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+{"text":"\\section*{SUPPLEMENTARY MATERIAL}\nMore computational details (with the definition of $N_c$ and $L_\\mathrm{p}$);\nevolution of $N_c$ for cMD at various temperatures;\npotential energy distribution for each replica in T-REMD and REST;\nadditional statistical data and MD snapshots\nfor assembly structures; replica traversal in temperature space for T-REMD;\nan electronic archive for the configuration\nand topology files of the present BTA system.\n\n\n\n\\vspace{-0.5cm}\n\\begin{acknowledgments}\nThe authors acknowledge support from JSPS Grant-in-Aid for Scientific Research\non Innovative Areas ``Dynamical ordering of biomolecular systems\nfor creation of integrated functions'' (Grant No. 25102002).\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{sec:Intro}\n\nDue to their large dipole moments \\cite{G94}, Rydberg atoms experience\nstrong long-range dipole-dipole interactions. Within the last decade,\nthis feature has been put forward as the key ingredient of different\npromising atomic quantum-processing scenarios \\cite{SWM10}. For instance,\nRydberg-Rydberg interactions can be used to perform two-qubit logic\noperations in individual-atom systems by shifting a transition off-resonance\nin an atom, depending on the internal state of another atom in its\nimmediate neighborhood \\cite{JCZ00,PRS02,SW05}. In a mesoscopic ensemble,\ndipole-dipole interactions are able to inhibit transitions into collective\nstates that contain more than one Rydberg excitation, thus leading\nto the so-called Rydberg blockade. First predicted in \\cite{LFC01},\nthis phenomenon was locally observed in laser-cooled atomic systems\n\\cite{TFS04,SRA04,CRB05,AVG98,VVZ06} and could in principle be used\nin the future to manipulate and entangle collective excitation states\nof mesoscopic ensembles of cold atoms which could therefore be run\nas quantum processors \\cite{LFC01,BMM07,BMS07,BPS08} or repeaters\n\\cite{ZMH10,HHH10,BCA12}. Rydberg atomic ensembles are also investigated\non quantum non-linear optical purposes \\cite{WA13}: converting photons\ninto so-called Rydberg polaritons which strongly interact through\ndipole-dipole interaction, it seems indeed possible to generate giant\nnon-linearities in the quantum regime, \\emph{i.e.\\@} to effectively\nimplement photon-photon interactions \\cite{PFL12,PBS12,MSP13}.\n\nThe exact calculation of the time-dependent state of an ensemble of\natoms resonantly laser-driven towards a Rydberg level constitutes\na highly non-trivial coupled many-body problem. Such a complex system,\nhowever, often shows an effective thermalization behavior \n\\cite{LOG10,AL12,BMFALA13}: \nobservables, such as the number of Rydberg excitations, indeed tend\nto quasistationary values which can be computed assuming the system\nis in a thermal equilibrium state, either in the canonical \\cite{LOG10}\nor microcanonical \\cite{AL12,BMFALA13} ensembles. Considering the\nsame system as in \\cite{AL12,BMFALA13}, we compare the predictions\nof the microcanonical ensemble assumption to a numerical simulation\nof the unitary evolution of a $100$-particle system, confirmed by\na simplified analytical treatment. The discrepancies we observe allow\nus to show the limitations of the equilibrium hypothesis and to precise\nits applicability conditions.\n\nThe paper is organized as follows. In Sec.~\\ref{sec:PhysicalModel},\nwe present the physical system and simplified model. In Sec.~\\ref{sec:Thermal},\nwe give an analytical description of the distribution of excitations\naccording to the microcanonical ensemble. In Sec.~\\ref{Calculations},\nwe numerically compute the distribution of excitations and apply a\nsimplified analytical treatment which allows us to satisfactorily\nreproduce the results of the full simulation, in the regime of at\nmost 2 excitations. In Sec.~\\ref{sec:Comparison}, we discuss and\ncompare the results obtained according to the different approaches,\nbefore concluding in Sec.~\\ref{sec:Conclusion}.