3)}] \\non\\\\\n&&\n-4{\\rm tr}[\\K_{\\mu(1}]{\\rm tr}\\Big[\\K_{7)\\mu} +\\sfrac{28}3\\K_2\\K_{5)\\mu}\n  +\\sfrac{70}3\\K_3\\K_{4)\\mu} +\\sfrac{140}9\\K_4\\K_{3)\\mu}\n  +\\sfrac{272}3\\K_2\\K_2\\K_{3)\\mu} \\non\\\\\n&&\\qquad\\qquad\\qquad\\quad\n  +\\sfrac{400}3\\K_2\\K_3\\K_{2)\\mu} +\\sfrac{432}5\\K_2\\K_2\\K_2\\K_{1)\\mu}\\Big] \\non\\\\\n&&\n-14{\\rm tr}[\\K_{\\mu(2}]{\\rm tr}\\Big[\\K_{6)\\mu} +\\sfrac{15}2 \\K_2\\K_{4)\\mu}\n                         +15\\K_3\\K_{3)\\mu} +48\\K_2\\K_2\\K_{2)\\mu} \\Big] \\non\\\\\n&&\n-\\sfrac45{\\rm tr}[5\\K_{\\mu(3} +6\\K_{\\mu(1}\\K_2]\n     {\\rm tr}[7\\K_{5)\\mu} +40\\K_2\\K_{3)\\mu} +30\\K_3\\K_{2)\\mu} +48\\K_2\\K_2\\K_{1)\\mu}]\\non\\\\\n&&\n-\\sfrac{35}2{\\rm tr}[\\K_{\\mu(4}+4\\K_{\\mu(2}\\K_2]{\\rm tr}[\\K_{4)\\mu} +4\\K_2\\K_{2)\\mu}] \\non\\\\\n&&\n-  4{\\rm tr}[\\K_{\\mu(1}]\\, (7\\K_6 +125\\{\\K_2\\K_4\\} +170\\K_3\\K_3\n                    +720\\K_2\\K_2\\K_2 )\\sp{\\mu\\nu} {\\rm tr}[\\K_{1)\\nu}] \\non\\\\\n&&\n-168{\\rm tr}[\\K_{\\mu(1}]\\,  (\\K_5 + 12\\{\\K_2\\K_3\\})\\sp{\\mu\\nu} {\\rm tr}[\\K_{2)\\nu}] \\non\\\\\n&&\n-\\sfrac{56}5{\\rm tr}[\\K_{\\mu(1}]\\, (5\\K_4 +36\\K_2\\K_2)\\sp{\\mu\\nu} \n    {\\rm tr}[5\\K_{3)\\nu}+6\\K_2\\K_{1)\\nu}] \\non\\\\\n&&\n-280{\\rm tr}[\\K_{\\mu(1}]\\,\\K_3\\sp{\\mu\\nu} {\\rm tr}[\\K_{4)\\nu} +4\\K_2\\K_{2)\\nu}]\\,\n- 42{\\rm tr}[\\K_{\\mu(2}]\\, (5\\K_4 +36\\K_2\\K_2)\\sp{\\mu\\nu} {\\rm tr}[\\K_{2)\\nu}] \\non\\\\\n&&\n- 24{\\rm tr}[\\K_{\\mu(1}]\\,\\K_2\\sp{\\mu\\nu} {\\rm tr}[7\\K_{5)\\nu} +40\\K_2\\K_{3)\\nu}\n                +30\\K_3\\K_{2)\\nu} +48\\K_2\\K_2\\K_{1)\\nu} ] \\non\\\\\n&&\n-112{\\rm tr}[\\K_{\\mu(2}]\\,\\K_3\\sp{\\mu\\nu} {\\rm tr}[5\\K_{3)\\nu} +6\\K_2\\K_{1)\\nu}]\\ \n-420{\\rm tr}[\\K_{\\mu(2}]\\,\\K_2\\sp{\\mu\\nu} {\\rm tr}[ \\K_{4)\\nu} +4\\K_2\\K_{2)\\nu}] \\non\\\\\n&&\n-\\sfrac{56}5{\\rm tr}[5\\K_{\\mu(3} +6\\K_{\\mu(1}\\K_2]\\,\\K_2\\sp{\\mu\\nu}\n          {\\rm tr}[5\\K_{3)\\nu} +6\\K_2\\K_{1)\\nu}] \n\\ena\nwhere the seven terms of fifth order were found from (\\ref{ZSloc}) and also \nfrom expanding $B$ through tenth order in the normal coordinates (see appendix\nE).\n\nTo return to conventional notation we use\n\\bea\n\\Z       &=& X +\\sfrac16 R \\non\\\\\n\\Z_\\mu    &=& X_{;\\mu}  +\\sfrac13 F_\\mu\\sp\\nu\\sb{;\\nu}+\\sfrac16 R_{;\\mu} \\\\\n\\Z_{(2)} &=& X_;\\sp\\mu\\sb\\mu + \\sfrac12 F^{\\mu\\nu} F_{\\mu\\nu}\n               +\\sfrac15 R_;\\sp\\mu\\sb\\mu + \\sfrac1{12} R^2 \n -\\sfrac1{30} R^{\\mu\\nu} R_{\\mu\\nu} +\\sfrac1{30} R^{\\kappa\\lambda\\mu\\nu} R_{\\kappa\\lambda\\mu\\nu}\\non \n\\ena\netc (these suffice for $\\sa_1$ and $\\sa_2$). In $\\sa_4$ respectively $\\sa_5$ \nwe have e.g.\n\\bea\n\\Z_{(1}\\do\\HZ\\Z_{1)}&=& \n\\Z_\\mu\\do (\\Z g^{\\mu\\nu} +F^{\\mu\\nu} -\\sfrac13 R^{\\mu\\nu})\\Z_\\nu \\\\\n\\Z_{(2}\\do\\HZ\\Z_{2)}&=& \\sfrac13 \\Z_{(2)}\\Z\\Z_{(2)} \n+\\sfrac23\\Z_{\\kappa\\mu}\\do (\\Z g^{\\mu\\nu} +2F^{\\mu\\nu} -\\sfrac23 R^{\\mu\\nu})\\Z_{\\kappa\\nu} \n+\\sfrac49\\Z_{\\kappa\\mu}\\do R^{\\kappa\\lambda\\mu\\nu}\\Z_{\\lambda\\nu}  \\non\n\\ena\nUsing such translations we have verified that our results for the first four \ncoefficients agree with earlier authors, in particular \\cite{DeW}, \\cite{Gi2} \nand \\cite{Avr}, after accounting for differences in conventions and the \noccasional typographical mistake.\n\nFinally, to illustrate the compactness of our notation, consider the special \ncase of a scalar field in a Ricci flat space. Then only the Weyl tensor and \nits covariant derivatives can appear in the heat kernel coefficients. \nFrom appendix D of \\cite{FKW} we can read off the total number of general \ncoordinate scalars in that case. Thus, for $j=2$, 3 and 4 we expect 1, 3 and 12\nterms in $\\sa_j$, respectively. In our notation, see (\\ref{trKj}), \nRicci-flatness implies that ${\\rm tr}[\\K_j]$ vanishes. Only a few terms remain then\nin (\\ref{ZSj}, \\ref{Zs2j}). In particular, $\\Z$ and $\\Z_1$ vanish and $\\sa_2$ \nthrough $\\sa_4$ contain only \n\\bea\n\\sa_2 & : & {\\rm tr}[\\K_{(2}\\K_{2)}] \\non\\\\\n\\sa_3 & : & {\\rm tr}[\\K_{(2}\\K_{4)}]\\ ,\\ {\\rm tr}[\\K_{(3}\\K_{3)}]\n       \\ ,\\ {\\rm tr}[\\K_{(2}\\K_2\\K_{2)}] \\non\\\\\n\\sa_4 & : & {\\rm tr}[\\K_{(2}\\K_{6)}]\\ ,\\ {\\rm tr}[\\K_{(3}\\K_{5)}]\\ ,\\ {\\rm tr}[\\K_{(4}\\K_{4)}]\n       \\ ,\\ {\\rm tr}[\\K_{(2}\\K_2\\K_{4)}]\\ ,\\ {\\rm tr}[\\K_{(2}\\K_3\\K_{3)}]\\ ,\\\\\n&\\phantom{:}& \n            {\\rm tr}[\\K_{(2}\\K_2\\K_2\\K_{2)}]\\ ,\\ \n{\\rm tr}[\\K_{\\mu(1}\\K^\\mu\\sb{1}]{\\rm tr}[\\K^\\nu\\sb{1}\\K_{1)\\nu}]\\ ,\\\n{\\rm tr}[\\K_{\\mu(1}\\K_2]{\\rm tr}[\\K_2\\K_{1)\\mu}]\\ ,\\ ({\\rm tr}[\\K_{(2}\\K_{2)}])^2 \\non\n\\ena\nHere there are 1, 3 and 9 terms respectively, so that, starting with $\\sa_4$, \nour notation is not only index-free, but also generates less terms. \nThis becomes even more pronounced for $\\sa_5$ where \\cite{FKW} informs us \nthat in the Ricci-flat case there are 67 terms, however in our notation there \nare only 17 terms.\n\\section{Conclusions}\n\\setcounter{equation}{0}\nWe have presented the explicit diagonal values of the first five (six) heat \nkernel coefficients for a general Laplace-type operator on a Riemannian \n(respectively flat) space.  To solve the pertinent recursion relations, we \nrelied not only on well known techniques (matrix notation for field indices, \nFock-Schwinger gauge and Riemann normal coordinates), but also used a new \nnotation free of spacetime indices. It is this latter compact notation which \nallows us to write down the fifth and sixth coefficient in the first place. \nInsisting also on manifest hermiticity of the results, the fifth (sixth) \ncoefficient has 26 (respectively 75) terms. They were presented here for the \nfirst time. Beyond these coefficients, the leading terms -- to the same order \nin derivatives or curvatures as needed for the fourth coefficient -- for any \nheat kernel coefficient were given in the general case. To determine the heat \nkernel coefficients, we have found it useful to proceed in a few steps, \nstarting from the simplest case of a flat spacetime without a gauge field. \nWe could show that `turning on a gauge field' and next `curving spacetime' is \ntaken care of by specific covariant substitutions in the flat space \ncoefficients. With this `dressing up' understood, the number of terms and their\nnumerical prefactors do not change in the process.\n\nIn our notation, a typical term in a heat kernel coefficient consists of a \nmaximally symmetrized (product of) covariant derivatives of the basic \ncurvatures, made into a scalar by contracting all indices. Consider e.g. the \nfollowing term\n\\bee\\label{K3K3K4}\n{\\rm tr}[\\K_{(3}\\K_3\\K_{4)}] \\= \\sfrac3{20} R^\\alpha\\sb{(\\kappa}\\sp\\beta\\sb{\\kappa;\\lambda}\nR^\\beta\\sb\\lambda\\sp\\gamma\\sb{\\mu;\\mu} R^\\gamma\\sb\\nu\\sp\\alpha\\sb{\\nu;\\rho\\r)}\n\\ene\nwhich appears in the fifth coefficient. Since we do not integrate over \nspacetime, partial integrations are not permitted. Furthermore, as long as we\ndo not write out the symmetrization, the Bianchi identity can not be used here.