diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdcqw" "b/data_all_eng_slimpj/shuffled/split2/finalzzdcqw" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdcqw" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe Bernoulli convolution has been around for over seventy years and has surfaced in several different areas of mathematics. This probability measure depends on a parameter $\\beta >1$ and is defined on the interval $\\big[ 0, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\big]$, where $\\lfloor \\beta \\rfloor$ is the largest integer not exceeding $\\beta$. The {\\bf symmetric Bernoulli convolution} is the distribution of $\\sum_{k=1}^{\\infty} \\frac{b_k}{\\beta_k}$ where the coefficients $b_k$ take values in the set $\\{ 0 ,1, \\ldots, \\lfloor \\beta \\rfloor \\}$, each with probability $\\frac{1}{\\lfloor \\beta \\rfloor +1}$. If instead the values $0, 1, \\ldots, \\lfloor \\beta \\rfloor$ are not taken with equal probabilities, then the Bernoulli convolution is called {\\bf asymmetric} or {\\bf biased}. See \\cite{PSS00} for an overview of results regarding the Bernoulli convolution up to the year 2000. Recently attention has shifted to the multifractal structure of the Bernoulli convolution. Jordan, Shmerkin and Solomyak study the multifractal spectrum for typical $\\beta$ in \\cite{JSS11}, Feng considers Salem numbers $\\beta$ in \\cite{Fen12} and Feng and Sidorov look at the Lebesgue generic local dimension of the Bernoulli convolution in \\cite{FS11}. In this paper we are interested in the local dimension function of the Bernoulli convolution and to study the local dimension we use a new approach.\n\nIf a point $x \\in \\big[ 0, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\big]$ can be written as $x = \\sum_{k=1}^{\\infty} \\frac{b_k}{\\beta^k}$ with $b_k \\in \\{0,1, \\ldots, \\lfloor \\beta \\rfloor\\}$ for all $k \\ge 1$, then this expression is called a $\\beta$-expansion of the point $x$. In \\cite{S03} (see also \\cite{DV05}) it is shown that Lebesgue almost every $x$ has uncountably many $\\beta$-expansions. In \\cite{DV05} a random transformation $K$ was introduced that generates for each $x$ all these possible expansions by iteration. The map $K$ can be identified with a full shift which allows one to define an invariant measure $\\nu_{\\beta}$ for $K$ of maximal entropy by pulling back the uniform Bernoulli measure. One obtains the Bernoulli convolution from $\\nu_{\\beta}$ by projection. In this paper we study the local dimension of the measure $\\nu_{\\beta}$. By projection, some of these results can be translated to the Bernoulli convolution. For now, our methods work only for a special set of $\\beta$'s called the generalised multinacci numbers. We have good hopes that in the future we can extend these methods to a more general class of $\\beta$'s.\n\nThe paper is organized as follows. In the first section we will give the necessary definitions. Next we study the local dimension of $\\nu_{\\beta}$. We give a formula for the lower and upper bound of the local dimension that holds everywhere using a suitable Markov shift. Moreover, we show that the local dimension exists and is constant a.e.~and we give this constant. We also show that on the set corresponding to points with a unique $\\beta$-expansion, the local dimension of $\\nu_{\\beta}$ takes a different value. Next we translate these results to a lower and upper bound for the local dimension of the symmetric Bernoulli convolution that holds everywhere. We then use a result from \\cite{FS11} to obtain an a.e.~value for the Bernoulli convolution in case $\\beta$ is a Pisot number. Finally we give the local dimension for points with a unique expansion. In the last section we consider one specific example of an asymmetric Bernoulli convolution, namely when $\\beta$ is the golden ratio. We give an a.e.~lower and upper bound for the local dimension of both the invariant measure for $K$ and the asymmetric Bernoulli convolution. This last section is just a starting point for more research in this direction.\n\n\n\\section{Preliminaries}\n\nThe set of $\\beta$'s we consider in this paper, the generalised multinacci numbers, are defined as follows. On the interval $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\big]$ the {\\bf greedy} $\\beta$-transformation $T_{\\beta}$ is given by\n\\[ T_{\\beta} x = \\left\\{\n\\begin{array}{ll}\n\\beta x \\, (\\text{mod }1), & \\text{if } x \\in \\big[0,1),\\\\\n\\beta x - \\lfloor \\beta \\rfloor, & \\text{if } x \\in \\big[1, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\big].\n\\end{array}\n\\right.\\]\nThe {\\bf greedy digit sequence} of a number $x \\in \\big[ 0, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\big]$ is defined by setting \n\\[ a_1=a_1(x) = \\left\\{\n\\begin{array}{ll}\nk, & \\text{if } x \\in \\big[\\frac{k}{\\beta}, \\frac{k+1}{\\beta} \\big), \\, k \\in \\{0, \\ldots, \\lfloor \\beta \\rfloor -1 \\},\\\\\n\\lfloor \\beta \\rfloor, & \\text{if } x \\in \\big[ \\frac{\\lfloor \\beta \\rfloor}{\\beta}, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\big],\n\\end{array}\n\\right.\\]\nand for $n \\ge 1$, $a_n=a_n(x) = a_1(T_{\\beta}^{n-1} x)$. Then $T_{\\beta}x = \\beta x - a_1(x)$ and one easily checks that $x = \\sum_{n=1}^{\\infty} \\frac{a_n}{\\beta^n}$. This $\\beta$-expansion of $x$ is called its {\\bf greedy $\\beta$-expansion}. A number $\\beta >1$ is called a {\\bf generalised multinacci number} if the greedy $\\beta$-expansion of the number 1 satisfies\n\\begin{equation}\\label{q:genmn}\n1 = \\frac{a_1}{\\beta} + \\frac{a_2}{\\beta^2} + \\cdots + \\frac{a_n}{\\beta^n},\n\\end{equation}\nwith $a_j \\ge 1$ for all $1 \\le j \\le n$ and $n \\ge 2$. (Note that $a_1 = \\lfloor \\beta \\rfloor$.) We call $n$ the {\\bf degree} of $\\beta$.\n\n\\begin{rem}{\\rm\nBetween 1 and 2 the numbers that satisfy this definition are called the multinacci numbers. The {\\bf $n$-th multinacci number} $\\beta_n$ satisfies\n\\[ \\beta^n_n = \\beta^{n-1}_n + \\beta^{n-2}_n + \\cdots + \\beta_n + 1,\\]\nwhich implies that $a_j=1$ for all $1 \\le j \\le n$ in (\\ref{q:genmn}). The second multinacci number is better known as the golden ratio.}\n\\end{rem}\n\n\\vskip .2cm\nFor the Markov shift we will construct later on, we need a suitable partition of the interval $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$. Consider the maps $T_k x = \\beta x -k$, $k =0, \\ldots, \\lfloor \\beta \\rfloor$. For each $x \\in \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$, either there is exactly one $k \\in \\{0, \\ldots, \\lfloor \\beta \\rfloor \\}$ such that $T_k x \\in \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$, or there is a $k$ such that both $T_k x$ and $T_{k+1} x$ are in $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$. In this way the maps $T_k$ partition the interval $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ into the following regions:\n\\[ \\begin{array}{ll}\n\\displaystyle E_0 = \\Big[0, \\frac{1}{\\beta} \\Big), & \\displaystyle E_{\\lfloor \\beta \\rfloor} = \\Big( \\frac{\\lfloor \\beta \\rfloor}{\\beta(\\beta -1)} +\\frac{\\lfloor \\beta \\rfloor-1}{\\beta}, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\Big],\\\\\n\\\\\n\\displaystyle E_k = \\Big( \\frac{\\lfloor \\beta \\rfloor}{\\beta(\\beta -1)} +\\frac{k-1}{\\beta}, \\frac{k+1}{\\beta}\\Big), & k \\in \\{ 0, 1, \\ldots, \\lfloor \\beta \\rfloor \\},\\\\\n\\\\\n\\displaystyle S_k = \\Big[ \\frac{k}{\\beta}, \\frac{\\lfloor \\beta \\rfloor}{\\beta(\\beta-1)} + \\frac{k-1}{\\beta} \\Big], & k \\in \\{1, \\ldots, \\lfloor \\beta \\rfloor \\}.\n\\end{array}\\]\nSee Figure~\\ref{f:2.5} for a picture of the maps $T_k$ and the regions $E_k$ and $S_k$ in case $2 < \\beta < 3$.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics{figure1kleur}\n\\caption{The maps $T_0 x = \\beta x$, $T_1 x = \\beta x-1$ and $T_2 x = \\beta x -2$ and the intervals $E_0$, $S_1$, $E_1$, $S_2$ and $E_2$ for some $2< \\beta <3$.}\n\\label{f:2.5}\n\\end{figure}\n\n\\vskip .2cm\nWrite $\\Omega = \\{0,1 \\}^{\\mathbb N}$. The {\\bf random $\\beta$-transformation} is the map $K$ from the space $\\Omega \\times \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ to itself defined as follows.\n\\[ K(\\omega, x) = \\left\\{\n\\begin{array}{ll}\n(\\omega, T_k x), & \\text{if } x \\in E_k, \\, k \\in \\{0, \\ldots, \\lfloor \\beta \\rfloor \\},\\\\\n(\\sigma \\omega, T_{k-1+\\omega_1} x), & \\text{if }x \\in S_k, \\, k \\in \\{ 1, \\ldots, \\lfloor \\beta \\rfloor\\},\n\\end{array}\n\\right.\\]\nwhere $\\sigma$ denotes the left shift on sequences, i.e., $\\sigma (\\omega_n)_{n \\ge 1} = (\\omega_{n+1})_{n \\ge 1}$. The projection onto the second coordinate is denoted by $\\pi$. Let $\\lceil \\beta \\rceil$ denote the smallest integer not less than $\\beta$. The map $K$ is isomorphic to the full shift on $\\lceil \\beta \\rceil$ symbols. The isomorphism $\\phi: \\Omega \\times \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big] \\to \\{0,1, \\ldots, \\lfloor \\beta \\rfloor\\}^{\\mathbb N}$ uses the digit sequences produced by $K$. Let\n\\[b_1(\\omega,x) = \\left\\{\n\\begin{array}{ll}\nk, & \\text{if } x \\in E_k, \\, k \\in \\{0, 1, \\ldots, \\lfloor \\beta \\rfloor \\},\\\\\n& \\ \\text{or if } x \\in S_k \\text{ and } \\omega_1=1, \\, k \\in \\{1, \\ldots, \\lfloor \\beta \\rfloor \\},\\\\\nk-1, & \\text{if } x \\in S_k \\text{ and } \\omega_1=0, \\, k \\in \\{ 1, \\ldots, \\lfloor \\beta \\rfloor \\}\n\\end{array}\n\\right.\\]\nand for $n \\ge 1$, set $b_n(\\omega,x) = b_1\\big(K^{n-1}(\\omega, x)\\big)$. Then\n\\[ \\phi(\\omega,x) = \\big(b_n (\\omega,x) \\big)_{n \\ge 1}.\\]\nThis map is a bimeasurable bijection from the set $Z=\\big\\{ (\\omega, x) \\, : \\, \\pi \\big(K^n (\\omega, x)\\big) \\in S \\, \\, \\text{i.o.} \\big\\}$ to its image. We have $\\phi\\circ K = \\sigma \\circ \\phi$. Let $\\mathcal F$ denote the $\\sigma$-algebra generated by the cylinders and let $m$ denote the uniform Bernoulli measure on $(\\{0,1, \\ldots, \\lfloor \\beta \\rfloor\\}^{\\mathbb N}, \\mathcal F)$. Then $m$ is an invariant measure for $\\sigma$ and $\\nu_{\\beta} = m \\circ \\phi$ is invariant for $K$ with $\\nu_{\\beta}(Z)=1$. The projection $\\mu_{\\beta} = \\nu_{\\beta} \\circ \\pi^{-1}$ is the Bernoulli convolution on $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$. For proofs of these facts and more information on the map $K$ and its properties, see \\cite{DK03} and \\cite{DV05}.\n\n\\vskip .2cm\nWe are interested in the local dimension of the measures $\\nu_{\\beta}$ and $\\mu_{\\beta}$. For any probability measure $\\mu$ on a metric space $(X,\\rho)$, define the {\\bf local lower} and {\\bf local upper dimension} functions by\n\\[ \\underline{d}(\\mu,x) = \\liminf_{r \\downarrow 0} \\frac{\\log \\mu\\big(B_{\\rho}( x,r)\\big)}{\\log r} \\quad \\text{and} \\quad \\overline{d}(\\mu,x) = \\limsup_{r \\downarrow 0} \\frac{\\log \\mu\\big(B_{\\rho}(x,r)\\big)}{\\log r},\\]\nwhere $B_{\\rho}(x,r)$ is the open ball around $x$ with radius $r$. If $\\underline{d}(\\mu,x) = \\overline{d}(\\mu,x)$, then the {\\bf local dimension} of $\\mu$ at the point $x \\in X$ exists and is given by\n\\[ d(\\mu,x)= \\lim_{r \\downarrow 0} \\frac{\\log \\mu\\big(B_{\\rho}(x,r)\\big)}{\\log r}.\\]\nOn the sets $\\{0,1, \\ldots, \\lfloor \\beta \\rfloor\\}^{\\mathbb N}$ and $\\Omega$ we define the metric $D$ by\n\\[ D \\big( \\omega, \\omega' \\big) = \\beta^{-\\min\\{k\\ge 0 \\, :\\, \\omega_{k+1} \\neq \\omega_{k+1}'\\}}.\\]\nWe will define an appropriate metric on the set $\\Omega \\times \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ later.\n\n\n\n\\section{Local dimension for $\\nu_{\\beta}$}\n\nWe will study the local dimension of the invariant measure $\\nu_{\\beta}$ of the map $K$ for $\\beta$'s that are generalised multinacci numbers. It is proven in \\cite{DV05} that for these $\\beta$'s the dynamics of $K$ can be modeled by a subshift of finite type. So, on the one hand there is the isomorphism of $K$ with the full shift on $\\lceil \\beta \\rceil$ symbols and on the other hand there is an isomorphism to a subshift of finite type. It is this second isomorphism that allows us to code orbits of points $(\\omega, x)$ under $K$ in an appropriate way for finding local dimensions. We give the essential information here.\n\n\\medskip\nWe begin with some notation. We denote the {\\bf greedy map} by $T_{\\beta}$ as before, and the {\\bf lazy map} by $S_{\\beta}$. More precisely, \n\\[ T_{\\beta} x = \\left\\{\n\\begin{array}{ll}\nT_0 x, & \\text{if } x \\in E_0,\\\\\n\\\\\nT_k x, & \\text{if }x \\in S_k\\cup E_k,\\\\\n& \\quad 1 \\le k \\le \\lfloor \\beta \\rfloor,\n\\end{array}\n\\right.\n\\quad \\text{and} \\quad S_{\\beta}x = \\left\\{\n\\begin{array}{ll}\nT_k x, & \\text{if } x \\in E_k\\cup S_{k+1},\\\\\n& \\quad 0 \\le k \\le \\lfloor \\beta \\rfloor-1,\\\\\n\\\\\nT_{\\lfloor \\beta \\rfloor} x, & \\text{if }x \\in E_{\\lfloor \\beta \\rfloor}.\n\\end{array}\n\\right.\\]\n\n\\noindent\nWe will be interested in the $K$-orbit of the points $(\\omega,1)$ and of their `symmetric counterparts' $(\\omega, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1}-1)$.\nProposition 2 (ii) in \\cite {DV05} tells us that the following set $F$ is finite:\n\\begin{multline}\\label{q:finite}\nF=\\Big\\{\\pi \\big(K^n(\\omega, 1)\\big), \\pi \\Big(K^n\\big(\\omega, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1}-1\\big)\\Big) \\, : \\, n \\ge 0,\\, \\omega \\in \\Omega\\Big\\}\\\\\n\\cup \\Big\\{\\frac{k}{\\beta}, \\frac{\\lfloor \\beta \\rfloor}{\\beta(\\beta -1)} + \\frac{k}{\\beta} : k \\in \\{0, \\ldots, \\lfloor \\beta \\rfloor \\} \\Big\\}.\n\\end{multline}\nThese are the endpoints of the intervals $E_k$ and $S_k$ and their forward orbits under all the maps $T_k$. The finiteness of $F$ implies that the dynamics of $K$ can be identified with a topological Markov chain. To find the Markov partition, one starts by refining the partition given by the sets $E_k$ and $S_k$, using the points from the set $F$. Let $\\mathcal C$ be the interval partition consisting of the open intervals between the points from this set. Note that when we say {\\bf interval partition}, we mean a collection of pairwise disjoint open intervals such that their union covers the interval $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ up to a set of $\\lambda$-measure 0, where $\\lambda$ is the one-dimensional Lebesgue measure. Write\n\\[ \\mathcal C=\\{C_1,C_2,\\ldots , C_L\\}.\\]\nLet $ S=\\bigcup_{1 \\le k \\le \\lfloor \\beta \\rfloor} S_k$. The property {\\bf p3} from \\cite{DV05} says that no points of $F$ lie in the interior of $S$, i.e., each $S_k$ corresponds to a set $C_j$ in the sense that for each $1 \\le k \\le \\lfloor \\beta \\rfloor$ there is a $1 \\le j \\le L$ such that $\\lambda(S_k \\setminus C_j)=0$. Let $s \\subset \\{1, \\ldots, L\\}$ be the set containing those indices $j$. Consider the $L\\times L$ adjacency matrix $A=(a_{i, j})$ with entries in $\\{0,1\\}$ defined by\n\\begin{equation}\\label{matrix}\na_{i,j}\\, =\\, \\left\\{ \\begin{array}{ll}\n1, & {\\mbox{ if }}\\; i\\not \\in s \\text{ and } \\lambda(C_j \\cap T_{\\beta}(C_i))=\\lambda(C_j),\\\\\n0, & \\text{ if }\\; i\\not \\in s \\text{ and } \\lambda(C_i\\cap T_{\\beta}C_j)= 0,\\\\\n1, & \\text{ if }\\; i \\in s \\text{ and } \\lambda(C_j \\cap T_{\\beta} C_i)=\\lambda(C_j) \\text{ or } \\lambda(C_j \\cap S_{\\beta}C_i)=\\lambda(C_j),\\\\\n0, & \\text{ if }\\; i \\in s \\text{ and } \\lambda(C_i\\cap T_{\\beta}C_j)= 0 \\text{ and } \\lambda(C_i\\cap S_{\\beta}C_j)= 0.\n\\end{array}\\right.\n\\end{equation}\n\\noindent Define the partition $\\mathcal P$ of $\\Omega \\times \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ by\n\\[ \\mathcal P=\\big\\{\\Omega \\times C_j : j \\not \\in s \\big\\} \\cup \\big\\{ \\{\\omega_1=i\\} \\times C_j: i \\in \\{0,1\\}, \\, j \\in s \\big\\}.\\]\nThen $\\mathcal P$ is a Markov partition underlying the map $K$. Let $Y$ denote the topological Markov chain determined by the matrix $A$. That is, $Y=\\{(y_n)_{n \\ge 1}\\in \\{1,\\dots ,L\\}^{\\mathbb N}:a_{y_n, y_{n+1}}=1 \\}$. Let $\\mathcal Y$ denote the $\\sigma$-algebra on $Y$ determined by the cylinder sets, i.e., the sets specifying finitely many digits, and let $\\sigma_Y$ be the left shift on $Y$. We use Parry's recipe (\\cite{Par64}) to determine the Markov measure $Q$ of maximal entropy for $(Y, \\mathcal Y, \\sigma_Y)$. By results in \\cite{DV05} we know that $\\nu_{\\beta}$ is the unique measure of maximal entropy for $K$ with entropy $h_{\\nu_{\\beta}}(K)=\\log \\lceil \\beta \\rceil$. By the identification with the Markov chain we know that $h_Q(\\sigma_Y) = \\log \\lceil \\beta \\rceil$. One then gets that the corresponding transition matrix $(p_{i,j})$ for $Y$ satisfies $p_{i,j}=a_{i,j}\\frac{v_j}{\\lceil \\beta \\rceil v_i}$, where $(v_1,v_2, \\ldots,v_L)$ is the right probability eigenvector of $A$ with eigenvalue $\\lfloor \\beta \\rfloor +1$. From this we see that if $[j_1 \\cdots j_m]$ is an allowed cylinder in $Y$, then\n\\begin{equation}\\label{q:qmeasure}\nQ([j_1 \\cdots j_m])=\\frac{v_{j_m}}{\\lceil \\beta \\rceil^{m-1}}.\n\\end{equation}\nProperty {\\bf p5} from \\cite{DV05} says that for all $i \\in s$, $a_{i,1}=a_{i,L}=1$ and $a_{i,j}=0$ for all other $j$. By symmetry of the matrix $(p_{i,j})$, it follows that\n\\begin{equation}\\label{q:ps1L}\np_{i, 1} = p_{i,L} = \\frac12 \\quad \\text{for all } i \\in s.\n\\end{equation}\nLet\n\\[ X = \\Omega \\times \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big] \\backslash \\big( \\bigcup_{n \\ge 0} K^{-n} F \\big).\\]\nThen $\\nu_{\\beta} (X) =1$. The isomorphism $\\alpha: X \\to Y$ between $(K, \\nu_{\\beta})$ and $(\\sigma_Y, Q)$ is then given by \n$$\\alpha_j(\\omega, x) = k \\mbox{ if } K^{j-1}(\\omega, x) \\in C_k.$$ \nSee Theorem 7 in \\cite{DV05} for a proof of this fact. In Figure~\\ref{f:diagram} we see the relation between the different systems we have introduced so far.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics{figure2}\n\\caption{The relation between the different spaces.}\n\\label{f:diagram}\n\\end{figure}\n\nTo study the local dimension of $\\nu_{\\beta}$, we need to consider balls in $X$ under a suitable metric. Define the metric $\\rho$ on $X$ by\n\\[ \\rho\\big((\\omega,x),(\\omega^{\\prime},x^{\\prime})\\big)=\\beta^{-\\min\\{k\\ge 0\\, : \\, \\omega_{k+1}\\not=\\omega^{\\prime}_{k+1} \\text{ or } \\alpha_{k+1}(\\omega,x)\\not=\\alpha_{k+1}(\\omega^{\\prime},x^{\\prime}) \\}}.\\]\nConsider the ball \n\\[ B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)=\\{(\\omega^{\\prime},x^{\\prime})\\, : \\, \\omega_i^{\\prime}=\\omega_i, \\mbox{ and } \\, \\alpha_i(\\omega^{\\prime},x^{\\prime})= \\alpha_i(\\omega,x), \\, i=1,\\cdots ,k\\}.\\]\nLet\n\\[ M_k(\\omega,x)=\\sum_{i=0}^{k-1}{\\bf 1}_{X \\cap \\Omega\\times S}\\big(K^i(\\omega,x)\\big) = \\# \\{ 1 \\le i \\le k \\, : \\, \\alpha_i(\\omega,x) \\in s \\}.\\]\nTo determine $\\nu_{\\beta}\\big(B_{\\rho}((\\omega,x),r)\\big)$ for $r \\downarrow 0$ we will calculate $Q \\Big( \\alpha \\Big(B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big) \\Big)\\Big)$. For all points $(\\omega',x')$ in the ball $B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)$ the $\\alpha$-coding starts with $\\alpha_1(\\omega, x) \\cdots \\alpha_k(\\omega, x)$ and $\\omega'$ starts with $\\omega_1 \\cdots \\omega_k$. From the second part we know what happens the first $k$ times that the $K$-orbit of a point $(\\omega', x')$ lands in $\\Omega \\times S$. Since $M_k(\\omega, x)$ of these values have been used for $\\alpha_1(\\omega, x)\\cdots \\alpha_k(\\omega,x)$, there are $k-M_k(\\omega,x)$ unused values left. Note that $M_k(\\omega', x') = M_k(\\omega, x)=M_k$ for all $(\\omega', x') \\in B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)$. Define the set\n\\[ Z = X \\cap \\bigcap_{n \\ge 1}\\bigcup_{i \\ge 1} K^{-i} \\big(\\Omega \\times S\\big).\\]\nAll points in $Z$ land in the set $\\Omega \\times S$ infinitely often under $K$. Since $Z$ is $K$-invariant, by ergodicity of $K$ we have $\\nu_{\\beta} (Z) =1$. So, all points $(\\omega', x') \\in B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big) \\cap Z$ make a transition to $S$ some time after $k$. Moreover, after this transition these points move to $C_1$ if $\\omega_{M_k+1} =1$ and to $C_L$ otherwise. The image of a point $(\\omega', x') \\in B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)$ under $\\alpha$ will thus have the form\n\\begin{multline*}\n\\alpha_1 \\cdots \\alpha_k \\, \\underbrace{a_{k+1} \\cdots a_{m_1-1}}_{\\not \\in s} \\, \\underbrace{a_{m_1}}_{\\in s} \\, \\underbrace{a_{m_1+1} \\cdots a_{m_2-1}}_{\\not \\in s} \\, \\underbrace{a_{m_2}}_{\\in s} \\\\\n\\cdots \\, \\underbrace{a_{m_{N-1}-1}\\cdots a_{m_N-1}}_{\\not \\in s} \\, \\underbrace{a_{m_N}}_{\\in s} \\, \\underbrace{a_{m_N+1}a_{m_N+2} \\cdots}_{\\text{tail}},\n\\end{multline*}\nwhere $a_{m_j+1} \\in \\{1,L\\}$, $m_{j+1}-m_j-2 \\ge 1$ and $N = k-M_k(\\omega, x)$. Note that by the ergodicity of $\\nu_{\\beta}$ we have\n\\[ Q \\Big( \\bigcup_{m \\ge 1} [a_1 \\cdots a_m] \\, : \\, a_m \\in s \\text{ and } a_i \\not \\in s , \\, i < m \\Big) = \\nu_{\\beta} \\Big( X \\cap \\bigcup_{m \\ge 1} K^{-m} (\\Omega \\times S) \\Big)=1.\\]\nSo, the transition from any state to $s$ occurs with probability 1. Then one of the digits $\\omega_j$, $M_k+1 \\le j \\le k$, specifies what happens in this event and by (\\ref{q:ps1L}) both possibilities happen with probability $\\frac12$. To determine the measure of all possible tails of sequences in $\\alpha \\Big(B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big) \\Big)$, note that again by {\\bf p5} of \\cite{DV05} this tail always belongs to a point in $\\Omega \\times C_1$ or $\\Omega \\times C_L$. Since the $\\nu_{\\beta}$-measure of these sets is the same, the $Q$-measure of the set of all possible tails is given by $\\nu_{\\beta}(\\Omega \\times C_1)= \\mu_{\\beta}(C_1)$. Putting all this together gives\n\\begin{equation}\\label{q:Qball}\nQ \\Big( \\alpha \\Big(B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big) \\Big)\\Big) = \\lceil \\beta \\rceil^{-(k-1)} v_{\\alpha_k(\\omega,x)} \\cdot \\underbrace{1 \\cdot \\frac{1}{2} \\cdot 1 \\cdot \\frac{1}{2} \\cdots 1\\cdot \\frac{1}{2}}_{k-M_k(\\omega, x) \\text{ times}} \\cdot \\, \\mu_{\\beta}(C_1),\n\\end{equation}\nand hence,\n\\begin{equation}\\label{q:nuball}\n \\nu_{\\beta}\\big(B_{\\rho}((\\omega,x),\\beta^{-k})\\big) = \\lceil \\beta \\rceil^{-(k-1)}v_{\\alpha_k(\\omega,x)}\\,2^{-(k-M_k(\\omega,x))}\\,\\mu_{\\beta}(C_1).\n\\end{equation}\nThis gives the following theorem.\n\\begin{thm}\\label{t:locdim}\nLet $\\beta >1$ be a generalised multinacci number. For all $(\\omega,x) \\in X$ we have\n\\begin{multline}\\label{q:locdim}\n\\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} +\\frac{\\log 2}{\\log \\beta}\\Big[ 1-\\limsup_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k} \\Big] \\le \\underline d\\big(\\nu_{\\beta}, (\\omega,x)\\big)\\\\\n\\le \\overline d\\big(\\nu_{\\beta}, (\\omega,x)\\big) \\le \\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} +\\frac{\\log 2}{\\log \\beta}\\Big[ 1-\\liminf_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k} \\Big].\n\\end{multline}\n\\end{thm}\n\n\\begin{proof}\nLet $\\frac{1}{\\beta^{k+1}} < r \\le \\frac{1}{\\beta^k}$. Set $v_{min} = \\min\\{v_1, \\ldots, v_L\\}$ and $v_{max} = \\max\\{v_1, \\ldots, v_L\\}$. Then, by (\\ref{q:nuball}),\n\\[ \\frac{\\log \\nu_{\\beta}\\big(B_{\\rho}((\\omega,x),r)\\big)}{\\log r} \\le \\frac{(k-1)\\log \\lceil \\beta \\rceil}{k \\log \\beta} + \\frac{(k-M_k(\\omega, x))\\log 2}{k\\log\\beta} - \\frac{\\log \\big(v_{min}\\, \\mu_{\\beta}(C_1)\\big)}{k \\log \\beta}.\\]\nHence,\n\\[ \\overline d\\big(\\nu_{\\beta}, (\\omega,x)\\big) = \\limsup_{k \\to \\infty} \\frac{\\log \\nu_{\\beta}\\big(B_{\\rho}((\\omega,x),r)\\big)}{\\log r} \\le \\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} + \\frac{\\log 2}{\\log \\beta}\\Big[ 1 - \\liminf_{k \\to \\infty} \\frac{M_k(\\omega,x)}{k} \\Big].\\]\nOn the other hand,\n\\[ \\frac{\\log \\nu_{\\beta}\\big(B_{\\rho}((\\omega,x),r)\\big)}{\\log r} \\ge \\frac{(k-1)\\log \\lceil \\beta \\rceil}{(k+1) \\log \\beta} + \\frac{(k-M_k(\\omega, x))\\log 2}{(k+1)\\log\\beta} - \\frac{\\log \\big(v_{max}\\, \\mu_{\\beta}(C_1)\\big)}{(k+1) \\log \\beta}.\\]\nSince $M_{k+1}(\\omega, x)-1 \\le M_k(\\omega,x) \\le M_{k+1}(\\omega, x)$, we have that\n\\[ \\underline d\\big(\\nu_{\\beta}, (\\omega,x)\\big) = \\liminf_{k \\to \\infty} \\frac{\\log \\nu_{\\beta}\\big(B_{\\rho}((\\omega,x),r)\\big)}{\\log r} \\ge \\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} + \\frac{\\log 2}{\\log \\beta}\\Big[ 1 - \\limsup_{k \\to \\infty} \\frac{M_k(\\omega,x)}{k} \\Big].\\]\nThis proves the theorem.\n\\end{proof}\n\n\\begin{rem}{\\rm From the proof of the previous theorem it follows that if $\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}$ exists, then $d\\big(\\nu_{\\beta}, (\\omega,x) \\big)$ exists and is equal to $\\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} +\\frac{\\log 2}{\\log \\beta}\\Big[ 1-\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}\\Big]$.\n}\\end{rem}\n\n\\begin{cor}\\label{c:locdimnubeta}\nLet $\\beta$ be a generalised multinacci number. The local dimension function $d\\big(\\nu_{\\beta}, (\\omega,x)\\big)$ is constant $\\nu_{\\beta}$-a.e.~and equal to\n\\[d\\big(\\nu_{\\beta}, (\\omega,x)\\big)=\\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} +\\frac{\\log 2}{\\log \\beta} \\big( 1-\\mu_{\\beta}(S) \\big).\\]\n\\end{cor}\n\n\\begin{proof}\nSince $\\nu_{\\beta}$ is ergodic, the Ergodic Theorem gives that for $\\nu_{\\beta}$-a.e.~$(\\omega, x)$,\n\\[ \\lim_{k \\to \\infty} \\frac{M_k(\\omega, x)}{k} = \\nu_{\\beta}\\big(\\Omega \\times S\\big) = \\mu_{\\beta}(S).\\]\nThis gives the result.\n\\end{proof}\n\n\\medskip\nRecall that $\\phi$ maps points $(\\omega, x)$ to digit sequences $\\big( b_n(\\omega,x) \\big)_{n \\ge 1}$. It is easy to see that $x = \\sum_{n \\ge 1} \\frac{b_n(\\omega,x)}{\\beta^n}$ for each choice of $\\omega \\in \\Omega$. Note that a point $x$ has exactly one $\\beta$-expansion if and only if for all $n \\ge 0$, $\\pi\\big( K^n (\\omega,x) \\big) \\not \\in S$. Let $\\mathcal A_{\\beta} \\subset \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ be the set of points with a unique $\\beta$-expansion. Then $\\mathcal A_{\\beta} \\neq \\emptyset$, since $0, \\frac{\\lfloor \\beta \\rfloor}{\\beta-1} \\in \\mathcal A_{\\beta}$ for any $\\beta>1$. By Proposition 2 from \\cite{DV05} all elements from $\\cup_{n \\ge 0}K^{-n}F$ will be in $S$ at some point and hence they will have more than one expansion. So, $\\mathcal A_{\\beta} \\subset X$. The next result also follows easily from Theorem~\\ref{t:locdim}. The measure $\\nu_{\\beta}$ is called {\\bf multifractal} if the local dimension takes more than one value on positive Hausdorff dimension sets. \n\n\\begin{cor}\nLet $\\beta$ be a generalised multinacci number. If $x \\in \\mathcal A_{\\beta}$, then $d\\big(\\nu_{\\beta}, (\\omega,x)\\big) =\\frac{\\log \\lceil \\beta \\rceil + \\log 2}{\\log \\beta}$ for all $\\omega \\in \\Omega$. The measure $\\nu_{\\beta}$ is multifractal.\n\\end{cor}\n\n\\begin{proof}\nIf $x \\in \\mathcal A_{\\beta}$, then $M_k(\\omega, x)=0$ for all $\\omega \\in \\Omega$ and $k \\ge 1$. Hence, by (\\ref{q:locdim}), $d\\big(\\nu_{\\beta}, (\\omega,x)\\big) =\\frac{\\log \\lceil \\beta \\rceil + \\log 2}{\\log \\beta}$. From standard results in dimension theory and our choice of metric it follows that $dim_H\\big(\\Omega \\times \\{x\\}\\big)=\\frac{\\log 2}{\\log \\beta}$ for all $x\\in \\mathcal A_{\\beta}$. Hence, $\\nu_{\\beta}$ is a multifractal measure.\n\\end{proof}\n\n\\begin{exa}\\label{r:goldenmean}\nWe give a example to show what can happen on points in $F$. Let $\\beta =\\frac{1+\\sqrt 5}{2}$ be the golden ratio. Then, $1 = \\frac{1}{\\beta} + \\frac{1}{\\beta^2}$. Figure~\\ref{f:gm} shows the maps $T_0$ and $T_1$ for this $\\beta$.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics{figure3kleur}\n\\caption{The maps $T_0 x = \\beta x$ and $T_1 x = \\beta x-1$ for $\\beta = \\frac{1+\\sqrt 5}{2}$. The region $S$ is colored yellow.}\n\\label{f:gm}\n\\end{figure}\nNote that $F = \\{ 0, \\frac{1}{\\beta}, 1, \\beta \\}$. The partition $\\mathcal C$ consists of only three elements and the transition matrix and stationary distribution of the corresponding Markov chain are\n\\[ P = \\left(\n\\begin{array}{ccc}\n1\/2 & 1\/2 & 0\\\\\n1\/2 & 0 & 1\/2\\\\\n0 & 1\/2 & 1\/2\n\\end{array} \\right) \\quad \\text{ and } \\quad v=(1\/3,1\/3,1\/3).\\]\nHence, $\\mu_{\\beta}(S)=1\/3$ and Corollary~\\ref{c:locdimnubeta} gives that for $\\nu_{\\beta}$-a.e.~$(\\omega,x) \\in \\Omega \\times [0, \\beta]$,\n\\[d\\big(\\nu_{\\beta},(\\omega,x)\\big)=\\frac{\\log 2}{\\log \\beta} \\Big[ 2-\\mu_{\\beta}(S) \\Big]=(2-1\/3) \\frac{\\log 2}{\\log \\beta}=\\frac{5\\log 2}{6 \\log \\beta}.\\]\n\n\\vskip .2cm\nNow consider the $\\alpha$-code of the points $(\\overline{10},1)$ and $(\\overline{01},1\/\\beta)$, where the bar indicates a repeating block:\n\\[\\alpha \\big((\\overline{10},1)\\big)=\\alpha \\big((\\overline{01},1\/\\beta)\\big)=(s,s,s,\\cdots),\\]\nwhich is not allowed in the Markov chain $Y$. Then for any point \n\\[(\\omega,x)\\in \\bigcup_{m=0}^{\\infty} K^{-m} \\big(\\{(\\overline{10},1),(\\overline{01},1\/\\beta)\\}\\big),\\]\none has $B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)$ is a countable set for all $k$ sufficiently large. For the local dimension this implies that\n\\[ d\\big(\\nu_{\\beta},(\\omega,x)\\big) = \\lim_{r \\downarrow 0} \\frac{\\log 0}{\\log r} = \\infty.\\]\n\\end{exa}\n\n\n\\section{Local dimensions for the symmetric Bernoulli convolution}\n\nConsider now the Bernoulli convolution measure $\\mu_{\\beta}=\\nu_{\\beta}\\circ \\pi^{-1}$ on the interval $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ for generalised multinacci numbers. In \\cite{FS11}, the local dimension of $\\mu_{\\beta}$ with respect to the Euclidean metric was obtained for all Pisot numbers $\\beta$. A {\\bf Pisot number} is an algebraic integer that has all its Galois conjugates inside the unit circle. It is well known that all multinacci numbers are Pisot numbers, but unfortunately not all generalised multinacci numbers are Pisot. In Remark~\\ref{r:pisot}(i) we list some classes of generalised multinacci numbers that are in fact Pisot numbers. Before stating the results from \\cite{FS11}, we introduce more notation. Let\n\\begin{equation}\\label{q:fs}\n\\mathcal N_k (x,\\beta)=\\#\\left\\{(a_1, \\ldots, a_k)\\in \\{0,1, \\ldots, \\lfloor \\beta \\rfloor\\}^k:\\exists (a_{k+n})_{n \\ge 1} \\text{ with } x=\\sum_{m=1}^{\\infty} \\frac{a_m}{\\beta^m} \\right\\}.\n\\end{equation}\nA straightforward calculation (see also Lemma 4.1 of \\cite{Kem12}) shows that\n\\[ \\mathcal N_k (x,\\beta)=\\int_{\\Omega} \\, 2^{M_k(\\omega,x)} \\, dm(\\omega),\\]\nwhere $m$ is the uniform Bernoulli measure on $\\{0,1, \\ldots, \\lfloor \\beta \\rfloor\\}^{\\mathbb N}$ as before. In \\cite{FS11}, it was shown that if $\\beta$ is a Pisot number, then there is a constant $\\gamma=\\gamma(\\beta,m)$ such that \n\\begin{equation}\\label{q:gamma}\n\\lim_{k \\to\\infty}\\frac{\\log \\mathcal N_k(x,\\beta)}{k}=\\gamma\n\\end{equation}\nfor $\\lambda$-a.e.~$x$ in $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$, where $\\lambda$ is the one-dimensional Lebesgue measure. Using this, Feng and Sidorov obtained that for $\\lambda$-a.e.~$x$,\n\\[ d(\\mu_{\\beta},x)=\\frac{\\log \\lceil \\beta \\rceil -\\gamma}{\\log \\beta}\\]\n In fact, the result they obtained was stronger, but we will use their result in this form. We will show that one has the same value for the local dimension when the Euclidean metric on $\\mathbb R$ is replaced by the Hausdorff metric. To this end, consider the metric $\\bar \\rho$ on $\\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$ defined by\n\\[ \\bar \\rho(x,y)=d_H\\big(\\pi^{-1}(x),\\pi^{-1}(y)\\big),\\]\nwhere $d_H$ is the Hausdorff distance given by\n\\begin{multline*}\nd_H \\big(\\pi^{-1}(x),\\pi^{-1}(y)\\big)\\\\\n=\\inf\\Big\\{\\epsilon >0:\\pi^{-1}(y)\\subset \\bigcup_{\\omega \\in \\Omega}B_{\\rho}\\big((\\omega,x),\\epsilon \\big) \\mbox{ and } \\pi^{-1}(x)\\subset \\bigcup_{\\omega \\in \\Omega}B_{\\rho}\\big((\\omega,y),\\epsilon\\big)\\Big\\}.\n\\end{multline*}\n\n\\begin{thm}\nLet $\\beta$ be a generalised multinacci number. Then for all $x \\in \\big[0,\\frac{\\lfloor \\beta \\rfloor}{\\beta-1}\\big]$,\n\\begin{multline*}\n\\frac{1}{\\log \\beta} \\Big[ \\log \\lceil \\beta \\rceil - \\limsup_{k \\to\\infty}\\frac{\\log \\mathcal N_k(x,\\beta)}{k} \\Big] \\le \\underline d(\\mu_{\\beta},x)\\\\\n\\le \\overline d(\\mu_{\\beta},x) \\le \\frac{1}{\\log \\beta} \\Big[ \\log \\lceil \\beta \\rceil - \\liminf_{k \\to\\infty}\\frac{\\log \\mathcal N_k(x,\\beta)}{k} \\Big].\n\\end{multline*}\n\\end{thm}\n\n\\begin{proof}\nLet $B_ {\\bar \\rho}(x,\\epsilon)=\\{y:\\bar \\rho(x,y)<\\epsilon\\}$. We want to determine explicitly the set $\\pi^{-1}\\big(B_{\\bar \\rho}(x,\\beta^{-k}) \\big)$. First note that for any $(\\omega,x)$, and any $k\\ge 0$, one has $(\\omega^{\\prime},y)\\in B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)$, and $B_{\\rho}\\big((\\omega,x),\\beta^{-k}\\big)=B_{\\rho}\\big((\\omega^{\\prime},y),\\beta^{-k}\\big)$ for any $(\\omega^{\\prime},y)\\in [\\omega_1\\cdots\\omega_k]\\times [\\alpha_1(\\omega,x)\\cdots \\alpha_k(\\omega,x)]$. We denote the common set by $B_{\\rho}\\big(([\\omega_1 \\cdots \\omega_k],x),\\beta^{-k}\\big)$. This implies that\n\\[\\pi^{-1}\\big(B_{\\bar \\rho}(x,\\beta^{-k})\\big)=\\bigcup_{[\\omega_1\\cdots \\omega_k]}B_{\\rho}\\big(([\\omega_1 \\cdots \\omega_k],x),\\beta^{-k}\\big),\\]\nwhere the summation on the right is taken over all possible cylinders of length $k$ in $\\Omega$. Again set $v_{min} = \\min\\{v_1, \\ldots, v_L\\}$ and $v_{max} = \\max\\{v_1, \\ldots, v_L\\}$. Then,\n\\begin{eqnarray*}\n\\mu_{\\beta}\\big(B_{\\bar \\rho}(x,\\beta^{-k})\\big) &=& \\sum_{[\\omega_1 \\cdots \\omega_k]}\\nu_{\\beta}\\big(B_{\\rho}\\big(([\\omega_1 \\cdots \\omega_k],x),\\beta^{-k} \\big)\\big)\\\\\n& \\le & \\sum_{[\\omega_1 \\cdots \\omega_k]}\\lceil \\beta \\rceil^{-(k-1)}v_{\\alpha_k(\\omega,x)}\\,2^{-(k-M_k(\\omega,x))}\\, \\mu_{\\beta}(C_1)\\\\\n& \\le & \\lceil \\beta \\rceil^{-(k-1)}v_{max} \\, \\mu_{\\beta}(C_1) \\sum_{[\\omega_1 \\cdots \\omega_k]} 2^{M_k(\\omega,x)} 2^{-k}\\\\\n& = & \\lceil \\beta \\rceil^{-(k-1)}v_{max}\\, \\mu_{\\beta}(C_1) \\int_{\\Omega} \\, 2^{M_k(\\omega,x)} \\, dm(\\omega)\\\\\n& = & \\lceil \\beta \\rceil^{-(k-1)} \\, v_{max}\\, \\mu_{\\beta}(C_1) \\, {\\mathcal N}_k(x,\\beta).\n\\end{eqnarray*}\nNow taking logarithms, dividing by $\\log \\beta^{-k}$, and taking limits we get\n\\[ \\underline d(\\mu_{\\beta},x)\\ge \\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} - \\frac{1}{\\log \\beta} \\limsup_{k \\to \\infty} \\frac{\\log \\mathcal N_k(x, \\beta)}{k}.\\]\nSimilarly we get that\n\\[ \\mu_{\\beta}\\big(B_{\\bar \\rho}(x,\\beta^{-k})\\big) \\ge \\lceil \\beta \\rceil^{-(k-1)} \\, v_{min} \\, \\mu_{\\beta}(C_1) \\, {\\mathcal N}_{k}(x,\\beta),\\]\nwhich gives\n\\[ \\overline d(\\mu_{\\beta},x)\\le \\frac{\\log \\lceil \\beta \\rceil}{\\log \\beta} - \\frac{1}{\\log \\beta} \\liminf_{k \\to \\infty} \\frac{\\log \\mathcal N_k(x, \\beta)}{k}. \\qedhere\\]\n\\end{proof}\n\n\\noindent By the results from \\cite{FS11} we have the following corollary.\n\\begin{cor}\nIf $\\beta$ is Pisot, then $d(\\mu_{\\beta},x)$ exists for $\\lambda$-a.e.~$x$ and is equal to $\\frac{\\log \\lceil \\beta \\rceil- \\gamma}{\\log \\beta}$, where $\\gamma$ is the constant from (\\ref{q:gamma}).\n\\end{cor}\n\n\\begin{rem}\\label{r:pisot}{\\rm\n(i) We give some examples of generalised multinacci numbers that are Pisot numbers. The generalised multinacci numbers in the interval $[1,2]$, i.e., the multinacci numbers, are all Pisot. In the interval $[2, \\infty)$ the numbers $\\beta$ that satisfy $\\beta^2 - k\\beta-1=0$, $k \\ge 2$, are all Pisot as well. Recall that if $1=\\frac{a_1}{\\beta} + \\cdots + \\frac{a_n}{\\beta^n}$ with $a_i \\ge 1$ for all $1 \\le i \\le n$, then $n$ is called the degree of $\\beta$. From Theorem 4.2 in \\cite{AG05} by Akiyama and Gjini we can deduce that all generalised multinacci numbers of degree 3 are Pisot numbers. Similarly, from Proposition 4.1 in \\cite{AG05} it follows that all generalised multinacci numbers of degree 4 with $a_4=1$ are Pisot. An example of a generalised multinacci number that is not Pisot is the number $\\beta$ satisfying\n\\[ 1=\\frac{3}{\\beta} + \\frac{1}{\\beta^2} + \\frac{2}{\\beta^3} + \\frac{3}{\\beta^4}.\\]\n(ii) In \\cite{Kem12} it is shown that for all $\\beta > 1$ and $\\lambda$-a.e.~$x$, $\\liminf_{k \\to \\infty} \\frac{\\log \\mathcal N_k(x, \\beta)}{k} \\ge \\mu_{\\beta}(S)\\log 2$, so we get\n\\[ \\overline d(\\mu_{\\beta},x)\\le \\frac{1}{\\log \\beta}\\Big[ \\log \\lceil \\beta \\rceil - \\mu_{\\beta}(S)\\log 2 \\Big].\\]\nKempton also remarks that this lower bound is not sharp.\n}\\end{rem}\n\n\\section{Asymmetric random $\\beta$-transformation: the golden ratio}\n\nIn the previous section, we considered the measure $\\nu_{\\beta}=\\nu_{\\beta,1\/2}$ which is the lift of the uniform Bernoulli measure $m=m_{1\/2}$ under the isomorphism $\\phi(\\omega, x)=\\big(b_n(\\omega,x)\\big)_{n \\ge 1}$. The projection of $\\nu_{\\beta}$ in the second coordinate is the symmetric Bernoulli convolution. In this section, we will investigate the asymmetric Bernoulli convolution in case $\\beta$ is the golden ratio.\n\n\\medskip\nLet $\\beta = \\frac{1+\\sqrt 5}{2}$ and consider the $(p,1-p)$-Bernoulli measure $m_p$ on $\\{0,1\\}^{\\mathbb N}$, i.e., with $m_p([0])=p$ and $m_p([1])=1-p$. Let $\\nu_{\\beta,p}=m_p\\circ \\phi$ on $\\Omega \\times [0, \\beta]$. Since $m_p$ is shift invariant and ergodic, we have that $\\nu_{\\beta,p}$ is $K$-invariant and ergodic. We first show that $\\nu_{\\beta,p}$ is a Markov measure with the same Markov partition as in the symmetric case (see Example~\\ref{r:goldenmean}), but the transition probabilities as well as the stationary distribution are different. This is achieved by looking at the $\\alpha$-code as well. The Markov partition is given by the partition $\\{E_0,S,E_1\\}$, and the corresponding Markov chain has three states $\\{e_0,s,e_1\\}$ with transition matrix\n\\[ P_p=\\left( \\begin{array}{ccc}\np & 1-p & 0 \\\\\np & 0 & 1-p \\\\\n0 & p & 1-p \\end{array} \\right)\\] \nand stationary distribution \n\\[ u=(u_{e_0},u_s,u_{e_1})=\\Big(\\frac{p^2}{p^2-p+1},\\frac{p(1-p)}{p^2-p+1},\\frac{(1-p)^2}{p^2-p+1} \\Big).\\]\nWe denote the corresponding Markov measure by $Q_p$, that is\n\\[ Q_p\\big([j_1 \\cdots j_k]\\big)=u_{j_1}p_{j_1,j_2}\\cdots p_{j_k,j_{k+1}},\\]\nand the space of realizations by \n\\[ Y=\\big\\{(y_1,y_2,\\ldots):y_i\\in \\{e_0,s,e_1\\}, \\mbox{ and } p_{y_i,y_{i+1}}>0\\big\\}.\\]\nConsider the map $\\alpha:\\Omega \\times [0,\\beta]$ of the previous section, namely\n\\[ \\alpha_j(\\omega,x) = \\left\\{ \\begin{array}{ll}\n e_0, & \\mbox{if $K^{j-1}(\\omega,x)\\in \\Omega \\times E_0$};\\\\\n s, & \\mbox{if $K^{j-1}(\\omega,x)\\in \\Omega \\times S$};\\\\\ne_1, & \\mbox{if $K^{j-1}(\\omega,x)\\in \\Omega \\times E_1$}.\\end{array} \\right. \\] \nDefine $\\psi:Y\\to \\{0,1\\}^{\\mathbb N}$ by\n\\[ \\psi(y)_j = \\left\\{ \\begin{array}{ll}\n 0, & \\mbox{if $y_j=e_0$ or $y_jy_{j+1}=se_1$};\\\\\n 1, & \\mbox{if $y_j=e_1$ or $y_jy_{j+1}=se_0$}.\\end{array} \\right. \\] \nIt is easy to see that $\\psi\\circ \\alpha=\\phi$. We want to show that $Q_p\\circ \\alpha =\\nu_{\\beta,p}$. Since $\\nu_{\\beta,p}=m_p\\circ \\phi$, we show instead the following.\n\n\\begin{prop}\nWe have $m_p=Q_p\\circ \\psi^{-1}$.\n\\end{prop}\n\n\\begin{proof}\nIt is enough to check equality on cylinders. To avoid confusion, we denote cylinders in $\\{0,1\\}^{\\mathbb N}$ by $[i_1 \\cdots i_k]$ and cylinders in $Y$ by $[j_1 \\cdots j_k]$. We show by induction that\n\\[ \\psi^{-1}([i_1 \\cdots i_k])=[j_1 \\cdots j_k]\\cup [j^{\\prime}_1,\\ldots, j^{\\prime}_{k+1}],\\]\nwhere $j_k=e_{i_k}$, and $j^{\\prime}_kj^{\\prime}_{k+1}=se_{1-i_k}$, and\n\\[ Q_p([j_1 \\cdots j_k])+Q_p([j^{\\prime}_1 \\cdots j^{\\prime}_{k+1}])=m_p\\big([i_1 \\cdots i_k]\\big) =p^{k-\\sum_{\\ell=1}^k i_{\\ell} }(1-p)^{\\sum_{\\ell=1}^k i_{\\ell}}.\\]\nConsider the case $k=1$. We have $\\psi^{-1}[0]=[e_0]\\cup [se_1]$ and $\\psi^{-1}[1]=[e_1]\\cup [se_0]$. Furthermore,\n\\[ Q_p([e_0]\\big)+Q_p([se_1])=\\frac{p^2}{p^2-p+1}+\\frac{p(1-p)^2}{p^2-p+1}=p=m_p([0]),\\]\nand\n\\[ Q_p([e_1])+Q_p([se_0])=\\frac{(1-p)^2}{p^2-p+1}+\\frac{p^2(1-p)}{p^2-p+1}=1-p=m_p([1])\\]\nas required. Assume now the result is true for all cylinders $[i_1 \\cdots i_k]$ of length $k$, and consider a cylinder $[i_1 \\cdots i_{k+1}]$ of length $k+1$. Then,\n\\[ \\psi^{-1}([i_1 \\cdots i_{k+1}])=[j_1 \\cdots j_{k+1}]\\cup [j^{\\prime}_1 \\cdots j^{\\prime}_{k+2}],\\]\nwhere\n\\[ \\psi^{-1}([i_2 \\cdots i_{k+1}])=[j_2 \\cdots j_{k+1}]\\cup [j^{\\prime}_2 \\cdots j^{\\prime}_{k+2}],\\]\nand \n\\[ j_1, j^{\\prime}_1= \\left\\{ \\begin{array}{ll}\n e_{i_1}, & \\mbox{if $i_1=i_2$ or $i_1\\not= i_2$ and $j_2=s$},\\\\\n s, & \\mbox{if $i_1\\not= i_2$ and $j_2\\not=s$},\\end{array} \\right. \\] \nBy the definition of $Q_p$, we have \n\\[ Q_p([j_1 \\cdots j_{k+1}])=\\frac{u_{j_1} p_{j_1,j_2}}{u_{j_2}} Q_p([j_2 \\cdots j_{k+1}]),\\]\nand\n\\[ Q_p([j^{\\prime}_1 \\cdots j^{\\prime}_{k+2}])=\\frac{u_{j^{\\prime}_1} p_{j^{\\prime}_1,j^{\\prime}_2}}{u_{j^{\\prime}_2}} Q_p([j^{\\prime}_2 \\cdots j^{\\prime}_{k+2}]).