\n\n\n\\section{Model and approximations}\n\n\\label{sec:PhysicalModel}\n\nWe consider a system of $N$ identical atoms located along a line\nof length $L$. The Hilbert space of each atom is assumed to be restricted\nto the ground state $\\left|g\\right\\rangle $ and a highly excited\n(so-called) Rydberg state $\\left|r\\right\\rangle $. In the ensemble\n``vacuum state'' $\\left|\\varnothing\\right\\rangle $, all atoms are\nin the ground state: $\\left|\\varnothing\\right\\rangle \\equiv\\left|g\\dots g\\right\\rangle $.\nDenoting by $\\sigma_{+}\\equiv\\left|e\\right\\rangle \\left\\langle g\\right|$\nand $\\sigma_{-}\\equiv\\sigma_{+}^{\\dagger}$ the usual raising and\nlowering operators for a two-level atom, one defines the ensemble\nstate $\\left|i\\right\\rangle \\equiv\\sigma_{+}^{i}\\left|\\varnothing\\right\\rangle =\\left|g_{1},\\dots,g_{i-1},r_{i},g_{i+1},\\dots,g_{N}\\right\\rangle $\nin which the $i^{\\text{th}}$ atom is Rydberg-excited while the others are\nin the ground state. In the same way, one can define the doubly excited\nstate $\\left|i,j\\right\\rangle \\equiv\\sigma_{+}^{\\left(i\\right)}\\sigma_{+}^{\\left(j\\right)}\\left|\\varnothing\\right\\rangle $\nwhich contains only two Rydberg excitations at positions $i$ and\n$j$, and more generally any arbitrary multiply excited state $\\left|i,j,k,\\dots\\right\\rangle \\equiv\\sigma_{+}^{\\left(i\\right)}\\sigma_{+}^{\\left(j\\right)}\\sigma_{+}^{\\left(k\\right)}\\dots\\left|\\varnothing\\right\\rangle $.\n\nThe atomic ensemble is subject to a laser beam which resonantly drives\nthe transition $\\left|g\\right\\rangle \\leftrightarrow\\left|r\\right\\rangle $:\nin the rotating wave approximation, this process is simply described\nby the Hamiltonian $H_{L}=\\hbar\\Omega\\sum_{k=1}^{n}\\left(\\sigma_{+}^{k}+\\sigma_{-}^{k}\\right)$,\nwhere $\\Omega$ denotes the laser Rabi frequency. Moreover, when lying\nin their Rydberg state, two atoms interact through the (strong) dipole-dipole\ninteraction (this interaction is negligible when at least one atom\nin the pair is in the ground state): the corresponding Hamiltonian\nis \n\\begin{equation}\nV_{dd}=\\hbar C_{6}\\sum_{k\\neq m}\\frac{n_{m}n_{k}}{d(m,k)^{6}}\\label{V_dd}\n\\end{equation}\nwhere $C_{6}$ is the van der Waals interaction coefficient, $n_{k}\\equiv\\sigma_{+}^{k}\\sigma_{-}^{k}$\nthe projector onto the Rydberg state for the $k^{\\text{th}}$ atom and $d(m,k)$\nis the distance between the $m^{\\text{th}}$ and the $k^{\\text{th}}$ atoms.\nFinally,\nthe full Hamiltonian governing the dynamics of the system is \n\\begin{align}\nH & =H_{L}+V_{dd}\\label{Ham}\n\\end{align}\n\n\nStarting in the ensemble vacuum state $\\left|\\varnothing\\right\\rangle $,\nin the absence of interatomic interactions, each atom in the sample\nwould independently undergo Rabi oscillations. Because of dipole-dipole\ninteractions, atoms actually get entangled during their evolution,\naccording to the so-called Rydberg blockade phenomenon \\cite{LFC01}.\nTo understand this mechanism, let us first consider the simple case\nof two atoms. If they are ``close enough'' so that their dipole-dipole\ninteraction overwhelms the laser Rabi frequency, their simultaneous\nexcitation into the Rydberg state becomes impossible since the doubly\nexcited state is strongly shifted out of resonance. As a rule of thumb,\none can define the typical distance $R_{b}$, called the blockade\nradius, at which the blockade starts being effective as the distance\nfor which the van der Waals interaction becomes comparable with the\nlaser excitation, \\emph{i.e} $R_{b}\\approx\\left(\\frac{C_{6}}{\\Omega}\\right)^{\\frac{1}{6}}$.\nNow turning to the full sample, it is clear that dipole-dipole interactions\nforbid the system to explore its full Hilbert space: too off-resonant\nconfigurations, \\emph{i.e.\\@} ensemble states in which two Rydberg excited\natoms are closer than the radius $R_{b}$, will indeed never be substantially\npopulated. In other words, due to the Rydberg blockade the system\nis bound to essentially evolve in the subspace of ``allowed states''\nin which excited atoms are separated at least by $R_{b}$ (Note that\nin a 3D geometric arrangement, each Rydberg excited atom creates an\n``exclusion'' sphere of radius $R_{b}$ often called a ``Rydberg\nbubble'').