\nThus our notation gives a certain degree of uniqueness to the appearance of the\nheat kernel coefficients, missing in a more conventional notation with \nexplicit spacetime indices.\n\nOf course, depending on the application one has in mind, our notation may or \nmay not be useful. We plan to calculate the chiral anomaly based on our \nresults, i.e. essentially evaluate the spinor-trace ${\\rm tr}[\\gamma_{2j+1}\\sa_j]$ in \n$d=2j$ dimensions.\nIn this case elegant and complete results are known \\cite{ZWZ}, using the \nlanguage of differential forms. It should be interesting to see how this can \nbe related to our use of symmetric tensors. Also the gravitational anomalies \n\\cite{AGW} in $d=4j+2$ are computable in our framework. Possibly, the absence \nof certain terms from the heat kernel coefficients found here, see \n(\\ref{overlap}), plays a role in this connection. \n\nAnother task, now in progress \\cite{BoV}, is to work out the functional trace \nof the diagonal heat kernel coefficients given here. In that case we can \nmaintain our index-free notation (e.g. the integral of (\\ref{K3K3K4}) is easily\nseen to vanish without writing out the indices).\nAlthough it would probably hold few surprises, {\\it cf} \\cite{FrT}, the \nresult for the integrated and traced fifth coefficient could be used to study \nfor the first time the one-loop short distance divergences of ten-dimensional \nsupergravity.\n\nAs this paper neared completion, we were informed by the authors of \n\\cite{FHSS1} that they had extended their results so as to include gauge \nfields. In \\cite{FHSS2} they present the functional trace of the first six \nheat kernel coefficients for this case in flat spacetime, reduced to a \nso-called minimal basis. It should be possible to compare their result to the\nsixth heat kernel coefficient as presented here, after taking its trace and \nintegrating it. \n\nFinally, to avoid misunderstanding, we should mention that a recent preprint \n\\cite{Kir} with the title ``The $a_5$ heat kernel coefficient on a manifold \nwith boundary'' is not concerned with the fifth heat kernel coefficient as\npresented here. Rather, on a manifold with boundary the expansion (\\ref{hkc})\nis in powers of $\\sqrt\\tau$ rather than $\\tau$ and therefore our $a_j$ corresponds\nto $a_{2j}$ of \\cite{Kir}. Thus, in the terminology of \\cite{Kir} we have \ndetermined here `the volume part of $a_{10}$'. \n\\vskip.5cm\nI would like to thank J.-P. B\\\"ornsen for his assistance in using Mathematica\nto verify some of the results presented here. I am also grateful to I. Avramidi\nfor useful correspondence.\n\\newpage\n\\newpage\n\\begin{appendix}\n\\section{Notation}\n\\setcounter{section}{1}\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{\\Alph{section}.\\arabic{equation}}\nWe use Greek letters for spacetime indices. Indices $j,\\,k,\\,\\el,\\dots$ \nsimply enumerate various objects. We take differential operators to act on \neverything to their right. If this is not intended, we use comma (semicolon) \nnotation for partial (covariant) derivatives\n\\bee\nF_{,\\mu} \\,\\equiv\\, [\\pa_\\mu, F] \\quad ,\\quad F_{;\\mu} \\,\\equiv\\,[\\de_\\mu, F]\n\\ene\nA semicolon denotes simultaneous gauge and gravitational covariant \ndifferentiation. We use $a\\equiv b$ to define $a$ in terms of $b$.\nOur curvature conventions are fixed by ($V_\\lambda$ is a gauge singlet)\n\\bee\\label{Rcon}\n[\\de_\\nu,\\de_\\mu] V_\\lambda \\= 2\\,V_{\\lambda;[\\mu\\nu]}\\= R^\\kappa\\sb{\\lambda\\mu\\nu} V_\\kappa \\quad ,\\quad\nR_{\\lambda\\nu}\\,\\equiv\\,R^\\kappa\\sb{\\lambda\\kappa\\nu} \\quad ,\\quad R\\,\\equiv\\, R^\\lambda\\sb\\lambda\n\\ene\nIn Fock-Schwinger gauge and Riemannian coordinates, see (\\ref{FSgauge}) and\n(\\ref{Rnc}), we have\\footnote{\\mbox{Parenthesis around $j$ explicit indices \ndenotes total symmetrization (with division by $j!$).}}\n\\bee\nF_{,\\mu_1\\dots\\mu_j}(0) = F_{;(\\mu_1\\dots\\mu_j)}(0) \\quad ,\n\\ene\nfor any, possibly matrix valued, general coordinate scalar $F(x)$. This \nproperty allows one to immediately covariantize the partial derivatives. Due \nto the total symmetry here it is convenient and sufficient to keep track of \nonly the number of indices, leading us to define the {\\it sans serif} symbols\n\\bee\n\\Phi_j\\= F_{;(\\mu_1\\dots\\mu_j)}(0)\n\\ene\nWe thus use an index-free notation for totally symmetrized tensors.\nAt the same time, such a symbol implies evaluation at the origin.\nFollowing Avramidi, we use parenthesis around an even enumerative label to \nindicate not only total symmetrization, but also full contraction as in\n\\bee\n\\Phi_{(2j)} \\= F_{;(\\mu_1\\mu_1\\dots\\mu_j\\mu_j)}(0)\n\\ene\nSince the metric at the origin is flat there is no need to write the indices\nin their covariant respectively contravariant positions.\n{\\it Par abus de language} an odd label enclosed in parenthesis denotes \nsymmetrization and contraction of all but one of the involved indices. This \nwill occur only if there is exactly one other factor with an odd label,\nas e.g. in\n\\bee\n\\Phi_{(3)} \\Gamma_{(3)}\\= F_{;(\\kappa\\mu\\m)}(0) G_{;(\\kappa\\nu\\n)}(0)\n\\ene\nWe generalize Avramidi's notation by allowing such a simultaneous \nsymmetrization and contraction to extend over several factors, e.g.\n\\bee\\label{FkGl}\n\\Phi_{(k}\\Gamma_{2j-k)}\\=F_{;(\\mu_1\\,\\dots\\,\\mu_k}(0)\\,\nG_{;\\mu_{k+1}\\,\\dots\\,\\mu_{2j})}(0)\n\\,\\delta^{\\mu_1\\mu_2}\\,\\dots\\,\\delta^{\\mu_{2j-1}\\mu_{2j}}\n\\ene\nIf desired, such an expression can be written out in such a way that the total \nsymmetrizations extend only over the individual factors, as e.g. in\n\\bee\n\\Phi_{(2}\\Gamma_{2)} \\= \\sfrac13 F_{;(\\mu\\m)}(0) G_{;(\\nu\\n)}(0)\n+\\sfrac23 F_{;(\\mu\\nu)}(0) G_{;(\\mu\\nu)}(0)\n\\ene\nIn general, expanding (\\ref{FkGl}) in this way yields at first $(2j-1)!!$ \nterms. However, due to the total symmetry of the covariant derivatives acting \non each factor and assuming that $k\\leq j$, there remain only $[k\/2]+1$ terms, \n$[k\/2]$ being the maximal number of self-contractions possible for $\\Phi_k$. \nFinding the coefficient of each term is a combinatorical problem with the \nfollowing solution\n\\bea\\label{combi}\n\\Phi_{(2j}\\Gamma_{2k)} &=& {1\\ov (2j+2k-1)!!} \\su_{\\el=0}^j \nP^{2j}_{j-\\el} P^{2k}_{k-\\el} (2\\el)! \\,\n\\Phi^{(2j-2\\el)}_{\\mu_1\\dots\\mu_{2\\el}}\\,\\Gamma^{(2k-2\\el)}_{\\mu_1\\dots\\mu_{2\\el}}\\non\\\\\n\\Phi_{(2j+1}\\Gamma_{2k+1)} &=& {1\\ov (2j+2k+1)!!} \\su_{\\el=0}^j \nP^{2j+1}_{j-\\el} P^{2k+1}_{k-\\el} (2\\el +1)!