\\]\nOne easily checks that\n\\[ \\frac{u_{j_1} p_{j_1,j_2}}{u_{j_2}}=\\frac{u_{j^{\\prime}_1} p_{j^{\\prime}_1,j^{\\prime}_2}}{u_{j^{\\prime}_2}}=p^{1-i_1}(1-p)^{i_1} =\\left\\{ \\begin{array}{ll}\n p, & \\mbox{if }i_1=0;\\\\\n 1-p, & \\mbox{if }i_1=1.\\end{array} \\right. \\] \nBy the induction hypothesis applied to the cylinder $[i_2 \\cdots i_{k+1}]$, we have $j_{k+1}=e_{i_{k+1}}$, and $j^{\\prime}_{k+1}j^{\\prime}_{k+2}=se_{1-i_{k+1}}$, and\n\\[ Q_p([j_2 \\cdots j_{k+1}])+Q_p([j^{\\prime}_2 \\cdots j^{\\prime}_{k+2}])=m_p([i_2\\cdots i_{k+1}])\n=p^{k-\\sum_{\\ell=2}^{k+1} i_{\\ell} }(1-p)^{\\sum_{\\ell=2}^{k+1} i_{\\ell}}.\\]\nThus,\n\\begin{eqnarray*}\nQ_p([j_1 \\cdots j_{k+1}])+Q_p([j^{\\prime}_1 \\cdots j^{\\prime}_{k+2}]) & = &\np^{1-i_1}(1-p)^{i_1} p^{k-\\sum_{\\ell=2}^{k+1} i_{\\ell} }(1-p)^{\\sum_{\\ell=2}^{k+1} i_{\\ell}}\\\\\n& = & p^{k+1-\\sum_{\\ell=1}^{k+1} i_{\\ell} }(1-p)^{\\sum_{\\ell=1}^{k+1} i_{\\ell}}\\\\\n& = & m_p([i_1\\cdots i_{k+1}]).\n\\end{eqnarray*}\nThis gives the result.\n\\end{proof}\n\n\\noindent As before, let $M_k(\\omega,x)=\\sum_{i=0}^{k-1}{\\bf 1}_{\\Omega \\times S}(K_{\\beta}^i(\\omega,x))$.\n\\begin{thm} For $\\nu_{\\beta,p}$-a.e.~$(\\omega,x)$ for which $\\displaystyle\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}$ exists, one has\n\\[ d\\big( \\nu_{\\beta,p},(\\omega, x)\\big)=\\frac{H(p)}{\\log \\beta} \\Big(2-\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}\\Big),\\]\nwhere $H(p)=-p\\log p -(1-p)\\log(1-p)$.\n\\end{thm}\n\n\\begin{proof} We consider the same metric $\\rho$ as in the previous section, namely\n\\[ \\rho\\big((\\omega,x),(\\omega^{\\prime},x^{\\prime})\\big)=\\beta^{-\\min\\{k\\ge 0 \\, : \\, \n\\omega_{k+1}\\neq \\omega^{\\prime}_{k+1} \\text{ or } \\alpha_{k+1}(\\omega,x)\\neq \\alpha_{k+1}(\\omega^{\\prime},x^{\\prime})\\}}.\\]\nConsider a point $(\\omega,x)$ such that $\\displaystyle\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}$ exists. By the same reasoning that led to (\\ref{q:Qball}) we have\n\\begin{multline*}\n\\nu_{\\beta,p}\\big(B_{\\rho}((\\omega,x),\\beta^{-k})\\big)\\\\\n= Q_p([\\alpha_1(\\omega,x),\\ldots,\\alpha_k(\\omega,x)]) p^{(k-M_k(\\omega,x))-\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} (1-p)^{\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} u_{e_{1-\\omega_k}}.\n\\end{multline*}\nLet $u_{\\max}=\\max(u_{e_0},u_{e_1})$ and $u_{\\min}=\\min(u_{e_0},u_{e_1})$, then $\\log \\nu_{\\beta,p}\\big(B_{\\rho}((\\omega,x),\\beta^{-k})\\big)$ is bounded from above by\n\\begin{gather*}\n\\log Q_p([\\alpha_1(\\omega,x),\\ldots,\\alpha_k(\\omega,x)]) + \\Big((k-M_k(\\omega,x)-\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i\\Big)\\log p\\\\\n + \\sum_{i=M_k(\\omega,x)+1}^k \\omega_i\\log (1-p) +\\log u_{\\max},\n\\end{gather*}\nand is bounded from below by\n\\begin{gather*}\n\\log Q_p([\\alpha_1(\\omega,x),\\ldots,\\alpha_k(\\omega,x)]) + \\Big((k-M_k(\\omega,x)-\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i\\Big)\\log p\\\\\n+ \\sum_{i=M_k(\\omega,x)+1}^k \\omega_i\\log (1-p) +\\log u_{\\min}.\n\\end{gather*}\nDividing by $-k\\log \\beta$ and taking limits, we have by the Shannon-McMillan-Breiman Theorem that\n\\[ \\lim_{k\\to\\infty}\\frac{\\log Q_p([\\alpha_1(\\omega,x) \\cdots \\alpha_k(\\omega,x)])}{-k\\log \\beta}= \\frac{H(p)}{\\log \\beta},\\]\nand by the Ergodic Theorem we have\n\\[ \\lim_{k\\to\\infty}\\frac{\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i}{-k\\log \\beta}= \\frac{-(1-p)\\big(1-\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}\\big)}{\\log \\beta},\\]\nboth for $\\nu_{\\beta}$-a.e.~$(\\omega,x)$. Thus, both the upper and the lower bounds converge to the same value, implying that\n\\[ d\\big(\\nu_{\\beta,p}, (\\omega, x)\\big)=\\frac{H(p)}{\\log \\beta} \\Big(2-\\lim_{k\\to\\infty}\\frac{M_k(\\omega,x)}{k}\\Big). \\qedhere\\]\n\\end{proof}\n\n\\begin{cor}\nFor $\\nu_{\\beta,p}$-a.e.~$(\\omega,x)$ one has \n\\[ d\\big(\\nu_{\\beta,p}, (\\omega, x)\\big)=\\frac{H(p)}{\\log \\beta} \\Big(2-\\nu_{\\beta,p}\\big(\\Omega \\times S\\big)\\Big) =\\frac{H(p)}{\\log \\beta} \\Big(2-\\frac{p(1-p)}{p^2-p+1}\\Big).\\]\n\\end{cor}\n\n\\vskip .5cm\nWe now turn to the study of the local dimension of the asymmetric Bernoulli convolution $\\mu_{\\beta,p}$, which is the projection in the second coordinate of $\\nu_{\\beta,p}$. Let ${\\mathcal N}_k(\\omega,x)$ be as given in equation (\\ref{q:fs}). In the symmetric case, it was shown that\n\\[\\mathcal N_{k}(x,\\beta)=\\int_{\\{0,1\\}^{\\mathbb N}} \\, 2^{M_k(\\omega,x)} \\, dm(\\omega)=\\sum_{[\\omega_1 \\cdots \\omega_k]} 2^{M_k(\\omega,x)} 2^{-k}.\\]\nWe now give a similar formula for the asymmetric case.\n\n\\begin{lem} \\label{counting}\n$\\mathcal N_{k}(x,\\beta)=\\sum_{[\\omega_1 \\cdots \\omega_k]} p^{(k-M_k(\\omega,x))-\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} (1-p)^{\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i}$.\n\\end{lem}\n\n\\begin{proof} We use a similar argument as the one used for the symmetric case (see \\cite{Kem12}). Define\n\\[ \\Omega(k,x)=\\big\\{\\omega_1 \\cdots \\omega_{M_k(\\omega,x)}: \\omega\\in \\Omega \\big\\}.\\]\nIf $x$ has a unique expansion, then $\\Omega(k,x)$ consists of one element, the empty word. We now have $|\\Omega(k,x)|= \\mathcal N_k(x,\\beta)$, and \n\n\\begin{eqnarray*}\n& &\n \\sum_{[\\omega_1 \\cdots \\omega_k]} p^{(k-M_k(\\omega,x))-\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} (1-p)^{\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} \\\\\n& = & \\sum_{[\\omega_1 \\cdots \\omega_k]} \\frac{p^{k-\\sum_{i=1}^k \\omega_i}(1-p)^{\\sum_{i=1}^k \\omega_i}}{p^{M_k(\\omega,x)-\\sum_{i=1}^{M_k(\\omega,x)} \\omega_i}(1-p)^{\\sum_{i=1}^{M_k(\\omega,x)} \\omega_i}}\\\\\n& = & \\int_{\\Omega}\\frac{1}{m_p([\\omega_1 \\cdots \\omega_{M_k(\\omega,x)}])} dm_p(\\omega) = \\sum_{j=0}^k \\int_{\\{\\omega: M_k(\\omega,x)=j\\}}\\frac{1}{m_p([\\omega_1 \\cdots \\omega_j])} dm_p(\\omega)\\\\\n& = & \\sum_{j=0}^k\\,\\,\\sum_{\\omega_1 \\cdots \\omega_j \\in \\Omega(k,x)}\\frac{1}{m_p([\\omega_1 \\cdots \\omega_j])} m_p([\\omega_1 \\cdots \\omega_j]) = |\\Omega(k,x)|=\\mathcal N_k(x,\\beta). \\quad \\quad \\qedhere\n\\end{eqnarray*}\n\\end{proof}\n\n\n\\begin{thm}\\label{bounds}\nFor $\\lambda$-a.e.~$x \\in [0, \\beta]$ one has\n\\[ \\frac{-\\big(\\log (\\max(p,1-p))+\\gamma \\big)}{\\log \\beta}\\le \\underline{d}(\\mu_{\\beta,p},x)\\le \\overline{d}(\\mu_{\\beta,p},x)\\le \\frac{-\\big(\\log (\\min(p,1-p))+\\gamma \\big)}{\\log \\beta},\\]\nwhere $\\displaystyle\\lim_{k \\to\\infty}\\frac{\\log \\mathcal N_k(x,\\beta)}{k}=\\gamma$ is the constant from (\\ref{q:gamma}).\n\\end{thm}\n\n\\begin{proof} We use the same metric $\\bar \\rho$ on $[0,\\beta]$ as in the previous section. Then\n\\begin{multline*}\n\\mu_{\\beta,p}\\big(B_{\\bar \\rho}(x,\\beta^{-k})\\big) = \\sum_{[\\omega_1 \\cdots \\omega_k]}\\nu_{\\beta}(B_{\\rho}\\big([\\omega_1 \\cdots \\omega_k],x),\\beta^{-k})\\big)\\\\ \n= \\sum_{[\\omega_1 \\cdots \\omega_k]}Q_p([\\alpha_1(\\omega,x) \\cdots \\alpha_k(\\omega,x)]) p^{(k-M_k(\\omega,x))-\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} \\\\\n\\cdot(1-p)^{\\sum_{i=M_k(\\omega,x)+1}^k \\omega_i} u_{e_{1-\\omega_k}}.\n\\end{multline*}\nNow,\n\\[ Q_p([\\alpha_1(\\omega,x) \\cdots \\alpha_k(\\omega,x)])=u_{\\alpha_1(\\omega,x)} p^{L_k(\\omega,x)}(1-p)^{k-L_k(\\omega,x)},\\]\nwhere\n\\[ L_k(\\omega,x)=\\#\\{1\\le j\\le k:\\alpha_j(\\omega,x)=e_0\\}+\\#\\{1\\le j\\le k:\\alpha_j(\\omega,x)\\alpha_{j+1}(\\omega,x)=e_1s\\},\\]\nand hence\n\\begin{multline*}\nk-L_k(\\omega,x)\\\\\n=\\#\\{1\\le j\\le k:\\alpha_j(\\omega,x)=e_1\\}+\\#\\{1\\le j\\le k:\\alpha_j(\\omega,x)\\alpha_{j+1}(\\omega,x)=e_0s\\}.\n\\end{multline*}\nLet $C_1=\\max(u_{e_0},u_s,u_{e_1})$ and $C_2=\\min(u_{e_0},u_s,u_{e_1})$. Then, from Lemma (\\ref{counting}) we have\n\\[ C_2\\big(\\min(p,1-p))\\big)^k \\mathcal N_k(x,\\beta)\\le \\mu_{\\beta,p}(B_{\\bar \\rho}(x,\\beta^{-k}))\\le C_1\\big(\\max(p,1-p))\\big)^k \\mathcal N_k(x,\\beta).\\]\nSince $\\beta$ is a Pisot number, $\\displaystyle\\lim_{k \\to\\infty}\\frac{\\log \\mathcal N_k(x,\\beta)}{k}=\\gamma$ exists $\\lambda$-a.e.~(see \\cite{FS11}) and we have\n\\[ \\frac{-[\\log (\\max(p,1-p))+\\gamma]}{\\log \\beta}\\le \\underline{d}(\\mu_{\\beta,p},x)\\le \\overline{d}(\\mu_{\\beta,p},x)\\le \\frac{-[\\log (\\min(p,1-p))+\\gamma]}{\\log \\beta}. \\qedhere\\]\n\\end{proof}\n\n\\begin{rem}{\\rm (i) If $p=1\/2$, then both sides of the inequality in Theorem (\\ref{bounds}) are equal to $\\displaystyle\\frac{\\log 2-\\gamma}{\\log \\beta}$ leading to \n\\[ d(\\mu_{\\beta,1\/2},x)= \\frac{\\log 2-\\gamma}{\\log \\beta}\\]\na.e.~as we have seen earlier.\\\\\n(ii) We now consider the extreme cases $x\\in \\{0,\\beta\\}$, the only two points with a unique expansion. We begin with $x=\\beta$. In this case\n\\[ Q_p([\\alpha_1(\\omega,\\beta) \\cdots \\alpha_k(\\omega,\\beta)])=Q_p([e_1 \\cdots e_1])=u_{e_1}(1-p)^k,\\]\nand $\\mathcal N_k(\\beta,\\beta)=1$, so that\n\\[ C_2(1-p)^k\\le \\mu_{\\beta,p}(B_{\\bar \\rho}(\\beta,\\beta^{-k}))\\le C_1(1-p)^k.\\]\nHence,\n\\[ d(\\mu_{\\beta,p}, \\beta)=\\frac{-\\log (1-p)}{\\log\\beta}\\]\nfor all $\\omega\\in \\Omega$. A similar argument shows that\n\\[ d(\\mu_{\\beta,p},0)=\\frac{-\\log (p)}{\\log\\beta}\\]\nfor all $\\omega\\in \\Omega$.\n}\\end{rem}\n\n\\bigskip\n\\footnotesize\n\\noindent\\textit{Acknowledgments.}\nThe second author was supported by NWO (Veni grant no. 639.031.140).\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{S:0}\nThe weak separation condition (WSC) was introduced by Lau and Ngai \\cite{Lau-Ngai_1999} to study the multifractal formalism of self-similar measures defined by iterated function systems of contractive similitudes with overlaps. Although strictly weaker than the well-known open set condition (OSC), the weak separation condition leads to many interesting results, such as the validity of the multifractal formalism (see, e.g. \\cite{Lau-Ngai_1999,Feng_2003,Feng_2005,Feng-Lau_2009, Shmerkin_2005,Ye_2005}), equality of Hausdorff, box, and packing dimensions of self-conformal sets and the computation of these dimensions \\cite{Deng-Ngai_2011,Ferrari-Panzone_2011,Lau-Ngai-Wang_2009}, and conditions on absolute continuity of self-similar and self-conformal measures \\cite{Lau-Ngai-Rao_2001,Lau-Wang_2004,Lau-Ngai-Wang_2009}. Equivalent forms of the weak separation conditions have also been studied extensive (see, e.g. \\cite{Zerner_1996, Lau-Ngai-Wang_2009}).\n\nThe finite type condition (FTC) was introduced by Ngai and Wang \\cite{Ngai-Wang_2001} to calculate the Hausdorff dimension of self-similar sets with overlaps. It was generalized independently by Jin-Yau \\cite{Jin-Yau_2005} and Lau-Ngai \\cite{Lau-Ngai_2007} to include (OSC). Lau and Ngai proved that (FTC) implies (WSC) for IFSs of contractive similitudes. This result was generalized by Lau \\textit{et al.} \\cite{Lau-Ngai-Wang_2009} to conformal iterated function systems (CIFSs).\n\nIn 2009, Lau \\textit{et al.} \\cite{Lau-Ngai-Wang_2009} formulated (WSC) for conformal iterated function systems on $\\mathbb{R}^{n}$, and proved the equality of the Hausdorff, box and packing dimensions of the associated self-conformal sets. They also studied the absolute continuity of the associated self-conformal measures. The first goal of this paper is to extend results in \\cite{Lau-Ngai-Wang_2009} to Riemannian manifolds with nonnegative Ricci curvature. Our second goal is to generalize the method of computing the Haudorff dimension of self-similar sets in \\cite{Ngai-Wang_2001,Jin-Yau_2005,Lau-Ngai_2007} to Riemannian manifolds that are locally Euclidean.\n\nLet $M$ be a complete $n$-dimensional smooth Riemannian manifold.\nAssume that $U\\subset M$ is open and connected, and $W\\subset U$ is a compact set with $\\overline{W^{\\circ}}=W$, where $\\overline{W^{\\circ}}$ is the closure of the interior of $W$. Assume that $\\{S_{i}\\}_{i=1}^{N}$ is a conformal iterated function system (CIFS) on $U$ defined as in Section \\ref{S:1}. Then there exists a unique nonempty compact set $K\\subset W$, called the \\textit{attractor} or \\textit{self-conformal set}, satisfying $K=\\bigcup_{i=1}^{N}S_{i}(K)$ (see \\cite{Hutchinson_1981}).\n\nGiven a probability vector $(p_{1},\\dots,p_{N})$, i.e., $p_{i}>0$ for any $i\\in\\{1,\\dots,N\\}$ and $\\sum_{i=1}^{N}p_{i}=1$, there exists a unique Borel probability measure $\\mu$, called the \\textit{self-conformal measure}, such that $\\mu=\\sum_{i=1}^{N}p_{i}\\mu\\circ S_{i}^{-1}$ and $K={\\rm supp}(\\mu)$ (see \\cite{Hutchinson_1981}).\n\nLet $\\Sigma:=\\{1,\\dots,N\\}$, where $N\\in\\mathbb{N}$ and $N\\geq2$. Let $\\Sigma^{\\ast}:=\\bigcup_{k\\geq1}\\Sigma^{k}$, where $k\\in\\mathbb{N}$. For $\\mathbf{u}=(u_1,\\dots,u_k)\\in\\Sigma^{k}$, let $\\mathbf{u}^{-}:=(u_{1},\\dots,u_{k-1})$ and write $S_{\\mathbf{u}}:=S_{u_1}\\circ\\cdots\\circ S_{u_k}$, $p_{\\mathbf{u}}:=p_{u_1}\\cdots p_{u_k}$. Define\n$$r_{\\mathbf{u}}:=\\inf_{x\\in W}|\\det S'_{\\mathbf{u}}(x)|^{\\frac{1}{n}},~r:=\\min_{1\\leq i\\leq N}r_{i},~R_{\\mathbf{u}}:=\\sup_{x\\in W}|\\det S'_{\\mathbf{u}}(x)|^{\\frac{1}{n}},~R:=\\max_{1\\leq i\\leq N}R_{i}.$$\nIf $S=S_{\\mathbf{u}}$ for some $\\mathbf{u}\\in\\Sigma^{\\ast}$, we let $R_{S}=R_{\\mathbf{u}}$. For $00$ and any bounded subsets $D\\subset W$ and $A\\subset M$, denote the diameter of $A$ by $|A|$, and let\n$$\\mathcal{A}_{a,A,D}:=\\{S\\in\\mathcal{A}_{a|A|}:S(D)\\cap A\\neq\\emptyset\\},\\quad\\gamma_{a,A}:=\\sup_{A\\subset M}\\#\\mathcal{A}_{a,A,D}.$$\nFor $S\\in\\mathcal{A}_{b}$, let $p_{S}:=\\sum\\{p_{\\mathbf{u}}:S_{\\mathbf{u}}=S,\\mathbf{u}\\in\\mathcal{A}_{b}\\}$.\n\nTheorems \\ref{thm(0.1)}--\\ref{thm(0.4)} generalize analogous results in \\cite{Lau-Ngai-Wang_2009}. In our proofs, the Lebesgue measure in \\cite{Lau-Ngai-Wang_2009} is changed to the more complicated Riemannian volume measure; properties such as the volume doubling property, need not hold. We assume that $M$ is a complete Riemannian manifold with nonnegative Ricci curvature. Under this assumption, the Bishop-Gromov comparison theorem implies that $M$ is a \\textit{doubling space} (see, e.g. \\cite{Baudoin_2011,Berger_2003}), i.e., any $2r$-ball in $M$ can be covered by a finite union of a bounded number of $r$-balls, a property that obviously holds on $\\mathbb{R}^{n}$. Another complication arises on manifolds; unlike Euclidean spaces, it is not easy to calculate the volume of a ball in Riemannian manifolds. For Riemannian manifolds with nonnegative Ricci curvature, we use the Bishop-Gromov inequality (see Lemma \\ref{lem(1.00)}), which says that the Riemannian volume of a ball can be controlled by a ball in $\\mathbb{R}^{n}$ with the same radius. This is crucial in the proof of Theorem \\ref{thm(0.1)}.\n\n\nFor a set $K\\subset M$, let $\\dim_{{\\rm H}}(K),\\dim_{{\\rm B}}(K)$ and $\\dim_{{\\rm P}}(K)$ be the Hausdorff, box and packing dimensions, respectively. Let $\\mathcal{H}^{\\alpha}(K)$ and $\\mathcal{P}^{\\alpha}(K)$ be the Hausdorff and packing measures of $K$, respectively.\n\n\n\n\n\\begin{thm}\\label{thm(0.1)}\nLet $M$ be a complete $n$-dimensional smooth orientable Riemannian manifold with non-negative Ricci curvature, and let $U\\subset M$ be open and connect. Assume that $\\{S_{i}\\}_{i=1}^{N}$ is a CIFS on $U$ satisfying (WSC), and $K$ is the associated attractor. Then $\\alpha:=\\dim_{{\\rm H}}(K)=\\dim_{{\\rm B}}(K)=\\dim_{{\\rm P}}(K)$ and\n$0<\\mathcal{H}^{\\alpha}(K)\\leq\\mathcal{P}^{\\alpha}(K)<\\infty$.\n\\end{thm}\n\nLau \\textit{et al.} \\cite{Lau-Ngai-Rao_2001} formulated a sufficient condition for a self-similar measure defined by an IFS satisfying (WSC) to be singular. Later, Lau and Wang \\cite{Lau-Wang_2004} established the necessary perfected the result on absolute continuity in \\cite{Lau-Ngai-Rao_2001}. We extend these results to manifolds.\n\n\n\n\n\\begin{thm}\\label{thm(0.2)}\nAssume the same hypotheses as in Theorem \\ref{thm(0.1)}. Let $K$ be the attractor\nwith $\\dim_{{\\rm H}}(K)=\\alpha$. Then a self-conformal measure $\\mu$ defined by $\\{S_{i}\\}_{i=1}^{N}$ is singular with respect to $\\mathcal{H}^{\\alpha}|_{K}$ if and only if there exist $0R_{S}^{\\alpha}$.\n\\end{thm}\n\n\n\n\\begin{thm}\\label{thm(0.3)}\nAssume the same hypotheses as in Theorem \\ref{thm(0.1)}. If the self-conformal measure $\\mu$ is absolutely continuous with respect to $\\mathcal{H}^{\\alpha}|_{K}$, then the\nRadon-Nikodym derivative of $\\mu$ is bounded.\n\\end{thm}\n\nIn the proof of Theorem \\ref{thm(0.3)}, we use an analogue of the Lebesgue density theorem in metric spaces (see, e.g. \\cite[Theorem 2.9.8]{Federer_1969} and \\cite[Lemma 2.1(i)]{Bedferd-Fisher_1992}), applying to the collection of Borel sets that forms a Vitali relation (see \\cite{Federer_1969} and a brief summary in Section \\ref{S:2}). By \\cite[Definition 2.8.9 and Theorem 2.8.18]{Federer_1969}, we have a collection of open balls in Riemannian manifolds that forms a Vitali relation.\n\n\n We refer the reader to Section \\ref{S:3} for the definition of (FTC).\n\n\\begin{thm}\\label{thm(0.4)}\nAssume the same hypotheses as in Theorem \\ref{thm(0.1)}, and let $W\\subset U$ be a compact set with $\\overline{W^{\\circ}}=W$. If $\\{S_{i}\\}_{i=1}^{N}$ is a CIFS on $U$ satisfying (FTC) on some open sets $\\Omega\\subset W$,\nthen $\\{S_{i}\\}_{i=1}^{N}$ satisfies (WSC).\n\\end{thm}\n\nDenote the Riemannian distance in $M$ by $d(\\cdot,\\cdot)$. Let $W\\subset M$ be a compact set.\nWe say that $\\{S_{i}\\}_{i=1}^{N}$ is an \\textit{IFS of contractions} on $W$ if for any $i\\in\\{1,\\dots,N\\}$, there exists $\\rho_{i}\\in(0,1)$ such that for any $x,y\\in W$,\n\\begin{equation}\\label{eq:IFS_contraction}\nd(S_{i}(x),S_{i}(y))\\le \\rho_{i}d(x,y).\n\\end{equation}\nIf equality in \\eqref{eq:IFS_contraction} holds for all $i\\in\\{1,\\dots,N\\}$ and all $x,y\\in W$, then\nsay that $\\{S_{i}\\}_{i=1}^{N}$ is an \\textit{IFS of contractive similitudes} on $W$ and call $\\rho_i$ the \\textit{contraction ratio} of $S_{i}$.\n\n\nWe say that a Riemannian manifold $M$ is \\textit{locally Euclidean} if every point of $M$ has a neighborhood which is isometric to an open subset of a Euclidean space (see e.g. \\cite{Kobayashi-Nomizu}). By \\cite[Lemma 2 of Theorem 3.6]{Kobayashi-Nomizu}, contractive similitudes only exist in Riemannian manifolds that are locally Euclidean. In Section \\ref{S:4}, we obtain the following formula for computing the Hausdorff dimension formula of self-similar sets defined by a finite type IFS of contractive similitudes on a locally Euclidean Riemannian manifold, extending a result in \\cite{Jin-Yau_2005} and \\cite{Lau-Ngai_2007} to locally Euclidean Riemannian manifolds (see Theorem \\ref{thm(0.5)}).\n\n\\begin{thm}\\label{thm(0.5)}\nLet $M$ be a complete $n$-dimensional smooth orientable Riemannian manifold that is locally Euclidean, $W\\subseteq M$ be a compact subset, and $\\{S_{i}\\}_{i=1}^{N}$ be an IFS of contractive similitudes on $W$ with attractor $K$. Let $\\lambda_{\\alpha}$ be the spectral radius of the associated weighted incidence matrix $A_{\\alpha}$. If $\\{S_{i}\\}_{i=1}^{N}$ satisfies (FTC), then $\\dim_{{\\rm H}}(K)=\\alpha$, where $\\alpha$ is the unique number such that $\\lambda_{\\alpha}=1$.\n\\end{thm}\n\n\nIn order to compute the Hausdorff dimension of certain attractors on Riemannian manifolds, it is necessary to study graph iterated function systems (see Example~\\ref{exam(5.4)}). We define graph-directed iterated function systems (GIFSs) and graph finite type condition (GFTC) on Riemannian manifolds in Section \\ref{S:41}. We obtain the following result for computing the Hausdorff dimension of graph self-similar sets (see Theorem \\ref{thm(41.1)}), extending a result in \\cite{Ngai-Wang-Dong_2010}.\n\n\n\\begin{thm}\\label{thm(41.1)}\nLet $M$ be a complete $n$-dimensional smooth orientable Riemannian manifold that is locally Euclidean. Assume that $G=(V,E)$ is a GIFS defined on $M$ satisfying (GFTC), and $K$ is the associated graph self-similar set. Let $\\lambda_{\\alpha}$ be the spectral radius of the associated weighted incidence matrix $A_{\\alpha}$. Then $\\dim_{{\\rm H}}(K)=\\alpha$, where $\\alpha$ is the unique number such that $\\lambda_{\\alpha}=1$.\n\\end{thm}\n\n\n\n\nThis paper is organized as follows. Section \\ref{S:1} introduces the definition of CIFSs, some properties of (WSC), and gives the proof of Theorem \\ref{thm(0.1)}. In Section \\ref{S:2}, we study the absolute continuity of self-conformal measures on Riemannian manifolds and prove Theorems \\ref{thm(0.2)} and \\ref{thm(0.3)}. Section \\ref{S:3} is devoted to the proof of Theorem \\ref{thm(0.4)}. In Section \\ref{S:4}, we study finite type IFSs of contractive similitudes and prove Theorem \\ref{thm(0.5)}. Section \\ref{S:41} is devoted to the proof of Theorem \\ref{thm(41.1)}. Finally, we present some examples of CIFSs and GIFSs satisfying (FTC) and (GFTC) on Riemannian manifolds, respectively.\n\n\\section{The weak separation condition}\\label{S:1}\n\nLet $M$ be a complete $n$-dimensional smooth Riemannian manifold, $U\\subset M$ be open and connected, and let $W\\subset U$ be a compact set with $\\overline{W^{\\circ}}=W$.\nRecall that a map $S:U\\longrightarrow U$ is called \\textit{conformal} if $S'(x)$ is a similarity matrix for any $x\\in U$. We say that $\\{S_{i}\\}_{i=1}^{N}$ is a \\textit{conformal iterated function system} (CIFS)\non $U$, if\n\\begin{itemize}\n\\item[$(a)$] for any $i\\in\\Sigma$, $S_{i}:U\\longrightarrow S_{i}(U)\\subset U$ is a conformal $C^{1+\\varepsilon}$ diffeomorphism for some $\\varepsilon\\in(0,1)$;\n\\item[$(b)$] $S_{i}(W)\\subset W$ for any $i\\in\\Sigma$;\n\\item[$(c)$] $0<|\\det S'_{i}(x)|<1$ for any $i\\in\\Sigma$ and $x\\in U$.\n\\end{itemize}\n\nSince $M$ is a manifold, we can find an open and connected set $U_{1}$ such that $\\overline{U_{1}}$ is compact and $W\\subset U_{1}\\subset \\overline{U_{1}}\\subset U$. According to \\cite{Patzschke_1997}, conditions $(a)$--$(c)$ together imply the \\textit{bounded distortion property} (BDP), without assuming any separation condition, i.e., there exists a constant $C_{1}\\geq1$ such that for any $\\mathbf{u}\\in\\Sigma^{*}$ and $x,y\\in U_{1}$,\n\\begin{equation}\\label{eq(1.1)}\nC_{1}^{-n}\\leq\\frac{|\\det S'_{\\mathbf{u}}(x)|}{|\\det S'_{\\mathbf{u}}(y)|}\\leq C_{1}^{n}.\n\\end{equation}\nIt follows that for any $\\mathbf{u}\\in\\Sigma^*$,\n\\begin{equation}\\label{eq:r_R_bound}\nC_{1}^{-1}\\leq\\frac{r_{\\mathbf u}}{R_{\\mathbf u}}\\leq\\frac{R_{\\mathbf u}}{r_{\\mathbf u}}\\leq C_{1}.\n\\end{equation}\nMoreover, there exists a constant $C_{2}\\geq1$ such that for any $\\mathbf{u}\\in\\Sigma^{\\ast}$ and $x,y\\in W$,\n\\begin{equation}\\label{eq(1.2)}\nC_{2}^{-1}R_{\\mathbf{u}}d(x,y)\\leq d(S_{\\mathbf{u}}(x),S_{\\mathbf{u}}(y))\\leq C_{2}R_{\\mathbf{u}}d(x,y).\n\\end{equation}\nLet $\\nu$ be the Riemannian volume measure, and let $A\\subset W$ be a measurable set. Denote the Jacobian determinant of a function $f$ by $\\mathbf{J}f$. Then\n\\begin{equation}\\label{eq(1.02)}\n\\nu\\big(S_{\\mathbf{u}}(A)\\big)=\\int_{A}\\big|\\mathbf{J}\\big(S_{\\mathbf{u}}(x)\\big)\\big|d\\nu=\\int_{A}\\big|\\det S'_{\\mathbf{u}}(x)\\big|d\\nu.\n\\end{equation}\n(see, e.g. \\cite[Proposition 8.1.10]{Abraham-Marsden-Ratiu_2007}).\n\nNote that for any $x\\in W$ and $\\mathbf{u},\\mathbf{v}\\in\\Sigma^{\\ast}$,\n$$|\\det S'_{\\mathbf{u}\\mathbf{v}}(x)|=|\\det S'_{\\mathbf{u}}(S_{\\mathbf{v}}(x))\\cdot S'_{\\mathbf{v}}(x)|=|\\det S'_{\\mathbf{u}}(S_{\\mathbf{v}}(x))|\\cdot|\\det S'_{\\mathbf{v}}(x)|.$$\nHence $R_{\\mathbf{u}\\mathbf{v}}\\leq R_{\\mathbf{u}}R_{\\mathbf{v}}$ and $r_{\\mathbf{u}\\mathbf{v}}\\geq r_{\\mathbf{u}}r_{\\mathbf{v}}$. In particular, for $S=S_{\\mathbf{u}}\\in\\mathcal{A}_{b},\\mathbf{u}=(u_{1},\\dots,u_{k})\\in\\mathcal{W}_{b}$,\n$$\\begin{aligned}\nbr&\\bigg(\\frac{rb}{C_{1}}\\bigg)^{n}\\quad\\text{(by (\\ref{eq(1.3)}))}.\n\\end{aligned}$$\nIt follows that\n$$\\int_{A}b^{n}d\\nu\\geq\\int_{A}|\\det S'_{\\mathbf{u}}(x)|d\\nu\\geq\\int_{A}\\bigg(\\frac{rb}{C_{1}}\\bigg)^{n}d\\nu,$$\nwhich proves $(a)$ by (\\ref{eq(1.02)}).\n\n$(b)$ Making use of part $(a)$, we have\n$$\\nu(S_{\\mathbf{u}}(A))\\leq b^{n}\\nu(A)\\leq\\bigg(\\frac{C_{1}}{r}\\bigg)^{n}\\nu(S_{\\mathbf{v}}(A)).$$\nSimilarly,\n$$\\nu(S_{\\mathbf{v}}(A))\\leq b^{n}\\nu(A)\\leq\\bigg(\\frac{C_{1}}{r}\\bigg)^{n}\\nu(S_{\\mathbf{u}}(A)),$$\nwhich proves $(b)$.\n\\end{proof}\n\nLet $\\mathcal{F}:=\\{S_{\\mathbf{v}}S_{\\mathbf{u}}^{-1}:\\mathbf{u},\\mathbf{v}\\in\\Sigma^{\\ast}\\}$. It is possible that $\\tau=S_{\\mathbf{v}}S_{\\mathbf{u}}^{-1}$ can be simplified to $S_{\\mathbf{v}'}S_{\\mathbf{u}'}^{-1}$. Thus the domain of\n$\\tau$ is $S_{\\mathbf{u}'}(W)$ (containing $S_{\\mathbf{u}}(W)$). Denote the domain of $\\tau$ by ${\\rm Dom}(\\tau)$. The proof of the following lemma is similar to that of \\cite[Lemma 2.2]{Lau-Ngai-Wang_2009} and is omitted.\n\n\\begin{lem}\\label{lem(1.2)}\nAssume the same hypotheses of Lemma \\ref{lem(1.1)}. Then for any $\\mathbf{u},\\mathbf{v}\\in\\Sigma^{\\ast}$ and any $x,y\\in{\\rm Dom}(S_{\\mathbf{v}'}S_{\\mathbf{u}'}^{-1})$, we have\n$$\\frac{|\\det (S_{\\mathbf{v}}S_{\\mathbf{u}}^{-1})'(x)|}{|\\det (S_{\\mathbf{v}}S_{\\mathbf{u}}^{-1})'(y)|}\\leq C_{1}^{2n}.$$\n\\end{lem}\n\n\n\\begin{lem}\\label{lem(1.3)}\nAssume the same hypotheses of Lemma \\ref{lem(1.1)}. Let $\\tau=S_{\\mathbf{v}}S_{\\mathbf{u}}^{-1}=S_{\\mathbf{v}'}S_{\\mathbf{u}'}^{-1}\\in\\mathcal{F}$ with ${\\rm Dom}(\\tau)=S_{\\mathbf{u}'}(W)$. Then the following hold.\n\\begin{itemize}\n\\item[$(a)$] For any measurable subset $A\\subset {\\rm Dom}(\\tau)$,\n$$\\bigg(\\frac{r_{\\mathbf{v}'}}{R_{\\mathbf{u}'}}\\bigg)^{n}\\nu(A)\\leq \\nu(\\tau(A))\\leq\\bigg(\\frac{R_{\\mathbf{v}'}}{r_{\\mathbf{u}'}}\\bigg)^{n}\\nu(A).$$\n\\item[$(b)$] For any $A,B$ belonging to some collection $\\mathcal{C}$ of measurable subsets of $W$, suppose $C\\geq1$ is a constant such that\n$$C^{-1}\\nu(B)\\leq\\nu(A)\\leq C\\nu(B).$$\nThen for any $A,B\\in\\mathcal{C}$ such that $A,B\\subset{\\rm Dom}(\\tau)$,\n$$C^{-1}C_{1}^{-2n}\\nu(\\tau(B))\\leq\\nu(\\tau(A))\\leq CC_{1}^{2n}\\nu(\\tau(B)).$$\n\\end{itemize}\n\\end{lem}\n\\begin{proof}\n$(a)$ For $x\\in {\\rm Dom}(\\tau)$, let $y=S_{\\mathbf{u}'}^{-1}(x)\\in W$. Then\n$$|\\det\\tau'(x)|=|\\det (S_{\\mathbf{v}'}S_{\\mathbf{u}'}^{-1})'(x)|=|\\det S'_{\\mathbf{v}'}(S_{\\mathbf{u}'}^{-1}(x))|\\cdot|\\det (S_{\\mathbf{u}'}^{-1})'(x)|=\\frac{|\\det S'_{\\mathbf{v}'}(y)|}{|\\det S'_{\\mathbf{u}'}(y)|}.$$\nHence\n$$\\bigg(\\frac{r_{\\mathbf{v}'}}{R_{\\mathbf{u}'}}\\bigg)^{n}\\leq\n|\\det\\tau'(x)|\\leq\\bigg(\\frac{R_{\\mathbf{v}'}}{r_{\\mathbf{u}'}}\\bigg)^{n}.$$\nThus,\n$$\\int_{A}\\bigg(\\frac{r_{\\mathbf{v}'}}{R_{\\mathbf{u}'}}\\bigg)^{n}d\\nu\\leq\n\\int_{A}|\\det\\tau'(x)|d\\nu\\leq\\int_{A}\\bigg(\\frac{R_{\\mathbf{v}'}}{r_{\\mathbf{u}'}}\\bigg)^{n}d\\nu,$$\nwhich proves $(a)$ by (\\ref{eq(1.02)}).\n\n$(b)$ Making use of part $(a)$ and (\\ref{eq(1.1)}), we have\n$$\\begin{aligned}\n\\nu(\\tau(A))&\\leq\\bigg(\\frac{R_{\\mathbf{v}'}}{r_{\\mathbf{u}'}}\\bigg)^{n}\\nu(A)\n\\leq C\\bigg(\\frac{R_{\\mathbf{v}'}}{r_{\\mathbf{u}'}}\\bigg)^{n}\\nu(B)\\\\\n&\\leq C\\bigg(\\frac{R_{\\mathbf{v}'}}{r_{\\mathbf{u}'}}\\bigg)^{2n}\n\\nu(\\tau(B))\\quad\\text{(by part $(a)$)}\\\\\n&\\leq CC_{1}^{2n}\\nu(\\tau(B))\\quad\\text{(by (\\ref{eq:r_R_bound}))}.\n\\end{aligned}$$\nOn the other hand,\n$$\\begin{aligned}\n\\nu(\\tau(A))&\\geq\\bigg(\\frac{r_{\\mathbf{v}'}}{R_{\\mathbf{u}'}}\\bigg)^{n}\\nu(A)\n\\geq C^{-1}\\bigg(\\frac{r_{\\mathbf{v}'}}{R_{\\mathbf{u}'}}\\bigg)^{n}\\nu(B)\\\\\n&\\geq C^{-1}\\bigg(\\frac{r_{\\mathbf{v}'}}{R_{\\mathbf{u}'}}\\bigg)^{2n}\n\\nu(\\tau(B))\\quad\\text{(by part $(a)$)}\\\\\n&\\geq C^{-1}C_{1}^{-2n}\\nu(\\tau(B))\\quad\\text{(by (\\ref{eq:r_R_bound}))}.\n\\end{aligned}$$\nThis proves $(b)$.\n\\end{proof}\n\n\n\\begin{lem}\\label{lem(1.30)} (Bishop-Gromov inequality (see, e.g. \\cite{Anderson_1990,Berger_2003,Bishop-Crittenden}))\\label{lem(1.00)}\nLet $M$ be a complete $n$-dimensional Riemannian manifold with non-negative Ricci curvature, and $B_{r}(x)$ be an $r$-ball in $M$. Then\n$$\\nu(B_{r}(x))\\leq c_{n}r^{n},$$\nwhere $c_{n}=\\pi^{\\frac{n}{2}}\/\\Gamma(\\frac{n}{2}+1)$ is the volume of the unit ball in $\\mathbb{R}^{n}$.\n\\end{lem}\n\nThe following proposition generalizes \\cite[Proposition 3.1]{Lau-Ngai-Wang_2009} to manifolds.\n\n\\begin{prop}\\label{prop(1.1)}\nLet $M$ be a complete $n$-dimensional orientable Riemannian manifold with non-negative Ricci curvature. Assume the same hypotheses of Lemma \\ref{lem(1.1)}. Then the following are equivalent:\n\\begin{itemize}\n\\item[$(a)$] $\\{S_{i}\\}_{i=1}^{N}$ satisfies (WSC);\n\\item[$(b)$] there exists $a>0$ and a nonempty subset $D\\subset W$ such that $\\gamma_{a,D}<\\infty$;\n\\item[$(c)$] for any $a>0$ and any nonempty subset $D\\subset W$, $\\gamma_{a,D}<\\infty$;\n\\item[$(d)$] for any $D\\subset W$ there exists $\\gamma=\\gamma(D)$ (depending only on $D$) such that for any $00$ and a nonempty subset $D_{0}\\subset W$ such that $\\gamma_{a_{0},D_{0}}=\\infty$. Hence there exists a sequence $\\{A_{k}\\}_{k=1}^{\\infty}$ of nonempty sets bounded in $M$ such that\n\\begin{equation}\\label{eq(1.5)}\n\\{S\\in\\mathcal{A}_{a_{0}|A_{k}|}:S(D_{0})\\cap A_{k}\\neq\\emptyset\\}\\geq k.\n\\end{equation}\nTo prove $(b)$ fails, we fix an arbitrary $a>0$ and nonempty subset $D\\subset W$. We will show $\\gamma_{a,D}=\\infty$. Let\n$$s:=\\sup_{x\\in D_{0},y\\in D}d(x,y)<\\infty.$$\nWe first claim that for any $S\\in\\mathcal{A}_{a_{0}|A_{k}|}$ and $\\delta_{k}:=s a_{0} C_{2}|A_{k}|$, $S(D_{0})\\cap A_{k}\\neq\\emptyset$ implies $S(D)\\cap (A_{k})_{\\delta_{k}}\\neq\\emptyset$, where $(A_{k})_{\\delta_{k}}=\\{x\\in M:{\\rm dist}(x,A_{k})\\leq\\delta_{k}\\}$ is the closed $\\delta_{k}$-neighborhood of $A_{k}$.\nTo prove the claim, we let $y\\in S(D_{0})\\cap A_{k}$. Then there exists $x\\in D_{0}$ such that $y=S(x)\\in S(D_{0})$. Now let $\\tilde{x}\\in D$ and $\\tilde{y}:=S(\\tilde{x})\\in S(D)$. It follows from (\\ref{eq(1.2)}) that\n$$d(\\tilde{y},y)=d(S(\\tilde{x}),S(x))\\leq C_{2}R_{S}d(\\tilde{x},x)\\leq s C_{2}R_{S}\\leq s a_{0}|C_{2}A_{k}|=\\delta_{k}.$$\nThis proves the claim. Note that $(A_{k})_{\\delta_{k}}$ is a set of diameter $2\\delta_{k}+|A_{k}|=(2sa_{0} C_{2}+1)|A_{k}|$. Since $M$ is a Riemannian manifold with non-negative Ricci curvature, $M$ has the doubling property. Hence we can cover $(A_{k})_{\\delta_{k}}$ by no more than $\\lambda$ sets of diameter $(a_{0}|A_{k}|)\/a$. Note that\n$\\mathcal{A}_{a_{0}|A_{k}|}=\\mathcal{A}_{(a_{0}|A_{k}|)\/a}$. It follows from (\\ref{eq(1.5)}) and the claim in the above that there exists $A_{k}^{\\ast}\\subset M$ with\n$|A_{k}^{\\ast}|=(a_{0}|A_{k}|)\/a$ such that\n$$\\{S\\in\\mathcal{A}_{a_{0}|A_{k}^{\\ast}|}:S(D_{0})\\cap A_{k}^{\\ast}\\neq\\emptyset\\}\\geq \\frac{k}{\\lambda}.$$\nSince $\\lambda$ is independent of $k$, we conclude that $\\gamma_{a,D}=\\infty$.\n\n\n$(c)\\Rightarrow(d)$ Let $D\\subset W$. Then for any $x\\in W$ and $00$ such that for any $0\\frac{b^{\\alpha}}{4C_{3}}\\bigg\\},$$\nwhere $C_{3}$ is as in Corollary \\ref{coro(1.1)}. Then $P(\\Lambda)>1\/2$ implies $P(\\widetilde{\\Lambda})>1\/4$.\n\\end{lem}\n\n\n\n\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm(0.2)}] The proof of Theorem \\ref{thm(0.2)} follows by using Proposition \\ref{prop(1.1)}, Lemma \\ref{lem(2.1)}, and a technique in the proofs of \\cite[Theorem 1.1]{Lau-Ngai-Rao_2001} and \\cite[Theorem 1.1]{Lau-Wang_2004}; we omit the details.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe first give the definition of Vitali relation (see \\cite[Definition 2.8.16]{Federer_1969}). Let $X$ be a metric space.\nAny subset of\n$$\n\\{(x,A): x\\in A\\subset X\\}\n$$\nis called a \\textit{covering relation}. Let $\\mathbf{C}$ be a covering relation and $Z\\subset X$, we let\n$$\n\\mathbf{C}(Z):=\\{A\\subset X:(x,A)\\in \\mathbf{C}\\text{ for some }x\\in Z\\}.\n$$\nWe say $\\mathbf{C}$ is \\textit{fine at $x$} if\n$$\n\\inf\\{|A|:(x,A)\\in \\mathbf{C}\\}=0.\n$$\nWe say $\\mathbf{C}(Z)$ is \\textit{fine on $Z$} if for any $x\\in Z$, $\\mathbf{C}(x)$ is fine at $x$. Let $\\phi$ be a regular Borel measure on $X$. A covering relation $\\mathbf{V}$ is called a\n$\\phi$ \\textit{Vitali relation} if $\\mathbf{V}(X)$ is a family of Borel sets, $\\mathbf{V}$ is fine on $X$, and moreover, whenever $\\mathbf{C}\\subset \\mathbf{V}, Z\\subset X$, and $\\mathbf{C}$ is fine on $Z$, then $\\mathbf{C}(Z)$ has a countable disjoint subfamily covering $\\phi$ almost all of $Z$.\n\nFor an $n$-dimension Riemannian manifold $M$, by making use of \\cite[Definition 2.8.9 and Theorem 2.8.18]{Federer_1969}, we see that $\\{B_{r}(x):x\\in M,00$, let $A=A(Q):=\\{t\\in K:f(t)>Q\\}$. Then for any $\\delta>0$, by (\\ref{eq(2.4)}), there exist $x\\in K$ and $b>0$ such that\n\\begin{equation}\\label{eq(2.5)}\n\\phi\\big(A\\cap B_{b\\delta}(x)\\big)>\\frac{1}{2}\\phi(B_{b\\delta}(x)).\n\\end{equation}\nThen\n\\begin{equation}\\label{eq(2.6)}\\begin{aligned}\n\\mu(B_{b\\delta}(x))&=\\int_{B_{b\\delta}(x)}f(t)d\\phi(t)\\\\\n&\\geq \\int_{A\\cap B_{b\\delta}(x)}f(t)d\\phi(t)\\\\\n&\\geq Q\\phi(A\\cap B_{b\\delta}(x))\\\\\n&>\\frac{1}{2}Q\\phi(B_{b\\delta}(x))\\quad\\text{(by (\\ref{eq(2.5)}))}.\n\\end{aligned}\\end{equation}\nLet $\\delta:=C_{2}|K|$ in the above inequality. Note that $x\\in K=\\bigcup_{S\\in\\mathcal{A}_{b}}S(K)$, and hence there exists $S\\in\\mathcal{A}_{b}$ such that $x\\in S(K)$.\nMaking use of (\\ref{eq(1.2)}), we have\n$$|S(K)|\\leq C_{2}R_{S}|K|\\leq b\\delta,$$\nwhich implies that $S(K)\\subset B_{b\\delta}(x)$. For $S=S_{u_{1}\\cdots u_{k}}\\in\\mathcal{A}_{b}$, we have $R_{S}=R_{\\mathbf{u}}\\leq b\\bigg(\\frac{r}{C_{1}C_{2}}\\bigg)^{\\alpha}b^{\\alpha}\\phi(K).\n\\end{equation}\nCombining \\eqref{eq(2.6)} and \\eqref{eq(2.8)}, we obtain\n\\begin{equation}\\label{eq(2.9)}\n\\mu(B_{b\\delta}(x))>\\frac{1}{2}Q\\bigg(\\frac{r}{C_{1}C_{2}}\\bigg)^{\\alpha}b^{\\alpha}\\phi(K).\n\\end{equation}\nOn the other hand,\n$$\\begin{aligned}\n\\mu(B_{b\\delta}(x))&=\\sum_{S\\in\\mathcal{A}_{b},S(K)\\cap B_{b\\delta}(x)\\neq\\emptyset}p_{S}\\mu\\circ S^{-1}(B_{b\\delta}(x))\\\\\n&\\leq\\sum_{S\\in\\mathcal{A}_{b},S(K)\\cap B_{b\\delta}(x)\\neq\\emptyset}p_{S}.\n\\end{aligned}$$\nLet $S\\in\\mathcal{A}_{b}$ such that $p_S$ is the maximum among all the summands in the above summation. Then\n\\begin{equation}\\label{eq(2.10)}\n\\mu(B_{b\\delta}(x))\\leq p_{S}\\#\\{S\\in\\mathcal{A}_{b}:S(K)\\cap B_{b\\delta}(x)\\neq\\emptyset\\}\\leq p_{S}\\gamma_{\\frac{1}{2\\delta},K}.\n\\end{equation}\nWe choose $Q$ such that\n\\begin{equation}\\label{eq(2.11)}\n\\frac{1}{2}Q\\bigg(\\frac{r}{C_{1}C_{2}}\\bigg)^{\\alpha}\\phi(K)>\\gamma_{\\frac{1}{2\\delta},K}.\n\\end{equation}\nIt follows from \\eqref{eq(2.9)}--\\eqref{eq(2.11)} that there exists $S\\in\\mathcal{A}_{b}$ such that\n$$b^{\\alpha}\\gamma_{\\frac{1}{2\\delta},K}<\\mu(B_{b\\delta}(x))\\leq p_{S}\\gamma_{\\frac{1}{2\\delta},K}.$$\nHence\n$$R_{S}^{\\alpha}\\leq b^{\\alpha}\\overline{m}_{k}$, there exists $\\mathbf{u}\\in\\mathcal{M}_{k}$ such that $\\mathbf{u}\\preceq\\mathbf{v}$;\n\\item[$(d)$] for any $\\mathbf{v}\\in\\Sigma^{\\ast}$ with $|\\mathbf{v}|<\\underline{m}_{k}$, there exists $\\mathbf{u}\\in\\mathcal{M}_{k}$ such that $\\mathbf{v}\\preceq\\mathbf{u}$;\n\\item[$(e)$] there exists a positive integer $L$, independent of $k$, such that for any $\\mathbf{u}\\in\\mathcal{M}_{k}$ and $\\mathbf{v}\\in\\mathcal{M}_{k+1}$ with\n$\\mathbf{u}\\preceq\\mathbf{v}$, we have $|\\mathbf{v}|-|\\mathbf{u}|\\leq L$.\n\\end{itemize}\n\\end{defi}\n\nNote that $\\mathcal{M}_{k}$ can intersect $\\mathcal{M}_{k+1}$, and $\\{\\Sigma^{k}\\}_{k=0}^{\\infty}$ is an example of a sequence of nested index sets.\nFor each integer $k\\geq0$, let $\\mathcal{V}_{k}$ be the set of \\textit{$k$-th level vertices} defined as\n$$\\mathcal{V}_{0}:=\\{(\\mathbf{u},0)\\}\\quad\\text{and}\\quad\\mathcal{V}_{k}\n:=\\{(S_{\\mathbf{u}},k):\\mathbf{u}\\in\\mathcal{M}_{k}\\}\\text{ for any }k\\geq1.$$\nWe write $\\omega_{{\\rm root}}:=(\\mathbf{u},0)$ and call it the \\textit{root vertex}. Let $\\mathcal{V}:=\\bigcup_{k\\geq0}\\mathcal{V}_{k}$ be the set of all vertices. For\n$\\omega=(S_{\\mathbf{u}},k)$, we define $S_{\\omega}:=S_{\\mathbf{u}}$. Let $W\\subset M$ be a compact set. For an IFS $\\{S_{i}\\}_{i=1}^{N}$ on $W$, let $\\Omega\\subset W$ be a nonempty open set that is invariant under $\\{S_{i}\\}_{i=1}^{N}$. Such a set exists if $\\{S_{i}\\}_{i=1}^{N}$ are contractions on $W$. We say that two $k$-th level vertices $\\omega,\\omega'\\in\\mathcal{V}_{k}$ are \\textit{neighbors} if $S_{\\omega}(\\Omega)\\cap S_{\\omega'}(\\Omega)\\neq\\emptyset$. Let\n$$\\Omega(\\omega):=\\{\\omega':\\omega'\\in\\mathcal{V}_{k}\\text{ is a neighbor of }\\omega\\},$$\nwhich is called the \\textit{neighborhood} of $\\omega$.\n\n\\begin{defi}\\label{defi(3.2)}\nFor any two vertices $\\omega\\in\\mathcal{V}_{k}$ and $\\omega'\\in\\mathcal{V}_{k'}$, let\n$$\\tau:=S_{\\omega'}S_{\\omega}^{-1}:\\bigcup_{\\sigma\\in\\Omega(\\omega)}S_{\\sigma}(W)\\longrightarrow W.$$\nWe say $\\omega$ and $\\omega'$ are \\textit{equivalent}, i.e., $\\omega\\sim\\omega'$, if the following conditions hold\n\\begin{itemize}\n\\item[$(a)$] $\\{S_{\\sigma'}:\\sigma'\\in\\Omega(\\omega')\\}=\\{\\tau S_{\\sigma}:\\sigma\\in\\Omega(\\omega)\\}$;\n\\item[$(b)$] for $\\sigma\\in\\Omega(\\omega)$ and $\\sigma'\\in\\Omega(\\omega')$ such that $S_{\\sigma'}=\\tau S_{\\sigma}$, and for any positive integer $k_{0}\\geq1$,\n$\\mathbf{u}\\in\\Sigma^{\\ast}$, $\\mathbf{u}$ satisfies $(S_{\\sigma}\\circ S_{\\mathbf{u}},k+k_{0})\\in\\mathcal{V}_{k+k_{0}}$ if and only if it satisfies\n$(S_{\\sigma'}\\circ S_{\\mathbf{u}},k'+k_{0})\\in\\mathcal{V}_{k'+k_{0}}$.\n\\end{itemize}\n\\end{defi}\nIt is easy to see that $\\sim$ is an equivalence relation. Denote the equivalent class of $\\omega$ by $[\\omega]$, and call it the \\textit{neighborhood types} of $\\omega$.\n\\par Let $\\omega=(S_{\\mathbf{u}},k)\\in\\mathcal{V}_{k}$ and $\\sigma=(S_{\\mathbf{v}},k+1)\\in\\mathcal{V}_{k+1}$. Suppose that there exists $\\mathbf{w}\\in\\Sigma^{\\ast}$ such that\n$$\\mathbf{v}=(\\mathbf{u},\\mathbf{w}).$$\nThen we connect a \\textit{directed edge} from $\\omega$ to $\\sigma$, and denote this as $\\omega\\stackrel{\\mathbf{w}}{\\longrightarrow}\\sigma$. We call $\\omega$ a \\textit{parent} of $\\sigma$ and $\\sigma$ an \\textit{offspring} of $\\omega$. Define a graph $\\mathcal{G}:=(\\mathcal{V},\\mathcal{E})$, where $\\mathcal{E}$ is the set of all directed edges. We first remove from $\\mathcal{G}$ all but the smallest (in the lexicographic order) directed edges going to a vertex. After that, we remove all vertices that do not have any offspring, together with all vertices and edges leading only to them. The resulting graph is called the \\textit{reduced graph}. Denote it by $\\mathcal{G}_{R}:=(\\mathcal{V}_{R},\\mathcal{E}_{R})$, where $\\mathcal{V}_{R}$ and $\\mathcal{E}_{R}$ are the sets of all vertices and all edges, respectively.\n\nThe proof of the following proposition is similar to that of \\cite[Proposition 2.4]{Lau-Ngai_2007}; we omit the details.\n\n\\begin{prop}\\label{prop(3.1)}\nLet $\\omega$ and $\\omega'$ be two vertices in $\\mathcal{V}$ with offspring $\\mathbf{u}_{1},\\dots,\\mathbf{u}_{m}$ and $\\mathbf{u}'_{1},\\dots,\\mathbf{u}'_{s}$ in $\\mathcal{G}_{R}$, respectively. Suppose $[\\omega]=[\\omega']$. Then\n$$\\big\\{[\\mathbf{u}_{i}]:1\\leq i\\leq m\\big\\}=\\big\\{[\\mathbf{u}'_{i}]:1\\leq i\\leq s\\big\\}$$\ncounting multiplicity. In particular, $m=s$.\n\\end{prop}\n\n\\begin{defi}\\label{defi(3.3)}\nLet $\\{S_{i}\\}_{i=1}^{N}$ be an IFS on $W$ consisting of injective contractions, and let $\\mathcal{V}\/_{\\sim}:=\\{[\\omega]:\\omega\\in\\mathcal{V}\\}$. We say that $\\{S_{i}\\}_{i=1}^{N}$ satisfies the \\textit{finite type condition} (FTC) if there exists a nonempty invariant open set $\\Omega\\subset W$ with respect to some sequence of nested index sets $\\{\\mathcal{M}_{k}\\}_{k=0}^{\\infty}$ and such that\n$$\\#\\mathcal{V}\/_{\\sim}<\\infty.$$\nSuch a set $\\Omega$ is called \\textit{a finite type condition set (FTC set)}.\n\\end{defi}\n\nObviously, if $\\{S_{i}\\}_{i=1}^{N}$ satisfies (OSC), then $\\#\\mathcal{V}\/_{\\sim}=1$, and thus $\\{S_{i}\\}_{i=1}^{N}$ satisfies (FTC). We assume that $\\{S_{i}\\}_{i=1}^{N}$ is a CIFS in the rest of this section.