\n\nThough simple in its form, the Hamiltonian Eq.~\\eqref{Ham} leads\nto complex many-body dynamics. In particular, besides the Rydberg\nblockade phenomenon qualitatively described above, it was shown to\nyield thermalization effects \\cite{LOG10}. The full computation of\nthe dynamics is intractable for large numbers of atoms and one must\nresort to approximations. Following \\cite{LOG10}, we shall make the\nhardcore Rydberg sphere assumption, that is we shall merely discard\nall atomic configurations in which two Rydberg excitations are closer\nthan $R_{b}$, while keeping the others; moreover, we shall make the\nsimplistic approximation that in the allowed subspace the dipole-dipole\nHamiltonian is zero. In this approximation the full Hamiltonian therefore\nbecomes \n\\begin{equation}\nH\\approx\\hbar\\Omega\\sum_{k=1}^{n}(\\tilde{\\sigma}_{+}^{k}+\\tilde{\\sigma}_{-}^{k})\\label{Happrox}\n\\end{equation}\nwhere $\\tilde{\\sigma}_{+}^{k}$ is the raising operator of the $k^{\\text{th}}$\natom restricted to the allowed configuration subspace, \\emph{i.e.\\@}\nthe operator which excites the $k^{\\text{th}}$ atom into the Rydberg\nstate provided that no other Rydberg atom is in the range $R_{b}$.\n\n\n\\section{Thermalized state: Analytical results from the microcanonical ensemble\nassumption }\n\n\\label{sec:Thermal}\n\\\nAnalytical \\cite{AGL12,JAG13} and numerical \\cite{LOG10} investigations\nof the approximate Hamiltonian Eq.~\\eqref{Happrox} both predict\nthermalization to occur in the system. Intuitively, this phenomenon\nresults from the destructive interferences between different frequency\ncomponents of the evolved vector state: for large times, due to the\ncomplexity of the Hilbert space and the high connectivity of the basis\nstates, observables, such as the number of Rydberg-excited atoms,\nare expected to stop oscillating and tend to quasistationary values.\nAccording to the microcanonical ensemble assumption \\cite{AL12,BMFALA13},\nthese values can be accounted for by assuming an effective thermal\nequilibrium-like state for the system which consists of an equiprobable\nstatistical mixture of all allowed states.\n\nThe common probability of all the components in this mixture is therefore\nsimply given by the inverse of the total number $\\mathcal{N}$ of\nallowed states. This number can be determined by summing all numbers\n$\\mathcal{N}(\\nu)$ of allowed configurations with exactly\n$\\nu$ excitations, which, as we show below, can be calculated through\na straightforward combinatorial argument. From $\\mathcal{N}(\\nu)$,\none easily computes the average number of excitations $\\left\\langle \\nu\\right\\rangle $\nand, in the limit of a continuous distribution, one can even deduce\na simple expression of the spatial density of Rydberg excitations.\n\nIn this section, we present our analytical calculations in detail\nand compare our results to numerical Monte Carlo simulations presented\nin \\cite{BMFALA13}.\n\n\n\\subsection{Number of allowed states and average excitation number: combinatorial\nanalysis}\n\nThe goal of this subsection is to compute the average number of Rydberg\nexcitations observed in the thermalized state according to the microcanonical\nensemble assumption. For sake of simplicity, we assume that the atoms\nare located at the nodes of a regular 1D lattice of step $a$. The\ndistance between the $i^{\\text{th}}$ and $j^{\\text{th}}$ atoms is therefore $d(i,j)=a\\left|i-j\\right|$\nwhile the total length of the line is given by $L=\\left(N-1\\right)a$.\nThe quantity $n_{b}\\equiv\\left\\lfloor \\frac{R_{b}}{a}\\right\\rfloor $,\nwhere $\\left\\lfloor \\cdot\\right\\rfloor $ denotes the lower integer\npart, represents the minimal number of ground-state atoms which must\nlie between two Rydberg excitations in an allowed atomic configuration\naccording to the hardcore Rydberg sphere assumption. Finally, we introduce\nthe real parameter $\\Lambda\\equiv\\frac{L}{R_{b}}$. Adding one to its integer part\ngives the maximum number of Rydberg excitations the sample can accommodate for: \n$\\nu_{\\max}=\\left\\lfloor \\Lambda\\right\\rfloor + 1$.