\\,\n\\Phi^{(2j-2\\el)}_{\\mu_1\\dots\\mu_{2\\el+1}}\\,\\Gamma^{(2k-2\\el)}_{\\mu_1\\dots\\mu_{2\\el+1}}\n\\non\\\\\nP^j_k &\\equiv & {j{\\cal H} 2k}(2k-1)!!\\quad ,\\quad k\\leq [j\/2]\\quad ,\\quad \n(-1)!! \\equiv 1\n\\ena\nHere $P^j_k$ is the number of ways in which one can choose $k$ {\\it pairs} out\nof $j$ objects. The covariant derivatives on $\\Phi$ respectively $\\Gamma$ on the \nright hand side are understood to have been totally symmetrized. In general\nthere will be more than two factors, but in practice these are easily taken \ncare of, as e.g. in\n\\bee\n\\Phi_{(2}\\Gamma_2\\H_{2)} \\= \\sfrac1{15} \\Phi_{(2)}\\Gamma_{(2)}\\H_{(2)}\n+ \\sfrac2{15}(\\Phi_{(2)}\\Gamma_{;(\\mu\\nu)}\\H_{;(\\mu\\nu)} + 2\\ {\\rm more})\n+ \\sfrac8{15}\\Phi_{;(\\mu\\nu)}\\Gamma_{;(\\nu\\rho)}\\H_{;(\\rho\\mu)} \n\\ene\nMultinomial coefficients are defined as usual by\n\\bee\n{j{\\cal H} k_1,\\dots,k_N}\\= {j!\\ov k_1 !\\dots k_{N+1}!}\\quad ,\\quad\nk_{N+1}\\,\\equiv\\,j\\,-\\su_{n=1}^N k_n\n\\ene\nIf the index $k_{N+1}$ appears in the summand of a sum involving this \nmultinomial coefficient, then it is understood to have the indicated value. \nThis convention applies also to alphabetic ordering as e.g. in\n\\bee\\label{alphaord}\n\\su_{k=0}^j {j{\\cal H} k} F_k G_\\el \\,\\equiv\\, \\su_{k=0}^j {j{\\cal H} k} F_k G_{j-k}\n\\ene\n\\section{Moving boxes}\n\\setcounter{equation}{0}\nIn the main text we gave an expression, eq (\\ref{Xres}), for the diagonal \nvalues of the heat kernel coefficients $\\sa_j$ through third order in the \nmatrix potential $\\Xi$ which holds only for $j\\leq 5$. We indicated in \n(\\ref{mobo}) how to obtain $\\sa_6$ as well without generating terms with \noverlapping derivatives. In the general case we rely on the following lemma to\nmove the boxes from $G$ to $F$\n\\bea\nF_{(2n} G_{2p)(2q-2p)}&=&\\su_{j=0}^n \\beta_j(n,p,q) F_{(2n-2j)(2j} G_{2q)} \n\\quad ,\\quad n\\leq p\\leq q  \\\\\n\\beta_j(n,p,q) &=& {q-p{\\cal H} n-j}\\ {f(j,p,q)\\ov f(n,q,p)} \\quad ,\\quad\nf(j,p,q) \\= (2j+2q-1)!!\\, P^{2p}_{p-j} \\non\n\\ena\nwith a similar expression for the case $F_{(2n+1} G_{2p+1)(2q-2p-2)}$.\nNote that this can only be done when all indices are contracted. \nTo prove the lemma, expand both sides using (\\ref{combi}) and compare.\nWe also note the following exceptional case\n\\bee\nF_{(k+2n} G_{k)(2n)} \\= F_{(2n)(k} G_{k+2n)} \n\\ene\n\\section{$\\sa_6$ in flat space}\n\\setcounter{equation}{0}\nExpressions (\\ref{deriv}), (\\ref{Xres}), (\\ref{Subs}) and (\\ref{Zres}) \nsuffice to write down the sixth heat kernel coefficient in a flat space, but\nwith a gauge connection. We obtain\n\\bea\n&&\\sa_6 =\\Z^6\n+\\sfrac57\\{\\Z^4\\Z_{(2)}\\} \n+\\sfrac87\\{\\Z^3\\Z_{(2)}\\Z\\}\n+\\sfrac97\\Z^2\\Z_{(2)}\\Z^2 \n+\\sfrac27\\Z_{(1}\\do\\HZ^3\\Z_{1)} \n+\\sfrac47\\{\\Z\\Z_{(1}\\do\\HZ^2\\Z_{1)}\\} \\non\\\\\n&&\n+\\sfrac67\\{\\Z^2\\Z_{(1}\\do\\HZ\\Z_{1)}\\}\n+\\sfrac87\\Z\\Z_{(1}\\do\\HZ\\Z_{1)}\\Z \n+\\sfrac87\\{\\Z^3\\Z_{(1}\\do\\Z_{1)}\\} \n+\\sfrac{12}7\\{\\Z^2\\Z_{(1}\\do\\Z_{1)}\\Z\\} \n+\\sfrac5{14}\\{\\Z^3\\Z_{(4)}\\} \\non\\\\\n&&\n+\\sfrac9{14}\\{\\Z^2\\Z_{(4)}\\Z\\}  \n+\\sfrac{12}7\\{\\Z\\Z_{(1}\\do\\Z_{3)}\\Z\\}\n+\\sfrac{15}{14}\\{\\Z^2\\Z_{(1}\\do\\Z_{3)}\\} \n+\\sfrac97\\{\\Z^2\\Z_{(3}\\do\\Z_{1)}\\} \n+\\sfrac57\\{\\Z\\Z_{(1}\\do\\HZ\\Z_{3)}\\} \\non\\\\\n&&\n+\\sfrac67\\{\\Z\\Z_{(3}\\do\\HZ\\Z_{1)}\\} \n+\\sfrac5{14}\\{\\Z_{(1}\\do\\HZ^2\\Z_{3)}\\}\n+\\sfrac97\\{\\Z^2\\Z_{(2}\\do\\Z_{2)}\\}\n+\\sfrac37\\{\\Z^2\\Z_{(2)}\\sp{2}\\}\n+\\sfrac3{14}\\Z_{(2}\\do\\HZ^2\\Z_{2)} \\non\\\\\n&&\n+\\sfrac37 \\Z_{(2)}\\Z^2\\Z_{(2)} \n+\\sfrac9{14}\\{\\Z\\Z_{(2}\\do\\HZ\\Z_{2)}\\} \n+\\sfrac47\\{\\Z\\Z_{(2)}\\Z\\Z_{(2)}\\} \n+\\sfrac{27}{14}\\Z\\Z_{(2}\\do\\Z_{2)}\\Z\n+\\sfrac47 \\Z\\Z_{(2)}\\sp{2}\\Z \\non\\\\\n&&\n+\\sfrac9{14}\\{\\Z_{(1}\\do\\HZ_1\\HZ\\Z_{2)}\\} \n+\\sfrac47\\{\\Z_{(1}\\do\\Z_{1)}\\Z\\Z_{(2)}\\}   \n+\\sfrac{12}7\\{\\Z\\Z_{(1}\\do\\HZ_1\\Z_{2)}\\} \n+\\sfrac87\\{\\Z\\Z_{(1}\\do\\Z_{1)}\\Z_{(2)}\\}  \\non\\\\\n&&\n+\\sfrac{27}{14}\\{\\Z\\Z_{(2}\\do\\HZ_1\\Z_{1)}\\} \n+\\sfrac47\\{\\Z\\Z_{(2)}\\Z_{(1}\\do\\Z_{1)}\\} \n+\\sfrac67\\{\\Z_{(1}\\do\\HZ\\HZ_1\\Z_{2)}\\} \n+\\sfrac27\\{\\Z_{(1}\\do\\HZ\\Z_{1)}\\Z_{(2)}\\}  \\non\\\\\n&&\n+\\sfrac{18}7\\{\\Z\\Z_{(1}\\do\\HZ_2\\Z_{1)}\\}  \n+\\sfrac97\\{\\Z_{(1}\\do\\HZ\\HZ_2\\Z_{1)}\\} \n+\\sfrac{27}{14}\\Z_{(1}\\do\\HZ_1\\HZ_1\\Z_{1)}  \n+\\sfrac47\\Z_{(1}\\do\\Z_{1)}\\,\\Z_{(1}\\do\\Z_{1)} \\non\\\\\n&&\n+\\sfrac5{42}\\{\\Z^2\\Z_{(6)}\\} \n+\\sfrac4{21}\\Z\\Z_{(6)}\\Z  \n+\\sfrac5{21}\\{\\Z_{(1}\\do\\HZ\\Z_{5)}\\}\n+\\sfrac{10}{21}\\{\\Z\\Z_{(1}\\do\\Z_{5)}\\} \n+\\sfrac47 \\{\\Z\\Z_{(5}\\do\\Z_{1)}\\} \\non\\\\\n&&\n+\\sfrac{25}{84}\\{\\Z_{(2}\\do\\HZ\\Z_{4)}\\}  \n+\\sfrac{25}{28}\\{\\Z\\Z_{(2}\\do\\Z_{4)}\\} \n+\\sfrac{20}{21}\\{\\Z\\Z_{(4}\\do\\Z_{2)}\\} \n+\\sfrac{20}{21}\\{\\Z\\Z_{(3}\\do\\Z_{3)}\\} \n+\\sfrac5{21}\\Z_{(3}\\do\\HZ\\Z_{3)} \\non\\\\\n&&\n+\\sfrac{25}{28}\\{\\Z_{(1}\\do\\HZ_1\\Z_{4)}\\}  \n+\\sfrac{10}7  \\Z_{(1}\\do\\HZ_4\\Z_{1)} \n+\\sfrac{10}7\\{\\Z_{(1}\\do\\HZ_2\\Z_{3)}\\} \n+\\sfrac{40}{21}\\{\\Z_{(1}\\do\\HZ_3\\Z_{2)}\\} \n+\\sfrac{20}{21}\\{\\Z_{(2}\\do\\HZ_1\\Z_{3)}\\} \\non\\\\\n&&\n+\\sfrac{40}{21} \\Z_{(2}\\do\\HZ_2\\Z_{2)} \n+\\sfrac3{14}\\{\\Z\\Z_{(4)}\\Z_{(2)}\\} \n+\\sfrac5{28}\\{\\Z_{(2)}\\Z\\Z_{(4)}\\} \n+\\sfrac5{28}\\{\\Z\\Z_{(2)}\\Z_{(4)}\\}\n+\\sfrac37\\{\\Z\\Z_{(3)}\\do\\Z_{(3)}\\} \\non\\\\\n&&\n+\\sfrac27\\Z_{(3)}\\do\\HZ\\Z_{(3)}\n+\\sfrac5{28}\\{\\Z_{(1}\\do\\Z_{1)}\\Z_{(4)}\\}  \n+\\sfrac9{14}\\{\\Z_{(1}\\do\\Z_{2)}\\Z_{(3)}\\} \n+\\sfrac37\\{\\Z_{(2}\\do\\Z_{1)}\\Z_{(3)}\\} \n+\\sfrac5{14}\\{\\Z_{(2)}\\Z_{(1}\\do\\Z_{3)}\\} \\non\\\\\n&&\n+\\sfrac37\\{\\Z_{(1}\\do\\Z_{3)}\\Z_{(2)}\\} \n+\\sfrac37\\{\\Z_{(2}\\Z_{2)}\\Z_{(2)}\\}  \n+\\sfrac17  \\Z_{(2)}\\sp{3} \n+\\sfrac1{42}\\{\\Z\\Z_{(8)}\\} \n+\\sfrac2{21}\\{\\Z_{(1}\\do\\Z_{7)}\\}\n+\\sfrac29\\{\\Z_{(2}\\do\\Z_{6)}\\} \\non\\\\\n&&\n+\\sfrac13\\{\\Z_{(3}\\do\\Z_{5)}\\} \n+\\sfrac13\\Z_{(4}\\do\\Z_{4)} \n+\\sfrac5{126}\\{\\Z_{(2)}\\Z_{(6)}\\} \n+\\sfrac5{42}\\{\\Z_{(3)}\\do\\Z_{(5)}\\} \n+\\sfrac3{14}\\Z_{(2)(2}\\do\\Z_{2)(2)} \n+\\sfrac1{462} \\Z_{(10)}\\non\\\\\n\\ena\nThe $\\Z$'s appearing here were defined in (\\ref{Zj}), (\\ref{Z2j}) and \n(\\ref{Zhatdef}). Note in particular that $\\Z\\=\\Xi$. There is a total of 75 \nterms. In curved space, $\\sa_6$ will look exactly the same, but in that case \nwe do not know the values for some of the $\\Z$'s.\n\\newpage\n\\section{Consequences of Fock-Schwinger gauge}\n\\setcounter{equation}{0}\nTo prove (\\ref{AF}), we start from\n\\bee\n(A_{(\\mu_1} F)_{,\\mu_2\\dots\\mu_j)}(0) \\= 0\n\\ene\nwhich holds for any function $F$ in Fock-Schwinger gauge, see (\\ref{symA}).\nNow take $j=2n+2$, contract {\\it all} indices and write out with respect to\nthe index of the gauge connection. It is essential for the proof that, due to \nthe full contraction, all indices here are dummies. \n\nA further consequence of (\\ref{symA}) is the following.