\n\n\\begin{lem}\\label{lem(3.1)}\nLet $\\{S_{i}\\}_{i=1}^{N}$ be a CIFS on a compact subset $W\\subset M$. Assume that $\\{S_{i}\\}_{i=1}^{N}$ satisfies (FTC) with $\\Omega\\subset W$ being an FTC set. Then there\nexists a constant $C_{4}\\geq1$ such that for any two neighboring vertices $\\omega_{1}$ and $\\omega_{2}$, we have\n$$C_{4}^{-1}\\leq\\frac{\\nu(S_{\\omega_{1}}(\\Omega))}{\\nu(S_{\\omega_{2}}(\\Omega))}\\leq C_{4}.$$\n\\end{lem}\n\\begin{proof}\nLet $\\mathcal{T}$ be a neighborhood type, and $\\omega$ be a vertex such that $[\\omega]=\\mathcal{T}$. Let\n$$\\Omega(\\omega)=\\{\\omega_{0},\\omega_{1},\\dots,\\omega_{m}\\},$$\nwhere $\\omega_{0}=\\omega$. Substituting $S_{\\omega_{0}}(\\Omega)=A$ and $S_{\\omega_{i}}S_{\\omega_{0}}^{-1}=\\tau$ into Lemma \\ref{lem(1.3)}$(a)$ and using \\eqref{eq:r_R_bound}, we see that there exists a constant $c_{1}\\geq1$ such that for any $i\\in\\{0,1,\\dots,m\\}$,\n\\begin{equation}\\label{eq(3.1)}\nc_{1}^{-1}\\nu(S_{\\omega_{0}}(\\Omega))\\leq\\nu(S_{\\omega_{i}}(\\Omega))\\leq c_{1}\\nu(S_{\\omega_{0}}(\\Omega)).\n\\end{equation}\nLet $\\omega\\sim\\omega'$, $\\tau=S_{\\omega'}S_{\\omega}^{-1}\\in \\mathcal{F}$, and\n$$\\Omega(\\omega')=\\{\\omega'_{0},\\omega'_{1},\\dots,\\omega'_{m}\\},$$\nwhere $\\omega'_{0}=\\omega'$. Without loss of generality, for any $i\\in\\{0,1,\\dots,m\\}$ we can assume $S_{\\omega'_{i}}=\\tau S_{\\omega_{i}}$. It follows from the definition of $\\tau$ that\n$S_{\\omega_{i}}(\\Omega)\\subset{\\rm Dom}(\\tau)$. Making use of (\\ref{eq(3.1)}) and substituting $S_{\\omega_{i}}(\\Omega)=A$, $S_{\\omega_{0}}(\\Omega)=B$ and $S_{\\omega'_{i}}S_{\\omega_{i}}^{-1}=\\tau$ into Lemma \\ref{lem(1.3)}$(b)$, we see that for any $i\\in\\{0,1,\\dots,m\\}$,\n$$c_{1}^{-1}C_{1}^{-2n}\\nu(S_{\\omega'_{0}}(\\Omega))\\leq\\nu(S_{\\omega'_{i}}(\\Omega))=\\nu(\\tau S_{\\omega_{i}}(\\Omega))\\leq c_{1}C_{1}^{2n}\\nu(S_{\\omega'_{0}}(\\Omega)).$$\nHence the lemma holds for any two neighboring vertices $\\omega_{1},\\omega_{2}$ with one of them being of type $\\mathcal{T}$. Since there are only finitely\nmany distinct neighborhood types, the result follows.\n\\end{proof}\n\n\\begin{lem}\\label{lem(3.2)}\nLet $\\{S_{i}\\}_{i=1}^{N}$ be a CIFS on a compact subset $W\\subset M$. Then for any $\\mathbf{u}\\in\\Sigma^{k}$ and $\\Omega\\subset W$, we have\n\\begin{equation}\\label{eq(3.01)}\nr^{kn}\\leq\\frac{\\nu(S_{\\mathbf{u}}(\\Omega))}{\\nu(\\Omega)}\\leq R^{kn}.\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nLet $x\\in W$. Then by the definition of $R$, we have\n$$\\begin{aligned}\n|\\det S'_{\\mathbf{u}}(x)|&=|\\det S'_{\\mathbf{u}^{-}}(S_{u_{k}}(x))|\\cdot|S'_{u_{k}}(x)|\\\\\n&\\leq R_{\\mathbf{u}^{-}}^{n}R_{u_{k}}^{n}\\\\\n&\\leq R_{u_{1}}^{n}\\cdots R_{u_{k}}^{n}\\\\\n&\\leq R^{kn}.\n\\end{aligned}$$\nFor any set $\\Omega\\subset W$, making use of (\\ref{eq(1.02)}), we have\n$$\\nu(S_{\\mathbf{u}}(\\Omega))=\\int_{\\Omega}|\\det S'_{\\mathbf{u}}(x)|d\\nu(x)\\leq R^{kn}\\nu(\\Omega).$$\nThis proves the right side of (\\ref{eq(3.01)}). On the other hand, if $x\\in W$, then by the definition of $r$, we have\n$$\\begin{aligned}\n|\\det S'_{\\mathbf{u}}(x)|&=|\\det S'_{\\mathbf{u}^{-}}(S_{u_{k}}(x))|\\cdot|S'_{u_{k}}(x)|\\\\\n&\\geq r_{\\mathbf{u}^{-}}^{n}r_{u_{k}}^{n}\\\\\n&\\geq r_{u_{1}}^{n}\\cdots r_{u_{k}}^{n}\\\\\n&\\geq r^{kn}.\n\\end{aligned}$$\nConsequently,\n$$\\nu(S_{\\mathbf{u}}(\\Omega))=\\int_{\\Omega}|\\det S'_{\\mathbf{u}}(x)|d\\nu(x)\\geq r^{kn}\\nu(\\Omega).$$\nThis proves the left side of (\\ref{eq(3.01)}).\n\\end{proof}\n\n\nWe now prove Theorem \\ref{thm(0.4)}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm(0.4)}]\nFor $00$ such that for any $j\\in\\{1,\\dots,m\\}$,\n\\begin{equation}\\label{eq(3.10)}\nc_{3}\\leq R^{|\\mathbf{v}_{j}|}.\n\\end{equation}\nIn particular, we can take $c_{3}=r^{L+1}\/(C_{1}^{3}C_{4}^{1\/n})$. Let $c_{4}:=\\lfloor\\log c_{3}\/\\log R\\rfloor+1$. Then $|\\mathbf{v}_{j}|\\leq c_{4}$. Combining these and (\\ref{eq(3.2)}) yields\n$$\\#\\{S\\in\\mathcal{A}_{b}:x\\in S(\\Omega)\\}\\leq c_{2}N^{c_{4}},$$\nwhich implies that $\\{S_{i}\\}_{i=1}^{N}$ satisfies (WSC).\n\\end{proof}\n\n\n\\section{Hausdorff dimension of self-similar sets}\\label{S:4}\n\nIn this section, we assume that $M$ is a complete $n$-dimensional smooth orientable Riemannian manifold that is locally Euclidean, i.e., every point of $M$ has a neighborhood which is isometric to an open subset of a Euclidean space.\nLet $W\\subset M$ be compact, and let $\\{S_{i}\\}_{i=1}^{N}$ be an IFS satisfying (FTC) of contractive similitudes on some open sets $\\Omega\\subset W$ with attractor $K\\subset W$. Recall that $\\rho_i$ denotes the contraction ratio of $S_{i}$. We define\n$$\\rho:=\\min\\{\\rho_{i}:1\\leq i\\leq N\\}, \\quad \\rho_{\\max}:=\\max\\{\\rho_{i}:1\\leq i\\leq N\\}.$$\nDenote the neighborhood types of $\\{S_{i}\\}_{i=1}^{N}$ by $\\{\\mathcal{T}_{1},\\dots,\\mathcal{T}_{q}\\}$. Fix a vertex $\\omega\\in\\mathcal{V}_{R}$ such that $[\\omega]\\in\\mathcal{T}_{i}$, where $i\\in\\{1,\\dots,q\\}$. Let $\\sigma_{1},\\dots,\\sigma_{m}$ be the offspring of $\\omega$ in $\\mathcal{G}_{R}$, and let $\\mathbf{w}_{k}$ be the unique edge in $\\mathcal{G}_{R}$ connecting $\\omega$ to $\\sigma_{k}$ for $1\\leq k\\leq m$. Define a \\textit{weighted incidence matrix} $A_{\\alpha}=(A_{\\alpha}(i,j))_{i,j=1}^{q}$ as\n$$A_{\\alpha}(i,j):=\\sum_{k=1}^{m}\\{\\rho_{\\mathbf{w}_{k}}^{\\alpha}:\n\\omega\\stackrel{\\mathbf{w}_{k}}{\\longrightarrow}\\sigma_{k},[\\sigma_{k}]=\\mathcal{T}_{j}\\}.$$\nWe remark that the definition of $A_{\\alpha}$ is independent of the choice of $\\omega$ above. We denote by $\\omega\\rightarrow_{R}\\sigma$ if $\\omega,\\sigma\\in\\mathcal{V}_{R}$ and $\\sigma$ is an offspring of $\\omega$ in $\\mathcal{G}_{R}$. We define an (infinite) \\textit{path} in $\\mathcal{G}_{R}$ to be an infinite sequence $(\\omega_{0},\\omega_{1},\\dots)$ such that for any $k\\geq0$,\n$$\\omega_{k}\\in\\mathcal{V}_{k}\\quad\\text{and}\\quad\\omega_{k}\\rightarrow_{R}\\omega_{k+1},$$\nwhere $\\omega_{0}=\\omega_{{\\rm root}}$. Let $\\mathbb{P}$ be the set of all paths in $\\mathcal{G}_{R}$. If the vertices $\\omega_{0}=\\omega_{{\\rm root}},\\omega_{1},\\dots,\\omega_{k}$ are such that\n$$\\omega_{j}\\rightarrow_{R}\\omega_{j+1}\\text{ for }1\\leq j\\leq k-1,$$\nthen we call the set\n$$I_{\\omega_{0},\\omega_{1},\\dots,\\omega_{k}}=\\{(\\sigma_{0},\\sigma_{1},\\dots)\\in\\mathbb{P}:\\sigma_{j}=\\omega_{j}\n\\text{ for any }0\\leq j\\leq k\\}$$\na \\textit{cylinder}. Since the path from $\\omega_{0}$ to $\\omega_{k}$ is unique in $\\mathcal{G}_{R}$, we {\\color{blue}}let\n$$I_{\\omega_{k}}:=I_{\\omega_{0},\\omega_{1},\\dots,\\omega_{k}}.$$\nFor any cylinder $I_{\\omega_{k}}$, where $\\omega_{k}\\in\\mathcal{V}_{k}$ and $[\\omega_{k}]=\\mathcal{T}_{i}$, let\n$$\\hat{\\mu}(\\omega_{{\\rm root}})=a_{1}=1\\quad\\text{and}\\quad\\hat{\\mu}(\\omega_{k})=\\rho_{\\omega_{k}}^{\\alpha}a_{i},$$\nwhere $[a_{1},\\dots,a_{q}]^{T}$ is a $1$-eigenvector of $A_{\\alpha}$, normalized so that $a_{1}=1$. We will show that $\\hat{\\mu}$ is a measure on $\\mathbb{P}$ in the following. Note that for two cylinders $I_{\\omega}$ and $I_{\\omega}'$ with $\\omega\\in\\mathcal{V}_{k},\\omega'\\in\\mathcal{V}_{\\ell}$ and $k\\leq\\ell$, $I_{\\omega}\\cap I_{\\omega}'\\neq\\emptyset$ iff either $\\omega'=\\omega$ in the case $k=\\ell$ or $\\omega'$ is a descendant of $\\omega$ in the case $k<\\ell$. Whatever, $I_{\\omega}'\\subset I_{\\omega}$. Let $\\omega\\in\\mathcal{V}_{R}$ and $\\mathcal{D}:=\\{\\sigma_{k}\\}_{k=1}^{m}$ denote the set of all offspring of $\\omega$ in $\\mathcal{G}_{R}$. For $k\\in\\{1,\\dots,m\\}$, let $\\omega\\stackrel{\\mathbf{w}_{k}}{\\longrightarrow}_{R}\\sigma_{k}$. Then\n$$\\begin{aligned}\n\\sum_{\\sigma\\in\\mathcal{D}}\\hat{\\mu}(I_{\\sigma})\n&=\\sum_{j=1}^{q}\\sum_{\\sigma\\in\\mathcal{D},[\\sigma]=\\mathcal{T}_{j}}\\hat{\\mu}(I_{\\sigma})\\\\\n&=\\sum_{j=1}^{q}\\sum_{\\sigma\\in\\mathcal{D},[\\sigma]=\\mathcal{T}_{j}}\\rho_{\\sigma}^{\\alpha}a_{j}\n\\\\\n&=\\rho_{\\omega}^{\\alpha}\\sum_{j=1}^{q}\\sum_{\\sigma\\in\\mathcal{D},[\\sigma]=\\mathcal{T}_{j}}\n\\rho_{\\mathbf{w}_{k}}^{\\alpha}a_{j}\\\\\n&=\\rho_{\\omega}^{\\alpha}\\sum_{j=1}^{q}A_{\\alpha}(i,j)a_{j}\\\\\n&=\\rho_{\\omega}^{\\alpha}a_{i}=\\hat{\\mu}(I_{\\omega}).\n\\end{aligned}$$\nCombining these with $\\hat{\\mu}(\\mathbb{P})=\\hat{\\mu}(\\omega_{{\\rm root}})=1$ shows that $\\hat{\\mu}$ is indeed a measure on $\\mathbb{P}$. Define $f:\\mathbb{P}\\longrightarrow W$ by letting $f(\\omega_{0},\\omega_{1},\\dots)$ be the unique point in $\\bigcap_{k=0}^{\\infty}S_{\\omega_{k}}(K)$. It is clear that $f(\\mathbb{P})=K$. Let $\\widetilde{\\mu}:=\\hat{\\mu}\\circ f^{-1}$. Then $\\widetilde{\\mu}$ is a measure on $K$.\n\nFor any bounded Borel set $F\\subset M$, let\n\\begin{equation}\\label{eq(4.1)}\n\\mathcal{B}(F):=\\{I_{\\omega_{k}}=I_{\\omega_{k},\\dots,\\omega_{k}}:\n|S_{\\omega_{k}}(\\Omega)|\\leq|F|<|S_{\\omega_{k-1}}(\\Omega)|\\text{ and }F\\cap S_{\\omega_{k}}(\\Omega)\\neq\\emptyset\\}.\n\\end{equation}\n\n\\begin{lem}\\label{lem(4.1)}\nThere exists a constant $C_{5}>0$, independent of $k$, such that for any bounded Borel set $F\\subset M$, we have $\\#\\mathcal{B}(F)\\leq C_{5}$.\n\\end{lem}\n\\begin{proof}\nDefine\n$$\\begin{aligned}\n\\widetilde{\\mathcal{B}}(F):&=\\{\\omega_{k}\\in\\mathcal{V}_{k}:\n|S_{\\omega_{k}}(\\Omega)|\\leq|F|<|S_{\\omega_{k-1}}(\\Omega)|\\text{ and }F\\cap S_{\\omega_{k}}(\\Omega)\\neq\\emptyset\\}\\\\\n&=\\{\\omega_{k}\\in\\mathcal{V}_{k}:\n\\rho_{\\omega_{k}}\\leq|F|\/|\\Omega|<\\rho_{\\omega_{k-1}}\\text{ and }F\\cap S_{\\omega_{k}}(\\Omega)\\neq\\emptyset\\}.\n\\end{aligned}$$\nSince $I_{\\omega_{k}}$ is one-to-one with $\\omega_{k}$, we have $\\#\\mathcal{B}(F)=\\#\\widetilde{\\mathcal{B}}(F)$. Let $b:=|F|\/|\\Omega|$ and $\\omega_{k}\\in\\widetilde{\\mathcal{B}}(F)$. Then there exists a unique $\\mathbf{u}\\in\\mathcal{M}_{k}$ such that $\\omega_{k}=(S_{\\mathbf{u}},k)$. Let $\\mathbf{u}'\\preccurlyeq\\mathbf{u}$ such that $S_{\\mathbf{u}'}\\in\\mathcal{A}_{b}$. Then\n$$\\rho_{\\mathbf{u}'}\\leq b=|F|\/|\\Omega|<\\rho_{\\omega_{k-1}}.$$\nThus $\\mathbf{u}'\\in\\mathcal{M}_{k-1}$ or $\\mathcal{M}_{k}$. Combining these and Definition \\ref{defi(3.1)}$(e)$, we have $|\\mathbf{u}|-|\\mathbf{u}'|\\leq L$.\nNote that $F\\cap S_{\\omega_{k}}(\\Omega)\\neq\\emptyset$ implies that $F\\cap S_{\\mathbf{u}'}(\\Omega)\\neq\\emptyset$. Since $S_{\\mathbf{u}'}\\in\\mathcal{A}_{b}$, we have\n$$|S_{\\mathbf{u}'}(\\Omega)|=\\rho_{\\mathbf{u}'}|\\Omega|\\leq b\\Omega.$$\nLet $\\delta:=2b|\\Omega|$ and fix any $x_{0}\\in F$. Then\n$$S_{\\mathbf{u}'}(\\Omega)\\subset B_{\\delta}(x_{0}).$$\nSince (FTC) implies (WSC), there exists a constant $\\gamma>0$ (independent of $b$) such that for all $x\\in U$,\n$$\\#\\{S\\in\\mathcal{A}_{b}:x\\in S(\\Omega)\\}\\leq\\gamma.$$\nNote that the contraction ratio of $S_{\\mathbf{u}}$ is $\\rho_{\\mathbf{u}}=|\\det S_{\\mathbf{u}}'(x)|^{\\frac{1}{n}}$ for any $x\\in W$. Let $A\\subset W$ be a measurable set. Then by Lemma \\ref{lem(1.1)}, we have\n$$\\nu(S_{\\mathbf{u}}(A))\\geq(b\\rho)^{n}\\nu(A).$$\nCombining these we have\n$$\\begin{aligned}\n(b\\rho)^{n}\\nu(\\Omega)\\#\\{S_{\\mathbf{u}'}:F\\cap S_{\\mathbf{u}'}(\\Omega)\\neq\\emptyset\\}&\\leq\n\\sum\\{\\nu(S_{\\mathbf{u}'}(\\Omega)):F\\cap S_{\\mathbf{u}'}(\\Omega)\\neq\\emptyset\\}\\\\\n&\\leq \\gamma\\nu(B_{\\delta}(x_{0}))\\\\\n&\\leq\\gamma c_{n}\\delta^{n}\\quad(\\text{by Lemma \\ref{lem(1.30)}})\\\\\n&:=\\gamma c_{1}b^{n},\n\\end{aligned}$$\nwhere $c_{n}$ is the volume of the unit ball in $\\mathbb{R}^{n}$ and $c_{1}:=c_{n}2^{n}|\\Omega|^{n}$. Thus,\n$$\\#\\{S_{\\mathbf{u}'}:F\\cap S_{\\mathbf{u}'}(\\Omega)\\neq\\emptyset\\}\\leq\\frac{\\gamma c_{1}}{\\rho^{n}\\nu(\\Omega)}:=c_2.$$\nHence\n$$\\#\\mathcal{B}(F)=\\#\\widetilde{\\mathcal{B}}(F)\\leq\nN^{L}\\#\\{S_{\\mathbf{u}'}:F\\cap S_{\\mathbf{u}'}(\\Omega)\\neq\\emptyset\\}\\leq c_{2}N^{L}.$$\nThe lemma follows by letting $C_{5}:=c_{2}N^{L}$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm(0.5)}] Use of Lemma \\ref{lem(4.1)} and the properties of\nmeasures $\\widetilde{\\mu}$ on $K$, as in \\cite[Theorem 1.2]{Lau-Ngai_2007}; we omit the details.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Hausdorff dimension of graph self-similar sets}\\label{S:41}\n\nIn this section, we define graph self-similar sets on Riemannian manifolds, and derive a formula for computing the Hausdorff dimension of such sets. We assume that $M$ is a complete $n$-dimensional Riemannian manifold that is locally Euclidean.\n\nLet $G=(V,E)$ be a graph, where $V=\\{1,\\dots,t\\}$ is the set of vertices and $E$ is the set of all directed edges. We assume that there is at least one edge between two vertices. It is possible that the initial and terminal vertices are same. A \\textit{directed path} in $G$ is a finite string $\\mathbf{e}=e_{1}\\cdots e_{p}$ of edges in $E$ such that the terminal vertex of each $e_{i}$ is the initial vertex of the next edge $e_{i+1}$. For such a path, denote the \\textit{length} of $\\mathbf{e}$ by $|\\mathbf{e}|=p$. For any two vertices $i,j\\in V$, and any positive integer $p$, $E^{i,j}$ be the set of all directed edges from $i$ to $j$, let $E_{p}^{i,j}$ be the set of all directed paths of length $p$ from $i$ to $j$, $E_{p}$ be the set of all directed paths of length $p$, $E^{*}$ be the set of all directed paths, i.e.,\n$$E_{p}:=\\bigcup_{i,j=1}^{p}E_{p}^{i,j}\\quad\\text{and}\\quad E^{*}:=\\bigcup_{p=1}^{\\infty}E_{p}.$$\nFor any edge $e\\in E$, we assume that there corresponds a contractive similitude $S_{e}$ with ratio $\\rho_{e}$ on $M$. For $\\mathbf{e}=e_{1}\\cdots e_{p}\\in E^{*}$, let\n$$S_{\\mathbf{e}}=S_{e_{1}}\\circ\\cdots\\circ S_{e_{p}}\\quad\\text{and}\\quad \\rho_{\\mathbf{e}}=\\rho_{e_{1}}\\cdots \\rho_{e_{p}}.$$\nThen there exists a unique family of nonempty compact sets $K_{1},\\dots,K_{t}$ satisfying\n$$K_{i}=\\bigcup_{j=1}^{t}\\bigcup_{e\\in E^{i,j}}S_{e}(K_{j}),\\quad i\\in\\{1,\\dots,t\\},$$\n(see e.g. \\cite{Mauldin-Williams_1988,Edgar-Mauldin_1992,Ngai-Wang-Dong_2010,Das-Ngai_2004}).\nDefine\n$$K:=\\bigcup_{i=1}^{t}K_{i}.$$\nWe call $K$ the \\textit{graph self-similar set} defined by $G=(V,E)$, and call $G=(V,E)$ the \\textit{graph-directed iterated function system} (GIFS) associated with $\\{S_{e}\\}_{e\\in E}$.\n\nSubstituting $E^{*}$ for $\\Sigma^{*}$ in Definition \\ref{defi(3.1)}, we define a sequence of nested index sets $\\{\\mathcal{F}_{k}\\}_{k=1}^{\\infty}$ of directed paths. Note that $\\{E_{k}\\}_{k=1}^{\\infty}$ is an example of a sequence of nested index sets of directed paths. Fix a sequence $\\{\\mathcal{F}_{k}\\}_{k=1}^{\\infty}$ of nested index sets. For $i,j\\in\\{1,\\cdots,t\\}$, we partition $\\mathcal{F}_{k}$ to $\\mathcal{F}_{k}^{i,j}$ as\n$$\\mathcal{F}_{k}^{i,j}:=\\mathcal{F}_{k}\\cap\\bigg(\\bigcup_{p\\geq1}E_{p}^{i,j}\\bigg)=\\{\\mathbf{e}=e_{1}\\cdots e_{p}\\in \\mathcal{F}_{k}:\\mathbf{e}\\in E_{p}^{i,j}~\\text{for some }p\\geq1\\}.$$\nNote that $\\mathcal{F}_{k}=\\bigcup_{i,j=1}^{t}\\mathcal{F}_{k}^{i,j}$. For $i,j\\in\\{1,\\cdots,t\\}$, $k\\geq1$, let $\\mathbb{V}_{k}$ be the set of \\textit{$k$-th level vertices} defined as\n$$\\mathbb{V}_{k}:=\\{(S_{\\mathbf{e}},i,j,k):\\mathbf{e}\\in\\mathcal{F}_{k}^{i,j},1\\leq i,j\\leq t\\}.$$\nFor $\\mathbf{e}\\in\\mathcal{F}_{k}^{i,j}$, we call $(S_{\\mathbf{e}},i,j,k)$ (or simply $(S_{\\mathbf{e}},k)$) a \\text{vertex}. For a vertex $\\omega=(S_{\\mathbf{e}},i,j,k)\\in\\mathbb{V}_{k}$ with $k\\geq1$, let\n$$S_{\\omega}=S_{\\mathbf{e}}\\quad\\text{and}\\quad \\rho_{\\omega}=\\rho_{\\mathbf{e}}.$$\nLet $\\mathcal{F}_{0}=\\{1,\\dots,t\\}$ and $\\mathbb{V}_{0}=\\{\\omega^{1}_{{\\rm root}},\\dots,\\omega^{t}_{{\\rm root}}\\}$, where $\\omega^{i}_{{\\rm root}}=(I,i,i,0)$ and $I$ is the identity map on $M$. Then we say $\\mathbb{V}_{0}$ is the set of \\textit{root vertices}, and $\\{\\mathcal{F}_{k}\\}_{k=0}^{\\infty}$ is a sequence of nested index sets if $\\{\\mathcal{F}_{k}\\}_{k=1}^{\\infty}$ is. Let $\\mathbb{V}:=\\bigcup_{k\\geq0}\\mathbb{V}_{k}$ be the set of all vertices, and $\\pi:\\bigcup_{k\\geq0}\\mathcal{F}_{k}\\longrightarrow \\mathbb{V}$ be defined as\n$$\\pi(\\mathbf{e}):=\\begin{cases}(S_{\\mathbf{e}},i,j,k),\\quad\\text{if }\\mathbf{e}\\in\\mathcal{F}_{k}^{i,j}, k\\geq1,\\\\\n\\omega^{i}_{{\\rm root}},\\quad\\text{if }\\mathbf{e}=i\\in\\mathcal{F}_{0}.\\end{cases}$$\n\nLet $\\omega\\in\\mathbb{V}_{k}$ and $\\omega'\\in\\mathbb{V}_{k+1}$. Suppose that there exist directed paths $\\mathbf{e}\\in\\mathcal{F}_{k},\\mathbf{e}'\\in\\mathcal{F}_{k+1}$ and $\\mathbf{k}\\in E^{*}$ such that $\\pi(\\mathbf{e})=\\omega$, $\\pi(\\mathbf{e}')=\\omega'$ and $\\mathbf{e}'=\\mathbf{e}\\mathbf{k}$. Then we connect a \\textit{directed edge} $\\mathbf{k}$ from $\\omega$ to $\\omega'$, and denote this as $\\omega\\stackrel{\\mathbf{k}}{\\longrightarrow}\\omega'$. We call $\\omega$ a \\textit{parent} of $\\omega'$ and $\\omega'$ an \\textit{offspring} of $\\omega$.\nDefine a graph $\\mathbb{G}:=(\\mathbb{V},\\mathbb{E})$, where $\\mathbb{E}$ is the set of all directed edges of $\\mathbb{G}$. Let $\\mathbb{G}_{R}:=(\\mathbb{V}_{R},\\mathbb{E}_{R})$ be the \\textit{reduced graph} of $\\mathbb{G}$, defined as in Section \\ref{S:3} similarly, where $\\mathbb{V}_{R}$ and $\\mathbb{E}_{R}$ are the sets of all vertices and all directed edges, respectively.\n\nLet $\\mathbf{\\Omega}=\\{\\Omega_{i}\\}_{i=1}^{t}$, where $\\Omega_{i}\\subset M$ is a nonempty bounded open set for any $i\\in\\{1,\\dots,t\\}$. We say that $\\mathbf{\\Omega}$ is \\textit{invariant} under the GIFS $G=(V,E)$ if\n$$\\bigcup_{e\\in E^{i,j}}S_{e}(\\Omega_{j})\\subset\\Omega_{i},\\quad i,j\\in\\{1,\\cdots,t\\}.$$\nSince $S_{e}$ is a contractive similitude for any $e\\in E^{i,j}$, such a family always exists. Fix an invariant family $\\mathbf{\\Omega}=\\{\\Omega_{i}\\}_{i=1}^{t}$ of $G=(V,E)$. Let $\\omega=(S_{\\mathbf{e}},i,j,k)\\in\\mathbb{V}_{k}$ with $\\mathbf{e}\\in E_{q}^{i,j}$ and $\\omega=(S_{\\mathbf{e}'},i',j',k)\\in\\mathbb{V}_{k}$ with $\\mathbf{e}'\\in E_{s}^{i',j'}$, where $q,s>0$ are integers. We say that two vertices $\\omega$, $\\omega'$ are \\textit{neighbors} (with respect to $\\mathbf{\\Omega}$) if\n$$i=i'\\quad\\text{and}\\quad S_{\\mathbf{e}}(\\Omega_{j})\\cap S_{\\mathbf{e}'}(\\Omega_{j'})\\neq\\emptyset.$$\nLet\n$$\\mathcal{N}(\\omega):=\\{\\omega':\\omega'\\in\\mathbb{V}_{k}\\text{ is a neighbor of }\\omega\\},$$\nwhich is called the \\textit{neighborhood} of $\\omega$ (with respect to $\\mathbf{\\Omega}$).\n\\begin{defi}\\label{defi(41.1)}\nFor any two vertices $\\omega=(S_{\\mathbf{e}_{\\omega}},i_{\\omega},j_{\\omega},k)\\in\\mathbb{V}_{k}$ and $\\omega'=(S_{\\mathbf{e}_{\\omega'}},i_{\\omega'},j_{\\omega'},k')\\in\\mathbb{V}_{k'}$, let $\\sigma=(S_{\\mathbf{e}_{\\sigma}},i_{\\omega},j_{\\sigma},k)\\in\\mathcal{N}(\\omega)$ and $\\sigma'=(S_{\\mathbf{e}_{\\sigma'}},i_{\\omega'},j_{\\sigma'},k')\\in\\mathcal{N}(\\omega')$. Assume that\n$$\\tau=S_{\\omega'}S_{\\omega}^{-1}:\\bigcup_{\\sigma\\in\\mathcal{N}(\\omega)}S_{\\sigma}(\\Omega_{j_{\\sigma}})\n\\longrightarrow \\bigcup_{i=1}^{t}\\Omega_{i}$$\ninduces a bijection $f_{\\tau}:\\mathcal{N}(\\omega)\\longrightarrow\\mathcal{N}(\\omega')$ defined by\n\\begin{equation}\\label{eq(41.1)}\nf_{\\tau}(\\sigma)=f_{\\tau}(S_{\\mathbf{e}_{\\sigma}},i_{\\omega},j_{\\sigma},k)=(\\tau\\circ S_{\\mathbf{e}_{\\sigma'}},i_{\\omega'},j_{\\sigma'},k').\n\\end{equation}\nWe say $\\omega$ and $\\omega'$ are \\textit{equivalent}, i.e., $\\omega\\sim\\omega'$, if the following conditions hold:\n\\begin{itemize}\n\\item[$(a)$] $\\#\\mathcal{N}(\\omega)=\\#\\mathcal{N}(\\omega')$ and $j_{\\sigma}=j_{\\sigma'}$ in \\eqref{eq(41.1)};\n\\item[$(b)$] for $\\sigma\\in\\mathcal{N}(\\omega)$ and $\\sigma'\\in\\mathcal{N}(\\omega')$ such that $f_{\\tau}(\\sigma)=\\sigma'$, and for any positive integer $k_{0}\\geq1$, a directed path $\\mathbf{e}\\in E^{*}$ satisfies $(S_{\\sigma}\\circ S_{\\mathbf{e}},k+k_{0})\\in\\mathbb{V}_{k+k_{0}}$ if and only if it satisfies\n$(S_{\\sigma'}\\circ S_{\\mathbf{e}},k'+k_{0})\\in\\mathbb{V}_{k'+k_{0}}$.\n\\end{itemize}\n\\end{defi}\nIt is easy to check that $\\sim$ is an equivalence relation. Denote the equivalent class of $\\omega$ by $[\\omega]$, and call it the \\textit{neighborhood types} of $\\omega$ (with respect to $\\mathbf{\\Omega}$).\n\nFor a graph $\\mathbb{G}=(\\mathbb{V},\\mathbb{E})$, we can prove that $\\mathbb{G}$ satisfies Proposition \\ref{prop(3.1)} as in \\cite[Propositin 2.5]{Ngai-Wang-Dong_2010}; we omit the details. We now define the graph finite type condition.\n\n\n\\begin{defi}\\label{defi(41.2)}\nLet $M$ be a complete $n$-dimensional Riemannian manifold that is locally Euclidean, let $G=(V,E)$ be a GIFS, and let $\\{S_{e}\\}_{e\\in E}$ be a family of contractive similitudes defined on $M$.\nIf there exists an invariant family of nonempty bounded open sets $\\mathbf{\\Omega}=\\{\\Omega_{i}\\}_{i=1}^{t}$ with respect to some sequence of nested index sets $\\{\\mathcal{F}_{k}\\}_{k=0}^{\\infty}$ such that\n$$\\#\\mathbb{V}\/_{\\sim}<\\infty,$$\nthen we say that $G=(V,E)$ satisfies the \\textit{graph finite type condition} (GFTC).\nWe call such an invariant family $\\mathbf{\\Omega}$ a \\textit{graph finite type condition family} of $G$.\n\\end{defi}\n\n\nBy assuming a GIFS satisfies (GFTC), we get a formula for $\\dim_{{\\rm H}}(K)$.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm(41.1)}] The proof is similar to that of \\cite[Theorem 1.1]{Ngai-Wang-Dong_2010} and the definition of a weighted incidence matrix is the same as that in Section \\ref{S:4}; we omit the details.\n\\end{proof}\n\n\n\\section{Examples}\\label{S:5}\nIn this section, we construct some examples of CIFSs and GIFSs on Riemannian manifolds satisfying (FTC) and (GFTC), respectively.\n\nLet $\\{f_{i}\\}_{i=1}^{N}$ be a contractive CIFS on a compact set $W_0\\subset\\mathbb{R}^{n}$, i.e., $f_{i}$ is $C^{1+\\varepsilon}$ where $0<\\varepsilon<1$, and there exists an open and connected set $U_{0}\\supset W_{0}$ such that for any $i\\in\\{1,\\dots,N\\}$, $f_{i}$ can be extended to an injective conformal map $f_{i}:U_{0}\\longrightarrow U_{0}$. Let $M$ be a complete $n$-dimensional smooth Riemannian manifold. Assume that there exists a diffeomorphism\n$$\\varphi:U\\longrightarrow U_{0},$$\nwhere $U\\subset M$ is open and connected.\nDefine\n\\begin{equation}\\label{eq(5.1)}\nS_{i}:=\\varphi^{-1}\\circ f_{i}\\circ\\varphi:U\\longrightarrow S_{i}(U)\\quad\\text{for any }i\\in\\{1,\\dots,N\\}.\n\\end{equation}\nBy \\cite[Proposition 7.1]{Ngai-Xu_2022}, $\\{S_{i}\\}_{i=1}^{N}$ is a contractive CIFS on $U$.\n\n\nFix a sequence of nested index sets $\\{\\mathcal{M}_{k}\\}_{k=0}^{\\infty}$. Let $\\widetilde{\\mathcal{V}}$ and $\\mathcal{V}$ be the sets of all vertices with respect to $\\{\\mathcal{M}_{k}\\}_{k=0}^{\\infty}$ of $\\{f_{i}\\}_{i=1}^{N}$ and $\\{S_{i}\\}_{i=1}^{N}$, respectively.\nFor $\\tilde{\\omega}=(f_{\\mathbf{u}},k)\\in\\widetilde{\\mathcal{V}}_{k}$ and $\\omega=(S_{\\mathbf{u}},k)\\in\\mathcal{V}_{k}$, where $\\mathbf{u}\\in\\mathcal{M}_{k}$, $k\\geq0$, we define $f_{\\tilde{\\omega}}:=f_{\\mathbf{u}}$ and $S_{\\omega}:=S_{\\mathbf{u}}$. It follows from \\eqref{eq(5.1)} that\n\\begin{equation}\\label{eq(5.01)}\nS_{\\omega}=\\varphi^{-1}\\circ f_{\\tilde{\\omega}}\\circ\\varphi.\n\\end{equation}\n\n\\begin{prop}\\label{prop(5.1)}\nLet $S_{i}$ be defined as in (\\ref{eq(5.1)}), where $i\\in\\{1,\\dots,N\\}$. If $\\{f_{i}\\}_{i=1}^{N}$ satisfies (FTC), then $\\{S_{i}\\}_{i=1}^{N}$ satisfies (FTC).\n\\end{prop}\n\\begin{proof}\nFor any $\\tilde{\\omega},\\tilde{\\omega}'\\in\\widetilde{\\mathcal{V}}$ and $\\tilde{\\omega}\\sim\\tilde{\\omega}'$, by Definition \\ref{defi(3.2)}, there exist $\\tilde{\\sigma}\\in\\Omega(\\tilde{\\omega})$ and $\\tilde{\\sigma}'\\in\\Omega(\\tilde{\\omega}')$ such that\n\\begin{equation}\\label{eq(5.2)}\nf_{\\tilde{\\omega}'}^{-1}\\circ f_{\\tilde{\\sigma}'}=f_{\\tilde{\\omega}}^{-1}\\circ f_{\\tilde{\\sigma}}.\n\\end{equation}\nFor any $\\omega,\\omega'\\in\\mathcal{V}$, $\\sigma\\in\\Omega(\\omega)$ and $\\sigma'\\in\\Omega(\\omega')$, we have\n$$\\begin{aligned}\nS_{\\omega'}^{-1}\\circ S_{\\sigma'}&=\\varphi^{-1}\\circ f_{\\tilde{\\omega}'}^{-1}\\circ\\varphi\\circ\\varphi^{-1}\\circ f_{\\tilde{\\sigma}'}\\circ\\varphi\\quad\\text{(by~(\\ref{eq(5.01)}))}\\\\\n&=\\varphi^{-1}\\circ f_{\\tilde{\\omega}}^{-1}\\circ f_{\\tilde{\\sigma}}\\circ\\varphi\\quad\\text{(by~(\\ref{eq(5.2)}))}\\\\\n&=\\varphi^{-1}\\circ f_{\\tilde{\\omega}}^{-1}\\circ\\varphi\\circ\\varphi^{-1}\\circ f_{\\tilde{\\sigma}}\\circ\\varphi\\\\\n&=S_{\\omega}^{-1}\\circ S_{\\sigma}.\n\\end{aligned}$$\nIt follows that $\\omega\\sim\\omega'$. Since $\\{f_{i}\\}_{i=1}^{N}$ satisfies (FTC),\nwe have $\\#\\widetilde{\\mathcal{V}}\/_{\\sim}<\\infty$. Thus, $\\#\\mathcal{V}\/_{\\sim}<\\infty$. This proves the proposition.\n\\end{proof}\n\nLet\n$$\\mathbb{S}^{n}:=\\left\\{(x_{1},\\dots,x_{n+1})\\in \\mathbb{R}^{n+1}:\\sum_{i=1}^{n+1}x_{i}^{2}=1\\right\\},\\quad\n\\mathbb{D}^{n}:=\\left\\{(x_{1},\\dots,x_{n})\\in \\mathbb{R}^{n}:\\sum_{i=1}^{n}x_{i}^{2}<1\\right\\}.$$\nLet $\\mathbb{S}^{n}_{+}$ be the upper hemisphere of $\\mathbb{S}^{n}$, and define the stereographic projection $\\varphi:\\mathbb{S}^{n}_{+}\\rightarrow \\mathbb{D}^{n}$ as\n$$\n\\varphi(x_{1},\\dots,x_{n+1})=\\frac{1}{1+x_{n+1}}(x_{1},\\dots,x_{n}):=(y_{1},\\dots,y_{n}).\n$$\nThen\n$$\\varphi^{-1}(y_{1},\\dots,y_{n})=\\frac{1}{|\\boldsymbol{y}|^{2}+1}\\big(2y_{1},\\dots,2y_{n},1-|\\boldsymbol{y}|^{2}\\big),$$\nwhere $|\\boldsymbol{y}|^{2}=y_{1}^{2}+\\cdots+y_{n}^{2}$.\n\nFor convenience, we consider the case of $n=2$. We will give some actual examples of Proposition \\ref{prop(5.1)}. The first example below satisfies (OSC), while the other two satisfy (FTC) but not (OSC).\n\n\\begin{exam}\\label{exam(5.1)}\nLet $\\{f_{i}\\}_{i=1}^{3}$ be a Sierpinski gasket on $\\mathbb{R}^{2}$, i.e., for $\\boldsymbol{x}\\in\\mathbb{R}^{2}$,\n$$f_{1}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(0,\\frac{1}{2}\\bigg),\\quad\nf_{2}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(-\\frac{1}{4},0\\bigg),\\quad\nf_{3}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},0\\bigg).$$\nLet $S_{i}$ be defined as in (\\ref{eq(5.1)}), where $i\\in\\{1,2,3\\}$. Then $\\{S_{i}\\}_{i=1}^{3}$ is a CIFS satisfying (OSC) on $\\mathbb{S}^{2}_{+}$ (see Figure \\ref{fig.1}(a)).\n\\end{exam}\n\n\\begin{exam}\\label{exam(5.2)}\nLet $\\{f_{i}\\}_{i=1}^{4}$ be a CIFS defined as in \\cite{Lau-Ngai_2007} satisfying (FTC), i.e., for $\\boldsymbol{x}\\in\\mathbb{R}^{2}$,\n$$\\begin{aligned}\nf_{1}(\\boldsymbol{x})&=\\rho\\boldsymbol{x}+\\bigg(\\frac{1}{2}\\rho,0\\bigg),\\qquad\nf_{2}(\\boldsymbol{x})=r\\boldsymbol{x}+\\bigg(\\rho-\\rho r+\\frac{1}{2}r,0\\bigg),\\\\\nf_{3}(\\boldsymbol{x})&=r\\boldsymbol{x}+\\bigg(1-\\frac{1}{2}r,0\\bigg),\\qquad\nf_{4}(\\boldsymbol{x})\n=r\\boldsymbol{x}+\\bigg(\\frac{1}{2}r,1-r\\bigg),\n\\end{aligned}$$\nwhere $0<\\rho,r<1$ and $\\rho+2r-\\rho r\\leq1$. Let $S_{i}$ be defined as in (\\ref{eq(5.1)}), where $i\\in\\{1,2,3,4\\}$. By Proposition \\ref{prop(5.1)}, $\\{S_{i}\\}_{i=1}^{4}$ is a CIFS satisfying (FTC) on $\\mathbb{S}^{2}_{+}$ (see Figure \\ref{fig.1}(b)).\n\\end{exam}\n\n\\begin{exam}\\label{exam(5.3)}\nLet $\\{f_{i}\\}_{i=1}^{3}$ be a golden Sierpinski gasket defined as in \\cite{Ngai-Wang_2001} satisfying (FTC), i.e., for $\\boldsymbol{x}\\in\\mathbb{R}^{2}$,\n$$\nf_{1}(\\boldsymbol{x})=\\rho \\boldsymbol{x},\\quad\nf_{2}(\\boldsymbol{x})=\\rho \\boldsymbol{x}+\\big(\\rho^{2},0\\big),\\quad\nf_{3}(\\boldsymbol{x})=\\rho^{2} \\boldsymbol{x}+\\big(\\rho,\\rho\\big),\n$$\nwhere $\\rho=\\big(\\sqrt{5}-1\\big)\/2$. Let $S_{i}$ be defined as in (\\ref{eq(5.1)}), where $i\\in\\{1,2,3\\}$. By Proposition \\ref{prop(5.1)}, $\\{S_{i}\\}_{i=1}^{3}$ is a CIFS satisfying (FTC) on $\\mathbb{S}^{2}_{+}$ (see Figure \\ref{fig.1}(c)).\n\\end{exam}\n\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n \\begin{tikzpicture}[scale=0.2,domain=0:180,>=stealth]\n \\coordinate (org) at (0,0);\n \\draw (0,0) circle[radius=8];\n \\draw (org) ellipse (8cm and 3cm);\n \\draw[help lines,dashed](9.5,0)--(0,0);\n \\draw[help lines,dashed](0,-9.5)--(0,0);\n \\draw[help lines,dashed](5.8,7.25)--(0,0);\n \\draw[white] plot ({8*cos(\\x)},{3*sin(\\x)});\n \\foreach\\i\/\\text in{{-6.8,-8.5}\/y,{-10.5,0}\/x,{0,10.5}\/{}}\n \\draw[help lines,->] (org)node[above right]{}--(\\i)node[above]{$\\text$};\n \\draw[help lines,->]node[left] at (0,10){$z$};\n \\draw[thin,black] (-4,0)--(4,0);\n \\draw[thin,black] (-4,0)--(-2.3,-2.875);\n \\draw[thin,black] (4,0)--(-2.3,-2.875);\n \\draw[thin,black] (-3.15,-1.4375)--(0.85,-1.4375);\n \\draw[thin,black] (0,0)--(0.85,-1.4375);\n \\draw[thin,black] 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[out=138,in=-21] (3.1,7.375);\n \\draw[fill=blue!30,opacity=.5](3.1,7.375) to [out=-125,in=70] (1.9,5) to [out=-20,in=150] (4.3,3.9) to [out=55,in=-130] (5.65,5.65) to [out=138,in=-21] (3.1,7.375)\n \\foreach \\i\/\\tex in {0\/(b)}\n \\draw(0,-10)node[below]{\\tex};\n \\foreach \\i\/\\tex in {0\/\\text{}}\n \\draw(0,-19)node[below]{\\tex};\n \\end{tikzpicture}\n \\quad\n \\begin{tikzpicture}[scale=.2,domain=0:180,>=stealth]\n \\coordinate (org) at (0,0);\n \\draw (0,0) circle[radius=8];\n \\draw (org) ellipse (8cm and 3cm);\n \\draw[help lines,dashed](9.5,0)--(0,0);\n \\draw[help lines,dashed](0,-9.5)--(0,0);\n \\draw[help lines,dashed](5.8,7.25)--(0,0);\n \\draw[white] plot ({8*cos(\\x)},{3*sin(\\x)});\n \\foreach\\i\/\\text in{{-6.8,-8.5}\/y,{-10.5,0}\/x,{0,10.5}\/{}}\n \\draw[help lines,->] (org)node[above right]{}--(\\i)node[above]{$\\text$};\n \\draw[help lines,->]node[left] at (0,10){$z$};\n \\draw[thin,black] (-8,0)--(-2.3,-2.875);\n \\draw[thin,black] (-2.3,-2.875)--(8,0);\n \\draw[thin,black] (8,0)--(-8,0);\n \\draw[thin,black] (-4.4375,-1.797)--(1.5625,-1.797);\n \\draw[thin,black] (-4.4375,-1.797)--(2,0);\n \\draw[thin,black] (-2,0)--(1.5625,-1.797);\n \\draw[top color=olive!60!white!30,bottom color=red!30,opacity=.5]\n (1.5625,-1.797)--(-2,0)--(8,0)--(1.5625,-1.797);\n \\draw[top color=olive!60!white!30,bottom color=red!30,opacity=.5]\n (-4.4375,-1.797)--(-2.3,-2.875)--(1.5625,-1.797)--(-4.4375,-1.797);\n \\draw[top color=olive!60!white!30,bottom color=red!30,opacity=.5](-8,0)--(-4.4375,-1.797)--(2,0)--(-8,0);\n \\draw[thin,cyan,densely dashed](0,-8)--(-8,0);\n \\draw[thin,cyan,dashed](0,-8)--(-2.3,-2.875);\n \\draw[thin,cyan,densely dashed](0,-8)--(8,0);\n \\draw[thin,blue] (-8,0) to [out=90, in=-180](0,8) to [out=0, in=90] (8,0) to [out=140, in=55] (-2.3,-2.875)\n to [out=125, in=-5] (-8,0);\n \\draw[thin,cyan,densely dashed](0,-8)--(-5.4,-0.51);\n \\draw[thin,cyan,densely dashed](0,-8)--(-3.8,7.05);\n \\draw[thin,cyan,densely dashed](0,-8)--(3.8,7.05);\n \\draw[thin,blue](-5.4,-0.51) to [out=45, in=150] (2.2,0.7);\n \\draw[thin,blue](3.8,7.05) to [out=-175, in=75] (-5.4,-0.51);\n \\draw[thin,blue](-3.8,7.05) to [out=-20, in=110] (2.2,0.7);\n \\draw[fill=blue!30,opacity=.5](-5.4,-0.51) to [out=45, in=150] (2.2,0.7) to [out=-158, in=53] (-2.3,-2.875) to [out=126, in=-19](-5.4,-0.51);\n \\draw[fill=blue!30,opacity=.5](8,0) to [out=90, in=0] (0,8) to [out=-180, in=26] (-3.8,7.05) to [out=-20, in=110] (2.2,0.7) to [out=20, in=140] (8,0);\n \\draw[fill=blue!30,opacity=.5](-8,0) to [out=90, in=180] (0,8) to [out=0, in=154] (3.8,7.05) to [out=-175, in=75] (-5.4,-0.51) to [out=162, in=-5] (-8,0);\n \\foreach \\i\/\\tex in {0\/(c)}\n \\draw(0,-10)node[below]{\\tex};\n \\foreach \\i\/\\tex in {0\/}\n \\draw(0,-19)node[below]{\\tex};\n \\end{tikzpicture}\n \\vspace{-3.2em}\n\\caption{Figures (a), (b) and (c) are respectively the IFSs in Examples \\ref{exam(5.1)}, \\ref{exam(5.2)} and \\ref{exam(5.3)}.} \\label{fig.1}\n\\end{center}\n\\end{figure}\n\n\n\nLet $M$ be a complete $n$-dimensional smooth Riemannian manifold that is locally Euclidean.\nNow, we construct an example of GIFS on $M$ satisfying (GFTC) to illustrate Theorem \\ref{thm(41.1)}. Let $G=(V,E)$ be a GIFS of contractive similitudes defined on $\\mathbb{R}^{n}$, and $\\{O_{i}\\}_{i=1}^{t}$, where $O_{i}\\subset\\mathbb{R}^{n}$, be an invariant family of nonempty bounded open sets under $G=(V,E)$. Let $O:=\\bigcup_{i=1}^{t}O_{i}$. For any edge $e\\in E$, there corresponds a contractive similitude $f_{e}:O\\longrightarrow O$. Assume that there exists a diffeomorphism\n$$\\varphi:O_{i}\\longrightarrow \\Omega_{i}\\quad\\text{for any }i\\in\\{1,\\dots,t\\},$$\nwhere $\\Omega_{i}\\subset M$ is open and connected. Let $\\Omega:=\\bigcup_{i=1}^{t}\\Omega_{i}$.\nFor any edge $e\\in E$, define\n\\begin{equation}\\label{eq(5.3)}\nS_{e}:=\\varphi^{-1}\\circ f_{e}\\circ\\varphi:\\Omega\\longrightarrow \\Omega.\n\\end{equation}\nAs in \\cite[Proposition 7.1]{Ngai-Xu_2022}, $\\{S_{e}\\}_{e\\in E}$ is a family of contractive maps on $M$. If for any $e\\in E$, $S_{e}$ is a similitude, then $G=(V,E)$, along with $\\{\\Omega_{i}\\}_{i=1}^{t}$ and $\\{S_{e}\\}_{e\\in E}$, forms a GIFS on $M$. The proof of the following proposition is similar to that of Proposition \\ref{prop(5.1)}; we omit it.\n\n\n\\begin{prop}\\label{prop(5.2)}\nUse the above notation and setup. Let $M$ be a complete $n$-dimensional smooth Riemannian manifold that is locally Euclidean. Let $G=(V,E)$ be a GIFS defined on $\\mathbb{R}^{n}$ satisfying (GFTC) with $\\{O_{i}\\}_{i=1}^{t}$ being a GFTC-family and $\\{f_{e}\\}_{e\\in E}$ being an associated family of contractive similitudes.\nFor any $e\\in E$, let $S_{e}$ be a similitude defined as in (\\ref{eq(5.3)}). Then the GIFS $G=(V,E)$ defined on $M$ satisfies (GFTC) with $\\{\\Omega_{i}\\}_{i=1}^{t}$ being a GFTC-family and $\\{S_{e}\\}_{e\\in E}$ being an associated family of contractive similitudes. Moreover, for such two GIFSs connected by a diffeomorphism, the neighborhood types, weighted incidence matrices, and the Hausdorff dimension of the corresponding graph self-similar sets are the same.\n\\end{prop}\n\n\n\nAssume that $G=(V,E)$ is a GIFS of contractive similitudes defined on $M$ satisfying (GFTC). Let $\\mathcal{T}_{1},\\dots,\\mathcal{T}_{q}$ be all the distinct neighborhood types with $\\mathcal{T}_{i}=[\\omega_{{\\rm root}}^{i}]$ for any $i\\in\\{1,\\dots,t\\}$. Fix a vertex $\\omega\\in\\mathbb{V}_{R}$ such that $[\\omega]\\in\\mathcal{T}_{i}$, where $i\\in\\{1,\\dots,q\\}$.\nLet $\\sigma_{1},\\dots,\\sigma_{m}$ be the offspring of $\\omega$ in $\\mathbb{G}_{R}$, let $\\mathbf{k}_{\\ell}$ be the unique edge in $\\mathbb{G}_{R}$ connecting $\\omega$ to $\\sigma_{\\ell}$ for $1\\leq \\ell\\leq m$, and let\n$$C_{ij}:=\\{\\sigma_{\\ell}:1\\leq \\ell\\leq m,~[\\sigma_{\\ell}]=\\mathcal{T}_{j}\\}.$$\nNote that for two edges $\\mathbf{k}_{\\ell}$ and $\\mathbf{k}_{\\ell'}$ connecting $\\omega$ to two distinct $\\sigma_{\\ell}$ and $\\sigma_{\\ell'}$ satisfying $[\\sigma_{\\ell}]=[\\sigma_{\\ell}']=\\mathcal{T}_{j}$, the contraction ratios $\\rho_{\\sigma_{\\ell}}$ and $\\rho_{\\sigma_{\\ell'}}$ may be different. We can partition $C_{ij}:=C_{ij}(1)\\cup\\cdots\\cup C_{ij}(n_{ij})$ by using $\\rho_{\\sigma_{\\ell}}$, where for $s=1,\\dots,n_{ij}$,\n$$C_{ij}(s):=\\{\\sigma_{\\ell}\\in C_{ij}:\\rho_{\\sigma_{\\ell}}=\\rho_{ijs}\\},$$\nand the $\\rho_{ijs}$ are distinct. Thus, for any entry $A_{\\alpha}(i,j)$ of the weighted incidence matrix,\n$$A_{\\alpha}(i,j)=\\sum_{s=1}^{n_{ij}}\\#C_{ij}(s)\\rho_{ijs}^{\\alpha}.$$\nMoreover, we can write symbolically\n$$\\mathcal{T}_{i}\\longrightarrow\\sum_{j=1}^{q}\\sum_{s=1}^{n_{ij}}\\#C_{ij}(s)\\mathcal{T}_{j}(\\rho_{ijs}),$$\nwhere the $\\mathcal{T}_{j}(\\rho_{ijs})$ are defined in an obvious way.\nWe say that $\\mathcal{T}_{i}$ generates $\\#C_{ij}(s)$ neighborhoods of type $\\mathcal{T}_{j}$ with contraction ratio $\\rho_{ijs}$.\n\nFor $\\boldsymbol{x}\\in[0,1]\\times[0,1]$, we consider the following iterated function system with overlaps:\n$$\nh_{1}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(0,\\frac{1}{4}\\bigg),\\ \\\nh_{2}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},\\frac{1}{4}\\bigg),\\ \\\nh_{3}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{2},\\frac{1}{4}\\bigg),\\ \\\nh_{4}(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},\\frac{3}{4}\\bigg).\n$$\nIterations of $\\{h_i\\}_{i=1}^4$ induce iterations on the $2$-torus $\\mathbb{T}^2=\\mathbb R^2\/\\mathbb Z^2$, generating an attractor on $\\mathbb{T}^2$. We are interested in computing the Hausdorff dimension of the attractor. However, the relations induced by $\\{h_i\\}_{i=1}^4$ on $\\mathbb{T}^2$ are not well-defined functions, making it awkward to apply the theory of IFSs developed in Section~\\ref{S:4}. To overcome this difficulty, we will use the GIFS framework and Theorem~\\ref{thm(41.1)}, as shown in the following example.\n\n\n\\begin{exam}\\label{exam(5.4)}\nLet $\\{h_i\\}_{i=1}^4$ be defined as above, and let $\\mathbb{T}^2=\\mathbb{S}^{1}\\times\\mathbb{S}^{1}$ be a $2$-torus, viewed as $[0,1]\\times[0,1]$ with opposite sides identified. Let $\\mathbb{T}^2$ be endowed with the Riemannian metric induced from $\\mathbb{R}^2$. Consider the IFS $\\{g_{i}\\}_{i=1}^{4}$ on $[0,1]\\times[0,1]$ under the Euclidean metric, where for $\\boldsymbol{x}\\in[0,1]\\times[0,1]$,\n$$\ng_{1}(\\boldsymbol{x})=h_1(\\boldsymbol{x}),\\quad\ng_{2}(\\boldsymbol{x})=h_2(\\boldsymbol{x}),\\quad\ng_{3}(\\boldsymbol{x})=h_3(\\boldsymbol{x}),\n$$\nand\n$$g_{4}(\\boldsymbol{x})=\\begin{cases}\\frac{1}{2}\\boldsymbol{x}+\\big(\\frac{1}{4},\\frac{3}{4}\\big),\\quad \\boldsymbol{x}\\in[0,1]\\times[0,\\frac{1}{2}],\\\\ \\frac{1}{2}\\boldsymbol{x}+\\big(\\frac{1}{4},-\\frac{1}{4}\\big),\\quad \\boldsymbol{x}\\in[0,1]\\times[\\frac{1}{2},1]\\end{cases}$$\n(see Figure \\ref{fig.3}(a)). $\\{g_{i}\\}_{i=1}^{4}$ induces four relations $\\{S_{i}\\}_{i=1}^{4}$ on $\\mathbb{T}^2$. We are interested in the Hausdorff dimension of the attractor generated by $\\{S_{i}\\}_{i=1}^{4}$.\nNote that the image of $S_4$ is a connected rectangle in $\\mathbb{T}^2$, but the image of $g_4$ is divided into two rectangles in $\\mathbb{R}^2$. It is easy to see that $\\{S_{i}\\}_{i=1}^{4}$ are not well-defined functions and are not contractive under the metric of $\\mathbb{T}^{2}$. As a result, we need to use the framework of a GIFS.\nConsider the GIFS $G=(V,E)$ defined on $\\mathbb{R}^{2}$ associated to $\\{g_{i}\\}_{i=1}^{4}$ with $V=\\{1,2\\}$ and $E=\\{e_{1},\\dots,e_{8}\\}$, where $\\mathbf{O}=\\{O_{1},O_{2}\\}$ with $O_{1}=(0,1)\\times(0,1\/2)$ and $O_{2}=(0,1)\\times(1\/2,1)$ is the invariant family, and\n$$e_{1},e_{2},e_{3}\\in E^{1,1},\\quad e_{4}\\in E^{1,2},\\quad e_{5},e_{6},e_{7}\\in E^{2,2},\\quad e_{8}\\in E^{2,1}$$\n(see Figure \\ref{fig.2}). The associated similitudes are defined as\n$$f_{e_{1}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(0,\\frac{1}{4}\\bigg),~~\nf_{e_{2}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},\\frac{1}{4}\\bigg),~~\nf_{e_{3}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{2},\\frac{1}{4}\\bigg),~~\nf_{e_{4}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},-\\frac{1}{2}\\bigg),~~\n$$\n$$f_{e_{5}}=\\frac{1}{2}\\boldsymbol{x},~~\nf_{e_{6}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},0\\bigg),~~\nf_{e_{7}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{2},0\\bigg),~~\nf_{e_{8}}=\\frac{1}{2}\\boldsymbol{x}+\\bigg(\\frac{1}{4},\\frac{3}{4}\\bigg).~~\n$$\nLet $\\Omega_{1}$ and $\\Omega_{2}$, viewed as $(0,1)\\times(0,1\/2)$ and $(0,1)\\times(1\/2,1)$, be the lower and upper pieces of the interior of $\\mathbb{T}^{2}$, respectively. Obviously, for $i=1,2$, and any $e\\in E$, there exists a diffeomorphism\n$\\varphi:O_{i}\\longrightarrow \\Omega_{i}$ such that $S_{e}$, defined as in \\eqref{eq(5.3)}, is a contractive similitude.\nLet $K$ and $K_{0}$ be the graph self-similar set of $G=(V,E)$ generated by $\\{S_{e}\\}_{e\\in E}$ and $\\{f_{e}\\}_{e\\in E}$, respectively. Then $\\dim_{{\\rm H}}(K)=\\dim_{{\\rm H}}(K_{0})=\\log(2+\\sqrt{2})\/\\log2=1.77155\\dots$.