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{Fig1_2}%\n\n{\\Large{}{}$\\overbrace{\\bigstar\\bigstar\\bigstar}^{n'_{0}}|\\overbrace{\\phantom{|}}^{n'_{1}}|\\overbrace{\\bigstar\\bigstar\\bigstar\\bigstar}^{n'_{2}}|\\overbrace{\\bigstar\\bigstar\\bigstar}^{n'_{3}}$}\n\\protect\\protect\\caption{\\label{fig:StarsBars}: Description of a configuration of the excitations\nand mapping to the ``Stars and bars'' problem. An excited atom with\nall the atoms on its right closer than $R_b$ correspond to a bar, except\nfor the last bar constituted only by the last excited atom. Each remaining\natom represent a star. The depicted configuration corresponds to $L=16$,\n$N=17$, $2a\\le R_{b}<3a$, $n_{b}=2$, $\\{n_{k}\\}_{k}=\\{3,2,6,3\\}$,\n$\\{n'_{k}\\}_{k}=\\{3,0,4,3\\}$.}\n\\end{figure}\n\n\nTo begin with, we compute the number of allowed states which comprise\na given number of excitations $\\nu$. In such a state, the $\\nu$\nRydberg excitations split the sample into $\\left(\\nu+1\\right)$ groups\nof $n_{k=0,\\dots,\\nu}$ ground-state atoms (see Fig.~\\ref{fig:StarsBars}),\nwith the convention that the zeroth and $\\nu^{\\text{th}}$ groups are on the\nleft and the right of the leftmost and rightmost excited atoms, respectively,\nand allowing $n_{0}$ and $n_{\\nu}$ to be zero. The state indeed\ncorresponds to an allowed configuration if it satisfies the hardcore\nRydberg sphere condition, \\emph{i.e.\\@} $n_{k}\\geq n_{b}$ for $1\\leq k\\leq\\left(\\nu-1\\right)$,\nunder the prescription $\\sum_{k=0}^{\\nu}n_{k}=N-\\nu$: finding the\nnumber of allowed states with $\\nu$ excitations is therefore equivalent\nto computing the number of sets of integers $\\left\\{ n_{k=0,\\dots,\\nu}\\right\\} $\nwhich satisfy the two previous conditions. A slight modification in\nthe formulation of this problem turns it into a standard combinatorial\ncalculation as we shall now show. We first note that an allowed atomic\nconfiguration can be uniquely determined by the alternative set of\nnumbers $\\left\\{ n{}_{k}'\\right\\} $ defined by \n\\begin{align*}\nn_{0}' & \\equiv n_{0}\\\\\nn_{k}' & \\equiv n_{k}-n_{b}\\;\\text{for }1\\leq k\\leq\\nu-1\\\\\nn_{\\nu}' & \\equiv n_{\\nu}\n\\end{align*}\nwhich satisfy the conditions $n_{k}'\\geq0$ and $\\sum_{k=0}^{\\nu}n'_{k}=N-1-(\\nu-1)(n_{b}+1)$.\nThis change of variables suggests to associate the original atomic\narrangement with an abstract linear distribution of $\\left[N-1-(\\nu-1)(n_{b}+1)\\right]$\n``stars'' split by $\\nu$ ``bars'' into $\\left(\\nu+1\\right)$\ngroups labelled by $k=0,\\dots,\\nu$ and respectively comprising $n_{k}'$\nelements. As shown in Fig.~\\ref{fig:StarsBars}, the first $\\left(\\nu-1\\right)$\nbars symbolize the first $\\left(\\nu-1\\right)$ Rydberg excited atoms\nwith their first $n_{b}$ (ground-state) right neighbors, while the\nlast bar represents the last Rydberg excited atom only; stars then\nsimply stand for the remaining ground state atoms. Calculating the\nnumber $\\mathcal{N}(\\nu)$ of such configurations is a\nstandard combinatorial problem whose solution is given by the binomial\ncoefficient \n\\begin{align}\n\\mathcal{N}(\\nu) & =\\binom{N-(\\nu-1)n_{b}}{\\nu}\\nonumber \\\\\n & =\\frac{N^{\\nu}}{\\nu!}\\prod_{i=0}^{\\nu-1}\\left(1-\\frac{(\\nu-1)n_{b}+i}{N}\\right)\\label{eq:Nnudiscrete}\n\\end{align}\nNote that $\\mathcal{N}(\\nu)=0$ when $\\nu-1\\geq\\frac{N}{\\left\\lfloor R_{b}\\right\\rfloor +1}=\\frac{L+1}{\\left\\lceil R_{b}\\right\\rceil }$.\nIn the limit of large $N$ and $R_{b}$, this essentially means that\nwe only have to consider configurations with a number of excitations\nsmaller than $\\nu\\lesssim\\Lambda$. In this limit, when $\\Lambda\\ll R_b,N$,\nwe can approximate equation \\eqref{eq:Nnudiscrete} by \n\\begin{equation}\n\\mathcal{N}(\\nu)=\\frac{N^{\\nu}}{\\nu!}\\left[1-\\frac{\\nu-1}{\\Lambda}\\right]_{+}^{\\nu}+O(N^{\\nu-1}),\\label{eq:apNnudiscrete}\n\\end{equation}\nwhere $[x]_{+}^{\\nu}=0$ if $x\\le0$ and $[x]_{+}^{\\nu}=x^{\\nu}$\nif $x\\ge0$.\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{ExcProba}\\protect\\caption{\\label{fig:ExcProba} Probability to have $\\nu$ excitation considering\nthe microcanonical ensemble as a function of $\\Lambda$ for $N=10^{4}$.