\nIf we define $\\check\\Z$ to act to its {\\it left} as follows\n\\bee\n\\Z_{(j}\\do{\\check\\Z}_n\\do\\,\\equiv\\,\n\\Z_{(j}\\do \\Z_n\\do \\,-\\,{2j\\ov n+1}\\Z_{\\nu(j-1}\\do\\Y_{n+1}\\sp\\nu \n\\ene\nwhere we left out inessential factors to the right, then we have the following\nalternative notations\n\\bea\n\\Z_{(j}\\do{\\check\\Z}_n\\do\\Z_{k)} &=& \\Z_{(j}\\do\\HZ_n\\Z_{k)} \\non\\\\\n\\Z_{(j}\\do{\\check\\Z}_p\\do\\HZ_n\\Z_{k)} &=& \\Z_{(j}\\do\\HZ_p\\HZ_n\\Z_{k)}\n\\ena\nIn the last case the `check' notation has the advantage that it generates less\nterms than the `hat' notation, namely\n\\bea\n\\Z_{(j}\\do{\\check\\Z}_p\\do\\HZ_n\\Z_{k)} &=& \\Z_{(j}\\do\\Z_p\\do\\Z_n\\Z_{k)} \n\\,-\\,{2j\\ov p+1}\\,\\Z_{\\mu(j-1}\\do\\Y_{p+1}\\sp\\mu\\Z_n\\Z_{k)} \n\\,+\\,{2k\\ov n+1}\\,\\Z_{(j}\\do\\Z_p\\do\\Y_{n+1}\\sp\\nu\\Z_{k-1)\\nu} \\non\\\\\n&&\\quad\n\\,-\\,{4jk\\ov (p+1)(n+1)}\\,\\Z_{\\mu(j-1}\\dg\\Y_{p+1}\\sp\\mu\\Y_{n+1}\\sp\\nu\\Z_{k-1)\\nu}\n\\ena\n\\section{How to get your $\\Z$'s}\n\\setcounter{equation}{0}\nHere we shall give some details on how to obtain (\\ref{ZSj}). The steps for \n(\\ref{Zmj}) are very similar so we omit them. Our starting point is \n(\\ref{Zscalar}). We take $j$ partial derivatives of this expression and \nevaluate at the origin ($m\\equiv j-k-\\el$) to obtain\n\\bee\n\\Zs_j = \\sfrac12\\su_{k=0}^j {j+1{\\cal H} k} \\sg^{\\mu\\nu}\\sb{(k}\\B_{j-k\\,\\mu)\\nu}\n-\\sfrac14\\su_{k=1}^{j-1}\\su_{\\el=1}^{j-k} {j{\\cal H} k,\\el}\n\\B_{\\mu(k}\\sg^{\\mu\\nu}\\sb{m}\\B_{\\el)\\nu} \n\\ene\nwhere we used that $B$ and its first derivative vanish there.\nSeparate off the terms with undifferentiated inverse metric\n\\bea\\label{diffm}\n\\Zs_j &=& \\sfrac12\\,\\B_{\\mu\\m\\,j}\n+\\sfrac12\\su_{k=2}^j {j+1{\\cal H} k}\\sg^{\\mu\\nu}\\sb{(k}\\B_{j-k\\,\\mu)\\nu} \\non\\\\\n&&\\quad\n-\\sfrac14\\su_{k=1}^{j-1} {j{\\cal H} k} \\B_{\\mu(k} \\B_{\\el)\\mu}\n-\\sfrac14\\su_{k=1}^{j-3} \\su_{\\el=1}^{j-k-2} {j{\\cal H} k,\\el}\n                                \\B_{\\mu(k}\\sg^{\\mu\\nu}\\sb{m}\\B_{\\el)\\nu} \n\\ena\nInserting (\\ref{invmet}) and (\\ref{Bj}) and collecting terms of the same order\nin the curvatures $\\K$ and $\\Y$, we obtain (\\ref{ZSj}). Up to here all \ncontractions were explicitly indicated and the parenthesis denoted \nsymmetrization only. If we now replace $j$ by $2j$ and fully contract,\nthe second term in (\\ref{diffm}) vanishes due to (\\ref{gsym0}) and we find\n\\bee\n\\Zs_{(2j)} = \\sfrac12\\,\\B_{(2j+2)}\n-\\sfrac14\\su_{k=1}^{2j-1} {2j{\\cal H} k} \\B_{\\mu(k} \\B_{\\el)\\mu}\n-\\sfrac14\\su_{k=1}^{2j-3} \\su_{\\el=1}^{2j-k-2} {2j{\\cal H} k,\\el}\n    \\B_{\\mu(k}\\sg^{\\mu\\nu}\\sb{m}\\B_{\\el)\\nu} \n\\ene\nThis yields (\\ref{Zs2j}). Thus we see that, since $B$ is at least of first \norder in curvature, finding the heat kernel coefficients through order $n$ \nin the curvature requires the inverse metric only through order $n-2$. \n\nWriting $G$ for the inverse metric, we find through sixth order in the \nnormal coordinates\n\\bea\\label{invmetexpl}\n&& G(x) \\= \\I +\\,2\\K_2 {x^2\\ov 2!}\\,+\\,\n2\\K_3 {x^3\\ov 3!}\\,+\\,\n2(\\K_4 +\\sfrac{36}{5}\\K_2\\sp{2}) {x^4\\ov 4!}\\,+\\,\n2(\\K_5 + 12\\{\\K_2\\K_3\\}) {x^5\\ov 5!} \\non\\\\\n&&\\quad \n+2(\\K_6 +\\sfrac{125}7 \\{\\K_2\\K_4\\} +\\sfrac{170}7 \\K_3\\sp{2}\n   +\\sfrac{720}7 \\K_2\\sp{3}) {x^6\\ov 6!}\\,+\\, O(x^7) \n\\ena\nFor the logarithm of the Van Vleck-Morette determinant we find through tenth \norder in the normal coordinates\n\\bea\\label{Bexpl}\n&&B(x) \\={\\rm tr}\\Big[\\,\n \\K_2 {x^2\\ov 2!}\\,+\\, \n \\K_3 {x^3\\ov 3!}\\,+\\,\n(\\K_4 +\\sfrac65\\K_2\\sp{2}) {x^4\\ov 4!}\\,+\\,\n(\\K_5 + 4 \\K_2\\K_3  )      {x^5\\ov 5!}      \\non\\\\\n&&\n+(\\K_6+\\sfrac{40}7\\K_2\\K_4+\\sfrac{30}7\\K_3\\sp{2}\n      +\\sfrac{48}7\\K_2\\sp{3}){x^6\\ov 6!}\\,\n+\\,(\\K_7+\\sfrac{15}2\\K_2\\K_5 + 15\\K_3\\K_4 + 48\\K_2\\sp{2}\\K_3) {x^7\\ov 7!}\\non\\\\\n&&\n+(\\K_8+\\sfrac{28}3\\K_2\\K_6 +\\sfrac{70}3\\K_3\\K_5+\\sfrac{140}9\\K_4\\sp{2}\n      +\\sfrac{272}3\\K_2\\sp{2}\\K_4 +\\sfrac{400}3\\K_2\\K_3\\sp{2}\n      +\\sfrac{432}5\\K_2\\sp{4}) {x^8\\ov 8!} \\non\\\\\n&&\n+(\\K_9\n+\\sfrac{ 56}5\\K_2\\K_7\n+\\sfrac{168}5\\K_3\\K_6 \n+ 56\\K_4\\K_5 \n+\\sfrac{756}5\\K_2\\sp{2}\\K_5 \n+ 584\\K_2\\K_3\\K_4 \n+ 144\\K_3\\sp{3} \\non\\\\\n&&\\qquad \n+\\sfrac{5184}5\\K_2\\sp{3}\\K_3) {x^9\\ov 9!}\\non\\\\\n&&\n+(\\K_{10}\n+\\sfrac{144}{11}\\K_2\\K_8   +\\sfrac{504}{11}\\K_3\\K_7 +\\sfrac{1008}{11}\\K_4\\K_6 \n+\\sfrac{630}{11}\\K_5\\sp{2} +\\sfrac{12768}{55}\\K_2\\sp{2}\\K_6 \n+\\sfrac{12180}{11}\\K_2\\K_3\\K_5 \\non\\\\\n&&\\qquad\n+\\sfrac{8000}{11}\\K_2\\K_4\\sp{2} +1080\\K_3\\sp{2}\\K_4\n+\\sfrac{26880}{11}\\K_2\\sp{3}\\K_4 +3576\\K_2\\sp{2}\\K_3\\sp{2} \n+\\sfrac{19320}{11}\\K_2\\K_3\\K_2\\K_3 \\non\\\\\n&&\\qquad\n+\\sfrac{20736}{11}\\K_2\\sp{5}) {x^{10}\\ov 10!}\\,\\Big]\\ +\\ O(x^{11})\n\\ena\nFor the gauge connection (with covariant index) it suffices to know that\n\\bea\\label{Aexpl}\nA(x) \\! &=&\\!  \\Y_1 x^1\\,  +\\Y_2 {x^2\\ov 2!}\\,\n+(\\Y_3 -{3\\ov 2}\\K_2\\Y_1) {x^3\\ov 3!}\\,\n+(\\Y_4 -{18\\ov 5}\\K_2\\Y_2 -{8\\ov 5}\\K_3\\Y_1) {x^4\\ov 4!} \\non\\\\\n&&\\quad\n+\\Big(\\Y_5 -{20\\ov 3}\\K_2\\Y_3 - 5\\K_3\\Y_2 -\\sfrac13(5\\K_4-9\\K_2\\sp{2})\\Y_1\\Big)\n {x^5\\ov 5!} \\non\\\\\n&&\\quad\n+\\Big(\\Y_6-{75\\ov 7}\\K_2\\Y_4-{80\\ov 7}\\K_3\\Y_3-{9\\ov 7}(5\\K_4-9\\K_2\\sp{2})\\Y_2\n\\non\\\\  \n&&\\quad\\qquad  -{12\\ov 7}(\\K_5-2\\K_3\\K_2-4\\K_2\\K_3)\\Y_1\\Big) {x^6\\ov 6!} \\ \n+\\ O(x^7)\n\\ena\nTo convert the above expansions to conventional notation, use (\\ref{defY}) \nand (\\ref{defK}). \n\\section{Symmetry factors}\n\\setcounter{equation}{0}\nBelow we give a table containing all symmetry factors $S\/P_N$, defined in \n(\\ref{symmfac}), for $N\\leq 4$. The list $k_1\\dots k_N$ in the \nsecond column is an abbreviation for ${\\rm tr}[\\K_{k_1}\\dots \\K_{k_N}]$. \nIt is to be understood that $k$, $\\el$, $m$ and $n$ are different integers,\neach having at least the value 2.