\n\\end{exam}\n\\begin{figure}[htbp]\n\\begin{center}\n\\tikzset{every picture\/.style={line width=0.75pt}}\n\n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n\n\\draw (170.44,123.61) .. controls (191.13,144.03) and (315.33,147.16) .. (351.66,123.43) ;\n\\draw [shift={(353.78,121.94)}, rotate = 142.77] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n\\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (164.83,118.04) .. controls (164.13,119.22) and (162.61,119.61) .. (161.44,118.91) .. controls (160.26,118.21) and (159.88,116.7) .. (160.57,115.52) .. controls (161.27,114.35) and (162.79,113.96) .. (163.96,114.65) .. controls (165.14,115.35) and (165.52,116.87) .. (164.83,118.04) -- cycle ;\n\\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (364.66,118.71) .. controls (363.96,119.88) and (362.45,120.27) .. (361.27,119.58) .. controls (360.1,118.88) and (359.71,117.36) .. (360.41,116.19) .. controls (361.1,115.01) and (362.62,114.62) .. (363.79,115.32) .. controls (364.97,116.02) and (365.36,117.54) .. (364.66,118.71) -- cycle ;\n\\draw (170.11,112.28) .. controls (196.73,89.81) and (320.37,97.44) .. (350.28,111.92) ;\n\\draw [shift={(352.78,113.28)}, rotate = 211.87] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n\\draw (158.78,109.28) .. controls (92.94,41.46) and (229.06,41.9) .. (167.05,107.49) ;\n\\draw [shift={(165.11,109.5)}, rotate = 314.63] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n\\draw (358.44,109.28) .. controls (292.61,41.46) and (428.08,41.9) .. (366.05,107.49) ;\n\\draw [shift={(364.11,109.5)}, rotate = 314.63] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n\\draw (370.27,114.04) .. controls (438.68,48.82) and (437.64,183.9) .. (372.62,121.28) ;\n\\draw [shift={(370.63,119.32)}, rotate = 45.15] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n\\draw (366.67,124.69) .. controls (432.19,192.81) and (297.1,192.37) .. (359.44,127.08) ;\n\\draw [shift={(361.39,125.08)}, rotate = 134.9] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n\\draw (154.9,120.5) .. controls (86.64,185.87) and (87.37,50.79) .. (152.53,113.26) ;\n\\draw [shift={(154.52,115.22)}, rotate = 225.02] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n\\draw (166.01,124.02) .. controls (231.52,192.15) and (96.44,191.7) .. (158.78,126.41) ;\n\\draw [shift={(160.73,124.42)}, rotate = 134.9] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n\n\\draw (122,65.9) node [anchor=north west][inner sep=0.75pt] {$e_{1}$};\n\\draw (82.5,112) node [anchor=north west][inner sep=0.75pt] {$e_{2}$};\n\\draw (122,157) node [anchor=north west][inner sep=0.75pt] {$e_{3}$};\n\\draw (248,82) node [anchor=north west][inner sep=0.75pt] {$e_{4}$};\n\\draw (248,145) node [anchor=north west][inner sep=0.75pt] {$e_{8}$};\n\\draw (390,65.9) node [anchor=north west][inner sep=0.75pt] {$e_{5}$};\n\\draw (428.5,112) node [anchor=north west][inner sep=0.75pt] {$e_{6}$};\n\\draw (390,157) node [anchor=north west][inner sep=0.75pt] {$e_{7}$};\n\\draw (158,132) node [anchor=north west][inner sep=0.75pt] {$\\mathbf{1}$};\n\\draw (358,132) node [anchor=north west][inner sep=0.75pt] {$\\mathbf{2}$};\n\n\\end{tikzpicture}\n\\vspace{-1.1em}\n\\caption{The GIFS in Examples \\ref{exam(5.4)}.} \\label{fig.2}\n\\end{center}\n\\end{figure}\n\\begin{proof}\nFor convenience, we write $f_{e_{i}}=f_{i}$ for any $i\\in\\{1,\\dots,8\\}$. Let $\\mathcal{F}_{k}=E_{k}$ for $k\\geq1$, and let $\\mathcal{T}_{1}$ and $\\mathcal{T}_{2}$ be the neighborhood types of the root neighborhoods $[O_{1}]$ and $[O_{2}]$, respectively. Iterations of the root vertices are shown in Figure \\ref{fig.3}(b, c). All neighborhood types are generated after three iterations.\nTo construct the weighted incidence matrix in the reduced graph $\\mathbb{G}_{R}$, we note that\n$$\\mathbb{V}_{1}=\\{(f_{1},1),\\dots,(f_{8},1)\\}.$$\nDenote by $\\omega_{1},\\dots,\\omega_{8}$ the vertices in $\\mathbb{V}_{1}$ according to the above order. Then $[\\omega_{4}]=\\mathcal{T}_{2}$ and $[\\omega_{8}]=\\mathcal{T}_{1}$. Let $\\mathcal{T}_{3}:=[\\omega_{1}]$, $\\mathcal{T}_{4}:=[\\omega_{2}]$, $\\mathcal{T}_{5}:=[\\omega_{3}]$, $\\mathcal{T}_{6}:=[\\omega_{5}]$, $\\mathcal{T}_{7}:=[\\omega_{6}]$, $\\mathcal{T}_{8}:=[\\omega_{7}]$. Then\n$$\\mathcal{T}_{1}\\longrightarrow\\mathcal{T}_{2}+\\mathcal{T}_{3}+\n\\mathcal{T}_{4}+\\mathcal{T}_{5}$$\nand\n$$\\mathcal{T}_{2}\\longrightarrow\\mathcal{T}_{1}+\\mathcal{T}_{6}\n+\\mathcal{T}_{7}+\\mathcal{T}_{8},$$\nwhere we write $\\mathcal{T}_{i}=\\mathcal{T}_{i}(\\frac{1}{2})$ for convenience. Since $f_{13}=f_{21}$, the edge $e_{1}e_{3}$ is removed in $\\mathbb{G}_{R}$. Hence $\\omega_{1}$ has three offspring, i.e.,\n$$(f_{11},2),(f_{12},2),(f_{14},2)\\in\\mathbb{V}_{2},$$\nwhich are of neighborhood types $\\mathcal{T}_{3},\\mathcal{T}_{4},\\mathcal{T}_{2}$, respectively. Iterating $(f_{1},1)$ gives\n$$\\mathcal{T}_{3}\\longrightarrow\\mathcal{T}_{2}+\\mathcal{T}_{3}+\\mathcal{T}_{4}.$$\nSince $f_{23}=f_{31}$, the edge $e_{2}e_{3}$ is removed in $\\mathbb{G}_{R}$. Hence $\\omega_{2}$ has three offspring, i.e.,\n$$(f_{21},2),(f_{22},2),(f_{24},2)\\in\\mathbb{V}_{2}.$$\nNote that $[(f_{22},2)]=\\mathcal{T}_{4}$ and $[(f_{24},2)]=\\mathcal{T}_{2}$. Let $\\omega_{9}:=(f_{22},2)$ and $\\mathcal{T}_{9}:=[\\omega_{9}]$. Then\n$$\\mathcal{T}_{4}\\longrightarrow\\mathcal{T}_{2}+\\mathcal{T}_{4}+\\mathcal{T}_{9}.$$\nNote that $\\omega_{3}$ has four offspring, i.e.,\n$$(f_{31},2),(f_{32},2),(f_{33},2),(f_{34},2)\\in\\mathbb{V}_{2},$$\nwith $[(f_{32},2)]=\\mathcal{T}_{4}$, $[(f_{33},2)]=\\mathcal{T}_{5}$ and $[(f_{34},2)]=\\mathcal{T}_{2}$. Let $\\omega_{10}:=(f_{31},2)$ and $\\mathcal{T}_{10}:=[\\omega_{10}]$. Then\n$$\\mathcal{T}_{5}\\longrightarrow\\mathcal{T}_{2}+\\mathcal{T}_{4}+\\mathcal{T}_{5}+\\mathcal{T}_{10}.$$\nSince $f_{213}=f_{221}$, the edge $e_{2}e_{1}e_{3}$ is removed in $\\mathbb{G}_{R}$. Hence $\\omega_{9}$ has three offspring, i.e.,\n$$(f_{211},2),(f_{212},2),(f_{214},2)\\in\\mathbb{V}_{3},$$\nwhich are of neighborhood types $\\mathcal{T}_{10},\\mathcal{T}_{4},\\mathcal{T}_{2}$, respectively. Iterating $(f_{22},2)$ gives\n$$\\mathcal{T}_{9}\\longrightarrow\\mathcal{T}_{2}+\\mathcal{T}_{4}+\\mathcal{T}_{10}.$$\nSince $f_{313}=f_{321}$, the edge $e_{3}e_{1}e_{3}$ is removed in $\\mathbb{G}_{R}$. Hence $\\omega_{10}$ has three offspring, i.e.,\n$$(f_{311},2),(f_{312},2),(f_{314},2)\\in\\mathbb{V}_{3},$$\nwhich are of neighborhood types $\\mathcal{T}_{10},\\mathcal{T}_{4},\\mathcal{T}_{2}$, respectively. Iterating $(f_{31},2)$ gives\n$$\\mathcal{T}_{10}\\longrightarrow\\mathcal{T}_{2}+\\mathcal{T}_{4}+\\mathcal{T}_{10}.$$\nLet $\\mathcal{T}_{11}:=[(f_{65},2)]$ and $\\mathcal{T}_{12}:=[(f_{75},2)]$. Using the same argument, it can be checked directly that\n$$\\mathcal{T}_{6}\\longrightarrow\\mathcal{T}_{1}+\\mathcal{T}_{6}+\\mathcal{T}_{7},~~\n\\mathcal{T}_{7}\\longrightarrow\\mathcal{T}_{1}+\\mathcal{T}_{7}+\\mathcal{T}_{11},$$\n$$\\mathcal{T}_{8}\\longrightarrow\\mathcal{T}_{1}+\\mathcal{T}_{7}+\\mathcal{T}_{8}+\\mathcal{T}_{12},~~\n\\mathcal{T}_{11}\\longrightarrow\\mathcal{T}_{1}+\\mathcal{T}_{7}+\\mathcal{T}_{12},~~\n\\mathcal{T}_{12}\\longrightarrow\\mathcal{T}_{1}+\\mathcal{T}_{7}+\\mathcal{T}_{12}.$$\nHence the weighted incidence matrix is\n$$\nA_{\\alpha}=\\Big(\\frac{1}{2}\\Big)^{\\alpha}\\left({\\begin{array}{cccccccccccc}\n0&1&1&1&1&0&0&0&0&0&0&0\\\\\n1&0&0&0&0&1&1&1&0&0&0&0\\\\\n0&1&1&1&0&0&0&0&0&0&0&0\\\\\n0&1&0&1&0&0&0&0&1&0&0&0\\\\\n0&1&0&1&1&0&0&0&0&1&0&0\\\\\n1&0&0&0&0&1&1&0&0&0&0&0\\\\\n1&0&0&0&0&0&1&0&0&0&1&0\\\\\n1&0&0&0&0&0&1&1&0&0&0&1\\\\\n0&1&0&1&0&0&0&0&0&1&0&0\\\\\n0&1&0&1&0&0&0&0&0&1&0&0\\\\\n1&0&0&0&0&0&1&0&0&0&0&1\\\\\n1&0&0&0&0&0&1&0&0&0&0&1\n\\end{array}}\\right)=:\\Big(\\frac{1}{2}\\Big)^{\\alpha}\\widetilde{A}_{\\alpha},\n$$\nand the maximal eigenvalue of $\\widetilde{A}_{\\alpha}$ is $2+\\sqrt{2}$.\nThe GIFS $G=(V,E)$ defined on $\\mathbb{R}^{2}$ satisfies (GFTC) with $\\{O_{i}\\}_{i=1}^{2}$ being a GFTC-family and $\\{f_{e}\\}_{e\\in E}$ being an associated family of contractive similitudes.\nBy Proposition \\ref{prop(5.2)}, the GIFS $G=(V,E)$ defined on $\\mathbb{T}^{2}$ satisfies (GFTC) with $\\{\\Omega_{i}\\}_{i=1}^{2}$ being a GFTC-family and $\\{S_{e}\\}_{e\\in E}$ being an associated family of contractive similitudes.\nBy Theorem \\ref{thm(41.1)}, $\\dim_{{\\rm H}}(K)=\\dim_{{\\rm H}}(K_{0})=\\log(2+\\sqrt{2})\/\\log2=1.77155\\dots$.\n\\end{proof}\n\n\n\\begin{figure}[htbp]\n\\centering\n\\subfigure[]\n{\\includegraphics[width=4.8cm]{0.png}\\label{fig.(a)}}\n\\qquad\\qquad\n\\subfigure[]\n{\\includegraphics[width=4.8cm]{1.png}\\label{fig.(b)}}\n\\\\\n\\subfigure[]\n{\\includegraphics[width=4.8cm]{2.png}\\label{fig.(c)}}\n\\qquad\\qquad\n\\subfigure[]\n{\\includegraphics[width=4.8cm]{G9.png}\\label{fig.(d)}}\n\n\\caption{(a) $\\{g_{i}\\}_{i=1}^{4}$. (b) The first iteration of $G=(V,E)$ on $\\mathbb{R}^{2}$ associated to $\\{g_{i}\\}_{i=1}^{4}$. (c) The second iteration. (d) The corresponding graph self-similar set.}\\label{fig.3}\n\\end{figure}\n\n\n\\begin{rema}\\label{rema(5.1)}\nUnlike $\\mathbb{T}^{2}$, the family of contractive similitudes associated to a GIFS defined on $\\mathbb{R}^{2}$ can be characterized explicitly. In fact, Proposition \\ref{prop(5.2)} is not needed in the proof of Example \\ref{exam(5.4)}. We may use Theorem \\ref{thm(41.1)} and a similar arguement as in the proof of Example \\ref{exam(5.4)} to obtain the same result.\n\\end{rema}\n\n\n\\noindent\\textbf{Acknowledgements} The authors are supported in part by the National Natural Science Foundation of China, grant 11771136, and Construct Program of the Key Discipline in Hunan Province. The first author is also supported in part by a Faculty Research Scholarly Pursuit Funding from Georgia Southern University.\\\\\n\n\n\\noindent\\textbf{Declaration}\\\\\n\n\n\\noindent\\textbf{Competing interests} The authors hereby declare that there is no conflict of interest regarding this work.\n\n\\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nNonequilibrium systems are common in nature because they are, in general, subject to thermal gradients, chemical potential gradients or may be triggered by time-dependent forces. Heat transport is one such example of nonequilibrium systems where the heat carriers could be electrons, phonons, magnons, etc. To study heat transport in phononic systems, one considers a finite junction part, which can be an insulator, connected with two heat baths that are maintained at different temperatures. In the past decade, the main focus was on the calculation of the steady state heat current or heat flux flowing through the junction part from the leads \n\\cite{Caroli,Meir,Rego-Kirczenow-1998,Segal,Mingo-Yang-2003,Yamamoto-2006,Dhar1,Wang-prb06,Wang-pre07,WangJS-europhysJb-2008}. For diffusive systems, the answer is given by Fourier's law \\cite{Dhar, Lepri, Lebowitz} which is true only in the linear response regime, i.e., when the temperature difference between the baths is small. However for harmonic or ballistic systems, the heat current is given by a Landauer-like formula \\cite{Rego-Kirczenow-1998,Dhar1,WangJS-europhysJb-2008} which was first derived for electronic transport. Landauer formula on the contrary to the Fourier's law is true for arbitrary temperature differences between the leads. No such explicit expression for current is known for transient states. In recent times, several works \\cite{eduardos-paper,eduardos-paper1} followed to answer what happens to current in the transient regime. This is an important question both from the theoretical and experimental points of view. \n\nMuch attention has been given to phonon transport, in particular on thermal devices and on controlling heat flow \\cite{baowen-review}. With the advent of technology it is now possible to study transport problems and observe a single mode of vibration in small systems with few degrees of freedom \\cite{oconell}. These systems shows strong thermal fluctuations which play an important role because thermal fluctuations can lead to instantaneous heat transfer from colder to hotter lead. It is therefore necessary to talk about the statistical distribution of heat flux for these systems. In the electronic literature the distribution $P(Q_L)$ of the charge $Q_L$, flowing from the left lead to the junction part, was answered by calculating the corresponding CGF, ${\\cal Z}(\\xi)= \\langle e^{i \\xi Q_L} \\rangle$, and is given by the celebrated Levitov-Lesovik formula \\cite{Levitov,Levitov1,Levitov-PRB}. This methodology is also known as the {\\it full counting statistics} \\cite{otherworks,otherworks1,otherworks2,Klich,Pilgram,Bagrets,Gogolin,Urban,Gutman,fluct-theorems}\nin the field of electronic transport. Experimentally the electron counting statistics has been measured in quantum-dot systems \\cite{Flindt-etal-2009,Gustavsson-etal-2006}. However few experiments have been done for phonons \\cite{measurement}. In the phononic case Saito and Dhar \\cite{Saito-Dhar-2007} gave an explicit expression of the CGF. Ren et al.\\ gave a result for two-level systems \\cite{ren-jie}. Full counting statistics of energy fluctuations in a driven quantum resonator is studied by Clerk \\cite{AAClerk-2011}. The main focus in these papers was on the long-time limit and SSFT \\cite{noneq-fluct,fluct-theorem-1,fcs-bijay,kundu,Esposito-review-2009, Hanggi2}. Using the NEGF method \\cite{schwinger-keldysh,rammer86} and two-time measurement \\cite{Esposito-review-2009, Hanggi2,Hanggi3,Hanggi4} concept, Wang~et al.\\ \\cite{fcs-bijay} gave an explicit expression for the CGF which is valid for both transient and steady state regimes. \n\nIn this paper, we extend our previous work in Ref.~\\onlinecite{fcs-bijay} and derive the CGF in a more general scenario, i.e., in the presence of both the temperature difference and the time-dependent driving force. We analyze the cumulants of heat $Q_L$ for three different initial conditions of the density operator and study the effects on both transient and steady state regimes. We also derive the CGF for the joint probability distribution of left and right lead heat $P(Q_L,Q_R)$ which help us to obtain the correlations between $Q_L$ and $Q_R$. By calculating CGF for $P(Q_L,Q_R)$ we can immediately obtain the CGF for the total entropy that flows to the leads. We present analytical expressions of the CGF's in the steady state and discuss the SSFT. Our method can be easily generalized for multiple heat baths.\n\nThe plan of the paper is as follows. We start in Sec.~II by introducing our model. Then in Sec.~III we define current and corresponding quantum heat operator followed by the definition of CGF for $Q_L$ using the two-time measurement concept, in Sec.~IV. In Sec.~V we derive the CGF using Feynman's path integral method and in Sec.~VI we use Feynman's diagrammatic technique to derive the CGF. We discuss the steady state result and fluctuation theorems in Sec.~VII. In Sec. VIII and IX we discuss how to calculate the CGF's numerically in transient regime and give numerical results for one-dimensional (1D) linear chain model, connected with Rubin heat baths, for three different initial conditions of the density operator. Then in Sec.~X we obtain the CGF for joint probability distribution of heat transferred $P(Q_L,Q_R)$ and discuss correlations and total entropy flow. In. Sec.~XI we give the long-time limit expression for the driven part of the full CGF. In Sec.~XII we discuss another definition of generating function due to Nazarov and discuss the corresponding long-time limit. We found that using this definition, the generating function \ndoes not obey the Gallavotti-Cohen (GC) fluctuation symmetry. \nWe conclude with a short discussion in Sec.~XIII. Few appendices give some details of technique nature. In particular, an electron system of a tight-binding model is treated using our method.\n \n\n\\section{The model}\nOur model consists of a finite harmonic junction part, which we denote by $C$, coupled to two heat baths, the left ($L$) and the right ($R$), kept at two different temperatures $T_L$ and $T_R$, respectively. To model the heat baths, we consider an infinite collection of coupled harmonic oscillators. We take the three systems to be decoupled initially and to be described by the Hamiltonians,\n\\begin{equation}\n{\\cal H}_\\alpha = \\frac{1}{2} p_\\alpha^T p_\\alpha + \\frac{1}{2} u_\\alpha^T K^\\alpha u_\\alpha, \\quad \\alpha=L, C, R,\n\\end{equation}\nfor the left, right, and the finite central region. The leads are assumed to be semi-infinite. Masses are absorbed by defining $u = \\sqrt{m}\\,\nx$. $u_\\alpha$ and $p_\\alpha$ are column vectors of coordinates and momenta. $K^\\alpha$ is the spring constant matrix of region $\\alpha$.\nCouplings of the center region with the leads are turned on either adiabatically from time $t=-\\infty$, or switched on abruptly at $t=0$.\nThe interaction Hamiltonian takes the form \n\\begin{equation}\n{\\cal H}_{{\\rm int}} = u_L^T \\, V^{LC} \\,u_{C} + u_R^T\\, V^{RC}\\, u_{C}.\n\\end{equation}\nFor $t>0$, an external time-dependent force is applied only to the center atoms, which is of the form\n\\begin{equation}\n{\\cal V}_C(t) =-f^T(t)\\,u_C,\n\\end{equation}\nwhere $f(t)$ is the time-dependent force vector. The driving force couples only with the position operators of the center. The force can be in the form of electromagnetic field. Coupling of this form helps us to obtain analytical solution for the CGF of heat flux.\nSo the full Hamiltonian for $t>0$ (in the Schr\\\"odinger picture) is\n\\begin{equation}\n{\\cal H}(t) = {\\cal H}(0^{-}) + {\\cal V}_C(t) = {\\cal H}_C + {\\cal H}_L+ {\\cal H}_R+ {\\cal H}_{\\rm int}+ {\\cal V}_C(t).\n\\end{equation}\nIn the next section we will define current operator and the corresponding heat operator based on this Hamiltonian.\n \n\\section{Definition of current and heat operators}\nIt is possible to define the current operator ${\\cal I}$ depending on where we want to measure the current. Here we consider the current flowing from the left lead to the center system and ${\\cal I}_{L}$ is defined (in Heisenberg picture) as\n\\begin{equation}\n{\\cal I}_{L}(t) = - \\frac{d{\\cal H}^{H}_L(t)}{dt}=\\frac{i}{\\hbar}[{\\cal H}^{H}_L(t), {\\cal H}_H(t)]= p_L^{T}(t)\\,V^{LC}\\,u_C(t),\n\\label{current}\n\\end{equation}\nwhere ${\\cal H}_H(t)$ is the (time-dependent) Hamiltonian in the Heisenberg picture at time $t$. The corresponding heat operator can be written down as\n\\begin{equation}\n\\label{eq-hatQ}\n{\\cal Q}_{L}(t)=\\int_0^t {\\cal I}_{L}(t')\\,dt' = {\\cal H}_L(0)-{\\cal H}^{H}_L(t),\n\\end{equation}\nwhere ${\\cal H}_L \\big[={\\cal H}_L(0)\\big]$ is the Schr\\\"odinger operator of the free left lead and \n\\begin{equation}\n{\\cal H}^{H}_L(t)= {\\cal U}(0,t)\\, {\\cal H}_L\\, {\\cal U}(t,0),\n\\end{equation}\nand ${\\cal U}(t,t')$ is the evolution operator corresponding to the full Hamiltonian ${\\cal H}(t)$ and satisfies the Schr\\\"odinger equation\n\\begin{equation}\ni\\hbar \\,{ \\partial {\\cal U}(t,t') \\over \\partial t} = {\\cal H}(t)\\, {\\cal U}(t,t').\n\\end{equation}\nThe formal solution of this equation is (assuming $t \\geq t'$)\n\\begin{equation}\n{\\cal U}(t,t')=T \\exp\\left\\{ - \\frac{i}{\\hbar} \\int_{t'}^t {\\cal H}(\\bar{t})\\, d\\bar{t} \\right\\},\n\\label{eq-unitary}\n\\end{equation}\nwhere $T$ is the time-order operator where time increases from right to left. Also ${\\cal U}^{\\dagger}(t,t')={\\cal U}(t',t)$. $Q$ of non-calligraphic font will be a classical variable. \n\nIn the following section we derive the CGF based on this definition of heat operator and using two-time measurement scheme.\n\n\\section{Definition of the generating function for heat operator}\nOur primary interest here is to calculate the moments or cumulants of the heat energy transferred in a given time interval $t_M$. Hence, it is advantageous to calculate the generating function instead of calculating moments directly. Since ${\\cal Q}_{L}$ is a quantum operator, there are subtleties as to how exactly the generating function should be defined. Naively we may use $\\langle e^{i \\xi {\\cal Q}_{L}} \\rangle $. But this definition fails the fundamental requirement of positive definiteness of the probability distribution.\n\nHere we will give two different definitions that are used to calculate the generating function for such problem. The first definition comes from the idea of two-time measurements and based on this concept the CGF can be written down as\n\\begin{equation}\n{\\cal Z}(\\xi)=\\langle e^{i \\xi {\\cal H}_L} \\, e^{-i \\xi {\\cal H}^{H}_L(t) } \\rangle'\n\\label{eq-Z-two-time}\n\\end{equation}\nwhich we will discuss in great detail in this section.\n\nThe second definition of the CGF is\n\\be\n{\\cal Z}_1(\\xi) = \\langle \\bar{T} e^{i\\xi {\\cal Q}_{L}\/2} T e^{i\\xi {\\cal Q}_{L}\/2} \\rangle,\n\\label{eq-Z1-Nazarov}\n\\ee\nwhere $\\bar{T}$ is the anti-time order operator. The time (or anti-time) order is meant to apply to the integrand when the\nexponential is expanded and ${\\cal Q}_{L}$ is expressed as integral over ${\\cal I}_{L}$ as in Eq.~(\\ref{eq-hatQ}). This definition is used by Nazarov et al.\\ \\cite{otherworks,otherworks1} mostly for the electronic transport case. In the last section we will show how this generating function can be derived starting from ${\\cal Z}(\\xi)$ under a particular approximation and will also give explicit expression for ${\\cal Z}_{1}(\\xi)$ in the long-time limit. \n\nIn the following we will discuss the idea of two-time measurement and derive the corresponding CGF ${\\cal Z}(\\xi)$.\n \n\\subsection{Two-time measurement}\nThe heat operator in Eq.~(\\ref{eq-hatQ}) depends on the left-lead Hamiltonian ${\\cal H}_{L}$ at time 0 and $t$. The concept of two-time measurement implies the measurement of a certain operator (in this case ${\\cal H}_{L})$ at two different times. Here the measurement is in the sense of quantum measurement\nof von Neumann \\cite{neumann}. \n\nLet us first assume that the full system is in a pure state $|\\Psi_0\\rangle$ at $t=0$.\nWe want to do measurement of the energy associated with the operator\n${\\cal H}_L$. According to quantum mechanics, the result of a measurement can\nonly be an eigenvalue of the (Schr\\\"odinger) operator ${\\cal H}_L$ and the wave function\ncollapses into an eigenstate of ${\\cal H}_L$. Let\n\\begin{equation}\n{\\cal H}_L | \\phi_a \\rangle = a | \\phi_a \\rangle,\\quad \\Pi_a = |\\phi_a\\rangle \\langle \\phi_a |,\n\\end{equation}\nwhere $\\Pi_a$ is the projector into the state $|\\phi_a\\rangle$ satisfying $\\Pi_a^2 = \\Pi_a$,\nand $\\sum_a \\Pi_a = 1$. We assume the eigenvalues are discrete (this is always so\nif the lattice system is finite). After the measurement at time $t=0$, the wave function\nis proportional to $\\Pi_a | \\Psi_0 \\rangle$ if the result of the measurement is the energy $a$ and the probability of such event happen is $\\langle \\Psi_0 | \\Pi_a^2 | \\Psi_0 \\rangle$. Let's propagate this state to time $t$ and do a second measurement of the lead\nenergy, finding that the result is $b$. The wave function now becomes proportional to $\\Pi_b \\, {\\cal U}(t,0) \\, \\Pi_a \\, | \\Psi_0 \\rangle$.\nThe joint probability of getting $a$ at time $0$ and $b$ at time $t$ is\nthe norm (inner product) of the above (unnormalized) state. \n\nIf the initial state is in a mixed state, we add up the initial probability classically, i.e., if \n\\begin{equation}\n\\rho(0) = \\sum_k w_k | \\Psi_0^k \\rangle\\langle \\Psi_0^k |,\\quad w_k > 0, \\quad \\sum_k w_k = 1,\n\\end{equation}\nthe joint probability distribution of two-time measurement output is\n\\begin{eqnarray}\nP(b,a) &=& \\sum_k w_k \\langle \\Psi_0^k | \\,\\Pi_a\\, {\\cal U}(0,t) \\,\n\\Pi_b \\,{\\cal U}(t,0) \\,\\Pi_a \\,| \\Psi_0^k \\rangle \\nonumber \\\\\n&=& {\\rm Tr} \\bigl[ \\Pi_a \\,\\rho(0) \\, \\Pi_a \\, {\\cal U}(0,t) \\, \\Pi_b \\, {\\cal U}(t,0)\\bigr].\n\\end{eqnarray}\nBy definition, we see that $P(b,a)$ is a proper probability in the sense that $P(b,a) \\ge 0$ and $\\sum_{a,b} P(b,a) = 1$.\nThen the generating function for $Q_L=a-b$ is defined as\n\\begin{eqnarray}\n{\\cal Z}(\\xi) &=& \\langle e^{i\\xi (a-b)} \\rangle = \\sum_{a,b} e^{i \\xi(a-b)} P(b,a) \\nonumber \\\\\n&=& \\sum_{a,b} e^{i \\xi (a-b)} {\\rm Tr} \\bigl[\\Pi_a \\, \\rho(0) \\, \\Pi_a \\, {\\cal U}(0,t) \\, \\Pi_b \\, {\\cal U}(t,0) \\bigr] \\nonumber \\\\\n& = & \\langle e^{i \\xi {\\cal H}_L} \\, e^{-i \\xi {\\cal H}^{H}_L(t) } \\rangle' \\nonumber \\\\\n\\label{eqZ2-def}\n& = & \\langle e^{i \\xi {\\cal H}_L\/2} \\, e^{-i \\xi {\\cal H}^{H}_L(t) } \\, e^{i \\xi {\\cal H}_L\/2 } \\rangle'.\n\\end{eqnarray}\nwhere we define \\cite{neumann}\n\\begin{equation}\n\\rho'(0) = \\sum_{a} \\Pi_a\\, \\rho(0) \\Pi_a.\n\\label{projected}\n\\end{equation}\nwhich we call as the {\\it projected} density matrix.\n\nIf the initial state at $t=0$ is a product state i.e., $\\rho(0)=\\rho(-\\infty)=\\rho_L \\otimes \\rho_C \\otimes \\rho_R $, where the left, center and right density\nmatrices are in equilibrium distributions corresponding to the \nrespective temperatures:\n$\\rho_\\alpha={e^{-\\beta_\\alpha {\\cal{H}}_\\alpha}}\/{{\\rm Tr} [e^{-\\beta_\\alpha\n {\\cal H}_\\alpha}]}$ for $\\alpha=L,C,R$ and $\\beta_{\\alpha}=1\/(k_{\\rm B}\nT_{\\alpha})$, then the projection operators $\\Pi_a$ do not play any role and $\\langle....\\rangle'={\\rm Tr}\\Bigl[\\rho(-\\infty)\\cdots \\Bigr]=\\langle....\\rangle$. \n\nHere we will derive the CGF for three different initial conditions:\n\\begin{itemize}\n\\item{Product initial state, i.e., $\\rho(-\\infty)$, which corresponds to sudden switch-on of the coupling between the leads and the center.}\n\\item{steady state as the initial state, i.e., $\\rho(0)$, which we can obtain, starting with the decoupled Hamiltonians at $t=-\\infty$, switch on the couplings between the center region and the leads, adiabatically upto time $t=0$.}\n\\item{{\\it projected} density matrix $\\rho'(0)$ considering $\\rho(0)$ as the steady state, i.e., taking the effects of measurements into account.}\n\\end{itemize} \n\nIn the following sections we will analytically show that the CGF's corresponding to different initial conditions reach the same steady state in the long-time limit and hence is independent of initial distributions. However for short time transient behavior depends significantly on initial conditions and also the measurements do play an important role. \n\n\\section{calculation for ${\\cal Z}(\\xi)$ for initial states $\\rho(0)$ and $\\rho'(0)$}\nIn this section we will give detail derivation for ${\\cal Z}(\\xi)$, using Feynman path-integral formalism, for two different initial density operators $\\rho(0)$ and $\\rho'(0)$. \n\n\\subsection{Removing the projection $\\Pi_a$ at $t=0$}\nThe projection by $\\Pi_a$ at $t=0$ Eq.~(\\ref{projected}) to the density matrix creates a problem\nfor formulation in path integrals. We can remove it following Ref.~\\onlinecite{Esposito-review-2009} by putting it into part of an evolution of ${\\cal H}_L$, just like the factor associated with the generating function variable $\\xi$, with a price we have to pay, introducing another integration variable $\\lambda$. The key observation is that we can represent the projector by the Dirac\n$\\delta$ function\n\\begin{eqnarray}\n\\Pi_a & \\propto & \\delta(a-{\\cal H}_L) \\nonumber \\\\\n&=& \\int_{-\\infty}^\\infty \\frac{d\\lambda}{2\\pi} \\,e^{-i\\lambda(a-{\\cal H}_L)}.\n\\label{eq-delta}\n\\end{eqnarray}\nFor this to make sense, we assume the spectrum of the energy of\n${\\cal H}_L$ is continuous, which is valid if we take the large size limit first.\nIdentifying $\\Pi_a$ as $\\delta(a-{\\cal H}_L)$ with a continuous variable\n$a$ introduces an constant proportional to the Dirac $\\delta(0)$ to $\\rho'(0)$, since\n$\\Pi_a$ is now normalized as $\\Pi_a \\Pi_b = \\delta(a-b) \\Pi_a$.\nHowever, this constant can be easily fixed by the condition\n${\\cal Z}(0)=1$. So using $\\Pi_a = \\delta(a -{\\cal H}_L)$ will not cause\ndifficulty. \n\nSubstituting the Fourier integral representation into $\\rho'$ we obtain\n\\begin{eqnarray}\n\\rho'(0) &\\propto&\\int da\\, \\Pi_a\\, \\rho(0) \\Pi_a \\\\\n&=& \\int \\frac{d\\lambda}{2\\pi} e^{i\\lambda {\\cal H}_L} \\rho(0) e^{-i\\lambda {\\cal H}_L}.\n\\end{eqnarray}\nUsing the symmetric form of ${\\cal Z}$, Eq.~(\\ref{eqZ2-def}), we have\n\\begin{eqnarray}\n{\\cal Z}(\\xi)&\\propto& \\int \\frac{d\\lambda}{2\\pi} {\\rm Tr} \\bigl\\{ \n\\rho(0) \\,{\\cal U}_{\\xi\/2-\\lambda}(0,t)\\, {\\cal U}_{-\\xi\/2-\\lambda}(t,0) \\bigr\\} \\nonumber \\\\\n&=& \\int \\frac{d\\lambda}{2\\pi}\\; {\\cal Z}(\\xi, \\lambda),\n\\label{eq-Uxilambda}\n\\end{eqnarray}\nwhere ${\\cal U}_{x}(t,t')$ is the modified evolution operator of an effective Hamiltonian given by\n\\begin{equation}\n{\\cal H}_{x}(t) = e^{i x {\\cal H}_L} {\\cal H}(t) e^{-i x {\\cal H}_L},\n\\end{equation}\nwhere $x$ is a real parameter which in this case is $\\xi\/2 -\\lambda$ and $-\\xi\/2 -\\lambda$. Finally ${\\cal U}_{x}(t,t')$ is given by ($t \\geq t'$)\n\\begin{eqnarray}\n{\\cal U}_{x}(t,t') &=& e^{i x {\\cal H}_L} {\\cal U}(t,t') e^{-i x {\\cal H}_L} \\nonumber \\\\\n&=& \\sum_{n=0}^\\infty \\left(-\\frac{i}{\\hbar} \\right)^n \\int_{t'}^t dt_1 \\int_{t'}^{t_1} dt_2 \\cdots \\int_{t'}^{t_{n-1}}dt_n\n\\nonumber \\\\ \n&& \\times e^{i x {\\cal H}_L} {\\cal H}(t_1) {\\cal H}(t_2) \\cdots {\\cal H}(t_n) e^{-i x {\\cal H}_L} \\nonumber \\\\\n\\label{eq-Ulambda}\n&=& T \\exp\\left\\{ - \\frac{i}{\\hbar} \\int_{t'}^t {\\cal H}_x(t') dt' \\right\\}.\n\\end{eqnarray}\nIt is important to note that substituting $\\lambda=0$ in ${\\cal Z}(\\xi,\\lambda)$ gives us the initial density matrix $\\rho(0)$. \n\nNow we will give an explicit expression of the modified Hamiltonian ${\\cal H}_x$ which helps us to calculate the CGF using path integral.\n\n\n\\subsection{The expression for ${\\cal H}_{x}$}\nThe modified Hamiltonian is the central quantity for calculating CGF. It is the Heisenberg evolution of the full Hamiltonian ${\\cal H}(t)$ (in Schr\\\"odinger picture) with respect to ${\\cal H}_L$. Since ${\\cal H}_L$ commutes with\nevery term $\\tilde{\\cal H}$ where ${\\cal H}(t)= {\\tilde {\\cal H}} + u_L^T V^{LC} u_C$, except the coupling term $u_L^T V^{LC} u_C$, we can write\n\\begin{eqnarray}\n{\\cal H}_{x}(t) &=& e^{i x {\\cal H}_L} {\\cal H}(t) e^{-i x {\\cal H}_L} \\nonumber \\\\\n& = & e^{i x {\\cal H}_L} \\bigl({\\tilde {\\cal H}} + u_L^T V^{LC} u_C \\bigr) e^{-i x {\\cal H}_L} \\nonumber \\\\\n& = & {\\cal H}(t) + \\bigl( u_L(\\hbar x) - u_L\\bigr)^T V^{LC} u_C,\n\\end{eqnarray}\nwhere $u_L(\\hbar x) = e^{i x {\\cal H}_L} u_L e^{-i x {\\cal H}_L}$ is the free left lead\nHeisenberg evolution to time $t = \\hbar x$. $u_L(\\hbar x)$ can be obtained\nexplicitly as\n\\begin{equation}\nu_L(\\hbar x) = \\cos(\\sqrt{K_L} \\hbar x) u_L + \\frac{1}{\\sqrt{K_L}} \\sin( \\sqrt{K_L} \\hbar x) p_L.\n\\end{equation} \nThe matrix $\\sqrt{K_L}$ is well-defined as the matrix $K_L$ is positive definite. $u_L$ and $p_L$ are the initial conditions at $t=0$.\nThe final expression for ${\\cal H}_x(t)$ is \n\\begin{equation}\n{\\cal H}_x(t) = {\\cal H}(t) + \\bigl[ u_L^T {\\cal C}(x) + p_L^T {\\cal S}(x) \\bigr]u_C,\n\\label{modified}\n\\end{equation}\nwhere \n\\begin{eqnarray}\n\\label{eq-C}\n{\\cal C}(x) &=& \\bigl(\\cos(\\hbar x \\sqrt{K_L}) -I\\bigr) V^{LC}, \\\\\n\\label{eq-S}\n{\\cal S}(x) &=& (1\/\\sqrt{K_L}) \\sin( \\hbar x \\sqrt{K_L}) V^{LC}.\n\\end{eqnarray}\nThe effective Hamiltonian now has two additional term with respect to the full ${\\cal H}(t)$. The term $u_L^T {\\cal C}(x)u_C $ is like the harmonic coupling term which modifies the coupling matrix $V^{LC}$. \n\n\nIn the following we calculate the two parameter generating function ${\\cal Z}(\\xi,\\lambda)$ using Eq.~(\\ref{eq-Uxilambda}).\n\n\\subsection{Expression for ${\\cal Z}(\\xi,\\lambda)$}\nThe expression for ${\\cal Z}(\\xi,\\lambda)$ can be written down on the contour as (see Fig.~\\ref{fig1})\n\\begin{equation}\n{\\cal Z}(\\xi,\\lambda) = {\\rm Tr} \\Bigl[ \\rho(0) T_c e^{-\\frac{i}{\\hbar} \\int_C {\\cal H}^{x}(\\tau) d\\tau} \\Bigr].\n\\label{eq-Z-contour} \n\\end{equation} \nwhere $T_c$ is the contour-ordered operator which orders operators according to their contour time argument, earlier contour time places an operator to the right. The contour function $x(\\tau)$ is defined as 0 whenever $t < 0$ or $t>t_M$, and when $0 < t < t_M$, i.e., within the measurement time interval, for upper branch of the contour $x^{+}(t) = -\\xi\/2 - \\lambda$, and for lower branch $x^{-}(t) = \\xi\/2 - \\lambda$. \n \nFor the moment, let us forget about the other lead and concentrate only on the left lead and center.\nThe effect of other lead simply modifies the self-energy of the\nleads additively, according to Feynman and Vernon \\cite{Feynman-Vernon-1963}. Using Feynman path integral technique we can write\n\\begin{equation}\n{\\cal Z}(\\xi,\\lambda)= \\int {\\cal D}[u_C] {\\cal D}[u_L] \\rho(-\\infty) e^{(i\/\\hbar) \n\\int_K d\\tau ( {\\cal L}_C + {\\cal L}_L + {\\cal L}_{LC} ) }.\n\\label{eq-Z-keldysh}\n\\end{equation}\nNote that in Eq.~(\\ref{eq-Z-contour}), the \ncontour $C$ is from 0 to $t_M$ and back, while that in Eq.~(\\ref{eq-Z-keldysh}) is on the\nKeldysh contour $K$, that is, from $-\\infty$ to $t_M$ and back to take into account\nof adiabatic switch on, replacing $\\rho(0)$ by $\\rho(-\\infty)$. Their relation\nis\n\\begin{equation}\n\\rho(0) = {\\cal U}(0, -\\infty) \\rho(-\\infty) {\\cal U}(-\\infty, 0).\n\\end{equation}\nWe can identify the Lagrangian's as \n\\begin{eqnarray}\n{\\cal L} &=& {\\cal L}_L + {\\cal L}_C + {\\cal L}_{LC}, \\nonumber \\\\\n{\\cal L}_L &=& \\frac{1}{2}\\dot{u}_L^2 - \\frac{1}{2} u_L^T K^L u_L, \\nonumber \\\\\n{\\cal L}_C &=& \\frac{1}{2} \\dot{u}_C^{2} +f^T u_C -\\, \\frac{1}{2} u_C^T \\bigl(K^C-{\\cal S}^T{\\cal S}\\bigr) u_C, \\nonumber \\\\\n{\\cal L}_{LC} &=& - \\dot{u}^T_L {\\cal S} u_C - u_L^T \\bigl(V^{LC} + {\\cal C}\\bigr) u_C.\n\\label{Lagrangian}\n\\end{eqnarray}\nFor notational simplicity, we have dropped the argument $\\tau$.\nThe vector or matrices $f$, ${\\cal C}$, and ${\\cal S}$ are parametrically dependent on\nthe contour time $\\tau$. They are zero except on the interval\n$0 < t < t_M$. $f$ is the same on the upper and lower branches,\nwhile ${\\cal C}$ and ${\\cal S}$ take different values depending on $x(\\tau)$.\n\nNow the lead part can be integrated out by performing Gaussian integral \\cite{Feynman-Vernon-1963}. Since the\ncoupling between the lead and the center is linear, it is plausible that the result will be\na quadratic form in the exponential, i.e., another Gaussian. To find exactly\nwhat it is, we convert the path integral back to the\ninteraction picture (with respect to ${\\cal H}_L$) operator form and evaluate the expression by standard perturbative expansion. The only difference is that now the coupling with the center involving both $u_L$ and $\\dot{u}_L$. The result for the influence\nfunctional is given by \\cite{Stockburger}\n\\begin{eqnarray}\nI_L[u^C(\\tau)] &\\equiv & \\int {\\cal D}[u_L] \\rho_L(-\\infty) \ne^{\\frac{i}{\\hbar} \\int d\\tau ({\\cal L}_L + {\\cal L}_{LC}) } \\nonumber \\\\\n&=& {\\rm Tr}\\Bigl[\\frac{e^{-\\beta_L H_L}}{Z_L} T_c e^{ -\\frac{i}{\\hbar} \\int d\\tau {\\cal V}_I(\\tau) } \\Bigr] \\nonumber \\\\\n&=& e^{-\\frac{i}{2\\hbar} \\int d\\tau \\int d\\tau' u_C^T(\\tau) \\Pi(\\tau, \\tau') u_C(\\tau') }.\\\n\\end{eqnarray}\n\n\\begin{figure}[t]\n\\includegraphics[width=0.5\\columnwidth]{fig1.eps\n\\caption{\\label{fig1}The complex-time contour in the Keldysh formalism. The path of the contour begins at time $t_{0}$, goes to time $t_M$, and then goes back to time $t=t_{0}$. $\\tau$ and $\\tau'$ are complex-time variables along the contour. $t_{0}=-\\infty$ and $0$ corresponds to Keldysh contour K and C, respectively.} \n\\end{figure}\nIn the influence functional, the contour function $u_C(\\tau)$ is not a \ndynamical variable but a parametric function. ${\\cal V}_I(\\tau)$ is the interaction picture operator with respect to the Hamiltonian ${\\cal H}_L$ and is given by\n\\begin{eqnarray}\n{\\cal V}_I(\\tau) &=& p_L^T {\\cal S} u_C + u_L^T(V^{LC} + {\\cal C}) u_C+\\frac{1}{2} u_C^T {\\cal S}^T {\\cal S} u_C \\nonumber \\\\\n\\label{eq-SSterm} \n&=&u_L^T\\bigl(\\tau + \\hbar x(\\tau)\\bigr)V^{LC}u_C + \\frac{1}{2} u_C^T {\\cal S}^T {\\cal S} u_C.\n\\end{eqnarray} \n\nThe important influence functional self-energy on contour is given by\n\\begin{eqnarray}\n\\Pi(\\tau,\\tau')= \\Sigma_L^A(\\tau,\\tau') &+& \\Sigma_L(\\tau,\\tau') + {\\cal S}^T{\\cal S}\\,\\delta(\\tau, \\tau'), \\\\\n\\Sigma_L^A(\\tau,\\tau') + \\Sigma_L(\\tau,\\tau') &= & V^{CL} g_L\\bigl(\\tau + \\hbar x(\\tau), \\tau'+\\hbar x(\\tau') \\bigr) V^{LC} \\nonumber \\\\\n\\label{eq-SAL}\n&=& \\Sigma_L\\bigl(\\tau + \\hbar x(\\tau), \\tau'+\\hbar x(\\tau') \\bigr),\n\\label{eq-shifted-self-energy}\n\\end{eqnarray}\nwhere we obtain a shifted self-energy $\\Sigma_L\\big(\\tau+\\hbar x(\\tau),\\tau'+\\hbar x(\\tau')\\big)$ which is the usual self-energy of the lead in contour time with arguments\nshifted by $\\hbar x(\\tau)$ and $\\hbar x(\\tau')$. We define the self-energy $\\Sigma^A_L$ as the difference between the shifted self-energy and the usual one $\\Sigma_L(\\tau,\\tau')$. $\\Sigma^{A}_L$ turns out to be a central quantity for this problem as we will show that, the CGF ${\\cal Z}$ can be concisely expressed in terms of the center Green's function $G_{0}$ and $\\Sigma^{A}_L$.\n\nSubstituting the explicit expression for the influence functionals of both the left and right leads to the path integral expression given in Eq.~(\\ref{eq-Z-keldysh}), we have\n\\begin{eqnarray}\n{\\cal Z}(\\xi,\\lambda)&=& \\int {\\cal D}[u_C] \\rho_C(-\\infty) e^{(i\/\\hbar) \n\\int d\\tau {\\cal L}_C } I_L[u_C] I_R[u_C]\\nonumber \\\\\n&=& \\int {\\cal D}[u_C] \\rho_C(-\\infty) e^{\\frac{i}{\\hbar}S_{\\rm eff}},\n\\end{eqnarray}\nwhere the effective action is given by\n\\begin{eqnarray}\nS_{\\rm eff} &=& \\int d\\tau \\Bigl[ \n\\frac{1}{2} \\dot{u}_C^2 - \\frac{1}{2} u_C^T K^C u_C \n+ f^T u_C \\Bigr] \\\\\n& \\!\\!-\\!& \\frac{1}{2}\\! \\int d\\tau \\!\\int d\\tau'\\! u_C^T(\\tau) \\bigl(\n\\Sigma(\\tau, \\tau') + \\Sigma_L^A(\\tau, \\tau') \\bigr) u_C(\\tau'), \\nonumber\n\\end{eqnarray}\nwhere $\\Sigma = \\Sigma_L +\\Sigma_R$, taking into account the effect of both the leads. The ${\\cal S}^T{\\cal S}$ term in $I_{L}[u_C]$ cancels exactly with\nthe one in ${\\cal L}_C$. We can perform an integration by part on the $\\dot{u}^2$ term, assuming that the surface term\ndoes not matter (since it is at $t=-\\infty$), we can write the expression in a standard quadratic form\n\\begin{eqnarray}\nS_{\\rm eff} &=& \\frac{1}{2} \\int d\\tau \\int d\\tau' u_C^T(\\tau) D(\\tau, \\tau') u_C(\\tau') \\nonumber \\\\\n&& \\qquad + \\int f^T(\\tau) u_C(\\tau) d\\tau. \n\\end{eqnarray}\n$D(\\tau,\\tau')$ is the differential operator and is given by\n\\begin{eqnarray}\nD(\\tau, \\tau') &=& -I \\frac{\\partial^2}{\\partial \\tau^2} \\delta(\\tau, \\tau')\n- K^C \\delta(\\tau, \\tau') \\nonumber \\\\\n&& -\\Sigma(\\tau, \\tau') - \\Sigma_L^A(\\tau, \\tau') \\nonumber \\\\\n&=& D_0(\\tau,\\tau') - \\Sigma^A_L(\\tau,\\tau').\n\\end{eqnarray}\nThe above equation defines the Dyson equation on Keldysh contour. The generating function is obtained by doing another Gaussian integration and is of the following form\n\\begin{equation}\n{\\cal Z} \\propto {\\rm det}(D)^{-1\/2} e^{-\\frac{i}{2\\hbar} f^T D^{-1} f}.\n\\label{eq-Z-det}\n\\end{equation}\n(The meaning of the determinant will be explained in Appendix~C). \nWe define the Green's function $G$ and $G_{0}$ by $DG = 1$, and $D_0 G_0 = 1$, or more precisely\n\\begin{equation}\n\\int D(\\tau, \\tau'') G(\\tau'', \\tau') d\\tau'' = I \\delta(\\tau, \\tau'),\n\\end{equation}\nand similarly for $G_0$. $G$ can be written in terms of $G_{0}$ in the following Dyson equation form\n\\begin{eqnarray}\n\\label{eq-Dyson-full}\nG(\\tau, \\tau') &=& G_0(\\tau,\\tau') \\\\\n&&\\> + \\int \\int d\\tau_1d \\tau_2 \nG_0(\\tau, \\tau_1) \\Sigma_L^A(\\tau_1, \\tau_2) G(\\tau_2, \\tau'). \\nonumber\n\\end{eqnarray}\n\nWe view the differential operator\n(integral operator) $D$ and $D^{-1}$ as matrices that are\nindexed by space $j$ and contour time $\\tau$. $f$ is a column vector. The\nexponential factor term can also be written as a trace, \n$f^T D^{-1} f = {\\rm Tr}_{(j,\\tau)} ( G f f^T)$.\nWe can fix the proportionality constant by noting that\n${\\cal Z}(\\xi=0, \\lambda=0) =1$. Since when $\\xi=0$, $\\lambda=0$, we have\n$x=0$ and thus $\\Sigma_L^A(\\tau,\\tau') = \\Sigma_L(\\tau+x, \\tau'+x') - \\Sigma_L(\\tau, \\tau') = 0$, so $D=D_0$. The properly normalized\nCGF is \n\\begin{equation}\n\\label{eq-Zxilam}\n{\\cal Z}(\\xi, \\lambda) = {\\rm det}\\bigl( D_0^{-1}D)^{-1\/2}e^{-\\frac{i}{2\\hbar} f^T D^{-1} f}.\n\\end{equation}\nWe don't need to do anything for the exponential factor because of the following reason.\nWe note\n\\begin{eqnarray}\nf^T G_0 f &=& \\int \\int d\\tau d\\tau' f(\\tau)^T G_0(\\tau, \\tau') f(\\tau') \\\\\n&=& \\sum_{\\sigma,\\sigma'} \\int \\int \\sigma dt\\, \\sigma' dt' f(t)^T G_0^{\\sigma\\sigma'}(t,t') f(t').\\nonumber\n\\end{eqnarray}\nSince the driven force $f$ does not depend on the branch indices, i.e., \n$f^{+}(t) = f^{-}(t)$, we can take the summation inside and obtain\n\\begin{equation}\n\\sum_{\\sigma\\sigma'} \\sigma\\sigma' G^{\\sigma\\sigma'} = G_0^t + G_0^{\\bar t} - G_0^> - G_0^< = 0.\n\\end{equation}\n\nFinally making use of the formulas for operators or matrices \n${\\rm det}(M) = e^{{\\rm Tr} \\ln M}$, and \n$\\ln (1 - y) = -\\sum_{k=1}^\\infty \\frac{y^k}{k}$ we can write the CGF in terms of $\\Sigma^{A}_L$ for the {\\it projected} initial condition case as,\n\\begin{eqnarray}\n\\ln {\\cal Z}(\\xi)&=& \\lim_{\\lambda \\to \\infty}\\ln {\\cal Z}(\\xi,\\lambda) \\nonumber \\\\\n &=& \\lim_{\\lambda \\to \\infty} \\left\\{-\\frac{1}{2} {\\rm Tr}_{j,\\tau} \\ln ( 1 - G_0 \\Sigma_L^A) - \\frac{i}{2\\hbar} {\\rm Tr}_{j,\\tau}( G f f^T) \\right\\} \\nonumber \\\\\n&=& \\lim_{\\lambda \\to \\infty} \\sum_{n=1}^\\infty \\frac{1}{2n}\n{\\rm Tr}_{(j,\\tau)} \\Bigl[(G_0 \\Sigma_L^A)^n \\Bigr] \\nonumber - \\frac{i}{2\\hbar} f^T G f \\nonumber \\\\\n&=& \\frac{1}{2} {\\rm Tr}_{(j,\\tau)}(G_0 \\Sigma_L^A) + \\frac{1}{4} {\\rm Tr}_{(j,\\tau)}(G_0 \\Sigma_L^A G_0 \\Sigma_L^A) + \\cdots \\nonumber \\\\\n&& -\\frac{i}{2\\hbar} f^T G_0 \\Sigma^A G_0 f + \\cdots.