\nThe successive peaks correspond to increasing value of $\\nu$. For\nexample, $P(\\nu=2)$ is close to 1 when $\\Lambda$ is between 1 and\n2. }\n\\end{figure}\n\n\nFrom $\\mathcal{N}(\\nu)$, one easily computes the total number of\nallowed configurations $\\mathcal{N}=\\sum_{\\nu}\\mathcal{N}(\\nu)$,\nthe probability to have $\\nu$ excitations in the sample $\\mathcal{P}(\\nu)=\\mathcal{N}(\\nu)\/\\mathcal{N}$\nand the average excitation number $\\langle\\nu\\rangle=\\sum_{\\nu}\\nu\\mathcal{P}(\\nu)$\nas well as its standard deviation $\\left\\langle \\sigma_{\\nu}\\right\\rangle $\nas a function of $\\Lambda$. The family of curves $\\left\\{ \\mathcal{P}(\\nu),\\nu=0,1,\\dots\\right\\} $\nis plotted on Fig.~\\ref{fig:ExcProba} as a function of $\\Lambda$\nfor $N=10^{4}$ and on Fig.~\\ref{fig:Probabilities} for $N=10^{2}$;\n$\\left\\langle \\nu\\right\\rangle $ and $\\left\\langle \\sigma_{\\nu}\\right\\rangle $\nare represented on Fig.~\\ref{fig:AvrgNu} as functions of $\\Lambda$\nfor $N=10^{4}$. Our results show perfect quantitative agreement with\n\\cite{BMFALA13}, as detailed in Appendix~\\ref{App:Comparison}\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{AvrgNu}\\protect\\caption{\\label{fig:AvrgNu} Average $\\nu$ and standard deviation in the microcanonical\npredictions as a function of $\\Lambda$ for $N=10^{4}$. The crosses\ncorrespond to the Monte-Carlo results from \\cite[fig 2c]{BMFALA13}.}\n\\end{figure}\n\n\n\n\\subsection{Spatial density of Rydberg excitations}\n\nWe can go further in our analysis and compute how Rydberg excitations\nare distributed along the line in average. Calculations turn to be\nmuch easier in the limit of a homogeneous and continuous atomic distribution,\nof constant linear density $\\delta\\equiv\\frac{1}{a}$ which is a good\napproximation of our model when $R_{b},L\\gg a$.\n\nLet us denote by $\\mathcal{N}(\\nu,l)$ the number of configurations\nwith $\\nu$ excitations on a line of length $l$ with the density\n$\\delta$. We have \n\\begin{align*}\n\\forall l\\ge0,\\;\\mathcal{N}(0,l) & =1\\\\\n\\forall l<0,\\forall\\nu,\\;\\mathcal{N}(\\nu,l) & =0\n\\end{align*}\nWith these notations, $\\mathcal{N}(\\nu)=\\mathcal{N}(\\nu,L)$\nand if $\\nu>0$, the number of configurations with the leftmost excited\natom at position $x$ is given by $\\mathcal{N}(\\nu-1,L-R_{b}-x)$.\nIntegrating over $x$, we get the recurrence relation \n\\begin{equation}\n\\mathcal{N}(\\nu+1,L)=\\int_{0}^{L}dx\\:\\delta\\:\\mathcal{N}(\\nu,L-R_{b}-x),\n\\end{equation}\nand \n\\begin{align}\n\\mathcal{N}(\\nu,L) & =\\frac{\\delta^{\\nu}}{\\nu!}\\left[l-(\\nu-1)R_b\\right]_{+}^{\\nu}\\nonumber \\\\\n & =\\frac{N^{\\nu}}{\\nu!}\\left[1-\\frac{\\nu-1}{\\Lambda}\\right]_{+}^{\\nu}\\label{eq:Nnu2}\n\\end{align}\nwhich is consistent with Eq.~\\eqref{eq:apNnudiscrete}.\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{ExcDistribution}\\protect\\caption{\\label{fig:ExcDistribution} Probability distribution of Rydberg excitations\nalong the chain as a function of $\\Lambda$ and the position for $N=10^{4}$\natoms. This figure quantitatively reproduces the Monte-Carlo simulation \nof \\cite[fig 2a]{BMFALA13}, as detailed in Appendix~\\ref{App:Comparison}.}\n\\end{figure}\n\n\nThe probability density to have the $n^{\\text{th}}$ excited atom out of $\\nu$\nat the position $x$ is: \n\\begin{align}\n  p(\\nu,n,x)= \n    & \\delta\\:\\frac{\\mathcal{N}(n-1,x-R_b)\\times\\mathcal{N}(\\nu-n,L-R_b-x)}{\\mathcal{N}(\\nu)} \n    \\notag\\\\\n  = & \\frac{\\nu!\\left[\\xi-\\tfrac{n-1}{\\Lambda}\\right]_{+}^{n-1}%\n          \\left[1-\\xi-\\tfrac{\\nu-n}{\\Lambda}\\right]_{+}^{\\nu-n}}%\n    {\\left(n-1\\right)!\\left(\\nu-n\\right)!\\left[1-\\tfrac{\\nu-1}{\\Lambda}\\right]_{+}^{\\nu}}\n    \\label{eq:excdensity}\n\\end{align}\nwhere we introduced the normalized dimensionless position $\\xi\\equiv\\frac{x}{L}$.\nNote that it does not depend on $N$; as seen above, however, $N$\nplays a role in the global probability for having $\\nu$ excitations.