\n\\vskip.5cm\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\\hline\n$N$ & $k_1\\dots k_N$    & $S$ & $S\/P_N$ \\\\ \\hline\\hline \n    & $k\\ k$            &  1  &  1\/2    \\\\ \\cline{2-4}\n\\raisebox{1.5ex}[-1.5ex]{2}\n    & $k\\ \\el$            &  2  &   1     \\\\ \\hline\n    & $k\\ k\\ k$         &  1  &  1\/6    \\\\ \\cline{2-4}\n  3 & $k\\ k\\ \\el$       &  3  &  1\/2    \\\\ \\cline{2-4}\n    & $k\\ \\el\\ m$       &  6  &   1     \\\\ \\hline\n    & $k\\ k\\ k\\ k$      &  1  &  1\/8    \\\\ \\cline{2-4}\n    & $k\\ \\el\\ k\\ \\el$  &  2  &  1\/4    \\\\ \\cline{2-4}\n    & $k\\ k\\ k\\ \\el$    &  4  &  1\/2    \\\\ \\cline{2-4}\n  4 & $k\\ k\\ \\el\\ \\el$  &  4  &  1\/2    \\\\ \\cline{2-4}\n    & $k\\ \\el\\ k\\ m$    &  4  &  1\/2    \\\\ \\cline{2-4}\n    & $k\\ k\\ \\el\\ m$    &  8  &   1     \\\\ \\cline{2-4}\n    & $k\\ \\el\\ m\\ n$    &  8  &   1     \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{appendix}\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Changing Look active galactic nuclei (CL AGNs) are objects which undergo dramatic variability of the emission line profiles and classification type, which can change from  type 1 (showing both broad and narrow lines) to type 1.9 (where the broad lines almost disappear) or vice versa within a short time interval (typically a few months). The most extreme examples of known CL AGNs exhibit very strong changes not only in their broad Balmer lines, but also in their more narrow high-ionization lines like [FeVII], [FeX] and [FeXIV] \\citep[see e.g., ][]{Parker2016}. There are currently only a few dozen cases of CL AGNs recorded, but their number is growing steadily all the time, in particular during last 5 years \\citep[see e.g., ][],{Shappee2014, Koay2016, MacLeod2016, Rinco2016, Rumbaugh2018, Yang2018}. More than 20 AGNs have been witnessed as changing their spectral look in the X-ray domain \\citep[see references in][]{Ricci2016}. The origin of the UV and optical variability in AGN, and its correlation with the X-ray variability, is not well understood. Therefore investigations of CL AGNs during such transitions can be very helpful for understanding the central structure of AGNs as well as the physics underpinning such dramatic changes.\n\nNGC~1566 is a galaxy with a very well-studied variable active nucleus. It is a nearly face-on spiral galaxy with morphological type SAB(s)bc. It is one of the brightest ($V\\approx$10.0 mag) and nearest galaxies with AGN in the South Hemisphere (the distance is still subject to large uncertainty and so we adopt the  median value given by NED{\\footnote{The NASA\/IPAC Extragalactic Database (NED) is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration}} , i.e. $D\\approx7.2$ Mpc). The NGC~1566 nucleus exhibits a Seyfert emission spectrum \\citep{Vaucouleurs1961, Shobbrook1966} that was later classified as type 1. Spectral variations had been witnessed in this object even in some of the earliest spectroscopic investigations in 1956 \\citep{Vaucouleurs1961} and in 1966 \\citep{Shobbrook1966, Pastoriza1970}. Other early observed spectra \\citep{Osmer1974} showed that the H$\\alpha$ and H$\\beta$ profiles had broad components with strong asymmetry towards the red. NGC~1566 can be classified as a CL object since it has exhibited strong variations in the past decades \\citep{Alloin1985, Kriss1991, Baribaud1992, Winkler1992}. During the 1970s and 1980s the object spent much time in a low-luminosity state where the broad permitted lines were nearly undetectable \\citep{Alloin1986}. However it had several recurrent brightening events when its type changed from Sy 1.9-Sy 1.8 to Sy 1.5-Sy 1.2 states. The most significant recorded past outbursts were observed in 1966 and 1992 \\citep{Silva2017}. The first multiwavelength investigations of the NGC~1566 variability from X-ray to IR, as well as first IR reverberation mapping in the object, were done by \\cite{Baribaud1992}. They found a possible delay in the variations of the IR $K$ band relative to those of H$_\\alpha$ of about $2\\pm1$ month. The lag of the $K$ band variability relative to $J$ was however elsewhere found to be less than 20 days \\citep{Oknyansky2001}. \\cite{Alloin1985} and \\cite{Silva2017} found the broad line region upper limit radius to be 20 light-days and 15 light-days respectively. These values together with the IR time delays may be used to estimate the location of the inner edge of the dust torus \\citep[see e.g.,][]{Clavel2000, Netzer2015}. In the X-ray regime, studies of the Fe$\\alpha$ emission line suggest that it is linked to reflection from the torus \\citep{Kawamuro2013}. \n\nOn 2018 June 12-19 data from the {\\it INTEGRAL} observatory showed that NGC~1566 was in outburst in hard X-rays \\citep{Ducci2018}, and led to follow-up observations with the {\\it Swift} observatory \\citep{Ferrigno2018, Grupe2018, Kuin2018}. An ASAS-SN $V$-band light curve for the period 2014--2018 shows that the nuclear started to brighten significantly around 2017 September  \\citep{Dai2018}. The 2014-2018 mid-infrared light curves of the object show that the nucleus brightened by 1 mag at 3.4 $\\mu$m and 1.4 mag at 4.6 $\\mu$m between 2017 January  and 2018 July \\citep{Cutri2018}.      \n\nIn this paper we report on follow-up optical and UV photometric observations (MASTER (Mobile Astronomical System of TElescope Robots); {\\it Swift}\/UVOT), as well as new X-ray ({\\it Swift}\/XRT) data. We also report on our optical spectroscopy at the South African Astronomical Observatory (SAAO) of what is one of the brightest and most nearby CL cases witnessed to date \\citep{Oknyansky2018B}.\n\n\n\\section{Observations, instruments and reduction}\n\n\\subsection{{\\it Swift}: optical, ultraviolet, and X-ray observations}\n\n\\begin{figure}\n\\includegraphics[scale=0.8,angle=0,trim=0 0 0 0]{FIG1.eps}\n \\caption {Multiwavelength observations of NGC~1566 spanning from 2007 Dec. 11 through 2018 Aug. 25. {\\it Top panel:} The {\\it Swift}\/XRT 0.5--10 keV  X-ray flux (in erg cm$^{-2}$ s$^{-1}$) -- (filled circles). {\\it Bottom panel:} Optical--UV photometric observations. The large open circles are  MASTER unfiltered optical photometry of NGC~1566 reduced to the $V$ system while the points are $V$ ASAS-SN (nightly means) reduced to the {\\it Swift} $V$ system. The open boxes are $V$ data obtained by {\\it Swift}. The filed circles are MASTER $V$ photometry. The small open boxes are $U$ and $UVW1$ UVOT {\\it Swift} photometry.}\n    \\label{fig1}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[scale=0.55,angle=0,trim=0 0 0 0]{FIG2.eps}\n \\caption{Optical--UV photometric observations of NGC~1566 shown for the last year only. The large open circles are MASTER unfiltered optical photometry of NGC~1566 reduced to the $V$ system while the points are $V$ ASAS-SN (nightly means) reduced to the {\\it Swift} $V$ system. The open boxes are $BV$ data obtained by {\\it Swift}. The filed circles are MASTER $BV$ photometry. The small open boxes are $U$ and $UVW1$ UVOT {\\it Swift} photometry.}\n    \\label{fig2}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[scale=0.45,angle=0, bb = 40 255 555 700]{FIG3.eps}\n \\caption {Energy spectra of NGC~1566 obtained with the {\\it Swift}\/XRT telescope in low state (MJD 56185--56210; open circles) and high state (MJD 58339.7;  squares). Solid lines correspond to the best-fitting models consisting of the power law in the high state and power-law plus Gaussian emission components in the low state.