\n\\label{projected_state}\n\\end{eqnarray}\nThis expression for CGF is valid for any transient time $t_M$ present in the self-energy\n$\\Sigma^A_{L}$ and is the starting point for the calculation in transient regime. The notation ${\\rm Tr}_{(j,\\tau)}$ means trace both in\nspace index $j$ and contour time $\\tau$ (see Appendix~C). In order to obtain ${\\cal Z}(\\xi)$ from ${\\cal Z}(\\xi,\\lambda)$ we have to take the limit $\\lambda \\rightarrow \\infty$ because ${\\cal Z}(\\xi, \\lambda)$ approaches a constant as $| \\lambda| \\to \\infty$ and hence\nthe value of the integral is dominated by the value at infinity. Since $\\Sigma^{A}_{L}(\\tau,\\tau')=0$ for $\\xi=0$ we have the correct normalization ${\\cal Z}(0)=1$.\n\nSimilarly, for the steady state initial condition $\\rho(0)$ the CGF is given by \n\\begin{equation}\n\\ln {\\cal Z}(\\xi)=\\lim_{\\lambda \\rightarrow 0} \\ln {\\cal Z}(\\xi,\\lambda)\n\\end{equation}\nThe difference in this two cases is in the matrix $\\Sigma^{A}_L$. \n\nSimilar relations also exist if we want to calculate the CGF for right lead heat operator ${\\cal Q}_R$. In this case one has to do two-time measurement on the right lead corresponding to the Hamiltonian ${\\cal H}_R$. The final formula for the CGF remains the same except $\\Sigma^{A}_L$ should be replaced by $\\Sigma^{A}_R$.\n \nNow in order to calculate the cumulants $\\langle \\langle Q_{\\alpha}^{n} \\rangle \\rangle$ with $\\alpha=L,R$ we need to go to the real time using Langreth's rule \\cite{rammer86}. In this case, it is more convenient to work with a Keldysh rotation (see Appendix~C) for the contour ordered functions while keeping\n${\\rm Tr}(ABC \\cdots D)$ invariant. The effect of the Keldysh rotation is to change any given matrix ${\\cal D}^{\\sigma\\sigma'}(t,t')$,\nwith $\\sigma, \\sigma'=\\pm$ for branch indices, to,\n\\begin{eqnarray}\n\\label{keldysh-rotation}\n\\breve{{\\cal D}} &=& \n\\left( \\begin{array}{cc}\n {\\cal D}^r & {\\cal D}^K \\\\\n {\\cal D}^{\\bar K} & {\\cal D}^a \n\\end{array} \\right) \\\\\n &=&\n\\frac{1}{2} \\left( \\begin{array}{cc}\n {\\cal D}^t - {\\cal D}^< + {\\cal D}^> - {\\cal D}^{\\bar t}, & {\\cal D}^t + {\\cal D}^{\\bar t} + {\\cal D}^< + {\\cal D}^> \\\\\n {\\cal D}^t + {\\cal D}^{\\bar t} - {\\cal D}^< - {\\cal D}^>, & {\\cal D}^< - {\\cal D}^{\\bar t} + {\\cal D}^{t} - {\\cal D}^{>} \n \\end{array} \\right). \\nonumber\n\\end{eqnarray}\nIn this case we define the quantities \n${\\cal D}^r$, ${\\cal D}^a$, ${\\cal D}^K$, and ${\\cal D}^{\\bar{K}}$ as above. In particular,\n${\\cal D}^{K} \\neq {\\cal D}^< + {\\cal D}^>$, as one usually might thought it is. \n\nUsing the above definition for the center Green's function $G_0$ we get \n\\begin{equation}\n\\breve{G_0} = \n\\left( \\begin{array}{cc}\n G_0^r & G_0^K \\\\\n 0 & G_0^a \n\\end{array} \\right). \n\\end{equation}\nThe $G_0^{\\bar K}$ component is 0 due to the standard relation\namong Green's functions. But the $\\bar K$ components are not zero for\n$\\Sigma_L^A$ and $G_0$, as we will compute later. \n\nIt is useful computationally to work in Fourier space even if \nthere is no time translational invariance. We define the two-frequency\nFourier transform by\n\\begin{equation}\n\\breve{A}[\\omega, \\omega'] =\n\\int_{-\\infty}^{+\\infty}\\!\\!\\! dt \\int_{-\\infty}^{+\\infty}\\!\\!\\! dt' \n\\breve{A}(t,t') e^{i(\\omega t + \\omega' t')}.\n\\label{two-time-FT}\n\\end{equation}\nSince $\\breve{G}_0$ is time-translationally invariant then,\n\\begin{equation}\n\\breve{G}_0[\\omega, \\omega'] = 2\\pi \\delta(\\omega+\\omega') \\breve{G}_0[\\omega],\n\\label{G0}\n\\end{equation}\nis ``diagonal'', where the single argument Fourier transform is similarly defined,\n\\begin{equation}\n\\label{eq-Fourier}\nA[\\omega] = \\int_{-\\infty}^{+\\infty} A(t,0) e^{i\\omega t} dt.\n\\end{equation}\n(The expressions for different components of $\\breve{G}_{0}[\\omega]$ and $\\breve{\\Sigma}[\\omega]$ are given in Appendix A). Using $\\breve{G}_0[\\omega]$, we can save one integration due to the $\\delta$ function, and have\n\\begin{eqnarray}\n\\label{eq-Zsteady}\n\\ln {\\cal Z}(\\xi) &=& - \\frac{1}{2} {\\rm Tr}_{j,\\sigma,\\omega}\n\\ln\\Bigl[ 1 - \\breve{G}_0[\\omega] \\breve{\\Sigma_L^A}[\\omega, \\omega'] \\Bigr]\n\\nonumber \\\\\n&& - \\frac{i}{2 \\hbar} {\\rm Tr}_ {j,\\sigma,\\omega} \\Bigl[\\breve{G}[\\omega,\\omega'] \\, \\breve{{\\cal F}}[\\omega', \\omega]\\Bigr],\n\\end{eqnarray}\nwhere $ \\breve{G}_0[\\omega] \\breve{\\Sigma_L^A}[\\omega, \\omega']$ is viewed\n\nas a matrix indexed by $\\omega$ and $\\omega'$. The trace is performed on\nthe frequency as well as the usual space and branch components. (The meaning of trace in frequency domain is discussed in Appendix~C). $\\breve{{\\cal F}}$ is given by\n\\begin{equation}\n\\label{force-matrix}\n\\breve{{\\cal F}}[\\omega,\\omega'] = \n\\left( \\begin{array}{cc}\n 0 & 2 f[\\omega]f[\\omega']^T \\\\\n 0 & 0 \n\\end{array} \\right). \n\\end{equation}\nIn the next section we derive the CGF for the product initial condition using Feynman diagrammatic technique. Because of the special form of the initial density matrix the calculation for the CGF simplifies greatly in this case. \n \n\\section{Product state $\\rho(-\\infty)$ as initial state}\nIn this section, we derive the CGF starting with a product initial state, i.e., the density matrix at time $t=0$ is given\nby $\\rho(-\\infty)=\\rho_C \\otimes \\rho_L \\otimes \\rho_R$. Since this density matrix commutes with the projection operator $\\Pi_a$, the initial projection does not play any role in this case. Working in the interaction picture with respect to the decoupled Hamiltonian ${\\cal H}(-\\infty)= \\sum_i {\\cal H}_i$, the \ninteraction part of the Hamiltonian on the contour $C=[0,t_M]$ is \n\\begin{eqnarray}\n{\\cal V}^{x}_I(\\tau) &=& -f^T(\\tau) u_C(\\tau) + u_R(\\tau) V^{RC} u_C(\\tau) \\nonumber \\\\\n&&+\\, u_L\\bigl(\\tau+\\hbar x(\\tau)\\bigr)V^{LC}u_C(\\tau).\n\\end{eqnarray}\nIn the last term for $u_L$, the argument is shifted by $\\hbar x$ where \n$x^+(t) = - \\xi\/2$, $x^{-}(t) = \\xi\/2$ for $0< t < t_M$.\n\nThe density matrix remains unaffected by the transformation to the interaction picture, because it commutes with ${\\cal H}(-\\infty)$. The CGF can now be written as \n\\begin{equation}\n{\\cal Z}(\\xi) = {\\rm Tr}\\Bigl[ \\rho(-\\infty) T_c \\,e^{-\\frac{i}{\\hbar} \\int_C {\\cal V}^{x}_I(\\tau)\\, d\\tau}\\Bigr].\n\\end{equation}\nExpanding the exponential, we generate various terms of product of\n$u_\\alpha$. These terms can be decomposed in pairs according to\nWick's theorem \\cite{rammer86}. Since the system is decoupled, each type of $u$ comes\nin an even number of times for a non-vanishing contributions because \n$\\langle u_C\\rangle = 0$, $\\langle u_C u_L\\rangle = 0$ and we know\n\\begin{equation}\n-\\frac{i}{\\hbar} \\langle T_C u_{\\alpha}(\\tau) u_{\\alpha'}(\\tau')^T \\rangle_{\\rho(-\\infty)} = \\delta_{\\alpha,\\alpha'} g_\\alpha(\\tau, \\tau').\n\\end{equation}\nWe use Feynman diagrammatic technique to sum the series. since ${\\cal V}_I$ contains only two-point couplings,\nthe graphs are all ring type. The combinatorial factors can be worked\nout as $1\/(2n)$ for a ring containing $n$ vertices.\nWe use a very general theorem which says $\\ln {\\cal Z}$ contains only \nconnected graphs, and the disconnected graphs cancel exactly when we\ntake the logarithm. The final result can be expressed as\n\\begin{equation}\n\\ln {\\cal Z}(\\xi) = - \\frac{1}{2} {\\rm Tr}_{j,\\tau} \\ln \\Big[ 1 - g_C \\Sigma^{\\rm tot} \\Big] \n-\\frac{i}{2\\hbar} f^T G f, \n\\end{equation}\nwhere\n\\begin{equation}\n\\Sigma^{\\rm tot} = \\Sigma_L(\\tau+x,\\tau'+x') + \\Sigma_R(\\tau,\\tau') = \\Sigma + \\Sigma_L^A,\n\\end{equation} \nand $\\Sigma$ is the total self-energy due to both the leads. $G(\\tau,\\tau')$ obeys the following Dyson's equation\n\\begin{eqnarray}\nG(\\tau,\\tau') &=& g_C(\\tau,\\tau') \\\\ \n&&\\> + \\int \\int d\\tau_1 d\\tau_2 g_C(\\tau,\\tau_1) \\Sigma^{\\rm tot}(\\tau_1,\\tau_2) G(\\tau_2,\\tau'). \\nonumber \n\\end{eqnarray} \n\n\nThe above expression for CGF can be written down more explicitly, by\ngetting rid of the vacuum diagrams. Let us define a new type of Dyson's equation\n\\begin{eqnarray}\n\\label{eq-Dyson-product}\nG_0(\\tau, \\tau') &=& g_C(\\tau,\\tau') \\\\\n&&\\> + \\int \\!\\int d\\tau_1d \\tau_2\\, \ng_C(\\tau, \\tau_1) \\Sigma(\\tau_1, \\tau_2) G_0(\\tau_2, \\tau'), \\nonumber\n\\end{eqnarray}\nwhere $g_C$ is the contour ordered Green's function of the isolated center. (The Green's functions for an isolated single \nharmonic oscillator is given is appendix A).\nThis expression looks formally the same as before except that $G_0$\nsatisfies a Dyson equation defined on the contour from 0 to $t_M$ and\nback, while $G$ is defined on the Keldysh contour from $-\\infty$ to\n$t_M$. Using this definition we can write\n\\begin{eqnarray}\n1 - g_C \\Sigma^{\\rm tot} &=& 1 - g_C (\\Sigma + \\Sigma_L^A) \\nonumber \\\\\n&=& (1 - g_C \\Sigma)\\, (1 - G_0 \\Sigma_L^A).\n\\end{eqnarray}\nThe two factors above are in matrix (and contour time) multiplication.\nUsing the relation between trace and determinant, \n$\\ln \\det(M) = {\\rm Tr} \\ln M$, and the fact, $\\det(AB) = \\det(A) \\det(B)$,\nwe find that the two terms give two factors for ${\\cal Z}$, and the factor\ndue to $1 - g_C \\Sigma$ is exactly 1. We have then\n\\begin{equation}\n\\label{eq-Zprod}\n\\ln {\\cal Z}(\\xi) = - \\frac{1}{2} {\\rm Tr}_{j,\\tau} \\ln \\Big[ 1 - G_0 \\Sigma_L^{A} \\Big]\n-\\frac{i}{2\\hbar} f^T G f,\n\\end{equation}\nwhere the $G(\\tau,\\tau')$ can now be expressed in terms of $G_{0}(\\tau,\\tau')$ as\n\\begin{equation}\nG^{-1} = G_0^{-1} - \\Sigma_L^A.\n\\end{equation}\nwhich is similar in form to Eq.~(\\ref{eq-Dyson-full}).\n\nThe expression for $\\ln {\\cal Z}(\\xi)$ is consistent with the earlier result,\nEq.~(\\ref{projected_state}), in the long-time limit. So we can conclude that the long-time limit is the same independent of the initial distributions. \n\nTo compute the cumulants $\\langle \\langle Q^n \\rangle \\rangle$, we\nneed to take derivative with respect to $\\xi$ $n$-times to $\\ln {\\cal Z}$.\nNote that the shifted self-energy for $0 < t < t_M$ is (for all three initial conditions)\n\\begin{eqnarray}\n\\Sigma_A^t (t,t') &=& 0, \\nonumber \\\\\n\\Sigma_A^{\\bar t}(t,t') &=& 0, \\nonumber \\\\\n\\Sigma_A^{<}(t,t') &=& \\Sigma_L^{<}(t-t'-\\hbar \\xi) - \\Sigma_L^{<}(t-t'), \\nonumber \\\\\n\\Sigma_A^{>}(t,t') &=& \\Sigma_L^{>}(t-t'+\\hbar \\xi) - \\Sigma_L^{>}(t-t').\n\\label{shifted-product}\n\\end{eqnarray}\nWe note $\\Sigma_L^A(\\xi=0)=0$. The derivatives at $\\xi=0$ can be obtained as\n\\begin{eqnarray}\n{\\partial^n \\Sigma_A^{<} \\over\n\\partial \\xi^n}\\Big|_{\\xi=0} &=& (-\\hbar)^n \\Sigma_L^{<,(n)}(t-t'), \\nonumber \\\\\n{\\partial^n \\Sigma_A^{>} \\over\n\\partial \\xi^n}\\Big|_{\\xi=0} &=& \\hbar^n \\Sigma_L^{>,(n)}(t-t'),\n\\end{eqnarray}\nwhere the superscript $(n)$ means derivatives with respect to the argument of\nthe function $n$ times. In the following sections we first show the explicit expression of the CGF in the long-time limit and then discuss the steady state fluctuation theorem. \n\n\n\\section{Long-time limit and Steady state fluctuation theorem}\nFor the long-time limit calculation we can use either Eq.~(\\ref{eq-Zsteady}) or Eq.~(\\ref{eq-Zprod}). \nFor convenience of taking the large time limit, i.e., $t_M$ large, we prefer\nto set interval to $(-t_M\/2, t_M\/2)$. In this way, when $t_M \\to \\infty$,\nthe interval becomes the full domain and Fourier transforms to all the\nGreen's functions and self-energy can be performed (where the translational\ninvariance is restored). Applying the convolution theorem to the\ntrace formula in Eq.~(\\ref{eq-Zprod}), we find that there is one more time integral left\nwith integrand independent of $t$. This last one can be set from $-t_M\/2$ to $t_M\/2$, obtaining an overall factor\nof $t_M$ and we have\n\\begin{equation}\n{\\rm Tr}_{(j,\\tau)}(AB \\cdots D) = t_M \n\\int \\frac{d\\omega}{2\\pi}{\\rm Tr} \\Bigl[\\breve {A}(\\omega)\\breve{B}(\\omega) \\cdots \\breve{D}(\\omega)\\Bigr].\n\\end{equation}\n \nIn the long-time limit, the shift given to the argument in $\\Sigma_L^A$ depends on the branches, and the \ntwo arguments $(t,t')$ becomes $t-t'$ and we have \n\\begin{eqnarray}\n\\Sigma_A^{\\sigma\\sigma'}(t,t') &=& \\Sigma^{\\sigma\\sigma'}_L(t\\! +\\!x^\\sigma\\!-\\! t'\\!-\\!x^{\\sigma'}) - \\Sigma^{\\sigma\\sigma'}_L(t\\!-\\!t'), \\\\\n\\Sigma_A^t &=& \\Sigma_A^{\\bar t} = 0,\\nonumber \\\\\n\\Sigma_A^<(t) &=& \\Sigma_L^{<}(t-\\hbar \\xi) -\\Sigma_L^{<}(t),\\\\\n \\Sigma_A^>(t) &=& \\Sigma_L^{>}(t+\\hbar \\xi)-\\Sigma_L^{>}(t). \\nonumber \n\\label{steady}\n\\end{eqnarray}\nFourier transforming the greater and lesser self-energy, we get\n\\begin{eqnarray}\n\\label{eq-a}\n\\Sigma_A^{>}[\\omega] = \\Sigma^{>}_L[\\omega] \\bigl(e^{-i\\hbar \\omega \\xi} - 1 \\bigr) = a, \\\\\n\\label{eq-b}\n\\Sigma_A^{<}[\\omega] = \\Sigma^{<}_L[\\omega] \\bigl(e^{i\\hbar \\omega \\xi} - 1 \\bigr) = b.\n\\end{eqnarray}\nWe note that $\\Sigma_L^A$ is supposed to depend on both $\\xi$ and $\\lambda$.\nHowever in the long-time limit, the $\\lambda$ dependence drops out which makes the steady state result independent of the initial distribution. \n\nFinally, we can express the generating function as\n\\begin{eqnarray}\n\\ln {\\cal Z}(\\xi) &= & - t_M \\int \\frac{d\\omega}{4\\pi} {\\rm Tr} \\ln \\Bigl[1 - \\breve{G}_0 [\\omega]\\breve{\\Sigma}_L^A [\\omega]\\Bigr] \\nonumber \\\\\n&& \\quad - \\frac{i}{\\hbar} \\int \\frac{d\\omega}{4\\pi}\n{\\rm Tr} \\Bigl[ \\breve{G}[\\omega] \\breve{\\cal F}[\\omega,-\\omega]\\Bigr],\n\\label{eq-lnZxi-1}\n\\end{eqnarray}\nwhere $\\breve{G}[\\omega]$ is obtained by solving the Dyson equation\nin frequency domain and in the long-time obeys time-translational invariance. So the full CGF can be written as the sum of contributions due to driving force and due to temperature difference between the leads, i.e.,\n\\begin{equation}\n\\ln {\\cal Z}(\\xi) = \\ln {\\cal Z}^{s}(\\xi) + \\ln {\\cal Z}^{d}(\\xi). \n\\end{equation}\nIn the following and subsequent sections we discuss about ${{\\cal Z}^{s}(\\xi)}$ and we will return to ${{\\cal Z}^{d}(\\xi)}$ in Sec. XI. \n\nIn order to obtain the explicit expression for $\\ln {\\cal Z}^{s}(\\xi)$ we need to compute the matrix product\n\\begin{eqnarray}\n\\breve{G}_0[\\omega] \\breve{\\Sigma}_L^A[\\omega] &=& \n\\frac{1}{2} \\left( \\begin{array}{cc}\nG_0^r & G_0^{K} \\\\\n0 & G_0^a \n\\end{array}\n\\right)\n\\left( \\begin{array}{cc}\na-b & a+b \\\\\n-(a+b) & b-a \n\\end{array}\n\\right).\n\\end{eqnarray}\nTo simplify the expression, we rewrite the term ${\\rm Tr} \\ln (1-M)$ as\na determinant and use the formula (assuming A to be an invertible matrix) \n\\begin{equation}\n{\\rm det} \\left( \\!\\!\\begin{array}{cc}\nA & B \\\\\nC & D \\end{array}\\!\\!\n\\right) = {\\rm det}(A) \\det(D - C A^{-1}B)\n\\end{equation}\nto reduce the dimensions of the determinant matrix by half. \nThe steady state solution for ${{\\cal Z}^{s}(\\xi)}$ is given by\n\\begin{eqnarray}\n\\ln {\\cal Z}^{s}(\\xi)&=&-t_M \\int \\frac{d\\omega}{4\\pi} \\,\\ln \\det \\Bigl\\{ I - G_{0}^r \\Gamma_L \nG_{0}^a \\Gamma_R \\Big[ (e^{i\\xi \\hbar \\omega}\\! -\\! 1) f_L \\nonumber \\\\ \n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+ ( e^{-i\\xi\\hbar \\omega} \\!-\\! 1) f_R + (\ne^{i\\xi \\hbar \\omega} \\!+\\! e^{-i\\xi\\hbar\\omega} \\!-\\!2 ) f_L f_R \\Big]\\Bigr\\}.\\qquad \n\\label{eq-steady1}\n\\end{eqnarray}\nwith $f_{\\alpha}=1\/(e^{\\beta_{\\alpha} \\hbar \\omega} - 1)$, $\\alpha=L,R,$ the Bose-Einstein distribution function and $\\Gamma_{\\alpha}[\\omega]=i\\big(\\Sigma_{\\alpha}^{r}[\\omega]-\\Sigma_{\\alpha}^{a}[\\omega]\\big)$. If we consider the full system as a one-dimensional linear chain, then because of the special form of $\\Gamma_{\\alpha}$ matrices (only one entry of the $\\Gamma$ matrices are non-zero) it can be easily shown that\n\\begin{equation}\n\\det [I-\\bigl(G_{0}^r \\Gamma_L G_{0}^a \\Gamma_R \\bigr) \\Xi(\\xi)]=1-{\\cal T}[\\omega]\\Xi(\\xi)\n\\end{equation}\nwhere $\\Xi(\\xi)$ is any arbitrary function of $\\xi$ and ${\\cal T}[\\omega]={\\rm Tr}(G_{0}^r \\Gamma_L G_{0}^a \\Gamma_R)$ is known as the transmission function and is given by the Caroli formula \\cite{Caroli,WangJS-europhysJb-2008}. The generating function ${{\\cal Z}^{s}(\\xi)}$ in the steady state obeys the following symmetry \n\\begin{equation}\n{{\\cal Z}^{s}}(\\xi)={{\\cal Z}^{s}}\\big(-\\xi + i\\,{\\cal A}\\big),\n\\end{equation}\nwhere ${\\cal A}=\\beta_R-\\beta_L$ is known as thermodynamic affinity. This relation is also known as Gallavotti-Cohen (GC) symmetry \\cite{noneq-fluct}. The immediate consequence of this symmetry is that the probability distribution for heat transferred $Q_{L}$ which is given by the Fourier transform of the CGF, i.e., $P(Q_L)=\\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} d\\xi \\, {\\cal Z}(\\xi) \\, e^{-i \\xi Q_L}$ obeys the following relation in the large $t_M$ limit,\n\\begin{equation}\nP_{t_M}(Q_L)=e^{{\\cal A}Q_L}\\,P_{t_M}(-Q_L).\n\\end{equation} \nThis relation is known as the steady state fluctuation theorem and was first derived by Saito and Dhar \\cite{Saito-Dhar-2007} in the phononic case. This theorem quantifies the ratio of positive and negative heat flux and second law violation. \n\nThe cumulants $\\langle \\langle {Q}^{n} \\rangle \\rangle $ can be obtained by taking derivative of $\\ln {\\cal Z}^{s}(\\xi)$ with respect to $i\\xi$ and setting $\\xi=0$. The first cumulant is given by \n\\begin{equation}\n\\frac{\\langle \\langle Q \\rangle \\rangle}{t_M} = \\int_{-\\infty}^{\\infty} \\frac{d\\omega}{4 \\pi} \\, \\hbar\\, \\omega \\,{\\cal T}(\\omega)(f_L-f_R),\n\\end{equation}\nwhich is known as the Landauer-like formula in thermal transport. Similarly the second cumulant $\\langle \\langle {Q}^{2} \\rangle \\rangle= \\langle {Q}^{2} \\rangle - \\langle {Q}\\rangle ^{2}$, which describes the fluctuation of the heat transferred, can be written as \\cite{Saito-Dhar-2007, Huag, Buttiker},\n\\begin{eqnarray}\n\\frac{\\langle \\langle Q^{2} \\rangle \\rangle}{t_M} &=&\\int_{-\\infty}^{\\infty} \\frac{d\\omega}{4 \\pi} \\, (\\hbar \\omega)^{2} \\Bigl \\{{\\cal T}^{2}(\\omega)\\, (f_L-f_R)^{2} \\nonumber \\\\\n&&+ {\\cal T}(\\omega) \\, (f_L+f_R+2\\,f_L f_R ) \\Bigr\\}.\n\\end{eqnarray}\nOur formalism can be easily generalized for multiple heat baths and for $N$ leads connected with the center $C$, we can generalize the above formula as\n\\begin{eqnarray}\n&&\\ln {\\cal Z}^{s}(\\xi)=\\!-\\!t_M \\!\\int \\frac{d\\omega}{4\\pi} \\,\\ln \\det \\Bigl\\{ I \\!-\\!\\sum_{m} G_{0}^r \\Gamma_L G_{0}^a \\Gamma_m \\Big[(e^{i\\xi \\hbar \\omega}\\! -\\! 1) \\times \\nonumber \\\\\n&& f_L\\!\\!+ ( e^{-i\\xi\\hbar \\omega} \\!-\\! 1) f_m \\!+\\! (\\!\ne^{i\\xi \\hbar \\omega} \\!\\!+\\! e^{-i\\xi\\hbar\\omega}\\! \\!-\\!2 )\\! f_L \\!f_m \\Big]\\Bigr\\}.\\qquad .\n\\end{eqnarray}\nIn the following section we will discuss the how to numerically calculate the CGF in the transient case for projected density matrix $\\rho'(0)$. We also discuss about solving the Dyson equation given in Eq.~(\\ref{eq-Dyson-product}).\n\n\n\\section{Transient Region}\nThe central quantity to calculate the CGF numerically is the shifted self-energy $\\Sigma_L^{A}$ which is given by \n\\begin{equation}\n\\Sigma_L^A(\\tau,\\tau') =\\Sigma_L\\bigl(\\tau + \\hbar x(\\tau), \\tau'+\\hbar x(\\tau') \\bigr)- \\Sigma_L\\big(\\tau,\\tau'\\big).\n\\end{equation}\nHere $\\tau$ is a contour variable which runs over Keldysh contour $K=(-\\infty,\\infty)$ and back, for the initial conditions $\\rho(0)$ and $\\rho'(0)$, whereas for $\\rho(-\\infty)$, $\\tau$ runs over the contour $C=[0,t_M]$ (see Fig.~1). The contour function $x(\\tau)$ is 0 whenever $t < 0$ or $t>t_M$, and for $0 < t < t_M$, $x^{+}(t) = -\\xi\/2 - \\lambda$, and $x^{-}(t) = \\xi\/2 - \\lambda$. \nDepending on the values of $t$, $t'$, and $\\lambda$ ($\\lambda \\rightarrow 0$ and $\\lambda \\rightarrow \\infty$ corresponds to steady state initial state and {\\it projected} initial state, respectively) $\\Sigma_L^A$ will have different functional form. If $ 0 < t,t' < t_M$ then $\\Sigma_L^A$'s are given by Eq.~(\\ref{shifted-product}). This is the region which dominates in the long-time limit and gives steady state result. If both $t$ and $t'$ lies outside the measurement time, i.e., $t,t'<0$ or $t,t'>t_M$ then $\\Sigma_L^A$ is zero. \n\nThe main computational task for a numerical evaluation of the cumulants\nis to compute the matrix series $-\\ln(1-M) = M + \\frac{1}{2} M^2 + \\cdots$.\nIt can be seen due to the nature of $\\Sigma_L^A$ that for the product initial\nstate, exact $n$ terms upto $M^n$ is required for the $n$-th culumants, as the infinite series terminates due to $\\Sigma_L^A(\\xi=0) = 0$.\nNumerically, we also observed for the projected state $\\rho'(0)$, exactly\n$3n$ terms is required (although we don't have a proof) if calculation is\nperformed in time domain. \n\nThe computation can be performed in time as well as in the frequency domain. However for projected and steady state initial condition since $G_{0}[\\omega]$ is time translational invariant it is advantageous to work in the frequency domain. But for the product state there is no such preference as $G_{0}$ in Eq.~(\\ref{eq-Dyson-product}) is not time translational invariant and one has to solve it numerically. \n\nIn the following we first discuss how to calculate $\\Sigma_{L}^{A}[\\omega,\\omega']$ for projected initial state, defined in Eq.~(\\ref{eq-Zsteady}) and then we will discuss how to solve the Dyson equation for the product initial condition case, given in Eq.~(\\ref{eq-Dyson-product}).\n\n\\subsection{calculation of $\\Sigma_L^{A}(\\omega,\\omega')$ }\nTo calculate $\\Sigma_L^{A}(\\omega,\\omega')$ for projected initial state $\\rho'(0)$ we define two types of theta functions $\\theta_{1}(t,t')$ and $\\theta_{2}(t,t')$.\n$\\theta_{1}(t,t')$ is non-zero when \n\\begin{equation}\n0 \\leq t \\leq t_M, \\,\\,{\\rm and} \\quad t' \\leq 0 \\quad {\\rm or} \\quad t'\\geq t_M,\n\\end{equation}\nor \n\\begin{equation}\n0 \\leq t' \\leq t_M, \\,\\,{\\rm and} \\quad t \\leq 0 \\quad {\\rm or} \\quad t \\geq t_M, \n\\end{equation}\nand $\\theta_{2}(t,t')$ is non-zero only in the regime where $ 0 \\le t,t' \\le t_M$ . For the regions where $\\theta_1(t,t')$ is non-zero the expression for $\\Sigma_L^{A}$ after taking the limit $\\lambda \\rightarrow \\infty$ is, (assuming all correlation functions decays to zero as $t \\rightarrow \\pm \\infty$)\n\\begin{equation}\n\\Sigma_{A}^{t,\\bar{t},<,>}(t,t')=-\\Sigma_{L}^{t,\\bar{t},<,>}(t-t').\n\\end{equation}\nSo using theta functions we may write $\\Sigma_L^{A}(t,t')$ in the full t,t' domain as \n\\begin{eqnarray}\n\\Sigma_{A}^{t,\\bar{t}}(t,t')&=&-\\theta_{1}(t,t')\\Sigma_{L}^{t,\\bar{t}}(t-t') \\nonumber \\\\\n\\Sigma_{A}^{<}(t,t')&=&-\\theta_{1}(t,t')\\Sigma_{L}^{<}(t-t')+ \\theta_2 (t,t') \\times \\nonumber \\\\\n&&\\>\\big[\\Sigma_{L}^{<}(t-t'-\\hbar \\xi) - \\Sigma_{L}^{<}(t-t')\\big] \\nonumber \\\\\n\\Sigma_{A}^{>}(t,t')&=&-\\theta_{1}(t,t')\\Sigma_{L}^{>}(t-t')+ \\theta_2 (t,t') \\times \\nonumber \\\\\n&&\\>\\big[\\Sigma_{L}^{>}(t-t'+\\hbar \\xi) - \\Sigma_{L}^{>}(t-t')\\big]\n\\end{eqnarray}\nBy doing Fourier transform it can be easily shown that\n\\begin{equation}\n\\Sigma_{A}^{t,\\bar{t}}[\\omega,\\omega']=-\\int_{-\\infty}^{\\infty}\\!\\!\\! \\frac{d\\omega_{c}}{2 \\pi} \\, \\theta_{1}\\bigl[\\omega\\!-\\!\\omega_{c},\\omega'\\!+\\!\\omega_{c}\\bigr]\\Sigma_{L}^{t,\\bar{t}}(\\omega_{c})\n\\end{equation}\nand \n\\begin{eqnarray}\n\\Sigma_{A}^{>,<}[\\omega,\\omega']&\\!=\\!& \\!-\\!\\int_{-\\infty}^{\\infty} \\!\\!\\!\\frac{d\\omega_{c}}{2 \\pi}\\theta_{1}\\bigl[\\omega\\!-\\!\\omega_{c},\\omega'\\!+\\!\\omega_{c}\\bigr]\\Sigma_{L}^{>,<}(\\omega_{c}) \\\\\n\\!+\\!\\int_{-\\infty}^{\\infty}\\! \\frac{d\\omega_{c}}{2 \\pi}\\!\\!\\!&&\\!\\theta_{2}\\bigl[\\omega\\!-\\!\\omega_{c},\\omega'\\!+\\!\\omega_{c}\\bigr]\\Sigma_{L}^{>,<}(\\omega_{c}) \n(e^{i \\omega_{c}\\eta \\xi}\\!-\\!1), \\nonumber\n\\end{eqnarray}\nwhere $\\eta=\\pm 1$. The positive sign is for $\\Sigma_{A}^{<}$ and negative sign for $\\Sigma_{A}^{>}$.\n\nThe theta functions are now given by\n\\begin{eqnarray}\n\\theta_{1}(\\omega_{a},\\omega_{b})&=&f(\\omega_{a}).g(\\omega_{b})+f(\\omega_{b}).g(\\omega_{a}), \\nonumber \\\\\n\\theta_{2}(\\omega_{a},\\omega_{b})&=&f(\\omega_{a}).f(\\omega_{b}),\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nf(\\omega)&=&\\frac{e^{i \\omega t_{M}}-1}{i \\omega}, \\nonumber \\\\\ng(\\omega)&=& \\frac{1}{i \\omega + \\epsilon} -\\frac{e^{i \\omega t_{M}-\\eta t_{M}}}{i \\omega -\\epsilon},\n\\end{eqnarray}\nwith $\\epsilon \\rightarrow 0^{+}$. The theta functions are of immense importance which carries all information about the measurement time $t_{M}$.\n\nIn the limit $t_{M} \\rightarrow \\infty$, the region $ 0 \\le t,t' \\le t_M$ dominates and corresponding theta function, i.e., $\\theta_2(\\omega,\\omega')$ reduces to \n\\begin{equation}\n\\theta_{2}(\\omega-\\omega_{c},\\omega'+\\omega_{c}) \\approx \\delta(\\omega-\\omega_{c}) \\delta(\\omega'+\\omega_{c}),\n\\end{equation}\nand is responsible for obtaining the steady state result.\n\nTo calculate all the cumulants we only need to take derivative of $\\Sigma_{A}(\\omega,\\omega')$ with respect to $i \\xi$ since $G_{0}$ does not have any $\\xi$ dependence. Also $\\Sigma^{A}$ has $\\xi$ dependence only for $ 0 \\le t,t' \\le t_M$ and hence the derivatives are given by \n\\begin{eqnarray}\n\\frac{\\partial^{n}\\Sigma_{A}^{>,<}}{{\\partial}(i \\xi)^{n}}[\\omega,\\omega']&=&\\int_{-\\infty}^{\\infty} \\frac{d\\omega_{c}}{2 \\pi} (\\eta \\hbar \\omega_{c})^{n} \\theta_{2}\\bigl[\\omega-\\omega_{c},\\omega'+\\omega_{c}\\bigr]\\nonumber \\\\\n&&\\Sigma_{L}^{>,<}(\\omega_{c}) e^{i \\omega_{c}\\eta \\xi}.\n\\end{eqnarray} \nHere $n$ refers to the order of the derivative. \n\n\\subsection{Dyson equation on contour C}\n\nLet us now discuss about solving the Dyson's equation for $G_{0}$ given in Eq.~(\\ref{eq-Dyson-product}) for product initial state $\\rho(-\\infty)$. In order to compute the matrix $\\breve{G}_0(t,t')$ we have to calculate two components $G_{0}^{r}$ and $G_{0}^{K}$ which are written in the integral form by applying Langreth's rule \\cite{Huag,rammer86}\n\\begin{eqnarray}\nG_{0}^{r}(t,t')&=&g_{C}^{r}(t\\!-\\!t') \\\\\n&&+\\!\\!\\int_{0}^{t_M}\\!\\!dt_1\\!\\!\\!\\int_{0}^{t_M}\\!\\!dt_2 \\, g_{C}^{r}(t\\!-\\!t_1)\\,\\Sigma^{r}(t_1\\!-\\!t_2)G_{0}^{r}(t_2,t'), \\nonumber \n\\end{eqnarray}\nand \n\\begin{eqnarray}\n&&G_{0}^{K}(t,t')=g_{C}^{K}(t\\!-\\!t') \\\\\n&&+\\!\\!\\int_{0}^{t_M}\\!\\!dt_1\\!\\!\\!\\int_{0}^{t_M}\\!\\!dt_2 \\, g_{C}^{r}(t\\!-\\!t_1)\\,\\Sigma^{r}(t_1\\!-\\!t_2) G_{0}^{K}(t_2,t') \\nonumber \\\\\n&&+\\!\\!\\int_{0}^{t_M}\\!\\!dt_1\\!\\!\\!\\int_{0}^{t_M}\\!\\!dt_2 \\, g_{C}^{r}(t\\!-\\!t_1) \\Sigma^{K}(t_1\\!-\\!t_2) G_{0}^{a}(t_2,t')\\nonumber \\\\\n&&+\\!\\!\\int_{0}^{t_M}\\!\\!dt_1\\!\\!\\!\\int_{0}^{t_M}\\!\\!dt_2 \\, g_{C}^{K}(t\\!-\\!t_1) \\Sigma^{a}(t_1\\!-\\!t_2)G_{0}^{a}(t_2,t'). \\nonumber \n\\end{eqnarray}\n\nNote that the argument for center Green's function $g_{C}$ and lead self-energy $\\Sigma$ are written as time difference $t-t'$ because they are Green's functions for isolated center part and leads respectively and hence are calculated at equilibrium. The analytical expressions for $\\Sigma$ and $g_{C}$ are known in frequency domain and are given in Appendix~A. To determine their time-dependence we numerically calculate their inverse Fourier transforms using trapezoidal rule \\cite{NR}. Then in order to solve above equations for any $t_M$ we discretize the time variable into $N$ total intervals of incremental length $\\Delta t=t_M\/N$ and thus converting the integral into a sum. After discretization, the above equations can be written in the matrix form which are indexed by space $j$ and discrete time $t$, as \n\\begin{eqnarray}\n\\tilde{G}_{0}^{r}&=& \\tilde{g}_C^{r} + \\tilde{g}_C^{r} \\tilde{\\Sigma}^{r} \\tilde{G}_{0}^{r}, \\nonumber \\\\\n\\tilde{G}_{0}^{K}&=& \\tilde{G}_{0}^{r}\\tilde{\\Sigma}^{K}\\tilde{G}_{0}^{a}+ ( I + \\tilde{G}_{0}^{r} \\tilde{\\Sigma}^{r}) \\tilde{g}_C^{K} (I+\\tilde{\\Sigma}^{a} \\tilde{G}_{0}^{a}).\n\\end{eqnarray}\nSo $\\tilde{G}_{0}^{r}$ can be obtained by doing an inverse of the matrix $(I-\\tilde{g}_{C}^{r} \\tilde{\\Sigma}^{r})$ and then multiplying by $\\tilde{g}_{C}^{r}$. $\\tilde{G}_{0}^{r}$ in this case also obeys time-translational invariance, so it can also be obtained by direct inverse Fourier transform. $\\tilde{G}_{0}^{a}$ can be obtained by taking transpose of $\\tilde{G}_{0}^{r}$. Once $\\tilde{G}_{0}^{r}$ and $\\tilde{G}_{0}^{a}$ are obtained we use the second equation to calculate $\\tilde{G}_{0}^{K}$ which is simply multiplying matrices.\n\nSimilarly $\\Sigma_L^{A}$ in Eq.~(\\ref{shifted-product}) are obtained by doing inverse Fourier transforms of the lead self-energy. We follow the same steps independently to calculate the cumulants for $Q_R$. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2.eps}\n\\caption{(Color online) The cumulants $\\langle \\langle {Q}_{L}^{n} \\rangle \\rangle$ and $\\langle \\langle {Q}_{R}^{n} \\rangle \\rangle$ for $n$=1, 2, 3, and 4 for one-dimensional linear chain connected with Rubin baths, for the projected initial state $\\rho'(0)$. The black and red curves corresponds to $\\langle \\langle {Q}_{L}^{n} \\rangle \\rangle$ and $\\langle \\langle {Q}_{R}^{n} \\rangle \\rangle$ respectively. The temperatures of the left and the right lead are 310 K and 290 K, respectively. The center (C) consists of one particle.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig3.eps}\n\\caption{(Color online) Same as in Fig.~2 except for product initial state $\\rho(-\\infty)$. The temperatures of the left, the center and the right lead are 310 K, 300 K and 290 K, respectively.}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig4.eps}\n\\caption{(Color online) Same as in Fig.~2 except for steady state initial state $\\rho(0)$.}\n\\end{figure}\n\n\\section{Numerical results}\nWe now present some numerical results. In Fig.~2 and 3, we show the results for first four cumulants for both ${\\cal Q}_L$ and ${\\cal Q}_R$ (measurement is on the right lead) for 1D linear chain connected with Rubin baths, starting with the projected initial state $\\rho'(0)$ and product state $\\rho(-\\infty)$ respectively. \nRubin baths \\cite{Rubin, Weiss} mean in our case a uniform linear\nchain with spring constant $K$ and a small onsite $K_0$ for all the atoms.\nOnly one atom is considered as the center. The atoms of the left and right side\nof the center are considered baths. We use $K=1$ eV\/(u\\AA$^2)$ and the onsite potential $K_0=0.1$ eV\/(u\\AA$^2)$ in all our calculations.\nFirst of all, cumulants greater than two are nonzero, which confirms that the distribution for $P(Q_L)$ or $P(Q_R)$ is not Gaussian. The generic features are almost the same in both the cases. However the fluctuations are larger for the product initial state $\\rho(-\\infty)$ as this state corresponds to the sudden switch on of the couplings between the leads and the center and hence the state is far away from the correct steady state distribution. On the contrary, for the initial state $\\rho'(0)$ the fluctuations are relatively small. For $\\rho'(0)$ due to the effect of the measurement, at starting time energy goes into the leads, which is quite surprising. But for $\\rho(-\\infty)$ although initial measurement do not play any role, energy still goes into the leads. This can also be shown analytically (see Appendix B). At the starting time the behavior of both ${\\cal Q}_L$ and ${\\cal Q}_R$ are very similar and can be understood since both the left and right leads are identical and the effect of temperature difference is not present. However at longer times the odd cumulants starts differing and finally grows linearly with time $t_M$ and agrees with the corresponding long-time predictions.\n\nIn Fig.~4 we show the results for the steady state initial condition, i.e., $\\rho(0)$ which can be obtained by mapping the projection operators as identity operator, i.e., taking the limit $\\lambda \\rightarrow 0$. So in this case measurement effect is ignored and the dynamics starts with the actual steady state for the full system. The first cumulant increases linearly from the starting, $\\langle Q \\rangle = t I$ and the slope gives the correct prediction with the Landauer-like formula. However, high order cumulants still have transient behavior. In this case the whole system achieve steady state much faster compared with the other two cases.\n\n\n\\section{correlation between left and right lead heat}\n\\subsection{Product initial state}\nIn this section, we derive the CGF for the joint probability distribution $P(Q_L,Q_R)$ for the product initial state $\\rho(-\\infty)$. In order to calculate the CGF we need to measure both ${\\cal H}_L$ and ${\\cal H}_R$ at time 0 and at time $t_M$. Since the Hamiltonians for the left and the right lead commute at the same instance of time i.e., $\\big[{\\cal H}_L, {\\cal H}_R \\big]=0$, such type of measurements are allowed in quantum mechanics and also Nelson's theorem \\cite{Nelson} gurentee's that $P(Q_L,Q_R)$ is a well-defined probability distribution. The immediate consequence of deriving such CGF is that, the correlations between the left and the right lead heat can be obtained and it is also possible to calculate the CGF for total entropy flow (defined below) to the reservoirs. To calculate the CGF we need two counting fields $\\xi_L$ and $\\xi_R$ and the CGF in this case can be written down as \\cite{Esposito-review-2009}\n\\begin{equation}\n\\mathcal {Z}(\\xi_L,\\xi_R)=\\langle e^{i\\,\\xi_L\\,{\\cal H}_L+i\\,\\xi_R\\,{\\cal H}_R}\\,\\, e^{-i\\,\\xi_L\\,{\\cal H}^{H}_L(t)-i\\,\\xi_R\\,{\\cal H}^{H}_R(t)} \\rangle',\n\\end{equation}\nwhere the average is defined as \n\\begin{equation}\n\\langle \\cdots \\rangle'=\\sum_{a,c}\\Pi^{L}_a \\, \\Pi^{R}_c \\, \\rho(0)\\, \\Pi^{L}_a \\,\\Pi^{R}_c.\n\\end{equation}\n$\\Pi^{L}_a$ and $\\Pi^{R}_c$ are the projectors onto the eigenstates of ${\\cal H}_L$ and ${\\cal H}_R$ with eigenvalues $a$ and $c$ respectively, corresponding to the measurements at $t=0$. Here we will consider only the product state $\\rho(-\\infty)$, then initial projections $\\Pi^{L}_a$ and $\\pi^{R}_c$ do not play any role. We can proceed as before and finally the CGF can be written down as\n\\begin{equation}\n \\ln {\\cal Z}(\\xi_L,\\xi_R) = \\sum_{k=1}^\\infty \\frac{1}{2k}\n{\\rm Tr}_{(j,\\tau)} \\Bigl[\\big( G_{0} (\\Sigma_L^{A}+\\Sigma_R^A) \\big)^k \\Bigr],\n\\label{eq-lead-lead}\n\\end{equation}\ni.e., in this case we need to shift the contour-time arguments for both left and right lead self-energies. In the long-time limit ${\\cal Z}(\\xi_L,\\xi_R)$ becomes a function of difference of counting field $\\xi_L$ and $\\xi_R$, i.e., $\\xi_L-\\xi_R$. The explicit expression for the CGF in the long-time limit is \n\\begin{eqnarray}\n&&\\ln {\\cal Z}(\\xi_L-\\xi_R)=-t_M \\int\\frac{d\\omega}{4\\pi} \\ln \\det \\Bigl\\{ I - G_{0}^r \\Gamma_L \nG_{0}^a \\Gamma_R \\nonumber \\\\ &&\\big[(e^{i(\\xi_L-\\xi_R) \\hbar \\omega}\\! -\\! 1) f_L \n+( e^{-i(\\xi_L-\\xi_R)\\hbar \\omega} \\!-\\! 1) f_R \\nonumber \\\\\n&&+(e^{i(\\xi_L-\\xi_R) \\hbar \\omega} \\!+\\! e^{-i(\\xi_L-\\xi_R)\\hbar\\omega} \\!-\\!2 ) f_L f_R \\big]\\Bigr\\}.\\qquad\n\\label{eq-lnZxi}\n\\end{eqnarray}\nwhere $G_{0}$ obeys the same type of Dyson equation as in Eq.~(\\ref{eq-Dyson-product}). This CGF in the steady state obeys the same type of GC fluctuation symmetry, which in this case is given by \n\\begin{equation}\n{\\cal Z}(\\xi_L-\\xi_R)={\\cal Z}(-\\xi_L+\\xi_R +i {\\cal A}).\n\\end{equation}\nNow performing Fourier transform of the CGF, the joint probability distribution is given by $P(Q_L,Q_R)=P(Q_L)\\,\\delta(Q_L+Q_R)$. The appearance of the delta function is a consequence of the energy conservation in the steady state, i.e., $I_L=-I_R$. In the steady state knowing probability distribution either for ${\\cal Q}_L$ or ${\\cal Q}_R$ is sufficient to know the joint probability distribution.\n\nThe cumulants can be obtained from the CGF by taking derivatives with respect to both $\\xi_L$ and $\\xi_R$, i.e.,$\\langle \\langle {Q}_L^{n} {Q}_R^{m} \\rangle \\rangle =\\partial^{n+m} \\ln {\\cal Z}\/\\partial (i\\xi_L)^n \\partial (i\\xi_R)^m,$ substituting $\\xi_L=\\xi_R=0.$ In the steady state the cumulants obey $\\langle \\langle {Q}_L^{n} {Q}_R^{m} \\rangle \\rangle= (-1)^{m} \\langle \\langle Q_L^{m+n} \\rangle \\rangle = (-1)^{n} \\langle \\langle Q_R^{m+n} \\rangle \\rangle$. The first cumulant give us the left and right lead correlation $\\langle \\langle {Q}_{L} {Q}_{R} \\rangle \\rangle=\\langle {Q}_{L} {Q}_{R} \\rangle - \\langle {Q}_{L} \\rangle \\langle {Q}_{R} \\rangle$ and in the steady state is equal to $-\\langle \\langle {Q}_L^{2} \\rangle \\rangle$.\n\nIn Fig.~5 we plot the first three cumulants for one dimensional linear chain connected with Rubin bath where the center consists of only one atom. Initially the cumulant $\\langle \\langle {Q}_{L} {Q}_{R} \\rangle \\rangle $ is positively correlated as both $Q_L$ and $Q_R$ are negative, however in the longer time since $Q_L=-Q_R$ the correlation becomes negative. We also give plots for $\\langle \\langle {Q}_L^{2} {Q}_R \\rangle \\rangle$ (black online) and $\\langle \\langle {Q}_R^{2} {Q}_L \\rangle \\rangle$ (red online) which are in the long-time limit \nnegative and positively correlated respectively and match with the long-time predictions.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig5.eps}\n\\caption{(Color online) First three cumulants of the correlations between left and right lead heat flux for one dimensional linear chain connected with Rubin baths, starting with product initial state $\\rho(-\\infty)$. The left graph corresponds to $\\langle \\langle {Q}_{L} {Q}_{R} \\rangle \\rangle$ and the right graph corresponds to cumulants $\\langle \\langle {Q}_L^{2} {Q}_R \\rangle \\rangle$ (Black curve) and $\\langle \\langle {Q}_R^{2} {Q}_L \\rangle \\rangle$ (Red curve). The left, center and right lead temperatures are 310 K, 290 K and 300 K respectively. The center (C) consists of one particle.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig6.eps}\n\\caption{The cumulants of entropy production $\\langle \\langle \\sigma^{n} \\rangle \\rangle$ for $n$=1, 2, 3, 4 for one dimension linear chain connected with Rubin baths, for product initial state $\\rho(-\\infty)$. The left, center and right lead temperatures are 510 K, 400 K, and 290 K respectively. The center (C) consists of one particle.}\n\\end{figure}\n\n\n\n\\subsection{Entropy flow to the reservoir}\nFrom the two parameter ($\\xi_L,\\xi_R$) CGF one can also obtain the total entropy that flows into the leads. The total entropy flow to the reservoirs can be defined as \\cite{ep1,ep2}\n\\begin{equation}\n{\\cal \\sigma}=-\\beta_{L}{\\cal Q}_{L} -\\beta_{R}{\\cal Q}_{R}.\n\\end{equation}\nIn order to calculate this CGF we just make the substitutions $\\xi_L \\rightarrow -\\beta_L \\mu$ and $ \\xi_R \\rightarrow - \\beta_R \\mu$ in Eq.~(\\ref{eq-lead-lead}).\nIn the long-time limit the expression for entropy-production is similar to $\\ln {\\cal Z}(\\xi_L,\\xi_R)$ with $\\xi_L-\\xi_R$ replaced by ${\\cal A}$ and becomes an explicit function of thermodynamic affinity $\\beta_R-\\beta_L$ \\cite{fluct-theorems}. The CGF in this case satisfies the following symmetry\n\\begin{equation}\n{\\cal Z}(\\mu)={\\cal Z}(-\\mu + i)\n\\end{equation}\nIn Fig.~6 we give results for the first four cumulants of the entropy flow. All cumulants are positive and in the long-time limit give correct predictions. \n\n\n\n\\section{Long-time result for $\\ln {\\cal Z}^d(\\xi)$}\n\nIn this section we derive the explicit expression for the long-time limit of the CGF $\\ln {\\cal Z}^d(\\xi)$ which is given by (Eq.~\\ref{eq-lnZxi-1})\n\\begin{equation}\n\\ln {\\cal Z}^d(\\xi)=- \\frac{i}{\\hbar} \\int \\frac{d\\omega}{4\\pi}\n{\\rm Tr} \\bigl[ \\breve{G}[\\omega] \\breve{{\\cal F}}[\\omega,-\\omega]\\bigr].\n\\end{equation}\nwhere $G[\\omega]$ obeys the Dyson equation given in Eq.~(\\ref{eq-Dyson-full}). It is possible to write down $\\breve{G}[\\omega]$ in terms of $\\breve{G}_{0}$ and $\\breve{\\Sigma}_L^{A}$ as $\\breve{G}[\\omega]=\\big(I-\\breve{G}_{0}\\breve{\\Sigma}^{A}_L\\big)^{-1} \\breve{G}_{0}[\\omega]$. This equation can be solved analytically. Next we assume that the product of $f(t)$ and $f(t')$ is a time-translationally invariant function, i.e., $f(t)f^{T}(t')=F(t-t')$ in order to get rid of $t+t'$ dependence term. In the Fourier domain this means $f[\\omega]f^{T}[\\omega']=2\\pi F[\\omega] \\delta(\\omega+\\omega')$. So from Eq.~(\\ref{force-matrix}) the matrix element ${\\cal F}_{12}$ is given by $\\breve{{\\cal F}}[\\omega,-\\omega]_{12} \\propto \\delta(0) F[\\omega]$. We write $\\delta(0)=t_M\/2\\pi$. Using these results the CGF can be expressed as\n\n\\begin{equation}\n \\ln {\\cal Z}^{d}(\\xi)= {i t_M} \\int \\frac{d\\omega}{4\\pi \\hbar} \\, \\frac{1} {{\\cal N}(\\xi)}{\\rm Tr} \\Big[G_{0}^{r}[\\omega] (a+b) G_{0}^{a}[\\omega] F[\\omega] \\Big],\n\\end{equation}\nwhere $a$ and $b$ are defined in Eq.