\nThis allows to plot the spatial distribution of excitation $\\mathcal{P}\\left(x\\right)=\\sum_{\\nu}\\sum_{n\\leq\\nu}p\\left(\\nu,n,x\\right)$,\nas in Fig.~\\ref{fig:ExcDistribution} for $N=10^{4}$, which quantitatively\nagrees with the Monte-Carlo simulations provided in \\cite[fig 2a]{BMFALA13,BMFALA13data}\n(see Appendix~\\ref{App:Comparison}).\nThe spatial distribution of excitations is also plotted in Fig.~\\ref{fig:exc_proba_1,5_}\n(red curve) for $N=100$ and $\\Lambda=1.5$.\n\n\n\\section{Numerical and simplified analytical calculations of the thermalized\nstate \\label{Calculations}}\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{Fig5_new}\\protect\\caption{\\label{fig:localisationExc2}: Numerically computed probability distribution\n$\\mathcal{P}_{k}$ of Rydberg excitations along the chain, as a function\nof $\\Lambda$ and the position (see section \\ref{sec:Numerics}) for\n$N=100$ atoms. The blue curve is the predicted position of the excitation\npeak by our simplified analytical treatment.}\n\\end{figure}\n\n\nIn this section, we present the results we obtained through a direct\nnumerical calculation of the thermalized state averages and recover\nsome of their interesting features through approximately diagonalizing\nthe Hamiltonian in a conveniently truncated basis. \n\n\n\\subsection{Numerical calculation of the thermalized state \\label{sec:Numerics}}\n\nOur numerical method consists in time-averaging the observables of\ninterest: if the average is performed on a very long (ideally infinite)\ntime, the obtained average must indeed coincide with the thermalized\nvalue. Due to computational complexity, we restricted our study to\na modest system of $N=100$ atoms equally spaced along the chain.\nEven with this relatively small value of $N$, the dimension $2^{N}$\nof the complete Hilbert space makes the full dynamical treatment intractable.\nWe therefore restricted ourselves to the regime $\\Lambda<2$, \\emph{i.e.\\@}\nthe chain is shorter than two Rydberg radii ($L\\leq2R_{b}$) and the\nmaximum number of excitations distributed along the chain is $2$.\nWe only need to take into account the states allowed by the Rydberg\nblockade whose number is given by Eq.~\\eqref{eq:Nnu2}:\n$\\sum_{\\nu\\leq2}\\mathcal{N}(\\nu)\\simeq\\mathcal{N}(\\nu=2)=\\frac{N^{2}}{2}\\left[1-\\frac{1}{\\Lambda}\\right]_{+}^{2}$.\nWe generate this set of allowed states through an arborescent\nsearch starting from $\\left|\\varnothing\\right\\rangle $ and adding\nallowed excitations.\n\nIn this subspace, we numerically diagonalize the Hamiltonian of\nEq.~\\eqref{Happrox}, yielding the (possibly degenerate) eigenenergies\n$E_{n}$ and the associated eigenvectors \n$\\ket{\\psi_{n}^{\\left(\\alpha_{n}\\right)}} $\nwhere $\\alpha_{n}=1\\dots d_{n}$, $d_{n}$ are the degeneracy index\nof the eigenenergy $E_{n}$. Fig.~\\ref{fig:DiffEnergy} presents the\nnumerical results of the diagonalization of $H$: more explicitly,\nthe red curve shows the absolute value $\\left|E_{n}\\right|$ versus\nthe rank of the corresponding eigenvectors \n$\\ket{\\psi_{n}^{\\left(\\alpha_{n}\\right)}}$,\narranged in increasing order of their eigenenergy; the blue curve\nrepresents the energy difference between two successive eigenvectors\nand therefore allows to check degeneracy. We take as a numerical criterion\nthat two energies coincide when their difference is less than $10^{-13}\\Omega$,\nconsistent with the precision of IEEE 754 floating-point arithmetics.\nOne first observes a wide central area corresponding to the highly\ndegenerate eigenenergy $E\\approx0$; in addition, on both\nsides of the spectrum, there exist two pairs of eigenstates with degenerate\nenergies.\n\nIf the system is initially prepared in \n$\\ket{\\Psi\\left(0\\right)} \\equiv \n \\sum_{n,\\alpha_{n}}c_{n}^{\\alpha_{n}}\\ket{\\psi_{n}^{\\left(\\alpha_{n}\\right)}}$,\nits state at time $t$ is given by \n$\\ket{\\Psi\\left(t\\right)} =\n  \\sum_{n,\\alpha_{n}}c_{n}^{\\alpha_{n}}e^{-\\mathrm{i}\\frac{E_{n}}{\\hbar}t}\n  \\ket{\\psi_{n}^{\\left(\\alpha_{n}\\right)}} $.