}\n    \\label{fig3}\n\\end{figure}\n\n\nThe {\\it Neil Gehrels Swift Observatory} \\citep{Gehrels2004} has monitored NGC~1566 regularly for years starting from 2007. Some results from these observations have been previously published \\citep{Kawamuro2013, Ferrigno2018, Grupe2018, Kuin2018}. In the current work all the data (both from the XRT and UVOT telescopes) were re-reduced uniformly \\citep[the same way as in ][]{Oknyansky2017A}, enabling us to trace the evolution of the behaviour of the source on a longer time-scale.\n\nThe XRT telescope \\mbox{\\citep{Burrows2005}} was operating both in photon counting and windowed timing  modes depending on the brightness of NGC~1566. The spectra were reduced using the standard online tools provided by the UK {\\it Swift} Science Data Centre \\citep[\\url{http:\/\/www.swift.ac.uk\/user_objects\/};][]{Evans2009}. Taking into account low count statistics we binned the spectra in the 0.5--10 keV range to have at least one count per energy bin and fitted them using the W-statistic \\citep{Wachter1979}.\\footnote{See {\\sc xspec} manual; \\url{https:\/\/heasarc.gsfc.nasa.gov\/xanadu\/} \\url{xspec\/manual\/XSappendixStatistics.html}}\nTo get the source flux in physical units we fitted all the spectra with a simple absorbed power-law model. The absorption parameter was free to vary, however we did not detect any significant absorption in the data either in high or in low flux states. The resulting light curve in the 0.5--10 keV band is shown in Fig.~\\ref{fig1}(a).  \n\n\nThe {\\it Swift} Ultraviolet\/Optical Telescope (UVOT) observes in different bands ($V$, $B$, $U$, $UVW1$, $UVW2$, $UVM2$) simultaneously with the XRT telescope, thus making it possible to get a broad-band view from the optical to X-rays.\nThe image analysis has been done following the procedure described on the web-page of the UK {\\it Swift} Science Data Centre. Photometry was performed with the {\\tt uvotsource} procedure with aperture radii of 5 and 10 arcsec for the source and background, respectively. The background was chosen with the centre about 1 arcmin away from the galaxy for all filters.\n\n\nThe XRT and UVOT observation results are presented in Fig.~\\ref{fig1}--\\ref{fig3} and will be discussed below  in Section 3.1. \n\n\n\n\\subsection{Observations with the \\uppercase{MASTER} network}\n\nMASTER \\citep{Lipunov2010} is a fully automated network of telescopes. The global MASTER robotic net for space monitoring was developed at the M.V. Lomonosov Moscow State University and consists of eight observatories. Here we are using data just from one of these observatories located at SAAO. This MASTER--SAAO observatory contains two 40-cm wide field telescopes (MASTER-IIs) with a combined field of view of 8  deg$^2$. Each MASTER II telescope is equipped with fast frame rate industrial GE4000 CCD cameras from AVT company (former Prosilica), which have a detector format of 4008$\\times$2672 pixels and an area of 24$\\times$36 mm. The MASTER-II instruments are able to provide surveys at a limiting celestial magnitude of 20 on dark, moonless nights. Observations can be made with Johnson $BVRI$ filters, or without a filter for integrated (white) light. Details of MASTER can be found in \\cite{Kornilov2012}. One of the goals of the MASTER is investigations of the transient variability of AGNs. One example is the investigation of the CL AGN NGC~2617 \\citep{Oknyansky2017A}. Here we present  the MASTER optical photometry of the CL AGN NGC 1566.\nTo minimize the differences with the {\\it Swift} UVOT observations, we performed photometry using a 5 arcsec radius aperture on all our data. We measured the background within an annulus of radii 35--45 arcsec. The calibration of $BV$ magnitudes was done relative to the comparison stars from\n\\url{http:\/\/www.astro.gsu.edu\/STARE\/ngc1566.html}. The unfiltered data were reduced to the system of the ASAS-SN \\citep[All-Sky Automated Survey for Supernovae;][]{Shappee2014, Kochanek2017, Dai2018} $V$--band  (using 14 common dates of observations), whereafter all these $V$ data were converted the same way to the $V$ UVOT {\\it Swift} system.\n\nThe MASTER observation results are presented in Fig.~\\ref{fig1} and \\ref{fig2} and will be discussed below in Section 3.2.\n\n\n\n\\subsection{ Optical  spectral observations and reductions}\n\n\\begin{figure}\n\t\\includegraphics[scale=0.49,angle=0,trim=0 0 0 0]{FIG4.eps}\n\n    \\caption{The isolated nuclear non-stellar spectrum (solid line) in NGC~1566 obtained by subtraction of the host galaxy spectrum (thin line) from the original spectrum (dashed line). (See details in the text.)}\n    \\label{fig4}\n\\end{figure}\n\n\n\n\\begin{figure*}\n\t\\includegraphics[scale=0.95,angle=0,trim=0 0 0 0]{FIG5.eps}\n\n    \\caption{The isolated nuclear non-stellar spectrum in NGC~1566 obtained by subtraction of the stellar spectrum (see details in the text.)}\n    \\label{fig5}\n\\end{figure*}\n\n\nLow-resolution spectra of NGC~1566 were obtained during the night 2018 August 2--3 with the 1.9 m telescope at SAAO in Sutherland. We used the Cassegrain spectrograph  with a \nlow-resolution (300 grooves per mm)  grating and a slit width of 2.7 arcsec  to give  spectral range from 3300  to 7500\\AA~ and  nominal resolution about 7 \\AA.  The spectrograph slit was oriented at a position angle of 90{$^\\circ$}. Two spectra of 600 s were taken and were later combined into a single spectrum using an average. The wavelength and flux calibration were achieved by bracketing the AGN spectra with Ar--spectra, and by means of an observation of the spectrophotometric standard star Feige 110 \\citep{Hamuy1994}. The spectrum of the nuclear region, displayed as a dashed line in Fig.~\\ref{fig4}, represents the flux collected within 5 arcsec east or west of the nuclear position.  This nuclear spectrum includes emission for the nearby HII region \\citep{Silva2017}  and contamination from the host galaxy. The HII emission is not relevant to our analysis since these lines are much more narrow  and weaker  \\citep[see figs 10--11 in][]{Silva2017} than the AGN narrow components of the emission lines. Their presence just represents a small additional constant contribution to the AGN emission lines which is smaller than the broad line flux uncertainty determined in Section 3.3. The host galaxy contribution to the object spectrum was estimated by scaling the off--nucleus spectrum measured between 6 and 13 arcsec from the nucleus (both to the east and to the west). The host galaxy spectrum therefore avoids an H~II region located approximately 1 arcsec from the nucleus \\citep{Silva2017}. This off-nuclear spectrum was then scaled by a factor that was adjusted until the stellar absorption features matched those in the object spectrum. The stellar spectrum is also shown in Fig.