~(\\ref{eq-a}) and Eq.~(\\ref{eq-b}). Using the expressions for the self-energy the CGF reduces to \n\\begin{equation}\n \\ln {\\cal Z}^{d}(\\xi)=\\!\\!\\int \\!\\frac{d\\omega}{4 \\pi \\hbar}\\,\\frac{\\cal{K}(\\xi)}{{\\cal N}(\\xi)}\\, {\\rm Tr} \\Big[G_{0}^{r}[\\omega] \\Gamma_{L}[\\omega] G_{0}^{a}[\\omega]F[\\omega] \\Big],\n\\label{eq-driven-CGF}\n\\end{equation}\nwith \n\\begin{equation}\n{\\cal{K}}(\\xi)=(e^{-i\\xi\\hbar \\omega}\\!-\\!1 )+f_L(e^{i\\xi\\hbar\\omega}\\!+\\! e^{-i\\xi\\hbar \\omega}\\!-\\!2),\n\\end{equation} \nand \n\\begin{eqnarray}\n{\\cal {N}}(\\xi)&=& {\\rm det} \\Big[I -\\big(G_{0}^r \\Gamma_L G_{0}^a \\Gamma_R\\big) \\Bigl \\{ (f_L e^{i\\xi \\hbar \\omega}\\!-\\!1) + f_R \\nonumber \\\\ \n&&( e^{-i\\xi\\hbar \\omega}\\!-\\!1) + (e^{i\\xi \\hbar \\omega}\\!+\\! e^{-i\\xi\\hbar\\omega}\\!-\\!2 ) f_L f_R \\Bigr\\}.\n\\end{eqnarray}\nIt is important to note that ${\\cal K}(\\xi)$ depends only on left lead temperature and satisfies the symmetry ${\\cal K}(\\xi)={\\cal K}(-\\xi-i\\beta_L)$. So we can immediately write ${\\cal Z}^{d}(-i\\beta_L)=1$ and this relation is completely independent of the information about the right lead. If we consider the two leads at the same temperature ($\\beta_L=\\beta_R=\\beta$), this form of symmetry is then closely related to the Jarzynski equality (JE) \\cite{JE,talkner2008} and ${\\cal Z}^{d}(-i\\beta)=1$ is one special form of JE. However since ${\\cal N}(\\xi)$ does not satisfy this particular symmetry of $\\xi$ at thermal equilibrium (it obeys the GC summery when the leads are at different temperatures) and the CGF $\\ln{\\cal Z}^{d}(\\xi)$ doesn't satisfy any such symmetry relation and hence JE is not satisfied. This does not violate JE as our definition of ${\\cal Z}^{d}(\\xi)$ is different from the one used to derive JE.\n \nLet us now come back to the general scenario with leads at different temperatures and give the explicit expression of first and second cumulant by taking derivative of $\\ln {\\cal Z}^{d}(\\xi)$ with respect to $i\\xi$. \n\nThe first cumulant or moment is given by \\cite{driven-bijay}\n\\begin{equation}\n\\frac{\\langle \\langle Q_{d} \\rangle \\rangle}{t_M} = - \\int \\frac{d\\omega}{4 \\pi} \\, \\omega \\, {\\cal S}[\\omega],\n\\end{equation}\nwhere we define ${\\cal S}[\\omega]$ as the transmission function for the driven case and is given by\n\\begin{equation}\n{\\cal S}[\\omega]={\\rm{Tr}} \\bigl[G_{0}^{r} \\Gamma_{L} G_{0}^{a} F \\bigr].\n\\end{equation}\nFrom the expression of ${\\cal S}[\\omega]$ it is clear that the average energy current due to driven force is independent of $\\hbar$ and since it contains $G_{0}^{r,a}$ and $\\Gamma_L$, which are independent of temperature we can conclude that the energy current is independent of the temperature of the heat baths in the ballistic transport case. However the second cumulant and similarly the higher ones do depend on temperature of the baths. The second cumulant can be written as\n\\begin{eqnarray}\n\\frac{\\langle \\langle Q_{d}^{2} \\rangle \\rangle}{t_M} &=& \\int \\frac{d\\omega}{4 \\pi \\hbar} \\, (\\hbar \\omega)^{2} \\, {\\cal S}[\\omega] \\Bigl[(1+2\\,f_{L})- \\nonumber \\\\\n&& 2 \\,{\\cal T}(\\omega) (f_L-f_R) \\Bigr].\n\\label{second-driven-cumulant}\n\\end{eqnarray}\nSimilarly all the higher cumulants can be obtained from the CGF and we can conclude that the distribution $P(Q_d)$ is not Gaussian.\n\n\n\\subsection{Classical limit}\nIn this section we will give the classical limit of the steady state expression for the CGF $\\ln {\\cal Z}^{s}(\\xi)$ and $\\ln {\\cal Z}^{d}(\\xi)$ given in Eq.~(\\ref{eq-steady1}) and Eq.~(\\ref{eq-driven-CGF}).\n\nFirst of all we note that retarded and advanced Green's functions, i.e., $G_{0}^{r}$ and $G_{0}^{a}$ are similar both for quantum and classical case, so they stay the same when $\\hbar \\to 0$. We know that in the classical limit $f_{\\alpha} \\rightarrow \\frac{k_{B}T_{\\alpha}}{\\hbar \\omega}$ and also $e^{ix}=1+ix + \\frac{(ix)^{2}}{2} + \\cdots$, where $x=\\xi \\hbar \\omega$. Using this we obtain from Eq.~(\\ref{eq-steady1}) the classical limit of ${\\cal Z}^{s}(\\xi)$. \n\\begin{eqnarray}\n\\ln{\\cal Z}^{s}_{\\rm cls}(\\xi)&=& \\frac{t_M}{4\\pi} \\int d\\omega\\,\\ln \\det \\Big[I-\\big(G_{0}^{r} \\Gamma_{L} G_{0}^{a} \\Gamma_{R}\\big) \\times \\nonumber \\\\\n&& \\, k_B T_L \\, k_B T_R \\,i\\xi (i\\xi +{\\cal A})\\Big].\n\\end{eqnarray}\nThis result reproduces that of Ref.~\\cite{kundu}.\nIn the classical case also the CGF obeys the GC symmetry, i.e., it remains invariant under the transformation $i\\xi \\rightarrow -i\\xi -{\\cal A}$.\n\nLet us now get the classical limit for $\\ln {\\cal Z}^{d}(\\xi)$ using Eq.~(\\ref{eq-driven-CGF}). Following above relations the function ${\\cal K}(\\xi)$ in the limit $\\hbar \\rightarrow 0$ reduces to\n\\begin{equation}\n{\\cal K}_{\\rm cls}(\\xi)= -\\hbar \\omega \\,\\Big(i\\xi + \\frac{\\xi^2}{\\beta_L}\\Big).\n\\end{equation} \nThe transmission function ${\\cal S}[\\omega]$ stays the same as it is independent of temperature and $\\hbar$. So in the classical limit $\\ln {\\cal Z}^{d}(\\xi)$ reduces to \n\\begin{equation}\n\\ln {\\cal Z}^{d}_{\\rm cls}(\\xi)= t_M \\int \\frac{d\\omega}{4 \\pi} \\, \\omega \\, {\\cal S}[\\omega] \\, \\frac{\\Big(i\\xi + \\frac{\\xi^2}{\\beta_L}\\Big)}{{\\cal N}_{\\rm cls}(\\xi)},\n\\end{equation}\nwhere \n\\begin{eqnarray}\n{\\cal N}(\\xi)_{\\rm cls}&=&\\det\\Big[I-\\big(G_{0}^{r} \\Gamma_{L} G_{0}^{a} \\Gamma_{R}\\big) \\,k_B T_L \\, k_B T_R \\nonumber \\\\\n&&i\\xi (i\\xi +{\\cal A})\\Big].\n\\end{eqnarray}\nHere we can easily see that ${\\cal Z}^{d}(-i\\beta_L)=1$.\n\nWe can also derive the fluctuation dissipation theorem from Eq.(\\ref{second-driven-cumulant}) if we assume the leads are at the same temperature, i.e., $\\beta_{L}=\\beta_{R}=\\beta$ then we can write the second cumulant $\\langle \\langle Q_{d}^{2} \\rangle \\rangle$ as \n\\begin{equation}\n\\frac{\\langle \\langle Q_{d}^{2} \\rangle \\rangle}{t_M} = \\int \\frac{d\\omega}{4 \\pi \\hbar} \\, (\\hbar \\omega)^{2} \\,{\\cal S}[\\omega] (1+2\\,f_{L}).\n\\end{equation}\nIn the high-temperature limit using $f_{L} \\rightarrow \\frac{k_{B}T_{L}}{\\hbar \\omega}$ and we obtain\n\\begin{equation}\n\\langle \\langle Q_{d}^{2} \\rangle \\rangle =\\frac{2}{\\beta_{L}} \\langle Q_{d} \\rangle.\n\\end{equation}\n\nIn the next section we discuss Nazarov's generating function and give long-time limit expression. \n\n\\section{Nazarov's definition of generating function}\nIn this section we will derive another definition of CGF given by Eq.~(\\ref{eq-Z1-Nazarov}), starting from the CGF, derived using two-time measurement concept, i.e., Eq.~(\\ref{eq-Z-two-time}). Eq.~(\\ref{eq-Z1-Nazarov}) can be obtained from Eq.~(\\ref{eq-Z-two-time}) in the small $\\xi$ approximation as follows. In the small $\\xi$ approximation the modified Hamiltonian given in Eq.~(\\ref{modified}) takes the following form \n\\be\n{\\cal H}_{x}(t)={\\cal H}(t)+\\hbar x {\\cal I}_L(0),\n\\ee\nbecause $\\lim_{x\\rightarrow 0}{\\cal C}(x)=0$ and $\\lim_{x\\rightarrow 0}{\\cal S}(x)=\\hbar x V^{LC}$. ${\\cal I}_L$ is defined in Eq.~(\\ref{current}).\nSo the modified unitary operator becomes\n\\be\n{\\cal U}_{x}(t,0)=T e^{-\\frac{i}{\\hbar} \\int_{0}^t [{\\cal H}(\\bar{t})+\\hbar x {\\cal I}_L(0)] d\\bar{t}}.\n\\ee\nWe can consider $\\hbar x {\\cal I}_L(0)$ as the interaction Hamiltonian and write the full unitary operator ${\\cal U}_x$ as a product of two unitary operators as following \n\\begin{equation}\n{\\cal U}_{x}(t,0)={\\cal U}(t,0) \\, {\\cal U}_{x}^{I}(t,0),\n\\label{u}\n\\end{equation}\nwhere \n\\begin{eqnarray}\n{\\cal U}(t,0)&=& T e^{-\\frac{i}{\\hbar} \\int_{0}^t {\\cal H}(t') \\, dt'}, \\nonumber \\\\\n{\\cal U}_{x}^{I}(t,0) &=& T e^{-\\frac{i}{\\hbar} \\int_{0}^t \\hbar x {\\cal I}_L(t') dt'},\n\\label{u1}\n\\end{eqnarray}\nwith ${\\cal I}_L(t')={\\cal U}^{\\dagger}(t',0)\\,{\\cal I}_L(0)\\,{\\cal U}(t',0)$ is the current operator in the Heisenberg picture. It is important to note that ${\\cal U}$ is the usual unitary operator which evolves with the full Hamiltonian ${\\cal H}(t)$ in Eq.~(\\ref{eq-unitary}) and has no $\\xi$ dependence.\n\nIf we use product state $\\rho(-\\infty)$ as the initial state the CGF is given by \n\\begin{equation}\n{\\cal Z}(\\xi)={\\rm Tr}\\big[\\rho(-\\infty) \\, {\\cal U}_{\\xi\/2}(0,t)\\, {\\cal U}_{-\\xi\/2}(t,0)\\big].\n\\end{equation} \nIn the small $\\xi$ approximation and using the expressions for ${\\cal U}_{x}$ we can write the CGF as\n\\begin{equation}\n{\\cal Z}_{1}(\\xi)=\\lim_{\\xi \\to 0} {\\cal Z}(\\xi)= {\\rm Tr}\\big[\\rho(-\\infty) \\, {\\cal U}_{\\xi\/2}^{I}(0,t)\\, {\\cal U}_{-\\xi\/2}^{I}(t,0)\\big],\n\\end{equation}\nwhere we use the property of unitary operator, i.e., ${\\cal U}^{\\dagger}(t,0) {\\cal U}(t,0)=1$. Finally using the definition of heat operator ${\\cal Q}_L$ given in Eq.~(\\ref{eq-hatQ}) and the CGF can be written down as \n\\begin{equation}\n{\\cal Z}_{1}(\\xi)=\\Big \\langle {\\bar T} e^{i\\xi {\\cal Q}_{L}(t)\/2} \\, T e^{i\\xi {\\cal Q}_{L}(t)\/2}\\Big \\rangle,\n\\end{equation}\nwhich is the same as in Eq.~(\\ref{eq-Z1-Nazarov}).\n\nIn the following we will give the long-time limit expression for this CGF.\n \nIn order to calculate the CGF, it is important to go to the interaction picture with respect to the Hamiltonian ${\\cal H}_{0}={\\cal H}_L+{\\cal H}_C+ {\\cal H}_R$, as we know how to calculate Green's functions for operators which evolves with ${\\cal H}_{0}$ and treat the rest part as the interaction ${\\cal V}_{x}={\\cal H}_{\\rm int}+ \\hbar x {\\cal I}_{L}(0)$. So the CGF on contour $C=\\big[0,t_M \\big]$ can be written as\n\\begin{equation}\n{\\cal Z}_{1}(\\xi)=\\Big \\langle T_c e^{-\\frac{i}{\\hbar} \\int {\\cal V}_{x}^{I}(\\tau) d\\tau } \\Big \\rangle,\n\\end{equation}\nwhere ${\\cal V}_{x}^{I}(\\tau)$ is now given by \n\\begin{eqnarray}\n{\\cal V}_{x}^{I}(\\tau)&=&u_{L}^T(\\tau) V^{LC} u_{C}(\\tau)+ u_{R}^T(\\tau) V^{RC} u_{C}(\\tau) \\nonumber \\\\\n&&+\\hbar x(\\tau) p_{L}(\\tau) V^{LC} u_{C}(\\tau),\n\\end{eqnarray} \nwhere $p_L=\\dot{u}_L$. The time-dependence $\\tau$ is coming from the free evolution with respect to ${\\cal H}_{0}$. $x(\\tau)$ has the similar meaning as before, i.e., on the upper branch of the contour $x^{+}(t)=-\\xi\/2$ and on the lower branch $x^{-}(t)=\\xi\/2$. Now using the same idea as before, we expand the series, use Wick's theorem and finally the CGF can be expressed as\n\\begin{equation}\n\\ln {\\cal Z}(\\xi)=- \\frac{1}{2} {\\rm Tr}_{j,\\tau} \\ln \\Big[ 1 - G_0 \\Sigma_L^{A} \\Big]. \n\\end{equation}\nHere $G_{0}$ is the same as before and is given by Eq.~(\\ref{eq-Dyson-product}). However the shifted self-energy $\\Sigma_{L}^{A}$ in this case is different and is given by (in contour-time argument) \n\\begin{eqnarray}\n\\Sigma_{L}^{A}(\\tau,\\tau')&=&\\hbar\\, x(\\tau)\\, \\Sigma_{p_L u_L}(\\tau,\\tau')+\\hbar\\, x(\\tau')\\, \\Sigma_{u_L p_L}(\\tau,\\tau') \\nonumber \\\\\n&&+\\hbar^{2}\\, x(\\tau)\\,x(\\tau')\\,\\Sigma_{p_L p_L}(\\tau,\\tau').\n\\end{eqnarray}\nThe notation $\\Sigma_{A B}(\\tau,\\tau')$ means\n\\begin{equation}\n\\Sigma_{A B}(\\tau,\\tau')= \\bigl(-\\frac{i}{\\hbar}\\bigr) V^{CL} \\, \\langle\\, T_{c} A(\\tau) B^{T}(\\tau')\\,\\rangle \\, V^{LC}.\n\\end{equation}\nThe average here is with respect to equilibrium distribution of the left lead. It is possible to express the correlation functions such as $\\Sigma_{p_L u_L}(\\tau,\\tau')$ in terms of the $\\Sigma_{u_L,u_L}(\\tau,\\tau')=\\Sigma_L(\\tau,\\tau')$ correlations. $\\Sigma_{p_L u_L}(\\tau,\\tau')$ and $\\Sigma_{u_L p_L}(\\tau,\\tau')$ is simply related with $\\Sigma_L(\\tau,\\tau')$ by the contour-time derivative whereas for $\\Sigma_{p_L p_L}(\\tau,\\tau')$ the expression is \n\\begin{equation}\n\\Sigma_{p_L p_L}(\\tau,\\tau')= \\frac{\\partial^{2} \\Sigma_{u_L u_L}(\\tau,\\tau')}{\\partial \\tau \\partial \\tau'} + \\delta(\\tau,\\tau') \\Sigma_{L}^{I}.\n\\end{equation}\nWhere $\\Sigma_L^{I}=V^{CL} V^{LC}$. Now in the frequency domain different components of $\\Sigma_L^{A}$ takes the following form\n\\begin{eqnarray}\n\\Sigma_A^{t}[\\omega]&=& \\frac{\\hbar^{2}\\xi^{2}\\omega^{2}}{4} \\Sigma_{L}^{t}[\\omega]+ \\frac{\\hbar^{2}\\xi^{2}}{4}\\Sigma_{L}^{I}, \\nonumber \\\\\n\\Sigma_A^{\\bar{t}}[\\omega]&=& \\frac{\\hbar^{2}\\xi^{2}\\omega^{2}}{4} \\Sigma_{L}^{\\bar{t}}[\\omega]- \\frac{\\hbar^{2}\\xi^{2}}{4}\\Sigma_{L}^{I}, \\nonumber \\\\\n\\Sigma_A^{<}[\\omega]&=& \\big( i \\hbar \\xi \\omega - \\frac{\\hbar^{2}\\xi^{2}\\omega^{2}}{4} \\big) \\Sigma_{L}^{<}[\\omega], \\nonumber \\\\\n\\Sigma_A^{>}[\\omega]&=& \\big(-i \\hbar \\xi \\omega - \\frac{\\hbar^{2}\\xi^{2}\\omega^{2}}{4} \\big) \\Sigma_{L}^{>}[\\omega]. \n\\end{eqnarray}\n\nFinally using the relations between the self-energy (see Appendix A), in the long-time limit the CGF can be written down as,\n\\begin{eqnarray}\n\\ln {\\cal Z}_{1}(\\xi) &=& - t_M \\int \\frac{d\\omega}{4\\pi} \\ln \\Big[1-(i\\xi \\hbar \\omega){\\cal T}[\\omega]\\,(f_L-f_R) \n\\nonumber \\\\ &&- \\frac{(i\\xi \\hbar \\omega)^{2}}{4}\\Big({\\cal T}[\\omega](1+2f_L)(1+2f_R)-G_{0}^{a}\\Sigma_{L}^{r} \\nonumber \\\\\n&&+G_{0}^{r}\\Sigma_{L}^{a} -G_{0}^{r}\\Gamma_{L}G_{0}^{a}\\Gamma_{L}\\Big)+ {\\cal J}(\\xi^{2},\\xi^{4})\\Big],\n\\end{eqnarray}\nwhere ${\\cal J}(\\xi^2,\\xi^4)$ is given by\n\\begin{eqnarray}\n{\\cal J}(\\xi^2,\\xi^4)&=&-\\frac{\\hbar^2 \\xi^2}{4}\\big(G_{0}^{a}+G_{0}^{r}\\big)\\Sigma_{L}^{I} - \\frac{1}{4} \\frac{(i\\xi \\hbar \\omega)^2}{2} \\frac{\\hbar^2 \\xi^2}{2} \\nonumber \\\\ \n&&+\\big(G_{0}^{r}\\Sigma_{L}^{a}G_{0}^{a}\\Sigma_L^{I}+G_{0}^{r}\\Sigma_{L}^{I}G_{0}^{a}\\Sigma_L^{r}\\big) + \\frac{1}{4} \\frac{(i\\xi \\hbar \\omega)^4}{4} \\nonumber \\\\\n&&G_{0}^{r}\\Sigma_{L}^{a}G_{0}^{a}\\Sigma_L^{r} + \\frac{1}{4} \\frac{(\\hbar^{4} \\xi^{4})}{4} G_{0}^{r}\\Sigma_{L}^{I}G_{0}^{a}\\Sigma_L^{I}.\n\\end{eqnarray}\n\nThis CGF does not obey the GC fluctuation symmetry. However it gives the correct first and second cumulant as it should because the definition of first and second cumulant turn out to be the same for both the generating functions ${\\cal Z}(\\xi)$ and ${\\cal Z}_{1}(\\xi)$ and is given by\n\\begin{eqnarray}\n&&\\langle \\langle Q \\rangle \\rangle=\\langle Q \\rangle = \\frac{\\partial \\ln {\\cal Z}(\\xi)}{\\partial {(i\\xi)}}=\\frac{\\partial \\ln {\\cal Z}_1(\\xi)}{\\partial {(i\\xi)}}= \\int_{0}^{t} dt_1 \\langle {\\cal I}_L(t_1) \\rangle, \\nonumber \\\\\n&&\\langle \\langle Q^{2} \\rangle \\rangle =\\langle Q^{2} \\rangle- \\langle Q \\rangle^{2} = \\frac{\\partial^{2} \\ln {\\cal Z}(\\xi)}{\\partial {(i\\xi)^{2}}}=\\frac{\\partial^{2} \\ln {\\cal Z}_1(\\xi)}{\\partial {(i\\xi)^{2}}}\\nonumber \\\\\n&& \\> = \\int_{0}^{t} dt_1 \\int_{0}^{t} dt_2 \\langle {\\cal I}_L(t_1) {\\cal I}_L(t_2) \\rangle-\\Big[\\int_{0}^{t}\\! dt_1\\! \\langle {\\cal I}_L(t_1) \\rangle \\Big]^{2}. \n\\end{eqnarray}\nExpressions for higher cumulants are different for the two generating functions and hence the final expressions for the CGF's are completely different from each other. \n\n\\section{Conclusion}\nIn summary, we present an elegant way of deriving the CGF for heat ${\\cal Q}_{L,R}$ transferred from the leads to the center for driven linear systems using the two-time measurement concept and with the help of the NEGF technique. The CGF is written in terms of the Green's function of the center and the self-energy $\\Sigma_{L}^A$ of the leads. \nThe counting of the energy is related to the shifting in time for the self-energy.\nThis expression is valid in both transient and steady state regimes, where the information about the measurement time $t_M$ is contained in $\\Sigma_{L}^A$. The form of the expression,\n$-(1\/2) {\\rm Tr} \\ln (1 - G_{0} \\Sigma_L^A)$, is the same whether we use a product initial state or a projected initial state, except that the meaning of the Green's function has to be adjusted accordingly. We consider three different initial conditions and show numerically for 1D linear chains connected with Rubin baths, that transient behaviors significantly differs from each other but eventually leads to the same steady state distribution in the long-time limit. We give explicit expressions of the CGF in the steady state invoking the symmetry of translational invariance in time. The CGF obeys the GC symmetry. We also give the steady state expression for the CGF in the presence of time-dependent driving forces. We obtain a two parameter CGF which is useful for calculating the correlations between heat flux and also the total entropy which flows to the leads. Our calculations can be easily generalized to arbitrary dimensions with any number of heat baths. We will show in the appendix that our method can be extended for the electronic calculations where we derive the CGF for a tight-binding model. It will be interesting to derive the CGF by taking magnetic field contribution into the Hamiltonian and also to study the cumulants in the presence of nonlinear interactions such as phonon-phonon interactions or electron phonon interactions.\n\n\n\\section*{Acknowledgments}\nWe are grateful to Juzar Thingna, Meng Lee Leek, Zhang Lifa, and Li Huanan for insightful discussions. This work is supported in part by a URC research grant R-144-000-257-112 of National University of Singapore.\n\n\n\n\\section*{Appendix}\n\\subsection{Expressions for different type of Green's functions}\nHere we give the explicit expressions for the center Green's function $G_{0}[\\omega]$ in the steady state, for a harmonic system which is connected with the leads. These formulas are required to derive the analytical form of the CGF given in Eq.~(\\ref{eq-steady1}). For the basic definitions of different types of Green's functions we refer to Ref.~\\onlinecite{WangJS-europhysJb-2008}. \n\nThe retarded Green's function $G_{0}^{r}[\\omega]$ is given by\n\\begin{equation}\nG_{0}^{r}[\\omega]=\\Big[(\\omega+i\\eta)^{2}-K^{C}-\\Sigma_{L}^{r}[\\omega]-\\Sigma_{R}^{r}[\\omega]\\Big]^{-1}.\n\\end{equation}\nHere $\\eta$ is an infinitesimal positive number which is required to satisfy the condition of causality i.e.,$G_{0}^r(t)=0$ for $t <0 $.\nThe advanced Green's function is $G_{0}^{a}[\\omega]=\\big[G_{0}^{r}[\\omega]\\big]^{\\dagger}$. The Keldysh Green's function $G_{0}^{K}[\\omega]$ can be obtained by solving the corresponding Dyson equation, Eq.~(\\ref{eq-Dyson-product}), and is given by\n\\begin{equation}\nG_{0}^{K}[\\omega]=G_{0}^{r}[\\omega]\\Sigma^{K}[\\omega]G_{0}^{a}[\\omega],\n\\end{equation}\nwhere $\\Sigma^{K}=\\Sigma_L^{K}+\\Sigma_R^{K}$ and $\\Sigma_{\\alpha}^{K}=\\Sigma_{\\alpha}^{<}+ \\Sigma_{\\alpha}^{>}$ with $\\alpha=L,R$. Alternatively, $G_{0}^{K}=G_{0}^< +G_{0}^>$. Another important identity is\n\\begin{equation}\nG_{0}^{r}[\\omega]-G_{0}^{a}[\\omega]=-i\\, G_{0}^{r}[\\omega] \\big(\\Gamma_L[\\omega]+\\Gamma_R[\\omega]\\big)G_{0}^{a}[\\omega],\n\\end{equation}\nwhere $\\Gamma_{\\alpha}[\\omega]=i\\big(\\Sigma_{\\alpha}^{r}[\\omega]-\\Sigma_{\\alpha}^{a}[\\omega]\\big)$, and $\\alpha=L,R$. The self-energy for the leads are given by\n\\begin{eqnarray}\n\\Sigma_{\\alpha}^{<}[\\omega]&=&f_{\\alpha}[\\omega]\\big(\\Sigma_{\\alpha}^{r}[\\omega]-\\Sigma_{\\alpha}^{a}[\\omega]\\big), \\nonumber \\\\\n\\Sigma_{\\alpha}^{>}[\\omega]&=&(1+f_{\\alpha}[\\omega])\\big(\\Sigma_{\\alpha}^{r}[\\omega]-\\Sigma_{\\alpha}^{a}[\\omega]\\big).\n\\end{eqnarray}\nwhere $f_{\\alpha}[\\omega]=1\/\\bigl(e^{\\beta_{\\alpha} \\hbar \\omega_{\\alpha}}-1\\bigr)$ is the Bose distribution function. \n\nExplicit expressions for $G_{0}^{r}[\\omega]$ and $\\Sigma_{L}^{r}[\\omega]$ can be obtained for 1D homogeneous linear chain, with inter particle force constant $K$ and onsite spring constant $K_{0}$ and which is divided into three parts: the center, the left and the right. The classical equation of motion for the atoms in all three regions is\n\\begin{equation}\n\\ddot{u}_j=K u_{j-1} + \\bigl (-2K -K_{0}\\bigr )u_{j} + Ku_{j-1},\n\\end{equation}\nwhere the index $j$ runs over all the atoms in the full system.\n\nThe retarded Green's function $G_{0}^{r}[\\omega]$ can be obtained by solving \\cite{Wang-pre07} $[(\\omega+i\\eta)^{2}-\\tilde{K}]G_{0}^{r}=I$, where matrix $\\tilde{K}$ which is infinite in both directions and is $2K+K_{0}$ on the diagonals and $-K$ on the first off-diagonals. The solution is translationally invariant in space index and is given by \n\\be\nG_{0,jk}^{r}[\\omega]=\\frac{\\lambda^{|j-k|}}{K(\\lambda-\\frac{1}{\\lambda})},\n\\ee \nwith $\\lambda=-\\frac{\\Omega}{2K}\\pm \\frac{1}{2K}\\sqrt{\\Omega^{2}-4K^{2}}$ and $\\Omega=(\\omega+i\\eta)^{2}-2K-K_{0}$, choosing between plus and minus sign by $|\\lambda|\\le 1$. \n\nThe surface Green's function $g_L^{r}[\\omega]$ can be similarly obtained in frequency domain and is given in terms of the self-energy $\\Sigma_{L}^{r}[\\omega]=-K \\lambda$. Since in equilibrium only one Green's function is independent, knowing $\\Sigma_{L}^{r}[\\omega]$ is sufficient to obtain all other Green's functions.\n\nHere we also give the expressions for Green's functions $g_C$ in time and frequency domain for an isolated single harmonic oscillator with frequency $\\omega_{0}$ (we have omitted the subscript $C$ in $g_C$) \\cite{Zeng,Brouwer}\n\\begin{eqnarray}\ng^{r}(t) & = & -\\theta(t) \\, \\frac{\\sin{\\omega_{0}t}}{\\omega_{0}},\\nonumber \\\\\ng^{r}[\\omega] & = & \\frac{1}{(\\omega+i\\eta)^{2}-\\omega_{0}^{2}},\\nonumber \\\\\ng^{<}(t) & = & \\frac{-i}{2\\omega_{0}}\\left[(1+f)e^{i\\omega_{0}t}+fe^{-i\\omega_{0}t}\\right],\\nonumber \\\\\ng^{<}[\\omega] & = & \\frac{-i\\pi}{\\omega_{0}}\\left[\\delta(\\omega+\\omega_{0})(1+f)+\\delta(\\omega-\\omega_{0})f\\right],\n\\end{eqnarray} \nwhere $f=f(\\omega_{0})=\\frac{1}{e^{\\beta\\hbar\\omega_{0}}-1}$. Other components can be obtained by exploiting the symmetry between the Green's functions such as $g^a(-t)=g^r(t)$ for $t > 0$ hence $g^r[\\omega]=g^a[-\\omega]$. The greater component is related with the lesser component via $g^>(t)=g^<(-t)$ which in the frequency domain satisfy $g^>[\\omega]=g^<[-\\omega]$.\n\n\n\n\n\\subsection{Current at short time for product initial state $\\rho(-\\infty)$}\nUsing the definition of current operator given in Eq.~(\\ref{current}) the energy current flowing from the left lead to the center is (here we assume that there is no driving force $f(t)$) \n\\begin{equation}\n\\langle {\\cal I}_{L}(t)\\rangle =-\\langle \\frac{d{\\cal H}_L(t)}{dt} \\rangle =\\frac{i}{\\hbar} \\langle \\big[{\\cal H}_{L}(t),{\\cal H} \\big] \\rangle ,\n\\end{equation}\nwhere the average is with respect to $\\rho(-\\infty)$. If $t$ is small we can expand ${\\cal H}_{L}(t)$ in a Taylor series and is given by ${\\cal H}_{L}(t)={\\cal H}_{L}(0)+t \\dot{{\\cal H}}_{L}(0) +\\cdots $ \n\nNow since $\\big[\\rho(-\\infty), {\\cal H}_{L}(0) \\big]=0$, then it immediately follows that $\\langle \\big[{\\cal H}_{L}(0),{\\cal H}\\big]\\rangle =0$ by using the cyclic property of trace. So in linear order of $t$ the current is given by\n\\begin{equation}\n\\langle {\\cal I}_{L}(t)\\rangle=t \\frac{i}{\\hbar} \\langle \\big[{\\dot{\\cal H}}_{L}(0),{\\cal H}\\big]\\rangle= -t \\frac{i}{\\hbar}\\langle \\big[p_{L}^{T}V^{LC}u_C, {\\cal H}\\big]\\rangle.\n\\end{equation}\nThe only term of full ${\\cal H}$ that will contribute to the is ${\\cal H}_{LC}=u_{L}^T V^{LC} u_{C}$. \n\nNow using the relation that $\\big[p_L,u_L\\big]=-i\\hbar$, for one-dimensional linear chain we can write\n\\begin{equation}\n\\langle {\\cal I}_{L}(t) \\rangle =-t \\, K^{2} \\langle (u^{C}_{1})^{2} \\rangle = -t \\, K^{2} \\frac{\\hbar}{\\omega_{0}} \\Big(f_{C}(\\omega_{0})+\\frac{1}{2}\\Big).\n\\end{equation}\nwhere $u^{C}_{1}$ is the first particle in the center which is connected with the first particle of the left lead with force constant $K$. Now since the average is with respect to $\\rho(-\\infty)$,$\\langle (u^{C}_{1})^{2} \\rangle $ can be easily computed. Here $f_{C}(\\omega_{0})$ is the Bose distribution function of the particle with characteristic frequency $\\omega_{0}$. So we can see that for short time the current is negative, i.e, it goes into the lead. It is now easy to see that similar expression should also hold for $\\langle {\\cal I}_{R}(t)\\rangle$. The negative sign in currents means that the energy flows into the leads initially irrespect to the temperature of the leads. This is consistent with the numerical results obtained by Cuansing et al.\\ \\cite{eduardos-paper,eduardos-paper1}.\n\n\\subsection{\\label{apdC} Convolution, trace, and determinant on Keldysh contour}\nHere we discuss the meaning of convolution, trace and determinant on the Keldysh contour which we used to derive the CGF's for heat flux. We define the convolution on contour in the following way.\n\\begin{eqnarray}\nA B \\cdots D &\\rightarrow& \\sum_{j_2,j_3, \\cdots, j_n}\\int d\\tau_2 \\cdots \\int d\\tau_n A_{j_1,j_2}(\\tau_1, \\tau_2)\\nonumber \\\\\n&& B_{j_2,j_3}(\\tau_2, \\tau_3) \\cdots D_{j_n,j_{n+1}}(\\tau_n, \\tau_{n+1}), \n\\end{eqnarray}\nFrom the convolution we define trace by substituting $\\tau_{n+1}=\\tau_1$, $j_{n+1}=j_1$ and integrate also over $\\tau_1$, sum over $j_1$ i.e., \n\\begin{eqnarray}\n{\\rm Tr}_{j,\\tau}(AB \\cdots D)&=&\\int d\\tau_1 \\int d\\tau_2 \\cdots \\int d\\tau_n \\\\\n&& {\\rm Tr}_j\\bigl[ A(\\tau_1, \\tau_2) B(\\tau_2, \\tau_3) \\cdots D(\\tau_n, \\tau_1) \\bigr], \\nonumber \n\\end{eqnarray}\nChanging from contour to real-time integration from $-\\infty$ to $+\\infty$, i.e., using $\\int d\\tau = \\sum_{\\sigma} \\int \\sigma dt $ we have\n\\begin{eqnarray}\n{\\rm Tr}_{j,\\tau}(AB \\cdots D) = \\!\\!\\!\\sum_{\\sigma_1,\\sigma_2, \\cdots, \\sigma_n}\\!\\!\\! \\int dt_1 \\int dt_2 \\cdots \\int dt_n \\qquad \\\\\n{\\rm Tr}_j \\bigl[ A^{\\sigma_1\\sigma_2}(t_1, t_2) \\sigma_2 B^{\\sigma_2\\sigma_3}(t_2, t_3) \\cdots \\sigma_n D^{\\sigma_n\\sigma_{n+1}}(t_n, t_{1}) \\bigr]. \\nonumber \n\\end{eqnarray}\nLet us absorb the extra $\\sigma$ into the definition of branch components,\ni.e., define\n\\begin{equation}\n\\bar{A}_{\\sigma\\sigma'} = \\sigma A^{\\sigma\\sigma'}, \\quad{\\rm or}\\quad\n\\bar{A} = \\sigma_z A,\n\\end{equation}\nwhere $A$ is viewed as $2\\times 2$ block matrix with the usual\n$+$, $-$ component, \n\\begin{equation}\nA = \\left( \\begin{array}{cc}\n A^{++} & A^{+-} \\\\\n A^{-+} & A^{--} \n \\end{array} \\right) = \n\\left( \\begin{array}{cc}\n A^t & A^< \\\\\n A^> & A^{\\bar t} \n \\end{array} \\right),\n\\end{equation}\nand $\\sigma_z$ is defined as\n\\begin{eqnarray}\n\\sigma_z &=& \\left( \\begin{array}{cc}\n 1 & 0 \\\\\n 0 & -1 \n \\end{array} \\right),\n\\end{eqnarray}\nthen it can be easily seen that \n\\begin{eqnarray}\n{\\rm Tr}_{j,\\tau}(AB \\cdots D)&=& \\int dt_1 \\int dt_2 \\cdots \\int dt_n {\\rm Tr}_{j} \\bigl[{\\bar A}(t_1,t_2) \\nonumber \\\\\n&&{\\bar B}(t_2,t_3) \\cdots {\\bar D}(t_n,t_1) \\bigr] \\nonumber \\\\\n&&={\\rm Tr}_{t,j,\\sigma}(\\bar{A}\\bar{B} \\cdots \\bar {D}).\n\\end{eqnarray}\nThen we can do a rotation, where the rotation matrix is given by \n\\begin{eqnarray}\nO &=& \\frac{1}{\\sqrt{2}} \\left(\n\\begin{array}{cc}\n 1 & 1 \\\\\n -1 & 1 \n\\end{array} \\right), \\quad OO^T = I.\n\\end{eqnarray}\nand we define for any matrix $A$, the rotated matrix as \n\\begin{equation}\n\\breve{A} = O^T \\sigma_z A O = O^{T} \\bar{A} O.\n\\end{equation}\nThis is known as Keldysh rotation. The effect of Keldysh rotation is given in Eq.~(\\ref{keldysh-rotation}). Since this is an orthogonal transformation the trace remains invariant and hence we can write\n\\begin{eqnarray}\n{\\rm Tr}_{t,j,\\sigma}(\\bar{A}\\bar{B} \\cdots \\bar {D})&=& {\\rm Tr}_{t,j,\\sigma}(\\breve{A}\\breve{B} \\cdots \\breve{D}).\n\\end{eqnarray}\nIf we now go to the frequency domain using the definition of two-time Fourier transform given in Eq.~(\\ref{two-time-FT}) \nthen we can compute the trace in frequency domain as \n\\begin{eqnarray}\n{\\rm Tr}_{(j,\\tau)}(AB \\cdots D) &=& \\int\\! \\frac{d\\omega_1}{2\\pi} \\!\n\\int\\! \\frac{d\\omega_2}{2\\pi}\\! \\cdots\n\\int\\! \\frac{d\\omega_n}{2\\pi} \\!\n{\\rm Tr} \\bigl\\{ \\nonumber \\\\\n && \\breve{A}[\\omega_1, -\\omega_2] \n\\breve{B}[\\omega_2, -\\omega_3] \\cdots \\breve{D}[\\omega_n, -\\omega_1] \\bigr\\}\n\\nonumber \\\\ \n&=& {\\rm Tr}_{j,\\sigma,\\omega} ( \\breve{A} \\breve{B} \\cdots \\breve{D} ).\n\\end{eqnarray}\nThe last line above define what we mean by trace over frequency domain given in Eq.~(\\ref{eq-Zsteady}).\nUnlike trace in time domain, the second argument of the each of the variables\nneed a minus sign. \n\nLet us now define what do we mean by 1 on contour. In the sense of convolution we define 1 as\n\\begin{equation}\nA \\, 1\\, D =A \\, D\n\\end{equation}\nwhich means\n\\begin{equation}\n\\int d\\tau_1 \\!\\! \\int d\\tau_2\\, A(\\tau,\\tau_1) I \\delta(\\tau_1,\\tau_2) D(\\tau_2,\\tau')= \\int d\\tau_1 A(\\tau,\\tau_1) D(\\tau_1,\\tau'). \n\\end{equation}\nNote that $\\delta(\\tau,\\tau')$ in the real time has the following form\n\\begin{equation}\n\\delta^{\\sigma,\\sigma'}(t,t')=\\sigma \\delta_{\\sigma,\\sigma'} \\delta(t-t').\n\\end{equation}\nThe inverse on the contour is defined as\n\\begin{equation}\n\\int d\\tau_1 A(\\tau,\\tau_1) B(\\tau_1,\\tau') = I \\delta(\\tau,\\tau'),\n\\end{equation}\nwhere the identity matrix $I$ takes care about the space index. Similar to the above we go to the real time and multiply the above equation with the branch index $\\sigma$ and we can write,\n\\begin{equation} \n\\int dt_1 {\\bar A}(t,t_1) {\\bar B}(t_1,t') = I \\bar{\\delta}(t-t').\n\\end{equation}\nwhere \n\\begin{eqnarray}\n\\bar{\\delta}(t-t')&=&\\sigma \\delta^{\\sigma,\\sigma'}(t,t')=\\sigma^{2} \\delta_{\\sigma,\\sigma'} \\delta(t-t') \\nonumber \\\\\n&&=\\delta_{\\sigma,\\sigma'} \\delta(t-t')\n\\end{eqnarray}\nIf we now discretize the time and write $\\delta(t_i,t_{i'})=\\delta_{i,i'}\/\\Delta t$ with $\\Delta t= |t_i-t_{i'}|$ then we have\n\\begin{equation}\n\\tilde{A} \\tilde{B} =\\tilde{I}.\n\\end{equation}\nwith $\\tilde{A}= A \\Delta t$ and similarly for other matrices. \n\nWith similar notions we can now write different types of Dyson's equation given in Eq.~(\\ref{eq-Dyson-full},\\ref{eq-Dyson-product}) as following. In contour time we have \n\n\\begin{eqnarray}\nG_0(\\tau, \\tau') &=& g_C(\\tau,\\tau') \\\\\n&&\\> + \\int \\!\\int d\\tau_1d \\tau_2\\, \ng_C(\\tau, \\tau_1) \\Sigma(\\tau_1, \\tau_2) G_0(\\tau_2, \\tau'), \\nonumber\n\\end{eqnarray}\nIn real time following the above arguments we write\n\\begin{eqnarray}\n\\bar{G}_{0}(t,t') &=& \\bar{g}_C(t,t') \\\\\n&&\\> + \\int \\!\\int dt_1 dt_2\\, \n{\\bar g}_C(t, t_1) {\\bar \\Sigma}(t_1, t_2) {\\bar G}_0(t_2, t'), \\nonumber\n\\end{eqnarray}\nAfter Keldysh rotation we can write \n\\begin{eqnarray}\n\\breve{G}_{0}(t,t') &=& \\breve{g}_C(t,t') \\\\\n&&\\> + \\int \\!\\int dt_1 dt_2\\, \n{\\breve g}_C(t, t_1) {\\breve \\Sigma}(t_1, t_2) {\\breve G}_0(t_2, t'). \\nonumber\n\\end{eqnarray}\nFinally in the discretize time $t$ we write \n\\begin{equation}\n\\tilde{G}_{0}=\\tilde{g}_C + \\tilde{g}_C \\tilde{\\Sigma} \\tilde{G}_{0},\n\\end{equation}\nwhich is a matrix equation. Similar equations can also be written down for Eq.~(\\ref{eq-Dyson-full}).\n \nNow we define determinant via the relation $\\det(A)=\\exp({\\rm Tr} \\ln A)$, i.e, the determinant is defined in terms of trace. In order for $\\ln A$ to be defined we have to assume a Taylor expansion. For example we can define $\\ln(1+M)=M-M^2\/2 + M^3\/3 + \\cdots $ where 1 means $\\delta_{jj'}\\delta(\\tau,\\tau')$ in contour space.\n\n\\subsection{A quick derivation of the Levitov-Lesovik formula for electrons using NEGF}\nThe generating function for the non-interacting electrons was first derived by Levitov and Lesovik \\cite{Levitov,Levitov1} using Landauer type of wave scattering approach. Klich \\cite{Klich} and Sch\\\"onhammer \\cite{otherworks2} re-derived the formula using a trace and determinant relation to reduce the problem from many-body problem to a single particle Hilbert space problem. Esposito et al.\\ gave an approach using the superoperator nonequilibrium Green's function \\cite{Esposito-review-2009}. A more rigorous treatment is given in Ref.~\\onlinecite{Bernard}.\n\nOur method for calculating CGF can be easily extended for the electron case. Here we will derive the CGF for the joint probability distribution for particle and energy without time-dependent driving force. \nThe Hamiltonian of the whole system can be written as (using tight-binding model)\n\\begin{equation}\n{\\cal H}^{e}=\\sum_{\\alpha=L,C,R} c_{\\alpha}^{\\dagger} h^{\\alpha} c_{\\alpha} + \\sum_{\\alpha=L,R} \\big(c_{\\alpha}^{\\dagger} V_{e}^{\\alpha C} c_{C} + {\\rm h.c.}\\big)\n\\end{equation}\nwhere $c_{\\alpha}$ is a column vector consisting of all the annihilation operator of region $\\alpha$. $c_{\\alpha}^{\\dagger}$ is a row vector of the corresponding creating operators. $h^{\\alpha}$ is the single particle Hamiltonian matrix. $V_{e}^{\\alpha C}$ has similar meaning as $V^{\\alpha C}$ in the phonon Hamiltonian and $V_{e}^{\\alpha C}=(V_{e}^{C\\alpha})^{\\dagger}$. \n\nWe are interested in calculating the generating function corresponding to the particle operator ${\\cal N}_L$ and energy operator ${\\cal H}_L$ of the left-lead where ${\\cal H}_L= c_{L}^{\\dagger} h^{L} c_L$ and ${\\cal N}_L= c_{L}^{\\dagger} c_L$ \\cite{Hanggi1}. One can easily generalize the formula for right lead also as we did in the phonon case. For electrons ${\\cal N}_L$ and ${\\cal H}_L$ can be measured simultaneously because they commute, i.e., $\\big[{\\cal H}_L, {\\cal N}_L \\big]=0$. In order to calculate the CGF we introduce two counting fields $\\xi_p$ and $\\xi_e$ for particle and energy respectively. Here we will consider the product initial state (with fixed temperatures and chemical potentials for the leads) and derive the long-time result.\n\nSimilar to the phonon case we can write the CGF as \n\\begin{equation}\n{\\cal Z}(\\xi_e,\\xi_p)= \\Big \\langle e^{i\\big(\\xi_e {\\cal H}_L + \\xi_p {\\cal N}_L \\big)}\\,e^{-i\\big(\\xi_e {\\cal H}^{H}_L + \\xi_p {\\cal N}^{H}_L \\big)} \\Big \\rangle,\n\\end{equation}\nwhere superscript $H$ means the operators are in the Heisenberg picture at time $t$. In terms of modified Hamiltonian the CGF can be expressed as\n\\begin{equation}\n{\\cal Z}(\\xi_e,\\xi_p)= \\Big \\langle {\\cal U}_{(\\frac{\\xi_e}{2},\\frac{\\xi_p}{2})} (0,t) \\, {\\cal U}_{(-\\frac{\\xi_e}{2},-\\frac{\\xi_p}{2})} (t,0) \\Big \\rangle,\n\\end{equation} \nwhere \n\\begin{eqnarray}\n{\\cal U}_{x,y}(t,0)&=& e^{i x {\\cal H}_L + i y {\\cal N}_L}\\, {\\cal U}(t,0)\\,e^{-i x {\\cal H}_L - i y {\\cal N}_L} \\nonumber \\\\\n&&= e^{-\\frac{i}{\\hbar} {\\cal H}_{x,y} t}\n\\end{eqnarray}\nwith $x=\\xi_e\/2$ and $y=\\xi_p\/2$ and ${\\cal U}(t,0)=e^{-i {\\cal H} t}$. ${\\cal H}_{x,y}$ is the modified Hamiltonian which evolves with both ${\\cal H}_L$ and ${\\cal N}_L$ and is given by\n\\begin{eqnarray}\n{\\cal H}_{x,y}&=& e^{i x {\\cal H}_L + i y {\\cal N}_L}\\, {\\cal H} \\,e^{-i x {\\cal H}_L - i y {\\cal N}_L} \\nonumber \\\\\n&&= {\\cal H}_L + {\\cal H}_C + {\\cal H}_R + \\big(e^{iy} c_{L}^{\\dagger}(\\hbar x) V_{e}^{LC} c_C +{\\rm h.c.}\\big)\\nonumber \\\\\n&& + \\big(c_R^{\\dagger} V_{e}^{RC} c_{C} + {\\rm h.c.}\\big),\n\\end{eqnarray}\nwhere we have used the fact that\n\\begin{eqnarray}\ne^{i x {\\cal H}_L} c_{L}(0) e^{-i x {\\cal H}_L} &=& c_{L}(\\hbar x), \\nonumber \\\\\ne^{i y {\\cal N}_L} c_{L}(0) e^{-i y {\\cal N}_L} &=& e^{-i y} c_{L}.\n\\end{eqnarray}\nSo the evolution with ${\\cal H}_L$ and ${\\cal N}_L$ is to shift the time-argument and produce a phase for $c_{L},c_{L}^{\\dagger}$ respectively. Next we go to the interaction picture of the modified Hamiltonian ${\\cal H}_{x,y}$ with respect to ${\\cal H}_{0}= \\sum_{\\alpha=L,C,R} {\\cal H}_{\\alpha}$ and the CGF then can be written on the contour running from 0 to $t_M$ and back as,\n\\begin{equation}\n{\\cal Z}(\\xi_e,\\xi_p)= {\\rm Tr}\\Big[\\rho(-\\infty) T_{c} e^{-\\frac{i}{\\hbar} \\int d\\tau {\\cal V}_{x,y}^{I}(\\tau)} \\Big],\n\\end{equation} \nwhere ${\\cal V}_{x,y}^{I}(\\tau)$ is written in contour time. \n\\begin{eqnarray}\n{\\cal V}_{x,y}^{I}(\\tau)&=&\\big ( e^{i y} c_{L}^{\\dagger}(\\tau+\\hbar x) V_{e}^{LC} c_C (\\tau)+ {\\rm h.c.} \\big)+ \\nonumber \\\\\n&& \\big(c_{R}(\\tau)^{\\dagger} V_{e}^{RC} c_C(\\tau) + {\\rm h.c.}\\big).\n\\end{eqnarray}\nNow we can expand the exponential in the generating function and use Feynman diagrams to sum the series and finally the CGF can be shown to be\n\\begin{equation}\n\\ln {\\cal Z}(\\xi_e,\\xi_p)={\\rm Tr}_{j,\\tau} \\ln \\Big[1-G^{e}_{0} \\Sigma_{L,e}^{A} \\Big],\n\\end{equation}\nwhere we define the shifted self-energy for the electron case as\n\\begin{equation}\n\\Sigma_{L,e}^{A}(\\tau,\\tau')=e^{i (y(\\tau')-y(\\tau))} \\Sigma_{L,e}(\\tau+\\hbar x, \\tau'+\\hbar x') -\\Sigma_{L,e}(\\tau,\\tau').\n\\end{equation}\nThe counting of the electron number is associated with factor of a phase, while\nthe counting of the energy is related to translation in time.\nNote that the CGF does not have the characteristic $1\/2$ pre-factor as compared to the phonon case because $c$ and $c^{\\dagger}$ are independent\nvariables. In the long-time limit following the same steps as we did for phonons, the CGF can be written down as (after doing Keldysh rotation) \n\\begin{equation}\n\\ln {\\cal Z}(\\xi_e,\\xi_p)=t_M \\int \\frac{dE}{2\\pi \\hbar} {\\rm Tr} \\ln \\big(I- \\breve{G}^{e}_{0}(E)\\breve{\\Sigma}_{L,e}^{A}(E)\\big).\n\\end{equation}\nIn the energy $E$ domain different components of the shifted self-energy are\n\\begin{eqnarray}\n\\Sigma_{A}^{t}(E)&=&\\Sigma_{A}^{\\bar{t}}(E)=0, \\nonumber \\\\\n\\Sigma_{A}^{<}(E)&=& \\big( e^{i (\\xi_p + \\xi_e E)} -1 \\big) \\Sigma_{L}^{<}(E), \\nonumber \\\\\n\\Sigma_{A}^{>}(E)&=& \\big( e^{-i (\\xi_p + \\xi_e E)} -1 \\big) \\Sigma_{L}^{>}(E). \n\\end{eqnarray}\nFinally the CGF can be simplified as\n\\begin{eqnarray}\n\\ln {\\cal Z}&=&t_M \\int \\frac{dE}{2\\pi\\hbar} \\,\\ln \\det \\Bigl\\{ I + G_{0}^r \\Gamma_L \nG_{0}^a \\Gamma_R \\Big[(e^{i\\alpha}\\! -\\! 1)f_L \\nonumber \\\\ \n&&+ ( e^{-i\\alpha} \\!-\\! 1) f_R - (\ne^{i\\alpha} \\!+\\! e^{-i\\alpha} \\!-\\!2 ) f_L f_R \\Big]\\Bigr\\}.\\qquad .\n\\end{eqnarray}\nwhere $\\alpha=\\xi_p+ \\xi_e E$ and $f_\\alpha$ is the Fermi distribution. Note the difference of the signs in the CGF as compared to the phonons. If we replace $\\alpha$ by $(E-\\mu_L) \\xi$, the resulting formula is for the counting of the heat \n${\\cal Q}_L = {\\cal H}_L - \\mu_L {\\cal N}_L$ transferred, where $\\mu_L$ is the chemical potential of the left lead.\nThe CGF obeys the following fluctuation symmetry \\cite{fluct-theorem-1}\n\\begin{equation}\n{\\cal Z}(\\xi_e,\\xi_p)= {\\cal Z}\\big(-\\xi_e + i(\\beta_R-\\beta_L), -\\xi_p -i (-\\beta_R \\mu_R -\\beta_L \\mu_L)\\big).\n\\end{equation}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n The century-old general relativity (GR) theory still successfully passes \n all local experimental tests. However, there are many reasons to consider \n this theory not as an ultimate theory of gravity but only as a reasonable \n approximation well working in a large but finite range of length and energy \n scales. Among such reasons are the old problem of unifying gravity with other \n physical interactions and the difficulties in attempts to quantize GR. \n Other reasons for dealing with modifications of GR are the well-known problems \n experienced by the theory itself: its prediction of space-time singularities\n in the most physically relevant solutions, actually showing situations where \n the theory does not work any more, and its inability to explain the\n main observable features of the Universe without introducing so far invisible \n forms of matter, dark matter (DM) and dark energy (DE), which \n add up to as much as 95\\,\\% of the energy content of the Universe.\n \n The existing modifications and extensions of classical GR can be divided into \n two large classes. The first one changes the geometric content of the theory\n and includes, in particular, $f(R)$ theories, multidimensional theories and \n non-Riemannian geometries. The second class introduces new fundamental, \n non-geometric fields and includes, in particular, scalar-tensor theories, \n Horndeski theory \\cite{horn} and vector-tensor theories. Of much interest are the \n cases where it is possible to establish connections between different representatives \n of the same class or even different classes of theories (possibly the most well-known \n example of such a connection is the equivalence of $f(R)$ theories with a \n certain subclass of scalar-tensor theories, see, e.g., \\cite{f(R),odin}). In the present paper \n we discuss such an equivalence between large families of models of k-essence theories \n and Rastall's non-conservative theories with a scalar field as a source of gravity. \n\n The k-essence theories, introducing a non-stan\\-dard form of the kinetic term \n of a scalar field \\cite{k1, k2}, evidently belong to the class of theories with\n non-standard fundamental fields coupled to gravity. They proved to be a way of \n obtaining both early inflation and the modern accelerated expansion of the \n Universe \\cite{k2, k3, k4} driven by a scalar kinetic term instead of a \n potential. Notably, a kind of k-essence structure also appears in string theories,\n for example, in the Dirac-Born-Infeld action, where the kinetic term of the \n scalar field has a structure similar to that of the Maxwell-like term in Born-Infeld \n electrodynamics \\cite{dbi}.