\nThe time-averaged probability $\\mathcal{P}_{k}$ to have a Rydberg\nexcitation in site $k$ is therefore given by $\\mathcal{P}_{k}=\\mathrm{Tr}\\left[\\bar{\\rho}\\sigma_{rr}^{\\left(k\\right)}\\right]$\nwhere the average state $\\bar{\\rho}$ is \n\\begin{align}\n\\bar{\\rho} & =\\overline{\\left|\\Psi\\left(t\\right)\\right\\rangle \\left\\langle \\Psi\\left(t\\right)\\right|}\\nonumber \\\\\n & =\\sum_{m,\\alpha_{m}}\\sum_{n,\\beta_{n}}c_{m}^{\\alpha_{m}}\\left(c_{n}^{\\beta_{n}}\\right)^{*}\\left|\\psi_{m}^{\\left(\\alpha_{m}\\right)}\\right\\rangle \\left\\langle \\psi_{n}^{\\left(\\beta_{n}\\right)}\\right|\\times\\overline{e^{-\\mathrm{i}\\frac{E_{m}-E_{n}}{\\hbar}t}}\\nonumber \\\\\n & =\\sum_{n,\\alpha_{n},\\beta_{n}}c_{n}^{\\alpha_{n}}\\left(c_{n}^{\\beta_{n}}\\right)^{*}\\left|\\psi_{n}^{\\left(\\alpha_{n}\\right)}\\right\\rangle \\left\\langle \\psi_{n}^{\\left(\\beta_{n}\\right)}\\right|\\label{EqState}.\n\\end{align}\nWe have used the time average $\\overline{e^{-\\mathrm{i}\\frac{E_{m}-E_{n}}{\\hbar}t}}=\\delta_{mn}$\nto simplify the double sum. \n\nThe probability distribution $\\mathcal{P}_{k}$ is represented on\nFig.~\\ref{fig:localisationExc2} as a function of $\\Lambda$. For\n$\\Lambda\\gtrsim1.2$, two one-atom-wide black lines appear, revealing\na strong localization of Rydberg excitations. In the next subsection,\nwe account for this phenomenon through the approximate diagonalization\nof the Hamiltonian in a conveniently truncated basis.\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{Fig6_new}\\protect\\caption{\\label{fig:DiffEnergy} In red: $|E_{n}|$ versus $\\left(n,\\alpha_{n}\\right)$ with $\\Lambda = 1.5$\n(eigenstates are arranged in increasing order of their eigenenergy).\nIn blue: difference between two successive eigenvalues, \\emph{i.e.\\@}\n$\\left|E_{n+1}-E_{n}\\right|$. The dashed line shows the degeneracy\nlimit: below this line, any values can be assumed to be zero, up to numerical artifacts (see text). \n}\n\\end{figure}\n\n\n\n\\subsection{Simplified analytical treatment \\label{SimpleAnalyticalTreatment}}\n\nWe are now looking for a simple description of the system which would\nretain the basic physical features of the model, in particular the\nlocalization effect observed in the previous subsection. To this end,\nwe shall try and restrict the basis of the whole Hilbert space to\nonly the relevant states, \\emph{i.e.\\@} those which get significantly\npopulated during the evolution.\n\nAs a first try, we consider the four-dimensional basis $\\left\\{ \\left|\\varnothing\\right\\rangle ,\\left|\\phi_{1}\\right\\rangle ,\\left|\\phi_{2}\\right\\rangle ,\\left|\\phi_{3}\\right\\rangle \\right\\} $\ndefined by\n\\begin{align}\n  \\left|\\phi_{1}\\right\\rangle  \n  & \\equiv\\frac{H\\left|\\varnothing\\right\\rangle }%\n\t\t{\\left\\Vert H\\left|\\varnothing\\right\\rangle \\right\\Vert } \\label{state1}\\\\\n\t\\left|\\phi_{2}\\right\\rangle  \n\t& \\equiv\\frac{\\Pi_{2}H\\left|\\phi_1\\right>}%\n\t\t\t{\\left\\Vert\\Pi_{2}H\\left|\\phi_1\\right>\\right\\Vert}\n\t\t=\\frac{\\Pi_{2}H^{2}\\left|\\varnothing\\right\\rangle \n\t\t\t{\\left\\Vert \\Pi_{2}H^{2}\\left|\t\\varnothing\\right\\rangle \\right\\Vert }%\n\t \\label{state2}\\\\\n\t\\left|\\phi_{3}\\right\\rangle  &\n\t \\equiv\\frac{H\\left|\\phi_2\\right>}%\n\t\t\t{\\left\\Vert H\\left|\\phi_2\\right>\\right\\Vert}\n\t\t=\\frac{H\\Pi_{2}H^{2}\\left|\\varnothing\\right\\rangle }%\n\t\t\t{\\left\\Vert H\\Pi_{2}H^{2}\\left|\\varnothing\\right\\rangle \\right\\Vert  }%\n\t  \\label{state3}\n\\end{align}\nwhere $\\Pi_{2}$ denotes the projector onto the subspace of states\nwith exactly two Rydberg excitations. The diagonalization of $H$\nin this subspace yields four eigenstates $\\left|\\psi_{i=1,2}^{s=\\pm}\\right\\rangle $\nand eigenenergies $\\pm E_{i=1,2}$, such that $H\\left|\\psi_{i}^{s}\\right\\rangle =s\\times E_{i}\\left|\\psi_{i}^{s}\\right\\rangle $,\nwhose explicit expressions can be found in Appendix~\\ref{App:DiagH2}. We conventionally\nchoose $E_{2}\\geq E_{1}\\geq0$. The eigenenergies $E_{i}$ are plotted\nas functions of $\\Lambda$ on Fig.