~\\ref{fig4}, and provides a good match to the Sb galaxy template given by \\cite{Kinney1996}. Note that the weak emission near H$\\alpha$ is normal in galaxies of this morphological class, which confirms that there is no significant contamination from the nuclear emission lines in the adopted stellar background spectrum. The isolated nuclear nonstellar spectrum  in NGC~1566 obtained by subtraction of the stellar spectrum  from the original spectrum is displayed as a solid line in Fig.~\\ref{fig4} and bigger in size in Fig.~\\ref{fig5}.  \n\n\\section{Resuts}\n\n\n \\subsection{{\\it Swift} XRT and UVOT results}\n\nThe resulting light curve (XRT) in the 0.5--10 keV band (spanning from 2007 Dec. 11 to 2018 Aug. 25) is shown in Fig.~\\ref{fig1}(a).\nOur analysis revealed a strong dependence of the spectral photon index on the source luminosity. This is clearly demonstrated in Fig.~\\ref{fig3}, where two spectra are shown for very different intensity states. The low-state spectrum (open circles), collected in 2012  September ({\\it Swift}\/XRT ObsIds 00045604004-00045604008, where Obs Id is a numeric value that uniquely identifies an observation),  is well fitted by a power-law model with a photon index of $1.54\\pm0.06$ and a flux of $F_{\\rm 0.5-10 keV}=(4.7\\pm0.2)\\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$. In addition to the power-law continuum component, the fit required the inclusion of a narrow emission line near 6 keV. This spectrum for the low state is similar to one presented by \\cite{Kawamuro2013}. In the bright state ({\\it Swift}\/XRT Obs Id 00035880015; squares in Fig.~\\ref{fig3}) the source flux increased by a factor of $\\sim20$ to $F_{\\rm 0.5-10 keV}=(8.2\\pm0.2)\\times10^{-11}$ erg cm$^{-2}$ s$^{-1}$, and the spectrum became significantly softer, with a photon index of $1.88\\pm0.03$. The low- and high-state spectra are very different in shape and luminosity, with the soft excess brightening by much more than the hard X-rays. The soft X-ray excess produces most of the ionizing photons, so its dramatic change must lead to a strong outburst of the broad line region, and could therefore be the driver of the changing-look phenomenon. Some excess flux at energies above $\\sim5$ keV seen in the bright state may correspond to the iron line or a reflection component \\citep[see e.g.,][]{Kawamuro2013}.\n\nLight curves in optical and UV bands ($UV$ and $UVW$) are shown in Fig.~\\ref{fig1}(b) for 2007-2018 and in $UBV$ and UV bands  in Fig.~\\ref{fig2} for just the last year. Given the very high correlation between variations in different UV bands, we present the light curve just for one of these bands, $UVW1$. \n\nAnalysis of the {\\it Swift}\/XRT data demonstrates a substantial increase in flux by 1.5 orders of magnitude following the brightening in the UV and optical bands during the past year, with maximum reached during the end of June -- beginning of 2018 July. The minimum recorded level of X-ray flux was about 2.2 $\\times$ $10^{-12}$~erg~cm$^{-2}$~s$^{-1}$ (MJD 56936), while the maximum flux measured was 1.1 $\\times$ $10^{-10}$~erg~cm$^{-2}$ s$^{-1}$ (MJD 58309), i.e. a ratio of nearly 50 times. Day-to-day fluctuations were also observed a few times on scales much higher than for UV and optical fast variations. The difference is in part due to a lower galaxy contribution in X-rays and can also be explained by the relatively smaller size of the X-ray radiation region. After the maximum , the fluxes in all bands were decreasing quite fast, with some fluctuations. As can be seen in the figures, all variations in the optical, UV and X-ray are correlated. The amplitude is largest for X-rays and decreases with increasing wavelength. Possible lags between these variations will be discussed in future publications.\n\n\n\n\\subsection{MASTER results in comparison with ASAS-SN and {\\it Swift} photometry}\n\nThe MASTER photometry  presented here includes old unfiltered archival data from 2014, new intensive monitoring in the $B$ and $V$ bands, as well as unfiltered $W$ band data starting from 2018 July 6 \n(Fig.~\\ref{fig1}b and Fig.~\\ref{fig2}). We show the light curve for  our unfiltered data  reduced  to   the $V$ UVOT {\\it Swift} system. The mean nightly (ASAS-SN) $V$-band light curves are also shown in Fig.~\\ref{fig1}b and Fig.~\\ref{fig2}. The nominal errors for these magnitudes are no more than 0.01 mag in that system, and about three times bigger after reducing these to a smaller aperture. The actual uncertainties may be deduced from the dispersion of the points in the light curve. As is seen from the light curves, our magnitudes are in good agreement with the ASAS-SN and {\\it Swift} $V$-band light curves for 2014--2018. The brightening at the beginning of the ASAS-SN \n$V$-band light curve in 2014--2015 is due to supernova ASASSN-14ha (SN), which does not affect our data as our aperture excluded the SN. From 2018 July 6 we started intensive observations in the $B$ and $V$ bands using the MASTER telescopes located in South Africa (SAAO).  We have combined these estimates in Fig.~\\ref{fig1}b and Fig.~\\ref{fig2}. We present only mean nightly values for the MASTER photometry in all bands. Usually, we obtained two to six observations in each band per night. The nominal errors of the plotted magnitudes are mostly less than the sizes of the shown points.\n\nOur data ({\\it Swift} and MASTER) show a roughly linear decline (after the maximum at the beginning of July) that is clearly seen in all bands. Here the average brightness drop rate is about 0.005$\\pm$0.001 mag in $V$, 0.009$\\pm$0.02 mag in $B$, 0.014$\\pm$0.002 mag in $U$ and 0.020$\\pm$0.002 mag in $UVW1$ per day, respectively. Fluctuations seen during July--August probably are real since they are visible in all bands. For example, the same local maximum was seen on MJD 58359 at all wavelengths from X-ray to $V$.\n\n\n\n\\subsection{Optical spectrum: new results}\nThe spectra in Fig.~\\ref{fig5} reveal a dramatic strengthening of the broad emission lines compared to past published ones. This confirms NGC~1566 to be a changing look Seyfert galaxy. \n \nThe Balmer lines exhibit a complex profile consisting of (a) a narrow component with the same profile as the nearby [OIII] and [NII] forbidden lines, (b) a broad component with a Gaussian profile and (c) irregularly shaped blue and red wings\/humps. In view of the relatively wide slit and low-resolution grating employed, and also because the broad lines are dominant, the Balmer narrow line strengths could not be established to significant accuracy. The hydrogen narrow components were therefore generated by adopting the [OIII]5007\\AA-to-H$\\beta$ and [NII]6584\\AA-to-H$\\alpha$ ratios determined by \\cite{Silva2017}. The peak wavelengths of the Gaussian broad components ($215\\pm30\\,km\\,s^{-1}$ for H$\\alpha$ and $470\\pm30$\\,km\\,s$^{-1}$ for H$\\beta$) are slightly lower than was measured by \\cite{Silva2017}, while the widths of these components remained consistent with the values determined in that study. Furthermore:\n\n1. H$\\beta$ is quite a lot brighter than [OIII]5007\\AA. The total H$\\beta$ to [OIII]5007\\AA line ratio is about 4.2$\\pm$0.4, corresponding to a Sy1.2 classification according to the criteria proposed by \\cite{Winkler1992}. This ratio is not significantly affected by the narrow component uncertainty because the narrow H$\\beta$ intensity is only of the order of 10\\% of [OIII]5007\\AA. H$\\beta$ has not been observed at this strength before in this object.