\n\n Rastall's theory \\cite{rastall} is one more generalization of GR, which relaxes the\n conservation laws expressed by the zero divergence of the stress-energy tensor (SET) \n $T\\mN$ of matter. In this theory, the quantity $\\nabla_\\nu T\\mN$ is linked to the \n gradient of the Ricci scalar, and in this way Rastall's theory may be viewed as \n a phenomenological implementation of some quantum effects in a curved background.\n Rastall's theory leads to results of interest in cosmology, e.g., the evolution of \n small DM fluctuations is the same as in the $\\Lambda$CDM model, but DE is able to \n cluster. This might potentially provide an evolution of DM inhomogeneities in the non-linear \n regime different from the standard CDM model \\cite{cosmo}. The whole success of the \n $\\Lambda$CDM model is reproduced at the background and linear perturbation levels, \n but new effects are expected in the non-linear regime, where the $\\Lambda$CDM model\n faces some difficulties \\cite{cusp, satt}. It has also been shown \n \\cite{oliver} that Rastall's theory with the canonical SET of a scalar field, in the context of \n cosmological perturbations, is only consistent if matter is present. An interesting \n observation in this analysis is that scalar field coupling with gravity leads to equations \n very similar to those in some classes of Galileon theories.\n\n A consideration of static, spherically symmetric solutions in k-essence theories with a power law\n kinetic function \\cite{denis}\n and similar solutions in Rastall's theory in the presence of a free or self-interacting scalar field\n \\cite{we-16} has shown that some exact solutions of these two theories describe quite the same \n geometries, although the properties of the scalar fields are different. \n Also, a no-go theorem concerning the possible emergence of Killing horizons, proved in the \n k-essence framework \\cite{denis}, has its counterpart in the Rastall-scalar field system\n \\cite{we-16}. These similarities indicate a deeper relationship between the two theories,\n to be analyzed in this paper. We will show that in the absence of other matter than the \n (possibly self-interacting) scalar fields, the two theories lead to completely coinciding \n geometries (we will call this {\\sl k-R duality}) under the assumptions that the relevant \n quantities depend on a single spatial or temporal coordinate and that the k-essence theory \n is specified by a power-law function; then, there emerges a simple \n relation between the numerical parameters of the theories. If there are other forms of matter, \n the situation is more involved and depends on how the non-conservation of the SET is \n distributed between different matter contributions in Rastall's theory. We will discuss two \n variants of such non-conservation in the case of isotropic spatially flat cosmological models\n and show that k-R duality generically takes place. \n \n The paper is organized as follows. In the next section we discuss vacuum solutions of \n k-essence and Rastall theories. In Section 3, isotropic cosmological models are analyzed \n with a matter source in the form of a perfect fluid. Some considerations on the speed of \n sound are presented in Section 4, while Section 5 is devoted to some special values of \n the numerical parameters of both theories. Some concrete cosmological configurations \n with dustlike matter are discussed in Section 6. Our conclusions are presented in Section 7.\n\n\\section{Scalar-vacuum space-times}\n\n\\subsection{k-essence}\n\n The k-essence theories can be defined as \\GR\\ with generalized forms \n of scalar fields minimally coupled to gravity. In the absence of matter \n nonminimally coupled to gravity, the most general Lagrangian is\n\\begin{eqnarray} \\lal \t\t\t\\label{L-k}\n\t{\\cal L} = \\sqrt{-g}[R + F(X,\\phi) + L_m],\n\\end{eqnarray}\n with\n\\begin{eqnarray} \\label{X}\n\tX = \\eta\\phi_\\mu\\phi^\\mu,\n\\end{eqnarray}\n where $\\phi_\\mu = \\partial_\\mu \\phi$, $F(X,\\phi)$ is an arbitrary function, and\n $\\eta = \\pm 1$ is used to make $X$ positive since otherwise in the cases\n like general power-law dependence $F$ will be ill-defined for $X < 0$;\n $L_m$ is the Lagrangian density of other kinds of matter having no direct \n coupling to the curvature or the $\\phi$ field.\n We are using the system of units where $c = 8\\pi G =1$\n\n Variation of the Lagrangian (\\ref{L-k}) with respect to the metric and\n the scalar field leads to the field equations\n\\begin{eqnarray} \\lal \t\t \\label{EE}\n\tG\\mN \\equiv R\\mN - {\\fract{1}{2}}\\delta\\mN R = -T\\mN [\\phi] - T\\mN[m],\n\\\\[5pt] \\lal \\label{T-phi}\n\tT\\mN [\\phi] \\equiv \\eta F_X \\phi_\\mu \\phi^\\nu\n\t\t- \\frac{1}{2} \\delta\\mN F ,\n\\\\[5pt] \\lal \t\t\t\t \t\t \\label{eq-phi}\n \\eta\\nabla_\\alpha (F_X \\phi^\\alpha) - {\\fracd{1}{2}}\\, F_\\phi = 0,\n\\end{eqnarray}\n where $G\\mN$ is the Einstein tensor, $F_X = \\partial F\/\\partial X$, $F_\\phi = \\partial F\/\\partial\\phi$,\n and $T\\mN[m]$ is the SET of matter due to $L_m$.\n\n Now, let us make the following assumptions: \\\\[5pt] {}\n{\\bf (i)} the k-essence Lagrangian is\n\\begin{equation} \\label{L-k1}\n F (X, \\phi) = F_0 X^n - 2V(\\phi),\n\\end{equation}\n where $n = {\\rm const} \\ne 0$ and $V(\\phi)$ is an arbitrary function (the potential).\n\\\\[5pt] {}\n{\\bf (ii)} $\\phi = \\phi(u)$, where $u$ is one of the coordinates, which may be temporal or spatial.\n\\\\[5pt] {}\n{\\bf (iii)} The metric has the form \n\\begin{equation} \\label{ds}\n ds^2 = \\eta {\\,\\rm e}^{2\\alpha(u)} du^2 + h_{ik} dx^i dx^k, \n\\end{equation} \n where $i, k$ are the numbers of coordinates other than $u$, and the determinant of $h_{ik}$\n has the factorized structure\n\\begin{equation} \\label{h_ik}\n \\det (h_{ik}) = {\\,\\rm e}^{2\\sigma(u)} h_1(x^i).\n\\end{equation}\n\n In this case, we have $X = {\\,\\rm e}^{-2\\alpha(u)} \\phi_u^2$ (the index $u$ means $d\/du$), \n and the SET of the $\\phi$ field has the following nonzero components:\n\\begin{eqnarray} \\lal \\label{uu-k}\n T^u_u [\\phi] = (n - {\\fract{1}{2}}) F_0 X^n + V,\n\\\\[5pt] \\lal \\label{ii-k}\n T^\\ui_i [\\phi] = - {\\fract{1}{2}} F_0 X^n + V \n\\end{eqnarray}\n (there is no summing over an underlined index). The scalar field equation has the form\n\\begin{eqnarray} \\lal \\label{eq-k1}\n {\\,\\rm e}^{-2n\\alpha} \\phi_u^{2n-2} \\big[(2n-1)\\phi_{uu} + \\sigma_u \\phi_u \n\\nonumber\\\\ \\lal \\hspace*{1cm}\n - (2n-1) \\alpha_u\\phi_u \\big] = - \\frac {1}{nF_0} \\frac {d V}{d\\phi}.\n\\end{eqnarray}\n\n\\subsection{Rastall's theory with a scalar field}\n\n Rastall's theory of gravity is characterized by the following equations \\cite{rastall}:\n\\begin{eqnarray} \\lal \\label{EE-Ra}\n\tR\\mN - \\frac{\\lambda}{2}\\delta\\mN R = - T\\mN,\n\\\\[5pt] \\lal \n\t\\nabla_\\nu T\\mN = \\frac{\\lambda - 1}{2}\\partial_\\mu R,\n\\end{eqnarray}\n where $\\lambda$ is a free parameter and $T\\mN$ is the SET of matter. \n At $\\lambda = 1$, GR is recovered.\n\n These equations can be rewritten as\n\\begin{eqnarray} \\lal \t\\label{EE-R1}\n\tG\\mN = - \\biggr\\{T\\mN - \\frac{b - 1}{2}\\delta\\mN T\\biggl\\} \\equiv - \\tT\\mN,\n\\\\[5pt] \\lal \\hspace*{-0.5em} \\label{cons-R} \n \\nabla_\\nu T\\mN = \\frac{b {-} 1}{2}\\partial_\\mu T,\\quad \n b: = \\frac{3\\lambda - 2}{2\\lambda - 1}, \\quad T = T^\\alpha_\\alpha.\n\\end{eqnarray}\n In this parametrization, GR is recovered if $b = 1$.\n\n Let us consider matter in the form of a minimally coupled scalar field $\\psi$, so that \n\\begin{equation} \\label{SET-psi}\n T\\mN[\\psi] = \\epsilon (\\psi_\\mu \\psi^\\nu - {\\fract{1}{2}} \\delta\\mN \\psi_\\alpha \\psi^\\alpha)\n + \\delta\\mN W(\\psi), \n\\end{equation}\n where $\\epsilon = \\pm 1$, indicating an ordinary ($+1$) or phantom ($-1$) nature \n of the $\\psi$ field, $\\psi_\\mu \\equiv \\partial_\\mu\\psi$, and $W(\\psi)$ is a potential. \n The scalar field equation follows from \\rf{cons-R} and has the form\n\\begin{equation} \\label{e-psi-R}\n \\Box\\psi + (b - 1)\\frac{\\psi^\\mu \\psi^\\nu \\nabla_\\mu \\psi_\\nu}\n {\\psi^\\alpha \\psi_\\alpha} = - \\epsilon (3 - 2b) \\frac {dW}{d\\psi}.\n\\end{equation}\n \n Let us now, in full similarity with what was done for k-essence theory, assume that \n $\\psi = \\psi(u)$ and the metric has the form \\rf{ds}. Then the nonzero components \n of the modified scalar field SET in the right-hand side of \\eqn{EE-R1} are\n\\begin{eqnarray} \\lal \\label{uu-R}\n \\tT^u_u [\\psi] = {\\fract{1}{2}} \\epsilon b \\eta {\\,\\rm e}^{-2\\alpha} \\psi_u^2 + (3-2b) W(\\psi),\n \\\\[5pt] \\lal \\label{ii-R}\n\t \\tT^{\\ui}_i [\\psi] = {\\fract{1}{2}} \\epsilon (b-2) \\eta {\\,\\rm e}^{-2\\alpha} \\psi_u^2 + (3-2b) W(\\psi),\n\\end{eqnarray} \n while the scalar field equation \\rf{e-psi-R} takes the form\n\\begin{equation} \\label{epsi-R2}\n {\\,\\rm e}^{-2\\alpha}\\big[b \\psi_{uu} + \\psi_u (\\sigma_u - b\\alpha_u)\\big]\n = - \\epsilon \\eta (3-2b) W_\\psi,\n\\end{equation}\n where $W_\\psi \\equiv dW\/d\\psi$ and, as before, $\\eta = \\mathop{\\rm sign}\\nolimits g_{uu}$.\n\n\\subsection{Comparison}\n\n We assume that in the k-essence system there is no other matter than the scalar \n field $\\phi$ and in the Rastall system there is no other matter than the scalar $\\psi$. \n Let us find out under which conditions the right-hand sides of the \\EE s \\rf{uu-k}\n and \\rf{ii-k} coincide with those of the effective \\EE s of Rastall's theory,\n \\rf{uu-R} and \\rf{ii-R}. This will guarantee that the solutions for the metric are \n also the same. \n\n To begin with, we identify the potentials:\n\\begin{equation} \\label {VW}\n V(\\phi) = (3-2b) W(\\psi).\n\\end{equation} \n Next, we equate the ratios of the kinetic parts of $T\\mN[\\phi]$ and $\\tT\\mN[\\psi]$,\n to obtain \n\\begin{equation} \\label{nb}\n \\frac{2n-1}{-1} = \\frac b {b-2} \\ \\ \\Rightarrow\\ \\ (2-b) n = 1. \n\\end{equation}\n Then, equating the kinetic parts themselves, we find that\n\\begin{eqnarray} \\lal \\label{phips}\n \\psi^2_u = \\epsilon\\eta n F_0 \\phi_u^{2n}{\\,\\rm e}^{2(1-n)\\alpha}.\n\\end{eqnarray} \n\n Under the three conditions \\rf{VW}--\\rf{phips}, the metric field equations of the \n two theories completely coincide, therefore their sets of solutions are also identical.\n Substituting \\rf{phips} to \\rf{epsi-R2}, one can easily verify that under these conditions \n the scalar field equations \\rf{eq-k1} and \\rf{epsi-R2} are also equivalent.\n\n This general result covers many static symmetries (spherical, plane, cylindrical, etc.),\n homogeneous cosmologies (FRW, all Bianchi types, \\KS) and even inhomogeneous \n ones if their metrics are of the form \\rf{ds}, \\rf{h_ik}.\n\n Here and in most of the paper we consider the generic values of the parameters $n$ and \n $b$ and exclude from consideration their special values that require a separate analysis, \n such as, for example, $b=0$, $b=3\/2$ and $n = 1\/2$. Some remarks on these special cases \n will be made in Section 5. \n\n\\section{Cosmology with matter}\n \n When, besides the scalar field, matter is present, it is better, for evident technical reasons, \n to restrict ourselves from the beginning to a certain type of metrics. We will consider \n cosmological FLRW spatially flat metrics\n\\begin{equation} \\label{ds1}\n\t ds^2 = dt^2 - a(t)^2[dx^2 + dy^2 + dz^2],\n\\end{equation}\n so that in \\rf{ds} and \\rf{h_ik} we have $\\eta=1$, ${\\,\\rm e}^\\alpha =1$, and ${\\,\\rm e}^\\sigma = a(t)^3$.\n Matter will be taken in the form of a perfect fluid, so that \n\\begin{equation} \\label{Tm}\n\t T\\mN [m] = \\mathop{\\rm diag}\\nolimits (\\rho, -p, -p, -p), \n\\end{equation} \n where $\\rho$ is the density and $p$ is the pressure.\n\n\\subsection{k-essence cosmology}\n \n In the FLRW metric \\rf{ds1} and with $\\phi=\\phi(t)$, the field equations \\rf{EE}--\\rf{eq-phi} \n with matter (where we denote $\\rho = \\rho_k,\\ p=p_k$) take the form\n\\begin{eqnarray} \\lal \\label{tt-k1}\n \t3H^2 = {\\fract{1}{2}}(2n - 1) F_0\\dot\\phi^{2n} + V(\\phi) + \\rho_k,\n\\\\[5pt] \\lal \\label{ii-k1}\n 2\\dot H + 3H^2 = - {\\fract{1}{2}} F_0 \\dot\\phi^{2n} + V(\\phi) - p_k,\n\\\\[5pt] \\lal \\label{phi-k1} \n \\dot\\phi^{2n-2} [(2n-1)\\ddot\\phi + 3H\\dot\\phi] = - \\frac{1}{nF_0}V_\\phi,\n\\end{eqnarray}\n where $H =\\dot a\/a$ is the Hubble parameter and $V_\\phi \\equiv dV\/d\\phi$. The SET \n of matter satisfies the conservation law $\\nabla_\\nu T\\mN[m] =0$, whence\n\\begin{equation} \\label{cons}\n\t\\dot\\rho_k + 3H (\\rho_k + p_k) =0.\n\\end{equation} \n \n\\subsection{Rastall cosmology with a scalar field and matter}\n\\def\\trho{{\\widetilde \\rho}}\n\\def\\tp{{\\widetilde p}}\n\n The Rastall equations have the form \\rf{EE-R1} and \\rf{cons-R}, where now \n $T\\mN$ is the total energy-momentum tensor,\n\\begin{eqnarray}\n T\\mN = T\\mN [\\psi] + T\\mN [m],\n\\end{eqnarray}\n with $T\\mN[\\psi]$ given by \\rf{SET-psi} and $T\\mN[m]$ by \\rf{Tm}. \n In Eqs.\\, \\rf{EE-R1}, the modified energy-momentum tensor $\\tT\\mN$ is then a sum\n of $\\tT\\mN[\\psi]$ given by \\rf{uu-R} and \\rf{ii-R} and $\\tT\\mN[m]$ with the components\n\\begin{eqnarray} \\lal \\label {redef}\n\t \\tT^t_t [m] = {\\fracd{1}{2}} [(3-b) \\rho + 3 (b-1) p] \\equiv \\trho,\n\\nonumber\\\\ \\lal \n\t \\tT^\\ui_i [m] = {\\fracd{1}{2}} [(1-b)\\rho + (3b-5)p] \\equiv - \\tp \n\\end{eqnarray} \n (we preserve the notation $\\rho$ and $p$ without indices for matter in Rastall gravity). \n Hence the Rastall equations read\n\\begin{eqnarray} \\lal \\label{tt-Ra}\n 3 H^2 = {\\fracd{1}{2}} \\epsilon b \\dot\\psi{}^2 + (3-2b) W + \\trho,\n\\\\[5pt] \\lal \\label{ii-Ra}\n\t 2\\dot H + 3H^2 = {\\fracd{1}{2}} \\epsilon (b-2) \\dot\\psi{}^2 + (3-2b) W - \\tp,\n\\end{eqnarray}\n while the equation for $\\psi$ depends of further assumptions on how the nonconservation \n of the full SET according to \\rf{cons-R} is distributed between $\\psi$ \n and matter. One can notice that \n\\begin{equation} \\label{NEC-R}\n\t\\trho + \\tp = \\rho + p,\n\\end{equation}\n and \n\\begin{equation} \\label{trace}\n \\trho - \\rho = p - \\tp = \\frac{1-b}{2} (\\rho - 3p). \n\\end{equation}\n\n Let us consider two (of an infinite number of) alternatives in incorporating matter \n to Rastall's theory with a scalar field: \n\\begin{description}\n\\item[R1:] \n The SETs of $\\psi$ and matter obey \\rf{cons-R} each separately, so there is\n no mixing between the two sources of gravity;\n\\item[R2:]\n The SET of matter is conservative, so that \n\\begin{equation} \n \\dot\\rho + 3H (\\rho+p) =0. \n\\end{equation}\n\\end{description}\n\n\\subsection{Case R1: No mixing of scalar field and matter} \n\n In this case we have\n\\begin{eqnarray} \\lal \\label{cons-i1}\n \\nabla_\\nu T\\mN[\\psi] = \\frac{b - 1}{2}\\partial_\\mu T[\\psi],\n\\\\[5pt] \\lal \\label{cons-i2}\n \\nabla_\\nu T\\mN[m] = \\frac{b - 1}{2}\\partial_\\mu T[m],\n\\end{eqnarray}\n The first of these conditions leads to the scalar field equation \\rf{epsi-R2}, which in \n the present case reads\n\\begin{equation} \\label{epsi-i} \n\tb \\ddot\\psi + 3H \\dot\\psi = -\\epsilon (3-2b) W_\\psi.\n\\end{equation}\n With \\rf{NEC-R}, the condition \\rf{cons-i2} has the form \n\\begin{equation} \\label{cons-i}\n\t\\dot\\trho + 3H (\\rho + p) =0.\n\\end{equation} \n The full set of equations consists of \\rf{tt-Ra}, \\rf{ii-Ra}, \\rf{epsi-i}, and \\rf{cons-i}, \n with the definitions \\rf{redef}.\n \n From \\rf{NEC-R} it follows that if matter satisfies the null energy condition (NEC),\n then the same is true for the ``effective'' density and pressure ($\\trho$ and $\\tp$) in \n Rastall's theory. However, from \\rf{redef} it can be verified that the positivity of \n the energy density (or pressure) is not guaranteed in the k-essence case if it is \n imposed in the Rastall theory, and vice versa.\n\n It is easy to see that the right-hand sides of Eqs.\\, \\rf{tt-Ra} and \\rf{ii-Ra} coincide with\n those of \\rf{tt-k1} and \\rf{ii-k1} if, in addition to the relationships \\rf{VW}--\\rf{phips}\n for scalar variables, we identify \n\\begin{equation} \\label{id-m}\n \\rho_k = \\trho, \\qquad p_k = \\tp.\n\\end{equation}\n The correctness of this identification is confirmed by the identity of the conservation \n laws \\rf{cons} and \\rf{cons-i}. Thus, as in the vacuum case, the parameters \n $n$ and $b$ are related by \\rf{nb}, that is, $n(2-b) =1$, and the scalar fields $\\phi$ and \n $\\psi$ are related by \\eqn{phips} which now reads\n\\begin{equation} \\label{psif}\n \\dot\\psi{}^2 = \\epsilon n F_0 \\dot\\phi{}^{2n}.\n\\end{equation}\n \n\\subsection{Case R2: Conservative matter}\n \n We now have $\\nabla_\\nu T\\mN [m] = 0$. This condition is particularly important \n for the structure formation in the universe for the case of a pressureless fluid \n since ordinary matter must agglomerate.\n\n In this case, for the scalar field SET we have \n\\begin{eqnarray}\n \\nabla_\\nu T\\mN [\\psi] ={\\fracd{1}{2}} (b - 1) (\\partial_\\mu T[\\psi] + \\partial_\\mu T[m]),\n\\end{eqnarray}\n which leads to the scalar field equation\n\\begin{eqnarray} \\lal \\label{epsi-ii} \n\t\\dot\\psi(b \\ddot\\psi + 3H \\dot\\psi) = -\\epsilon (3-2b) W_\\psi \\dot\\psi\n\\nonumber\\\\ \\lal \\hspace*{1in}\n\t+ {\\fracd{1}{2}}\\epsilon (b-1)(\\dot\\rho - 3\\dot p). \n\\end{eqnarray}\n The full set of equations consists of \\rf{tt-Ra}, \\rf{ii-Ra}, \\rf{epsi-ii}, and \\rf{cons}, \n with the definitions \\rf{redef}. Note that \\eqn{epsi-ii} mixes the scalar field and the \n matter fluid even though the fluid is conserved as in GR.\n\n This conservation makes us identify the matter SET components in the k-essence\n and Rastall theories: $\\rho = \\rho_k$, $p = p_k$. As a result, identification of the other \n parts of the total SET is only partly the same as in the previous case. \n\n Identifying, as before, the potentials according to \\rf{VW} (that is, $V = (3-2b)W$)\n and comparing the Friedmann-like equations \\rf{tt-Ra} and \\rf{ii-Ra} with \n their k-essence counterparts \\rf{tt-k1} and \\rf{ii-k1}, we obtain, as before, \n\\begin{equation} \\label{psif-ii}\n \\epsilon\\dot\\psi^2 = nF_0\\dot\\phi^{2n}.\n\\end{equation}\n The correctness of this identification is verified by substituting \\rf{psif-ii} into \n the scalar field equation \\rf{epsi-ii}: indeed, since we have now, due to \\rf{trace}, \n\\begin{equation} \\label{nb-ii}\n \\epsilon\\dot\\psi^2 \\Big[b - \\frac{2n-1}{n}\\Big] = (b-1) (\\rho-3p), \n\\end{equation}\n this substitution leads precisely to the scalar field equation \\rf{phi-k1} of the \n k-essence theory.\n\n It is important that in the case of conservative matter, a comparison \n between the two theories does not lead to a direct relationship like \n \\rf{nb} between their numerical parameters $n$ and $b$. Instead, we have \n the equality \\rf{nb-ii}, from which \\rf{nb} is restored only in the special case \n $\\rho=3p$ (zero trace of the matter SET, radiation).\n\n\\subsection{Further consequences of matter conservation}\n\n The relation \\rf{nb-ii} creates a connection between the temporal behavior of $\\psi$ and the \n matter content. Indeed, inserting \\rf{nb-ii} to \\rf{epsi-ii} with zero or constant potential, \n we find\n\\begin{equation}\n \\dot F + \\frac{6n}{2n - 1}HF = 0,\n\\end{equation}\n where $F = \\dot\\psi^2$. This leads to\n\\begin{equation} \\label{sol1}\n \\dot\\psi^2 = \\psi_0a^{-6n\/(2n - 1)},\n\\end{equation}\nwhere $\\psi_0$ is an integration constant.\n\n From Eq.\\, \\rf{nb-ii} it is clear that the matter density and pressure must also evolve by a \n power law as functions of the scale factor. Hence, only an equation of state (EoS) of the type \n $p = w\\rho$, with $w = {\\rm const}$, is possible. In this case, \n\\begin{equation}\n \\rho = \\rho_0 a^{-3(1 + w)}, \\qquad \\rho_0 = {\\rm const}, \n\\end{equation}\n implying the relation between $w$ and $n$\n\\begin{equation} \\label{rela}\n w = \\frac{1}{2n - 1}.\n\\end{equation}\n We see that a substitution of $p = w\\rho$ into the scalar field equation relates the EoS factor\n $w$ with the k-essence power $n$, while the Rastall constant $b$ remains arbitrary. \n Moreover, Eqs.\\, \\rf{tt-k1} and \\rf{ii-k1} show that the pure k-essence scalar field $\\phi$ behaves \n as a perfect fluid with the same EoS factor \\rf{rela} (see also \\cite{carla}).\n\n In other words, assuming a zero or constant potential $V = (3-2b)W$ and conservative matter \n in the Rastall framework, we obtain that {\\sl the k-R duality is only possible if matter \n is a perfect fluid with the linear EoS $p = w\\rho$, coinciding with the effective EoS\n of the scalar field $\\phi$. }\n\n With $p = w\\rho$ \\eqn{nb-ii} gives \n\\begin{eqnarray} \\lal\n \\epsilon \\dot\\psi^2 = k\\rho,\n\\qquad\n\tk = \\frac{n(b- 1)(1 - 3w)}{bn-2n+1}.\n\\end{eqnarray}\n Inserting this to the Friedmann-like equation \\rf{tt-Ra}, we obtain in terms of $n$ or $w$\n\\begin{eqnarray}\n 3H^2 \\al =\\al V + \\frac{\\rho}{2n - 1}\n\t\\biggr\\{\\frac{2nb(b - 1)(n - 2)}{bn -2n + 1}\n\\nonumber\\\\ \\lal \\hspace*{1cm}\\cm\n\t + 2(2b - 3) + 2n(3 - b)\\biggr\\}\n\\\\[5pt] {} \n \\al =\\al V + \\frac{\\rho}2 \\biggl\\{\\frac{b(b-1)(1+w)(1-3w)}{(b-2)(1+w)+2w}\n\\nonumber\\\\ \\lal \\hspace*{1cm}\\cm\n + 3-b + 3w(b-1)\\biggr\\}.\n\\end{eqnarray}\n The right-hand side must be positive. Therefore, given $n$ and $V$ (or alternatively $w$ \n and $V$), we obtain a restriction on $b$. For example, if $V=0$ and $w = 0$ \n (dust, $n \\to \\infty$),\\footnote\n\t{This relation makes sense even if the Lagrangian formulation becomes ill-defined,\n\t see \\cite{carla}.} \n we have either $b < 3\/2$ or $b > 2$ (provided $\\rho > 0$). For $w = 1$ (stiff matter, $n = 1$), \n there is no restriction on $b$, and we obtain $H^2 = V = {\\rm const} >0$, hence a de Sitter \n expansion, $a(t) \\propto{\\,\\rm e}^{Ht}$. In this case, stiff matter precisely cancels the contribution \n from the scalar field $\\psi$ or $\\phi$. \n\n If we introduce a variable potential or a more complex equation of state, the situation becomes \n much more involved. It must be stressed that the EoS $p = w\\rho$ with $w={\\rm const}$\n covers most of the interesting cases in cosmology. Moreover, we expect that the \n perturbative behavior may be very different in the two theories even in this case.\n\n There emerge two more natural questions. First, we have obtained that in k-essence theory\n there are simultaneously a scalar field and a perfect fluid with the same EoS and hence the same \n time evolution of their densities and pressures. Can we unify them by, for example, redefining\n the scalar field? A probable answer is ``no'' because these two kinds of matter are expected to\n behave quite differently at the perturbative level.\n\n Another question is: how is it possible to have a completely definite situation in k-essence \n theory but an arbitrariness in the parameter $b$ in the dual solution of Rastall's theory? An \n answer is that this arbitrariness is compensated by the corresponding non-conservative \n behavior of the scalar field $\\psi$. \n\n\\section{Perturbations and the speed of sound}\n\n A power law k-essence model with $V = 0$ is equivalent in the cosmological \n framework (such that $(\\partial_\\mu \\phi)^2 > 0$) to a perfect fluid with the equation \n of state $p = w \\rho$, where the constant $w$ is related to the power $n$:\n\\begin{equation} \\label{w}\n\tw = \\frac{1}{2n - 1}.\n\\end{equation}\n In a fluid, adiabatic perturbations propagate as a sound with the speed $v_s$ such that\n\\begin{equation} \\label{v-fl}\n\tv_s^2 = dp\/d\\rho = w.\n\\end{equation}\n \n Scalar field perturbations for general Lagrangians of the form $F(X,\\phi)$\n have been treated in detail in \\cite{neven, k3}. It has been shown there that\n a k-essence theory implies \n\\begin{equation}\n v_s^2 = \\frac{F_X}{F_X + 2X F_{XX}},\n\\end{equation}\n and this expression is valid even if there is an arbitrary potential term $V(\\phi)$.\n In particular, for the theory \\rf{L-k1}, where $F(X) = F_0 X^n - 2V(\\phi)$, we find again\n\\begin{equation} \\label{ss1}\n v_s^2 = \\frac{1}{2n - 1}\n\\end{equation}\n in full agreement with \\rf{v-fl}. Thus there is a complete equivalence between \n a perfect fluid and k-essence without a potential not only for a cosmological \n background but even on the perturbative level as far as adiabatic perturbations\n are concerned. In particular, if $w < 0$, corresponding to $n < 1\/2$, the model \n is perturbatively unstable since it implies $v_s^2 < 0$. This is true both for a perfect \n fluid and for k-essence. Moreover, although the presence of a potential \n changes the scalar field dynamics, the propagation speed of its perturbations, coinciding \n with the derived speed of sound \\cite{neven}, is still the same as with $V=0$. \n \n In Rastall's theory things may be different. The speed of sound for a scalar field is \n given by \\cite{oliver, liddle}\n\\begin{equation} \\label{ss2}\n v_s^2 = \\frac{2 - b}{b}.\n\\end{equation}\n In scalar vacuum and in the R1 case (matter obeys the non-conservation equation \n \\rf{cons-i1}), we have the relation \\rf {nb}, $(2-b)n = 1$, which makes (\\ref{ss1}) \n and (\\ref{ss2}) identical. Furthermore, the fluids in the corresponding models \n obey different equations of state, see \\rf{id-m}. However, in the Rastall model we \n can still characterize the fluid by the ``effective'' density and pressure, $\\trho$ and \n $\\tp$, the SET written in their terms is conservative, hence the squared speed of \n sound of the Rastall fluid is equal to $d\\tp\/d\\trho = d p_k\/d\\rho_k$.\n Thus we can conclude, even without performing a complete perturbation analysis, \n that the models belonging to the two theories coincide not\n only at the background level but also at the level of adiabatic perturbations.\n\n In the case R2 (conserved matter), we have another relation \\rf{nb-ii} between the \n parameters $b$ and $n$, without such a simple connection. As a consequence, in principle \n it is possible that an unstable model in a k-essence model may correspond to a stable model \n in Rastall's theory, or vice versa, since, as shown above, Eq.\\, \\rf{nb} does not hold,\n $b$ being now essentially independent of $n$ up to some possible restrictions on their range. \n In fact, in this case, even non-adiabatic perturbations may appear, due to the coupling\n between the scalar field and matter. \n\n\\section{Some special cases} \n\n\\subsection{$n=1\/2$} \n \n In this case, the k-essence scalar field equation takes the form\n\\begin{equation} \\label{n-half}\n\t\t3H = -2 F_0^{-1} V_\\phi. \n\\end{equation}\n Thus if $V={\\rm const}$, we have $H=0$, hence $a = {\\rm const}$, and flat space-time is obtained. \n One can certainly obtain $H=0$ in Rastall gravity under special assumptions, but the question\n of k-R duality looks meaningless in this trivial case. \n \n If $V=V(\\phi)$, the $\\phi$ field has no dynamics of its own, but \\eqn{n-half} expresses it\n in terms of $H$, and the Friedmann equations \\rf{tt-k1} and \\rf{ii-k1} are meaningful.\n \n The Rastall counterpart in the scalar-vacuum and R1 cases is then obtained with $b=0$, \n $V = 3W$ (according to \\rf{nb} and \\rf{VW}) and \n\\begin{equation} \\label{psif-half}\n\t\t2\\epsilon \\dot\\psi{}^2 = F_0 \\dot\\phi. \n\\end{equation}\n\n In the case R2 (conserved matter), $b$ remains arbitrary, but k-R duality still holds.\n Indeed, if we substitute \\rf{psif-half} into \\eqn{epsi-ii} and use the Friedmann-like \n equations \\rf{tt-Ra}, \\rf{ii-Ra} to calculate $\\dot\\rho-3\\dot p$, we obtain \\eqn{n-half}.\n\n The main feature in this case is that nontrivial solutions with $n = 1\/2$ and k-R duality\n are only achieved in the presence of a variable potential.\n\n\\subsection{$b=3\/2$}\n\n With this value of $b$, the potential disappears from Rastall's gravity, hence the k-R duality \n implies $V=0$, and we deal with zero potentials. In other respects, the situation is \n described as in the general case. \n\n A feature of interest is that with $b = 3\/2$ \\eqn{redef} leads to $\\trho = 3\\tp$. \n Therefore, in the scalar-vacuum and R1 cases, the dual k-essence counterpart of \n this Rastall model contains matter with $\\rho_k= 3p_k$ (see \\rf{id-m}). \n Due to \\rf{nb}, in addition, $n=2$, so that the $\\phi$ field also behaves as radiation.\n\n In the case R2 (conserved matter), the relations \\rf{id-m} and \\rf{nb} are no more \n valid, and the general description is applicable. \n \n\\subsection{$b=2$}\n\n Equations \\rf{w} and \\rf{ss2} give zero values of pressure and the speed of sound of\n a scalar field in Rastall's theory. The corresponding expression \\rf{ss1} in k-essence \n theory leads to $n \\to \\infty$ according to the general relation \\rf{nb}.\n \n If there is conserved matter (case R2), then (unless this conserved matter is pure\n radiation, $\\rho = 3p$) \\eqn{nb} is no more valid, so that the speeds of sound \n of scalar fields are different in the two theories. It means that k-R duality does not\n exist for perturbations even though it does exist for the isotropic background.\n\n\\subsection{$b=0$}\n\n In the scalar-vacuum and R1 cases we return to the above description for $n = 1\/2$.\n \n With conserved matter (case R2), the Rastall scalar field takes the form\n\\begin{equation} \\label{psi-b0}\n 3H \\epsilon \\dot\\psi{}^2 = -3 \\dot W -{\\fracd{1}{2}} (\\dot\\rho-3\\dot p),\n\\end{equation}\n looking like a constraint equation since it contains only the first-order derivative.\n However, the k-R duality still works, as before: thus, a substitution of \\rf{psif} \n (what is important, with arbitrary $n \\ne 0$) and $\\rho-3p$ from Eqs.\\, \\rf{tt-Ra}, \n \\rf{ii-Ra} into \\rf{psi-b0} leads to \\rf{phi-k1}, which is a second-order equation \n unless $n = 1\/2$. \n\n\\section{Examples}\n\n Let us now consider some specific examples of the equivalence discussed above,\n assuming a zero or constant potential and dust as a possible matter contribution.\n\n\\subsection{Scalar vacuum}\n\n Consider scalar vacuum with zero potential. The k-essence equations give\n\\begin{eqnarray} \\label{sfe}\n\t\\dot\\phi \\al =\\al \\phi_0 a^{- 3\/(2n - 1)}, \\qquad \\phi_0 = {\\rm const},\n\\\\[5pt] {}\n H^2 \\al =\\al \\frac{2n - 1}{6}F_0 \\phi_0^{2n}a^{-6n\/(2n - 1)}.\n\\end{eqnarray}\n In terms of cosmic time we obtain\n\\begin{eqnarray}\n a \\al =\\al a_0 t^{2\/[3(1 + w)]}, \\qquad a_0 = {\\rm const},\n\\\\[5pt] {}\n\t \\dot\\phi \\al =\\al \\phi_1 t^{- 2w\/(1 + w)},\n\\end{eqnarray}\n where $\\phi_1$ is a combination of the previous constants, and we \n have written $w = 1\/(2n - 1)$, thus identifying the k-essence with a perfect \n fluid with the EoS $p = w\\rho$. \n\n In Rastall's theory, the dual solution contains the same $a(t)$, while the scalar field is given by\n\\begin{equation} \\label {sfr}\n \\dot\\psi \\propto a^{- 3(1 + w)\/2} = a^{-3\/b} \\propto t^{-1},\n\\end{equation}\n where now we should put $w = (2-b)\/b$. We notice that while the k-essence scalar field \n behavior depends on the EoS parameter $w$, the Rastall scalar is simply $\\psi = \\log t + {\\rm const}$.\n\n Addition of a constant potential, equivalent to a cosmological constant, does not change the \n scalar field evolution laws \\rf{sfe} and \\rf{sfr} in terms of $a$ but makes the time \n dependences more complex, not to be considered here.\n\n In the presence of matter, as we saw above, the form of k-R duality depends on how matter \n couples to the scalar field. \n\n\\subsection{Dust and Rastall-R1 models}\n\n Suppose that in k-essence theory, in addition to the scalar field $\\phi$, there is \n pressureless fluid (dust), so that\n\\begin{equation} \\label{dust-k} \n p_k =0, \\qquad \\rho_k = \\rho_1 a^{-3}, \\qquad \\rho_1={\\rm const}.\n\\end{equation}\n then, in the R1 version of Rastall's theory, according to \\rf{id-m}, we have the conditions\n\\begin{eqnarray} \\label{ex1}\n\t {\\fracd{1}{2}}\\Big[(3 - b)\\rho + 3(b-1)p\\Big] \\al =\\al \\trho = \\rho_k,\n\\nonumber\\\\ {}\n {\\fracd{1}{2}} \\Big[(b-1)\\rho + (5-3b)p \\Big] \\al =\\al \\tp = 0,\n\\end{eqnarray}\n leading to the following relations for the density and pressure \n\\begin{eqnarray} \\label{ex2}\n \\rho \\al =\\al \\frac{5-3b}{2(3-2b)} \\rho_k,\n\\nonumber\\\\ {}\n\tp \\al =\\al \\frac{1-b}{5-3b} \\rho = \\frac{1-b}{2(3-2b)} \\rho_k,\n\\end{eqnarray}\n both evolving as $\\rho \\propto p \\propto a^{-3}$. Thus in Rastall cosmology the \n fluid acquires pressure (except for the GR value $b = 1$). The scalar fields in both \n models satisfy the same relations as in the vacuum case, valid for any values of \n $n$ and $b$ such that $n(b-2)=1$. \n\n Adding a constant potential $V = (3-2b)W$ does not change the relations \\rf{ex1}\n and \\rf{ex2} and introduces an effective cosmological constant. We then obtain \n a three-component model with matter, a cosmological constant and a scalar field \n whose behavior is determined by $n$ or, equivalently, by $w = 1\/(2n-1)$. In the \n dual Rastall model, we have an effective pressure even though in the k-essence \n model $p_k=0$. \n\n If, on the contrary, we introduce matter with $p=0$ in Rastall's (R1) theory, \n then in the dual k-essence model we have \n\\begin{equation}\n \\rho_k = \\trho = {\\fracd{1}{2}} (3-b)\\rho, \\qquad p_k = \\tp = {\\fracd{1}{2}} (b-1)\\rho, \n\\end{equation}\n and their evolution law agreeing with \\eqn{cons-i} reads\n\\begin{equation}\n \\rho_k \\propto \\rho \\propto a^{-6\/(3-b)} = a^{-3(1+w_k)}, \n\\end{equation}\n where the EoS parameter $w_k$ of the fluid in the k-essence model is \n\\begin{equation}\n w_k = \\frac{b-1}{3-b} = \\frac{n-1}{n+1} \n\\end{equation}\n (not to be confused with $w = 1\/(2n-1)$ characterizing the $\\phi$\n field behavior); as before, the relation $n(2-b) =1$ holds. The model thus obtained\n is quite different from the one with dust introduced in k-essence theory.\n\n\\subsection{Dust and Rastall-R2 models}\n\n Let us again assume \\eqn{dust-k} but now consider version R2 of Rastall's theory,\n so that now $\\rho \\propto a^{-3}$ and \n\\begin{equation} \\label{nb-ex}\n\t\\epsilon \\dot\\psi^2\\Big[b - \\frac{2n - 1}{n}\\Big] = (b - 1)\\rho \\propto a^{-3}.\n\\end{equation}\n Then for $b \\neq 1$ we find according to \\rf{psif-ii}\n\\begin{eqnarray}\n\t\\dot\\psi &\\propto& a^{- 3\/2},\n\\nonumber\\\\ {}\n \\dot\\phi^n &\\propto& a^{-3\/2} \\ \\ \\ \\Rightarrow\\ \\ \\ \\dot\\phi \\propto a^{-3\/(2n)}.\n\\end{eqnarray}\n Combining this with the relation $\\epsilon \\dot\\psi{}^2 = nF_0 \\dot\\phi{}^{2n}$ and the \n field equation \\rf{phi-k} with $V_\\phi=0$, we find that this situation corresponds to\n the limit $n \\to \\infty$, which is, however, well defined. In this way we obtain \n $a \\propto t^{2\/3}$ as in the pure dust model of GR. \n For the scalar fields it follows in this limit\n\\begin{eqnarray} \\lal \n \\dot\\phi ={\\rm const} \\ \\ \\Rightarrow\\ \\ \\phi \\propto t,\n\\nonumber\\\\ \\lal \n \\dot\\psi \\propto a^{-3\/2} \\propto 1\/t.\n\\end{eqnarray}\n The condition for $b$ is obtained from \\rf{nb-ex}: writing $\\dot\\psi = \\psi_0\/t $ and\n $\\rho = \\rho_0\/t^2$, we arrive at\n\\begin{equation}\n\t\\epsilon\\psi_0^2 (b - 2) = (b - 1)\\rho_0.\n\\end{equation}\n Thus the value of $b$ is determined by the relative contributions of matter and the scalar field. \n Moreover, the speed of sound of the scalar field now does not follow the adiabatic relation \n verified in the R1 case.\n\n A cosmological constant can be easily introduced in the form of $V = (3-2b) W = {\\rm const}$. \n The scalar field again follows the law \\rf{nb-ex}, and the whole configuration reduces to the\n $\\Lambda$CDM model where $\\Lambda$ is given by the constant potential and the matter \n component consists of the scalar field and ordinary matter. All background relations of the \n $\\Lambda$CDM model are preserved in this case, but the degeneracy between the scalar \n field and usual matter is broken at the perturbative level. Due to the fact that the \n $\\Lambda$CDM model is subject to problem at the perturbative level in the non-linear regime \n (see, e.g., \\cite{cusp,satt}), such a more complex configuration in k-essence and Rastall \n models may lead to interesting results, to be studied in the future.\n\n\\section{Conclusion}\n\n We have studied the conditions of equivalence between the k-essence and Rastall theories of \n gravity in the presence a scalar field (k-R duality). These two theories have actually emerged in \n very different contexts, the k-essence theory being based on a generalization of the kinetic term \n of a scalar field, suggested by some fundamental theories, while Rastall's theory is a \n non-conservative theory of gravity which can be seen as a possible phenomenological \n implementation of quantum effects in gravitational theories. Such equivalence has been \n revealed in the case of static spherically symmetric models \\cite{denis, we-16}, and it \n has been more explicitly stated here for all cases where the metric and scalar fields essentially\n depend on a single coordinate, and the k-essence theory is specified by a power-law function \n of the usual kinetic term, to which a potential term can be added. This generalization covers \n diverse static and cosmological models, including all homogeneous cosmologies.\n\n We have discussed cosmological configurations with scalar fields and matter in the form of \n a perfect fluid whose evolution in Rastall's theory can follow one of two possible laws: one (R1) \n assumes no mixing between matter and the scalar field, each of them separately obeying \n the non-conservation law \\rf{cons-R}, and the other (R2) ascribes the whole non-conservation \n to the scalar field while matter is conservative ($\\nabla_\\nu T\\mN[m] =0$). Let us summarize\n the main results obtained in this context:\n\\begin{enumerate}\n\\item \n\tk-R duality has been established for version R1 of Rastall's theory with an arbitrary \n \tEoS of matter. It has been found that the EoS of matter is different in the mutually dual\n \tk-essence and Rastall models; however, it is argued that the respective speeds of \n\tsound are the same. Since the speeds of sound characterizing the scalar fields ($\\phi$ in \n\tk-essence theory and $\\psi$ in Rastall's) also coincide, we conclude that k-R duality\n is maintained not only for the cosmological backgrounds but also for adiabatic \n perturbations.\n\\item \n\tFor version R2 of Rastall's theory, it has been found that k-R duality exists only with \n\tfluids having the EoS $p = w\\rho,\\ w={\\rm const}$, which is the same for k-essence and \n\tRastall models. Moreover, in the k-essence model the scalar field obeys the same \n\teffective EoS. However, on the perturbative level the mutually dual models \n\tbehave, in general, differently. \n\\item \n\tSome special cases have been discussed, showing how there emerge some restrictions \n\ton the free parameters of each theory. \n\\item \n\tAn example has been considered in which a cosmological model completely equivalent \n to the $\\Lambda$CDM model of GR is obtained at the background level, but different \n\tfeatures must appear at the perturbative level. \n\\end{enumerate}\n\n The equivalence between the two theories discussed here is somewhat surprising because of\n their basically different origin. A curious aspect is that the k-essence theory has a Lagrangian\n formulation unlike Rastall's theory. It is possible that the equivalence studied here may lead to\n a restricted Lagrangian formulation of Rastall's theory in the minisuperspace in terms of \n metric functions depending on a single variable. If this is true, it might suggest how to recover \n a complete Lagrangian formulation for Rastall's theory in a more general framework.\n\n\\subsection*{Acknowledgments} \n\n We thank CNPq (Brazil) and FAPES (Brazil) for partial financial support.\n KB thanks his colleagues from UFES for kind hospitality and collaboration.\n The work of KB was performed within the framework of the Center \n FRPP supported by MEPhI Academic Excellence Project \n (contract No. 02.a03. 21.0005, 27.08.2013),\n within the RUDN University program 5-100, and RFBR project No. 16-02-00602. \n\n\\small\n\n\\subsection*{Acknowledgment} #1}\n\n\t\n\n\\def.95{.95}\n\\def.95{.95}\n\\def.95{.95}\n\\def.05{.05}\n\\def.95{.95}\n\\def.95{.95}\n\n\\newcommand{\\EFigure}[2]{\\begin{figure} \\centering \n \\framebox[85mm]{\\epsfxsize=80mm\\epsfbox{#1}}\n \\caption{\\protect\\small #2}\\medskip\\hrule\n\t\\end{figure}}\n\\newcommand{\\REFigure}[2]{\\begin{figure} \\centering\n \\framebox[85mm]{\\epsfysize=80mm\\rotate[r]{\\epsfbox{#1}}}\n \\caption{\\protect\\small #2}\\medskip\\hrule\n\t\\end{figure}}\n\\newcommand{\\WEFigure}[2]{\\begin{figure*} \\centering\n \\framebox[178mm]{\\epsfxsize=170mm\\epsfbox{#1}}\n \\caption{\\protect\\small #2}\\medskip\\hrule\n\t\\end{figure*}}\n\n\t\n\n\\def\\Jl#1#2{#1 {\\bf #2},\\ }\n\n\\def\\ApJ#1 {\\Jl{Astroph. J.