\n\n2. The H$\\alpha$ to H$\\beta$ ratio for the Gaussian broad components is 2.7$\\pm$0.3 . This is consistent with other recent studies \\citep[e.g. ][]{Silva2017}, that also found that the obscuration of the broad line region is negligible.\n\n3. The HeII4686\\AA ~emission feature is bright and broad. No such strong HeII was visible in the past published spectra \\citep[see e.g., ][]{Kriss1991, Winkler1992}.\n \n4. The [FeX]6374\\AA ~coronal emission line is stronger than [OI]6300\\AA, something that has not been seen before in NGC~1566. The ratio is about 1.4$\\pm$0.2 after contamination from [OI]6363\\AA ~is removed using the assumption that the ratio of [OI]6300\\AA~ to [OI]6363\\AA ~is 2.997 \\citep{Storey2000}. One previous measurement of the coronal line, in a spectrum from 1988 January, recorded this ratio as 0.43 \\citep{Winkler1992}. Furthermore, [FeVII]6086\\AA ~is quite prominent in the spectrum, and is also stronger than previously recorded. The variability of high-ionization coronal lines is not surprising since these would be expected to arise in the very inner part of the NLR (see e.g. discussion by \\citep{Peterson1988} and \\citep{Ulrich1997}) or at the inner torus wall \\citep{Rose2015}. It was noted that the [FeX]6374\\AA ~in some AGNs is broader than for other forbidden lines \\citep{Netzer1974, Osterbrock1982}. The variability of coronal lines has also been detected in a number of CL AGNs: NGC~4151, NGC~5548, NGC~7469, 3C~390.3 and others \\citep{Oknyansky1982, Veilleux1988, Oknyansky1991, Landt2015, Landt2015b, Parker2016}.\n\n5. A strong UV continuum is clearly seen in our spectrum, and was far more prominent than what is visible in spectra collected during earlier low states.\n\n6. FeII emission is evidently much stronger now than in recent years. For example, the multiplet 42 lines of FeII are clearly seen superimposed on the [OIII] lines in our spectrum, but these were not detected in NGC~1566's low state \\citep[see fig. 10 in ][]{Silva2017}.\n\nMore details will be presented in a future publication where we will examine the spectral development and profile variations in additional spectra arranged to be collected in the coming season.\n\n\n\\section{Discussion}\n\nNGC~1566 is one of the clearest cases of Seyfert spectra ranging from type 1.2 to type 1.9 AGNs, all being confirmed in the same object at different epochs. In view of the galaxy's relative proximity and brightness, it also offers one of the best opportunities for studying this phenomenon. What must happen to make such a dramatic change possible? CL AGNs like NGC~1566 present problems for the simplest unification models in which type 2 and type 1 AGNs arise under the same processes, and the difference in type is only due to the orientation of the observer. In this model we see a type 2 AGN if the BLR and accretion disc are blocked from our view by obscuring dust surrounding the AGN perpendicular to the axis of symmetry \\citep{Keel1980}. However, orientation cannot change on the time-scale of the observed type changes, and hence some other explanation is needed.\n\nWe have shown using spectroscopy and multiwavelength photometry that NGC~1566 is in a high state with strong broad emission lines. The duration of the high state and the continuing variability are not consistent with potential explanations where the type change is due to tidal disruption or a once-off event such as a supernova. The recurrent brightening with CL events seen in NGC~1566 is probably common in CL AGNs, as such behaviour has also been noted for some other well-known CL AGNs such as NGC~4151 \\citep{Oknyansky2016d}, NGC~5548 \\citep{Bon2016} and NGC~2617 \\citep{Oknyansky2017A, Oknyansky2018A}.\n\nWe propose that the change of type may be the result of increased luminosity causing the sublimation of dust in the line of sight which previously obscured part of the broad line region. This leads to a much more direct view of the central regions. The greater luminosity would also increase the intensity of the Balmer lines as well as highly ionized lines like [FeX] and [FeVII].\n\nWhat is the reason for these recurrent outbursts in the object? The main problem with invoking a TDE (tidal disruption event) or supernova near SMBH (supermassive black hole) in some AGN is that they are extremely rare and cannot explain the comparatively greater rate of CL cases. Repeat tidal stars stripping \\citep{Ivanov-Chernyakova06, Campana15} could lead to more frequent events \\citep{Komossa17}, but this possibility is not sufficiently investigated yet. Understanding the physical process remains elusive at this stage.\n\n\n\\section{Summary}\n\nWe have shown, using spectroscopy (1.9 m SAAO) and multi-wavelength photometry (MASTER, {\\it Swift} Ultraviolet\/Optical and  XRT Telescopes), that NGC~1566 has just experienced a dramatic outburst in all wavelengths, including a considerable strengthening of broad permitted and high-ionization [FeX]6374\\AA~lines, as well as substantial changes in the shape of the optical and X-ray continua. These confirm a new CL case for NGC~1566 where the AGN achieved brightness levels comparable to historical outbursts in about 1966 and 1992. We suspect that these strong outbursts may be recurrent events with a quasi-period of about 26 years. One possible interpretation for these outbursts involves tidal star stripping. More work is needed to determine the plausibility of this and alternative mechanisms.\n\n\\section*{Acknowledgements}\nHW and FVW thank the SAAO for the generous allocation of telescope time which also resulted in the spectrum presented in this paper. We also express our thanks to the {\\it Swift} ToO team for organizing and executing the observations. This work was supported in part by the Russian Foundation for Basic Research through grant 17-52-80139 BRICS-a and by the BRICS Multilateral Joint Science and Technology Research Collaboration grant 110480. MASTER work was supported by Lomonosov Moscow State University Development Programme and RSF grant 16-12-00085. DB is supported by the National Research Foundation of South Africa. We are grateful to S.~Komossa for useful discussions. \n\n\n\n\\bibliographystyle{mnras}\n\\expandafter\\ifx\\csname natexlab\\endcsname\\relax\\def\\natexlab#1{#1}\\fi\n\n\\interlinepenalty=10000\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}