}{#1}}\n\\def\\CQG#1 {\\Jl{Class. Quantum Grav.}{#1}}\n\\def\\DAN#1 {\\Jl{Dokl. AN SSSR}{#1}}\n\\def\\GC#1 {\\Jl{Grav. Cosmol.}{#1}}\n\\def\\GRG#1 {\\Jl{Gen. Rel. Grav.}{#1}}\n\\def\\JETF#1 {\\Jl{Zh. Eksp. Teor. Fiz.}{#1}}\n\\def\\JETP#1 {\\Jl{Sov. Phys. JETP}{#1}}\n\\def\\JHEP#1 {\\Jl{JHEP}{#1}}\n\\def\\JMP#1 {\\Jl{J. Math. Phys.}{#1}}\n\\def\\NPB#1 {\\Jl{Nucl. Phys. B}{#1}}\n\\def\\NP#1 {\\Jl{Nucl. Phys.}{#1}}\n\\def\\PLA#1 {\\Jl{Phys. Lett. A}{#1}}\n\\def\\PLB#1 {\\Jl{Phys. Lett. B}{#1}}\n\\def\\PRD#1 {\\Jl{Phys. Rev. D}{#1}}\n\\def\\PRL#1 {\\Jl{Phys. Rev. Lett.}{#1}}\n\n\t\n\n\\def&\\nhq{&\\hspace*{-0.5em}}\n\\def&&\\nqq {}{&&\\hspace*{-2em} {}}\n\\defEq.\\,{Eq.\\,}\n\\defEqs.\\,{Eqs.\\,}\n\\def\\begin{equation}{\\begin{equation}}\n\\def\\end{equation}{\\end{equation}}\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\begin{eqnarray} \\lal{\\begin{eqnarray} &&\\nqq {}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\def\\nonumber \\end{eqnarray}{\\nonumber \\end{eqnarray}}\n\\def\\nonumber\\\\ {}{\\nonumber\\\\ {}}\n\\def\\nonumber\\\\[5pt] {}{\\nonumber\\\\[5pt] {}}\n\\def\\nonumber\\\\ \\lal {\\nonumber\\\\ &&\\nqq {} }\n\\def\\nonumber\\\\[5pt] \\lal {\\nonumber\\\\[5pt] &&\\nqq {} }\n\\def\\\\[5pt] {}{\\\\[5pt] {}}\n\\def\\\\[5pt] \\lal {\\\\[5pt] &&\\nqq {} }\n\\def\\al =\\al{&\\nhq =&\\nhq}\n\\def\\al \\equiv \\al{&\\nhq \\equiv &\\nhq}\n\\def\\sequ#1{\\setcounter{equation}{#1}}\n\n\n\\def\\displaystyle{\\displaystyle}\n\\def\\textstyle{\\textstyle}\n\\def\\fracd#1#2{{\\displaystyle\\frac{#1}{#2}}}\n\\def\\fract#1#2{{\\textstyle\\frac{#1}{#2}}}\n\\def{\\fracd{1}{2}}{{\\fracd{1}{2}}}\n\\def{\\fract{1}{2}}{{\\fract{1}{2}}}\n\n\n\\def{\\,\\rm e}{{\\,\\rm e}}\n\\def\\partial{\\partial}\n\\def\\mathop{\\rm Re}\\nolimits{\\mathop{\\rm Re}\\nolimits}\n\\def\\mathop{\\rm Im}\\nolimits{\\mathop{\\rm Im}\\nolimits}\n\\def\\mathop{\\rm arg}\\nolimits{\\mathop{\\rm arg}\\nolimits}\n\\def\\mathop{\\rm tr}\\nolimits{\\mathop{\\rm tr}\\nolimits}\n\\def\\mathop{\\rm sign}\\nolimits{\\mathop{\\rm sign}\\nolimits}\n\\def\\mathop{\\rm diag}\\nolimits{\\mathop{\\rm diag}\\nolimits}\n\\def\\mathop{\\rm dim}\\nolimits{\\mathop{\\rm dim}\\nolimits}\n\\def{\\rm const}{{\\rm const}}\n\\def\\varepsilon{\\varepsilon}\n\\def\\epsilon{\\epsilon}\n\n\\def\\ \\Rightarrow\\ {\\ \\Rightarrow\\ }\n\\newcommand{\\mathop {\\ \\longrightarrow\\ }\\limits }{\\mathop {\\ \\longrightarrow\\ }\\limits }\n\\newcommand{\\aver}[1]{\\langle \\, #1 \\, \\rangle \\mathstrut}\n\\newcommand{\\vars}[1]{\\left\\{\\begin{array}{ll}#1\\end{array}\\right.}\n\\def\\sum\\limits{\\sum\\limits}\n\\def\\int\\limits{\\int\\limits}\n\n \n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nGraphs are ubiquitous in the real world, which can easily express various and complex relationships between objectives. In recent years, extensive studies have been conducted on deep learning methods for graph-structured data. There are several approaches on analyzing the graph, including network embedding \\cite{perozzi2014deepwalk, tang2015line, grover2016node2vec}, which only uses the graph structure, and graph neural networks (GNNs), which consider graph structure and node features simultaneously. GNNs have shown powerful ability on modeling the graph-structured data in a variety of graph learning tasks such as node classification \\cite{2018Large, hamilton2017inductive, yang2016revisiting, kipf2016semi, velivckovic2017graph, wu2019simplifying}, link prediction \\cite{zhang2017weisfeiler, zhang2018link, cai2020multi} and graph classification \\cite{gilmer2017neural, lee2019self, ma2019graph, xu2018powerful, ying2018hierarchical, zhang2018end}. GNNs have also been applied to a range of applications, including social analysis \\cite{qiu2018deepinf, li2019encoding}, recommender systems \\cite{ying2018graph, monti2017geometric}, traffic prediction \\cite{guo2019attention, li2019predicting}, drug discovery \\cite{zitnik2017predicting} and fraud detection \\cite{liu2019geniepath}. \n\nGNNs usually have different design paradigms, which include the spectral graph convolutional networks \\cite{bruna2013spectral, defferrard2016convolutional, kipf2016semi}, message passing framework \\cite{gilmer2017neural, hamilton2017inductive}, and neighbor aggregation via recurrent neural networks \\cite{li2015gated, dai2018learning}. By using the idea of message passing framework, GNNs are to design various graph convolutional layers to update each node representation by aggregating the node representations from its neighbors. \n\nHowever, most GNNs only consider the immediate neighbors for each node, which impedes their ability to extract the information of high-order neighbors. More layers usually lead to the performance degradation, which is caused by over-fitting and over-smoothing, of which the former is due to the increasing number of parameters when fitting a limited dataset whereas the latter is the inherent issue of the graph learning. How to make use of the high-order information of neighbors as well as achieving better performance remains a challenge. We need more insights to understand what GCN does and why over-smoothing occurs.\n\\begin{figure*}[t]\n\t\\centering \n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{original.pdf}}\\hspace{-4mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{mlp.pdf}}\\hspace{-2mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{GCN-2.pdf}}\\hspace{-2mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{GCN-6.pdf}}\\hspace{-2mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{GCN-8.pdf}}\\hspace{-2mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{GCN+8.pdf}}\\hspace{-2mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{GCN+16.pdf}}\\hspace{-2mm}\n\t\\subfigure{\n\t\t\\includegraphics[width=0.12\\textwidth]{GCN+32.pdf}}\n\t\\caption{t-SNE Visualization of learned node representations, which include original features, MLP, different layers of GCN and different hops of GCN+ on \\textit{Cora}. Colors represent node classes.}\n\t\\label{model_vs}\n\\end{figure*}\n\nSeveral studies \\cite{li2018deeper, xu2018representation, klicpera2018predict, chen2020simple, liu2020towards} have noticed over-smoothing, that is after multiple propagations, the final output of vanilla multi-layer GCN converges to a vector which only carries the information of the degree of graph and the node features are indistinguishable. Fig. \\ref{model_vs} shows the node representations of vanilla multi-layer GCN on a small citation network \\textit{Cora}. We can observe that 2-layer GCN learns a meaningful embeddings which distinguish the different classes whereas more layers degrade the performance and lead to indistinguishable features.\n\nDifferent from previous studies, we interpret the current graph convolutional operations from an optimization perspective, and argue that over-smoothing is mainly caused by the naive first-order approximation of the solution to the optimization problem. By solving it and applying the first-order approximation, we get the standard GCN kernel. This suggests that the original GCN kernel can be viewed as a simplified version of the solution. We argue that this simplification loses necessary information which is crucial to tackle the over-smoothing to some extent. Based on this observation, two metrics are proposed to measure the smoothness of connected and disconnected pairwise node features respectively. Furthermore, we set three constraints: (a) the embedding learned by GCNs should not be too far off of the original features; (b) the connected nodes should have similar embeddings; (c) the disconnected nodes are assumed to have different embeddings. \n\nAs a result, we build a universal theoretical framework of GCN from an optimization perspective which smooths the node features and regularizes the (disconnected) node feature simultaneously. We consider two different cases of our framework, where the first case contains the current popular GCN \\cite{kipf2016semi}, SGC \\cite{wu2019simplifying} and PPNP \\cite{klicpera2018predict}, and the second case regularizes the pairwise distance of disconnected nodes.\n\nThe contributions of this work are summarized as follow:\n\\begin{itemize}\n\t\\item We provide a universal theoretical framework of GCN from an optimization perspective where the popular GCNs can be viewed as a special case of it. Furthermore, we derive a novel convolutional kernel named GCN+, which relieves the over-smoothing inherently and has lower parameter amount.\n\t\\item We propose two quantitative metric to measure the smoothness and over-smoothness of the final nodes representations, which provides new insight to analyze the over-smoothing.\n\t\\item We conduct extensive experiments on several public real-world datasets. Our results demonstrate the superior performance of GCN+ over state-of-the-art baseline methods.\n\\end{itemize} \n\n\\section{Notations}\nGiven an undirected graph $G=(V,E,X)$, $V$ is node set with $|V|=n$, $E$ is edge set. Let $A\\in \\mathbb{R}^{n \\times n}$ denote the adjacency matrix, where $A_{ij}=1$ if there is an edge between node $i$ and node $j$ otherwise 0. Let $D\\in \\mathbb{R}^{n \\times n}$ denote the diagonal degree matrix where $D_{ii}=\\sum_{j}A_{ij}$. Each node is associated with $d$ features, and $X \\in \\mathbb{R}^{n\\times d}$ is the feature matrix of nodes. each row of $X$ is a signal defined over nodes. The graph Laplacian matrix is defined as $L=D-A$. Let $\\tilde{A}=A+I$ and $\\tilde{D}=D+I$ denote the adjacency and degree matrices of the self-loop graph respectively. We denote $\\tilde{A}_{\\textit{sym}}=\\tilde{D}^{-1\/2} \\tilde{A} \\tilde{D}^{-1\/2}$ and $\\tilde{A}_{\\textit{rw}}=\\tilde{D}^{-1} \\tilde{A}$. Assume that each node $v_i$ is associated with a class label $y_i \\in Y$ where $Y$ is a set of $c$ classes. Let $N(v)$ denote the neighbors of $v$ in graph, that is $N(v)=\\{u\\in V|\\{u,v\\}\\in E\\}$ and $\\tilde{N}(v)=N(v) \\cup \\{v\\}$.\n$L'$ is the Laplacian matrix of the graph $G'(V',E',X)$, which is the complement of $G$, that means $G'$ has the same nodes as $G$ whereas if $\\{u,v\\}\\in E$, then $\\{u,v\\}\\notin E'$. Let $A'$ and $D'$ denote the corresponding adjacency and degree matrix respectively. We have $A'+A=J_n-I_n$ and $D'+D=(n-1)I$ where $J_n$ is a matrix whose element are all 1. Let $\\text{num}(E)$ and $\\text{num}(E')$ denote the numbers of edges in $G$ and $G'$ respectively, we have $\\text{num}(E)+\\text{num}(E')=\\frac{n(n-1)}{2}$.\n\\section{Perspectives of GCN}\nHere we provide three views to derive or understand the vanilla GCNs.\n\\subsection{Spectral Graph Convolution}\n\\citet{bruna2013spectral} define the spectral convolutions on graph by applying a filter $g_\\theta$ in the Fourier domain to a graph signal. ChebNet \\cite{defferrard2016convolutional} suggests that the graph convolutional operation can be further approximated by the $k$-th order Chebyshev polynomial of Laplacian. \\citet{kipf2016semi} simplify the ChebNet and obtains a reduced version of ChebNet by the renormalization trick:\n\\begin{equation} \\label{gcn}\nH^{(l+1)}\\!\\!=\\!\\!\\sigma(\\!\\tilde{A}_{\\textit{sym}}H^{(l)}W^{(l)})\\!\\!=\\!\\!\\sigma(\\tilde{D}^{-\\frac{1}{2}}\\tilde{A}\\tilde{D}^{-\\frac{1}{2}}H^{(l)}W^{(l)}),\n\\end{equation}\nwhere $\\sigma$ denote the activation function such as ReLU. $W^{(l)}$ is a layer-specific trainable weight matrix. $H^{(l)}$ is the feature matrix of $l$-th layer and $H^{(0)}=X$.\n\n\\subsection{Message Passing}\nMessage passing \\cite{gilmer2017neural} means that a node on the graph aggregates the message from neighbors and update its embedding:\n\\begin{equation} \\label{mpnn}\nh_v^{(l)}=U_l\\bigg(h_v^{(l-1)}, \\sum_{u \\in N(v)}M_l\\big(h_u^{(l-1)}, h_v^{(l-1)}, e_{uv}\\big)\\bigg),\n\\end{equation}\nwhere $M_l(\\cdot)$ and $U_l(\\cdot)$ are message aggregation function and vertex update function, respectively. $h_v^{(l)}$ denotes the hidden state of node $v$ at $l$-th layer, and $e_{uv}$ is the edge features.\n\nIn this way, GCN layer can be decomposed into two steps, including the neighbors' message aggregation and update:\n\\begin{equation} \\label{gcn-mpnn}\nh_v^{(l)}=\\sigma\\bigg(W^{(l)}\\sum_{u\\in \\tilde N (v)} \\frac{h^{(l-1)}_u}{\\sqrt{|N(v)||N(u)|}}\\bigg).\n\\end{equation}\n\nHere a GCN layer can be viewed as a weighted average of all neighbors' message where the weighting is proportional to the inverse of the number of neighbors.\n\n\\subsection{Graph Regularized Optimization} \\label{GRO}\nLet $\\bar{X} \\in \\mathbb{R}^{n\\times d}$ denote the final node embeddings matrix, and $\\bar{x}_i$ is the $i$-th row of $\\bar{X}$. We consider the following optimization problem:\n\\begin{equation} \\label{gcn-opt}\nf = \\min_{\\bar X} \\bigg(\\sum_{i \\in V}\\|\\bar{x}_i -x_i\\|_{\\tilde{D}}^2 + \\alpha \\sum_{\\{i,j\\} \\in E}\\|\\bar{x}_i -\\bar{x}_j\\|_2^2\\bigg),\n\\end{equation}\nwhere $(x,y)_{\\tilde D}=\\sum_{i\\in V}d(i)x(i)y(i)$, if $x=y$, we have$\\quad \\|x\\|_{\\tilde D}=\\sqrt{(x,x)_{\\tilde D}}$.\n\nThe first term in the above optimization problem is the fitting constraint, which means the output features (also called embeddings) should not be too far off of the input features, while the second term is the smoothness constraint, which means the connected nodes should have similar embeddings. $\\alpha > 0$ is a hyperparameter to balance the importance of two objections. It is worth noting that there is no limit to the specific transformation from $X$ to $\\bar{X}$.\n\nBefore solving the optimization problem, we have the following lemma.\n\\begin{lemma}\n\t$\\tilde{A}_{\\textit{rw}}$ and $\\tilde{A}_{\\textit{sym}}$ always have the same eigenvalues $|\\lambda|\\leq1$.\n\\end{lemma}\n\n\\begin{corollary} \\label{inverse}\n\t$(I_n-\\alpha\\tilde{A}_{\\textit{rw}})$ and $(I_n-\\alpha\\tilde{A}_{\\textit{sym}})$ are invertible if $\\alpha \\in[0,1)$. \n\\end{corollary}\n\n\\begin{lemma}\n\tGiven a graph with adjacency matrix $A$, the powers of $A$ give the number of walks between any two vertices.\n\\end{lemma}\n\n\\begin{corollary} \\label{high-order}\n\t$A^k$ includes the information of high-order neighbors.\n\\end{corollary}\nNext, we derive the closed-form solution of Eq. \\ref{gcn-opt}. Specifically, we rewrite Eq. \\ref{gcn-opt} as \n$$\nf=\\min_{\\bar X} \\bigg(\\text{Tr}\\big((\\bar X-X)(\\bar X-X)^T\\tilde{D}\\big) + \\alpha\\text{Tr}({\\bar X}^TL \\bar X)\\bigg).\n$$\nDifferentiating $f$ with respect to $\\bar{X}$, we have \n$$\n\\frac{df}{d\\bar{X}}=\\tilde{D}(\\bar{X}-X)+\\alpha L\\bar{X}=0.\n$$\nNotice Corollary \\ref{inverse}, we have\n$$\n\\bar{X}=(1-\\mu) (I-\\mu\\tilde {A}_{\\textit{rw}})^{-1}X,\n$$\nwhere $\\mu=\\frac{\\alpha}{1+\\alpha}$.\n\nActually, the solution is also the personalized PageRank \\cite{page1999pagerank}' s limiting distribution. If we set $\\mu=0.5$, we get $\\bar X=(2I-\\tilde {A}_{\\textit{rw}})^{-1}X$, \nand $\\tilde {A}_{rw}X$ is the first-order Taylor approximation. \nBy replacing $\\tilde {A}_{rw}$ with $\\tilde {A}_{\\textit{sym}}$, we get standard graph convolution kernel. In other words, we lose the information from high-order neighbors, which is contained in the error series of the Taylor expansion. (See Corollary \\ref{high-order}).\n\nIn a nutshell, we obtains the well-known kernel or resemble form of the graph convolution from different ways. \n\\section{Over-smoothing in Vanilla Deep GCN}\n\nNeural network usually performs better when stack more layers while graph neural network does not benefit from the depth. On the contrary, more layers often result in significant degradation in performance. \n\nPrevious work illustrates the over-smoothing by computing the limiting distribution of $A_{k}$ when $k \\rightarrow \\infty$, Actually, this is not identical with vanilla deep GCN, which contains non-linear transformation among different layers. Litter work considers the non-linearity in multi-layer GCN. \\citet{oono2019graph} extend the linear analysis to the non-linearity firstly, which considers the ReLU activation function. They suggest that the node features of a $k$-layer GCNs will converge to a subspace and incur information loss, which makes the node feature indistinguishable.\n\nAt first, one main reason we introduce the deep architecture in GCN is that we want to use the long-range neighbor's information. We argue that vanilla deep GCN is not the correct way to capture this information. However, It does not mean that deep architecture is useless. \\citet{chen2020simple} and \\citet{liu2020towards} have shown that more layers can boost the performance of GCN on several datasets and tasks.\n\nTo quantify the over-smoothing in vanilla deep GCN, we compute the overall pairwise distance of node embeddings as follows:\n\\begin{equation} \n\\nonumber\n\\begin{aligned}\n\\!M_{\\textit{\\!overall}}&\\!=\\!\\!\\!\\sum_{i,j\\in\\!V}\\!\\!\\|\\bar{x}_i\\!-\\!\\bar{x}_j\\|_2^2\\!=\\!\\!\\!\\!\\sum_{\\{i,j\\}\\in\\!E}\\!\\!\\! \\|\\bar{x}_i\\!-\\!\\bar{x}_j\\|_2^2\\!\\!+\\!\\!\\!\\!\\sum_{\\{i,j\\}\\notin\\!E}\\!\\!\\|\\bar{x}_i\\!-\\!\\bar{x}_j\\|_2^2 \\\\\n&= \\text{Tr}({\\bar X}^TL \\bar X)+ \\text{Tr}({\\bar X}^TL' \\bar X).\\\\\n\\end{aligned}\n\\end{equation}\n\\begin{figure}[t]\n\t\\subfigure[Overall distance]{\n\t\t\\label{overall}\n\t\t\\includegraphics[width=0.22\\textwidth]{m_overall.pdf}}\\hspace{-1mm}\n\t\\subfigure[Fraction of diatsnce]{\n\t\t\\label{fraction}\n\t\t\\includegraphics[width=0.24\\textwidth]{smooth.pdf}}\n\t\\caption{ $M_{\\textit{\\!overall}}, M_{\\textit{\\!smooth}}$ and $M_{\\textit{\\!non-smooth}}$ of the output node embeddings of Vanilla GCN with increasing layers on \\textit{Cora}.}\n\t\\label{boxplot}\n\\end{figure}\n\nFig. \\ref{overall} depicts the pairwise distance distribution of vanilla GCN with increasing layers on \\textit{Cora}. We can see that $M_{overall}$ decreases as the model goes deeper. Revisit the two parts of $M_{overall}$, we propose two fine quantitative metrics to measure the over-smoothing of graph representation.\n\\begin{equation} \n\\begin{aligned}\nM_{\\textit{smooth}} &= D_{\\textit{smooth}}\/D_{\\textit{overall}},\\\\\nM_{\\textit{non-smooth}} &= D_{\\textit{non-smooth}} \/D_{\\textit{overall}},\\\\\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation} \n\\begin{aligned}\nD_{\\textit{smooth}} &= \\text{Tr}({\\bar X}^TL \\bar X)\/\\text{num}(E) ,\\\\\nD_{\\textit{non-smooth}} &= \\text{Tr}({\\bar X}^TL' \\bar X)\/\\text{num}(E'),\\\\\nD_{\\textit{overall}}&=D_{\\textit{smooth}}+D_{\\textit{non-smooth}}.\n\\end{aligned}\n\\end{equation}\n\nHere, $\\text{num}(E)$ and $\\text{num}(E')$ in the denominator are used to eliminate the impact of unbalanced edge numbers in $G$ and $G'$. $M_{\\textit{smooth}}$ measures the smoothness of the graph representation of connected pair nodes while $M_{\\textit{non-smooth}}$ measures the smoothness of the graph representation of disconnected pair nodes. \n\nFig. \\ref{fraction} compares $M_{\\textit{smooth}}$ and $M_{\\textit{non-smooth}}$. We see that $M_{\\textit{smooth}}$ contributes to quite a few parts of the overall distance, which seems counter-intuitive. We will discuss this two metrics of GCN+ in Section \\ref{oversmooth-of-gcn+} again.\n\n\\section{A General Framework of GCN}\nRecall the graph regularized optimization problem, we add a negative term to constrain the sum of distances between disconnected pairs as follow:\n\\begin{equation} \\label{full-opt}\n\\nonumber\n\tf \\!\\!=\\!\\min_{\\bar X}\\!\\!\\bigg(\\!\\sum_{i \\in V}\\|\\bar{x}_i \\!-\\!x_i\\|_{\\tilde{D}}^2 \\!+\\! \\alpha\\!\\!\\!\\!\\! \\sum_{\\{i,j\\} \\in E} \\! \\|\\bar{x}_i \\!-\\!\\bar{x}_j\\|_2^2 \\!-\\!\\beta\\!\\!\\!\\!\\! \\sum_{\\{i,j\\} \\notin E} \\!\\|\\bar{x}_i \\!-\\!\\bar{x}_j\\|_2^2\\!\\bigg)\\!,\n\\end{equation}\nwhere $\\alpha$ and $\\beta$ are hyperparameters to balance the importance of the corresponding terms.\n\n\nWe consider two cases: $\\beta=0$ and $\\beta \\neq0$.\n\\subsection{Case 1:$\\beta=0$}\nIn this situation, $\\bar{X}=(1-\\mu)(I_n-\\mu\\tilde{A}_{\\textit{rw}})^{-1}X$ where $\\mu=\\frac{\\alpha}{1+\\alpha}\\in(0,1)$. Directly calculating such an intractable expression is not only computationally inefficient but also results in a dense $\\mathbb{R}^{n \\times n}$matrix. It would lead to a high computational complexity and memory requirement when we apply such operator on large graphs. We can achieve linear computational complexity via power iteration.\n\nWe use $\\tilde{A}$ to denote $\\tilde{A}_{\\textit{sym}}$ and $\\tilde{A}_{\\textit{rw}}$. Here we consider a more general expression $(1-\\mu)(I_n-\\mu \\tilde{A})^{-1}H$ where $H=H=f_{\\theta}(X)$.\n\\begin{theorem} \\label{theorem1}\n\t$(I_n-\\mu\\tilde{A})$ is invertible. Consider the following iterative scheme\n\t\\begin{equation} \\label{iterative1}\n\t\\begin{aligned}\n\tZ^{(0)}&=H, \\\\\n\tZ^{(k)}&=\\mu\\tilde{A}Z^{(k-1)}+(1-\\mu) H,\n\t\\end{aligned}\n\t\\end{equation} where $\\mu \\in (0,1)$.\n\tWhen $k \\rightarrow \\infty$, \n\t\\begin{equation}\nZ^{(\\infty)}=(1-\\mu)(I_n-\\mu\\tilde{A})^{-1}H.\n\t\\end{equation} \n\\end{theorem}\n\n\\begin{proof}\n\tUsing corollary \\ref{inverse}, we can see that $(I_n-\\mu\\tilde{A})$ is invertible. Combining the two equation of \\ref{iterative1}, we have \n\t$$\n\tZ^{(k)}=\\bigg(\\mu^k \\tilde{A}^k+(1-\\mu) \\sum_{i=0}^{k-1} \\mu^i \\tilde{A}^i\\bigg)H.\n\t$$\n\tNotice that\n\t\\begin{equation} \n\t\\begin{aligned}\n\t\\lim_{k \\rightarrow \\infty}\\mu^k \\tilde{A}^k&=0, \\\\\n\t\\lim_{k \\rightarrow \\infty}\\sum_{i=0}^{k-1} \\mu^i \\tilde{A}^i&=\\big(I-\\mu\\tilde{A}\\big)^{-1}.\n\t\\end{aligned}\n\t\\end{equation}\n\tHence, the proof is finished. \n\n\\end{proof}\n\nActually, the prevalent GCN, SGC and APPNP can be viewed as the special variant of Case 1.\n\\subsection{Case 2:$\\beta \\neq 0$} \n\nIn this situation, $\\bar{X}=Q^{-1}$ if $Q=\\big((1+\\alpha+\\beta)I-(\\alpha+\\beta)\\tilde D^{-1}\\tilde A-\\beta n\\tilde D^{-1}+\\beta \\tilde D^{-1} \\mathbf J_n\\big)$ is invertible when we choose a suitable $\\beta$. We will introduce the conditions later.\n\n\nFirst we use the first-order Taylor approximation of above convolutional kernel ($\\text{GCN}^*$) directly without any tricks such as Batch Normalization \\cite{ioffe2015batch} or residual connection \\cite{he2016deep} on two small citation datasets \\textit{Cora} and \\textit{Citeseer}. We compare the performance of the vanilla deep GCN and $\\text{GCN}^*$ as the model layer increases. Fig. \\ref{figure1} shows the result of GCN and $\\text{GCN}^*$. Dashed lines illustrate the performance of GCN, which shows that deep GCN suffers from performance drop. We can see that the performance decay with $\\text{GCN}^*$ kernel is much slower.\n\n\\citet{oono2019graph} have proved that the node feature of vanilla $k$-layer GCN will converges to an invariant subspace which only carry the information of the connected component and node degree. The convergence speed is proportional to the $\\lambda^k$, where $\\lambda$ is the supremum of eigenvalue of $\\tilde{A}$. In GCN*, $\\lambda>1$(see the proof of Theorem \\ref{theorem2}), which implies that $\\lambda^k$ is large, thus the information loss and over-smoothing are relieved.\n\nAlthough the modified graph kernel relieves over-smoothing to some extent, more layers do not boost the performance, which is not our focus. However the above result demonstrates that it is an efficient way to tackle the over-smoothing issue. We can achieves linear computational complexity via power iteration similar to Case 1.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.9\\columnwidth]{cora}\n\t\\caption{Performance comparison of vanilla deep GCN vs. $\\text{GCN}^*$ with increasing layers on two small datasets.}\n\t\\label{figure1}\n\\end{figure}\n\n\\begin{theorem} \\label{theorem2}\n\t$(I_n-\\mu \\hat{A})$ is invertible when $\\beta<\\frac{1}{n}$ where $\\mu=\\frac{\\alpha+\\beta}{1+\\alpha+\\beta}$ and $\\hat{A}=\\tilde{A}+\\frac{\\beta n\\tilde D^{-1}-\\beta \\tilde D^{-1} \\mathbf J_n}{\\alpha+\\beta}$. Consider the following iterative scheme\n\t\\begin{equation} \\label{iterative2}\n\t\\begin{aligned}\n\tZ^{(0)}&=H, \\\\\n\tZ^{(k)}&=\\mu\\hat{A}Z^{(k-1)}+ (1-\\mu)H,\n\t\\end{aligned}\n\t\\end{equation} where $\\mu \\in (0,1)$.\n\tWhen $k \\rightarrow \\infty$, \n\t\\begin{equation}\n\tZ^{(\\infty)}=(1-\\mu)(I_n-\\mu\\hat{A})^{-1}H.\n\t\\end{equation} \n\\end{theorem}\n\\begin{proof}\n\tLet $M=\\frac{\\beta n\\tilde D^{-1}-\\beta \\tilde D^{-1} \\mathbf J_n}{\\alpha+\\beta}=\\frac{\\beta \\tilde D^{-1} }{\\alpha+\\beta}(nI-J_n)$. Note that $(nI-J_n)$ has the largest eigenvalue $n$. Suppose that $\\lambda$ is the eigenvalue of $M$, we have $\\lambda \\leq \\frac{\\beta n}{\\alpha+\\beta} $. Then eigenvalue of $ \\hat{A}$ is less than $1+\\frac{\\beta n}{\\alpha+\\beta}$. \n\t $(I_n-\\mu \\hat{A})$ is invertible iff $\\frac{1}{\\mu}$ is not an eigenvalue of $ \\hat{A}$. Note that $\\frac{1}{\\mu}=\\frac{1+\\alpha+\\beta}{\\alpha+\\beta}=1+\\frac{1}{\\alpha+\\beta}$, when $\\beta<\\frac{1}{n}$ we have $\\frac{1}{\\mu}>1+\\frac{\\beta n}{\\alpha+\\beta}$, hence $\\frac{1}{\\mu}$ cannot be an eigenvalue of $\\hat{A}$ and $(I_n-\\mu \\hat{A})$ is invertible. The proof of the iterative scheme follows the similar procedure of case 1 with a slight difference, as it is trivial, we omit the proof.\n\\end{proof}\n\n\\subsection{Why GCN+ relieve the over-smoothing?}\nWe have no assumptions on the specific transformation from $X$ to $\\bar{X}$. In our implementation, the mathematical expression of GCN+ is defined as \n\\begin{equation} \\label{gcn+}\n\\begin{aligned}\nZ^{(0)}&=H=\\sigma(XW_1), \\\\\nZ^{(k)}&=\\mu\\hat{A}Z^{(k-1)}+ (1-\\mu)H,\\\\\nX_{out}&=\\text{softmax}(Z^{(k)}W_2),\\\\\n\\end{aligned}\n\\end{equation} \nwhere $W_1\\in \\mathbb{R}^{d \\times m}$ and $W_2\\in \\mathbb{R}^{m \\times c}$ are learnable weight matrices, $k$ is the dimension of the hidden layers. \n\n\nWe interpret the anti-oversmoothing of GCN+ from two ways. First, note that in the power iterative scheme, a fraction of initial node features $H$ is always preserved in each iteration, which can be viewed as a flexible version of residual connection. In addition, we can also understand GCN+ from the frequency of graph signal. In Section \\ref{GRO} , we have shown that the original GCN is corresponding to the first-order Taylor approximation of the optimization solution, that means we lost the high frequency part of the signal which contains the high-order information. Actually we omit the error series when we approximate the GCN.\n\n\nRecall the current representative methods: DAGNN and JKNet, which shows promising improvement than the original GCN. The core formulas of them are as follows:\n\\begin{equation} \n\\nonumber\n\\begin{aligned}\n\\text{DAGNN:}&\\\\\n&Z=\\text{stack}(H, Z^{(1)}, ..., Z^{(k)}), \\quad Z^{(k)}=\\hat{A}^{(k)}H,\\\\\n\\text{JKNet:\\quad}&\\\\\n&Z=\\text{Aggr}(Z^{(1)}, ...,Z^{(k)}), \\\\\n\\end{aligned}\n\\end{equation} \nwhere \\text{Aggr} includes \\textit{Concatenation}, \\textit{Max-pooling} and \\textit{LSTM-attention}.\n\nActually, DAGNN and JKNet both make use of the information which from the immediate and high-order neighbors while GCN+ also benefit from this. Moreover, we provide the theoretical and empirical evidence of GCN+.\n\n\\subsection{Parameters Amount}\nIt is worth noting that the power iterative schemes are parameter-free in two versions of GCN+, which is similar to APPNP \\cite{klicpera2018predict}. In particular, GCN+ ($\\beta=0$) adopts the same scheme as APPNP, where we re-implement it and achieve more impressive results. \n\\section{Experiments}\nIn this section, we evaluate the performance of GCN+ on several benchmark datasets against various graph neural networks on semi-supersized node classification tasks.\n\t\n\\subsection{Experimental Setup}\n\\subsubsection{Datasets}\nWe conduct extensive experiments on the node-level tasks on two kinds commonly used networks: Planetoid: \\textit{Cora}, \\textit{CiteSeer}, \\textit{Pubmed} \\cite{sen2008collective} and recent \\textit{Open Graph Benchmark} (OGB) \\cite{hu2020open}:\\textit{ogb-arxiv}, \\textit{ogb-proteins}. The statistics of datasets are summarized in Table \\ref{dataset}. It is worth nothing that OGB includes enormous challenging and large-scale datasets than Planetoid. We refer readers to \\cite{hu2020open} for more details about OGB datasets.\n\\begin{table} \n\t\\small\n\t\\setlength{\\tabcolsep}{1mm}{\n\t\\begin{tabular}{{cccccc}}\n\t\t\\toprule\n\t\tDataset & Nodes & Edges & Classes & Features &Metric\\\\ \n\t\t\\midrule\n\t\tCora & 2708 & 5429 & 7 & 1433 & Accuracy \\\\ \n\t\tCiteseer & 3327 & 4732 & 7 & 2703 & Accuracy \\\\ \n\t\tPubmed & 19717 & 44338 & 3 & 500 & Accuracy \\\\ \n\t\togb-arxiv & 169343 & 1166243 & 40 & 128 & Accuracy \\\\ \n\t\togb-proteins & 132534 & 39561252 & 112 & 8 & ROC-AUC \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}}\n\\caption{Dataset statistics.} \n\\label{dataset}\n\\end{table}\n\\subsubsection{Implementations}\nWe choose the optimizer and hyperparameters of GNN models as follows. We use the Adam optimizer \\cite{kingma2014adam} to train all the GNN models with a maximum of 1500 epochs. We set the number of hidden units to 64 on \\textit{Cora}, \\textit{Citeseer} and \\textit{Pubmed} , 256 on \\textit{ogb-arxiv} and \\textit{ogb-proteins}. For SGC, we vary number of layer in \\{1, 2, ..., 10, 15, ..., 60\\} and for GCN and GAT in \\{2, 4, ..., 10, 15, ..., 30\\}. For $\\alpha$ in APPNP, we search it from \\{0.1, 0.2, 0.3, 0.4, 0.5\\}. For DAGNN and JKNet, we search layers from \\{2, 3, ..., 10\\}. For learning rate, we choose from \\{0.001, 0.005, 0.01\\}. For dropout rate, we choose from \\{0.1, 0.2, 0.3, 0.4, 0.5\\}. We perform a grid search to tune the hyperparameters for other models based on the accuracy on the validation set. We run each experiment 10 times and report the average. \n\nIn practice, we use Pytorch \\cite{paszke2019pytorch} and Pytorch Geometric \\cite{Fey\/Lenssen\/2019} for an efficient GPU-based implementation of GCN+.\nAll experiments in this study are conducted on NVIDIA TITAN RTX 24GB GPU.\n\n\\begin{table*}[t]\n\t\\small\n\t\\centering\n\t\\setlength{\\tabcolsep}{1.5mm}{\n\t\\begin{tabular}{ccccccc}\n\t\t\\toprule\n\t\t\\multirow{2}{*}{model} & \\multicolumn{2}{c}{\\textit{Cora}} & \\multicolumn{2}{c}{\\textit{Citeseer}} & \\multicolumn{2}{c}{\\textit{Pubmed}} \\\\\n\t\t& Fixed & Random & Fixed & Random & Fixed & Random \\\\\n\t\t\\midrule\n\t\tMLP & $61.6\\pm0.6$ & $59.8\\pm2.4$& $61.0\\pm1.0$ & $58.8\\pm2.2$ & $74.2\\pm0.7$ & $70.1\\pm2.4$ \\\\\n\t\tGCN\\cite{kipf2016semi} & $81.3\\pm0.8$ & $79.1\\pm1.8$& $71.1\\pm0.7$ & $68.2\\pm1.6$ & $78.8\\pm0.6$ & $77.1\\pm2.7$ \\\\\n\t\t GAT\\cite{velivckovic2017graph} &$83.1\\pm0.4$ &$80.8\\pm1.6$ &$70.8\\pm0.5$ &$68.9\\pm1.7$ & $79.1\\pm0.4$ &$77.8\\pm2.1$ \\\\\n\t\t SGC\\cite{wu2019simplifying} & $81.1\\pm0.5$ & $80.4\\pm0.3$& $71.9\\pm0.3$ & $71.8\\pm0.3$ & $78.9\\pm0.0$ & $77.8\\pm0.6$ \\\\\n\t\t JKNet\\cite{xu2018representation}& $80.7\\pm0.9$ &$79.2\\pm0.9$ &$70.1\\pm0.6$&$68.3\\pm1.8$ & $78.1\\pm0.6$ & $77.9\\pm0.9$ \\\\\n\t\t APPNP\\cite{klicpera2018predict} &$83.3\\pm0.5$ &$81.9\\pm1.4$ &$71.8\\pm0.4$ &$69.8\\pm1.7$ & $80.1\\pm0.2$ &$79.5\\pm2.2$ \\\\\n\t\t DAGNN\\cite{liu2020towards} &$84.4\\pm0.5$ &$\\textbf{83.7}\\pm\\textbf{1.4}$ &$73.3\\pm0.6$ &$71.2\\pm1.4$ & $\\textbf{80.5}\\pm\\textbf{0.5}$ &$80.1\\pm1.7$ \\\\\n\t\t\\midrule\n\t\t GCN+($\\beta=0$) &$85.2\\pm0.5$ &$83.3\\pm1.1$ &$73.3\\pm0.5$ &$72.3\\pm0.7$ & $80.4\\pm0.6$ &$80.1\\pm0.6$ \\\\\n\t\t GCN+($\\beta \\ne 0$) &$\\textbf{85.6}\\pm\\textbf{0.4}$ &$83.6\\pm1.3$ &$\\textbf{73.5}\\pm\\textbf{0.4}$ &$\\textbf{72.5}\\pm\\textbf{0.9}$ & $80.5\\pm0.6$ &$\\textbf{80.3}\\pm\\textbf{0.7}$ \\\\\n\t\t\\bottomrule \n\t\\end{tabular}}\n\\caption{Summary of classification accuracy(\\%) on Planetoid datasets of semi-supervised node classification.}\n\\label{Plantoid-semi}\n\\end{table*}\n\n\\begin{table}[t]\n\t\\centering\n\t\\setlength{\\tabcolsep}{3.5mm}{\n\t\\begin{tabular}{{ccccc}}\n\t\t\\toprule\n\t\tDataset & \\textit{ogb-arxiv} & \\textit{ogb-proteins} \\\\ \n\t\t\\midrule\n\t\tGCN & $71.74\\pm0.29$ & $72.51\\pm0.35$ \\\\ \n\t\tGraphSAGE & $71.49\\pm0.25$ & $77.68\\pm0.20$ \\\\ \n\t\t\\midrule\n\t\tGCN+($\\beta=0$) & $71.85\\pm0.23$&$78.63\\pm0.28$ \\\\ \n\t\tGCN+($\\beta \\ne0$) &$\\textbf{71.95}\\pm\\textbf{0.28}$ & $\\textbf{79.07}\\pm\\textbf{0.34}$ \\\\ \n\t\t\\bottomrule\n\t\\end{tabular}}\n\t\\caption{Summary of classification performance(\\%) on OGB datasets. For \\textit{ogb-arxiv}, it indicates accuracy and for \\textit{ogb-proteins}, it indicates ROC-AUC.} \n\t\\label{ogb-semi}\n\\end{table}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.9\\columnwidth]{constant}\n\t\\caption{$M_{\\textit{\\!non-smooth}}$ of GCN+ with increasing hops on \\textit{Cora}.}\n\t\\label{oversmooth}\n\\end{figure}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.9\\columnwidth]{sys_rw}\n\t\\caption{Performance comparison of different propagation matrices $\\tilde{A}_{sym}$ vs. $\\tilde{A}_{rw}$ in GCN+ with increasing hops on \\textit{Cora}.}\n\t\\label{choice}\n\\end{figure}\n\\begin{figure*}[t]\n\t\\centering\n\t\\subfigure[$\\alpha=9$]{\n\t\t\\includegraphics[width=0.24\\textwidth]{alpha_9.pdf}}\\hspace{-1mm}\n\t\\subfigure[$\\alpha=4$]{\n\t\t\\includegraphics[width=0.24\\textwidth]{alpha_4.pdf}}\\hspace{-1mm}\n\t\\subfigure[$\\alpha=2$]{\n\t\t\\includegraphics[width=0.24\\textwidth]{alpha_2.pdf}}\\hspace{-1mm}\n\t\\subfigure[$\\alpha=1$]{\n\t\t\\includegraphics[width=0.24\\textwidth]{alpha_1.pdf}}\\hspace{-1mm}\n\t\\caption{Test accuracy of different propagation steps and $\\alpha$ on \\textit{Cora}.}\n\t\\label{alpha and beta}\n\\end{figure*}\n\n\n\\subsection{Comparison with SOTA}\nThe evaluate metric of various datasets are listed in Table \\ref{dataset}. Actually it is commonly used to evaluate the model by the community.\n\\subsubsection{Planetoid}\nWe use standard fixed and random training\/validation\/testing splits. Specifically, we use 20 labeled nodes per class as the training set, 500 nodes as the validation set, and 1000 nodes as the test set for all models. For fixed split, we follow the experimental setup in \\cite{yang2016revisiting}. We compare Multiplayer Perception (MLP) ,GCN \\cite{kipf2016semi}, GAT \\cite{velivckovic2017graph}, SGC \\cite{wu2019simplifying}, DAGNN \\cite{liu2020towards} and APPNP \\cite{klicpera2018predict} with GCN+. Although DropEdge \\cite{rong2019dropedge}, PairNorm \\cite{zhao2019pairnorm} are proposed to tackle over-smoothing issue recently, our baseline methods don't include them as they do not help to boost the performance on node classification task. \nTable \\ref{Plantoid-semi} compares the average test accuracy of 10 runs for each model on Planetoid dataset. As shown, GCN+ outperforms better than the representative baselines. Note that the shallow model APPNP achieves better performance than GCN and GAT. Recent deeper model named DAGNN shows competitive result and robustness on these datasets and GCN+ performs slightly better than it.\n\\subsubsection{OGB}\nWe adopt the setting of \\cite{hu2020open}, which is more challenging and realistic. We consider the following representative models GCN \\cite{kipf2016semi}, GraphSAGE \\cite{hamilton2017inductive} and GCNII \\cite{chen2020simple} as our baselines. In particular, we use the reported metric of the leaderboards of OGB team, which provide an open benchmark on several tasks and datasets. \n\nTable \\ref{ogb-semi} compares the average test accuracy\/ROC-AUC on OGB datasets. As shown, GCN+ outperforms the GCN and GraphSAGE. It is clear that our proposed GCN+ outperform SOTA in two middle scale datasets.\n\nIn summary, GCN+ achieves superior performance on several benchmarks, which shows that considering the information of high-order neighbors makes sense and we need more reasonable way to deepen GCNs or make use of the high-order neighbors. Note that GCN+ ($\\beta \\ne0$) is slightly better than GCN+ ($\\beta =0$) which is benefit from the third term of Eq. (\\ref{full-opt}). \n\\subsection{Over-smoothing Analysis} \\label{oversmooth-of-gcn+}\nWe employ the two proposed metrics to measure the node embeddings learned by GCN+. The results on \\textit{Cora} are shown in Fig. \\ref{oversmooth}. We can observe that as the number of hops increases, the $M_{\\textit{smooth}}$ values nearly remains a small constant which is lower than vanilla deep GCN. This implies that GCN+ use the information of long-range neighbors and do not suffer from over-smoothing. \n\nFig. \\ref{model_vs} also compares the final output embeddings of GCN+ with multiple hops, which shows different behaviors with GCN. GCN+ relieves the over-smoothing and learns the meaningful embeddings with the increasing hops.\n\n\\subsection{Hyperparameter Analysis}\nIn the previous sections, we use $\\tilde{A}$ to refer the $\\tilde{A}_{\\textit{sym}}$ and $\\tilde{A}_{\\textit{rw}}$. Here we compare the different choices of propagation matrix $\\tilde{A}$. Fig. \\ref{choice} depicts the test accuracy achieved by varying the hops of different propagation matrices. The result illustrates that \n$\\tilde{A}_{\\textit{sym}}$ is slightly better than $\\tilde{A}_{\\textit{rw}}$. \n\nWe consider three hyperparameter of GCN+, that is $\\alpha$, $\\beta$ and number of power iteration steps $k$. Fig. \\ref{alpha and beta} compares the effect of these hyperparameters on \\textit{Cora}. We can see that $k=16,32$ is suitable and more steps does not boost the performance significantly. For \\textit{Cora}, when $\\alpha=9$ (that means the fraction of retained initial node features is 0.1.), GCN+ achieve the best performance. The value of $\\alpha$ varies by different datasets. More results and details listed in the supplementary material.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Related Work}\n\\subsection{Graph Neural Networks}\nGraph neural networks (GNNs) have been extensively studied for the past years. There are different views on designing new architecture, including the spectral-based, spatial-based and other types, such as understand the GNN using dynamic system \\cite{xhonneux2019continuous}. Numerous methods are proposed to model the graph-structure data and apply on a wide range of applications. Besides the GCNs, there are also other types of GNNs, such as attention-based GNN \\cite{velivckovic2017graph} which use multiple attention to aggregate information from neighbors, autoencoder-based GNN \\cite{kipf2016variational}, which use a GCN encoder and decoder to learn meaningful embeddings, and dynamic GNNs \\cite{seo2018structured, hajiramezanali2019variational, yan2020sgrnn} which learn the node embedding over time.\n\\subsection{Deep GCN and Over-smoothing}\nMost GNNs are shallow models as deep architecture suffers from over-smoothing. Several studies explore deep GCNs. \\citet{xu2018representation} introduce Jumping Knowledge Networks, which uses residual connection to combine the output of each layer. \\citet{klicpera2018predict} use Personalized PageRank, which consider the information of root node to replace the graph convolution operator to solve the over-smoothing. DropEdge \\cite{rong2019dropedge} suggests that randomly removing the edge of original graph impede over-smoothing. PairNorm \\cite{zhao2019pairnorm} is another scheme which uses a normalization layer to scale the node features after the convolution layer. \\citet{li2019deepgcns} build on ideas from ResNet to train very deep GCNs. \\citet{li2020deepergcn} further propose MsgNorm, which boosts the performance on several datasets. \\citet{yang2020revisiting} present NodeNorm to scale the node features. \\cite{chen2020simple} propose a deep GCN models which use initial residual connection and identity mapping. \n\nA few work analyzes the cause and behaviors of over-smoothing theoretically. \\citet{oono2019graph} investigate the asymptotic behaviors of GCNs as the layer size tends to infinity and reveals the information loss in deep GCNs. \\citet{cai2020note} further extend analysis of \\cite{oono2019graph} from linear GNNs to the nonlinear architecture.\n\n\n\\section{Conclusion}\nWe summarize the existing different views on the mechanism of GCNs, which help us understand and design the graph convolutional kernel. We further provide a general optimization framework named GCN+. Based on this framework, we derive two forms of GCN+ and propose two metrics to measure the smoothness of output node representations. Extensive empirical studies on several real-world datasets demonstrate that GCN+ compares favorably to state of the art with a small amount of parameters. For future work, we will consider different optimization objectives which encode the graph structure and node features adaptively. As we do not limit the transformation from $X$ to $\\bar{X}$, another reasonable formulas can be further explored.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}