~\\ref{fig:EvsLambda}. Note that\nfor $\\Lambda>1$, all four eigenenergies $\\pm E_{i=1,2}$ take different\nvalues, there is hence no degeneracy. \n\nSince the eigenstates $\\left|\\psi_{i}^{s}\\right\\rangle $ describe\nconfigurations where excitations are delocalized (see Appendix~\\ref{App:Localization}),\nthe probability $\\mathcal{P}_{k}$ computed from the time-averaged\nstate Eq.~(\\ref{EqState})\n\\begin{align*}\n\\bar{\\rho} & =\\sum_{i=1,2}\\sum_{s=\\pm}\\left|c_{i}^{s}\\right|^{2}\\left|\\psi_{i}^{s}\\right\\rangle \\left\\langle \\psi_{i}^{s}\\right|\n\\end{align*}\nwill not exhibit the observed strong localization effect. \nNote that the four eigenstates\n$\\left|\\psi_{i=1,2}^{s=\\pm}\\right\\rangle $ contribute to the statistical\nmixture $\\bar{\\rho}$ with the respective weights \n$\\left|c_{i}^{s}\\right|^{2}\\equiv\\left|\\Braket{\\psi_{i}^{s}|\\Psi(0)}\\right|^{2}$\ndetermined by the initial state vector \n$\\ket{\\Psi(0)}=\\ket{\\varnothing}$. \n\nTo correctly account for the observed localization phenomenon, we\nmust therefore slightly extend the basis. To this end, we consider\nthe family of states $\\left\\{ \\ket{\\varphi_{k=1,\\dots,N-n_b-1}^{s=\\pm}}\\right\\} $\ndefined by\n\\begin{equation}\n  \\Ket{\\varphi_{k=1,\\dots,N-R_b-1}^{\\pm}} \\equiv\n    \\frac{\\ket{\\Phi_{k}^{\\left(1\\right)}} \\pm \\ket{\\Phi_{k}^{\\left(2\\right)}}}%\n         {\\sqrt{2}} \\label{state}\n\\end{equation}\nwith\n\\begin{align*}\n\\left|\\Phi_{k}^{\\left(1\\right)}\\right\\rangle  & \\equiv\\frac{\\left|k\\right\\rangle +\\left|N-k\\right\\rangle }{\\sqrt{2}}\\\\\n\\left|\\Phi_{k}^{\\left(2\\right)}\\right\\rangle  & \\equiv\\sum_{l=0}^{N-n_b-k-1}\\frac{\\left|k,N-l\\right\\rangle +\\left|N-k,l\\right\\rangle }{\\sqrt{2\\left(N-n_b-k\\right)}}\n\\end{align*}\nNote that $\\ket{\\Phi_{k}^{\\left(1\\right)}}$ describes\na configuration with exactly one Rydberg excited atom, localized either\nat position $k$ or $\\left(N-k\\right)$; \n$\\ket{\\Phi_{k}^{\\left(2\\right)}} $\ndescribes a configuration with two Rydberg excitations, one being\nlocalized in $k$ or $\\left(N-k\\right)$ while the other is fully\ndelocalized along the chain. The states $\\left|\\varphi_{k}^{s}\\right\\rangle $\nare therefore coherent superpositions of states with either one or\ntwo excitations, one being localized with certainty either at position\n$k$ or $\\left(N-k\\right)$.\n\nThe states $\\left|\\varphi_{k}^{s}\\right\\rangle $ are found to be\napproximately orthogonal to $\\left|\\psi_{k}^{s}\\right\\rangle $, \\emph{i.e.\\@}\n\\[\n\\left\\langle \\varphi_{k}^{s}\\middle|\\psi_{k'}^{s'}\\right\\rangle =O\\left(\\frac{1}{\\sqrt{N}}\\right).\n\\]\nThey are, moreover, only very weakly coupled to $\\left|\\psi_{k}^{s}\\right\\rangle $\nby the Hamiltonian, \\emph{i.e.\\@}\n\\begin{equation}\n\\left\\langle \\varphi_{k}^{s}\\middle| H\\middle|\\psi_{k'}^{s'}\\right\\rangle =O\\left(\\frac{1}{\\sqrt{N}}\\right).\\label{HamDiag}\n\\end{equation}\nFinally, for any $k=1,\\dots,\\left(N-R_b-1\\right)$ and $s=\\pm$, one\nhas \n\\[\n\\left\\langle \\varphi_{k}^{s}\\left|H\\right|\\varphi_{k}^{s}\\right\\rangle =s\\times\\varepsilon_{k}.\n\\]\nThe expression of $\\varepsilon_{k}$ is given in Appendix~\\ref{App:Localization}.\nFig.~\\ref{fig:EvsLambda} shows the quasi-continuum formed by the\ndifferent $\\varepsilon_{k}$'s plotted as functions of $\\Lambda$.\n\nIf the system starts in a superposition of $\\ket{\\Psi_{k}^{s}}$,\n\\emph{i.e.\\@} $\\ket{\\psi\\left(0\\right)} =\\sum_{k,s}c_{k}^{s}\\ket{\\psi_{k}^{s}}$,\none could be tempted, due to Eq.~(\\ref{HamDiag}), to assume that\nnone of the states $\\ket{\\varphi_{k}^{s}}$ ever gets\nsubstantially populated and to discard the whole family \n$\\left\\{\\ket{\\varphi_{k}^{s}} \\right\\} $\nfrom our description. This would actually be incorrect: it may indeed\nhappen that, for a given $k=K$, $\\left|\\varphi_{K}^{s}\\right\\rangle $\nbecomes resonant with $\\left|\\psi_{1}^{s}\\right\\rangle $, \\emph{i.e.\\@}\n$\\varepsilon_{K}=E_{1}$ (as can be checked on Fig.~\\ref{fig:EvsLambda},\nsuch a resonance exists only for $\\Lambda\\geq\\frac{7}{6}$; \nin Appendix~\\ref{App:Localization}, this\nresult is also analytically deduced from the expressions of $E_{1}$\nand $\\varepsilon_{k}$). In that case, though very weak,\nthe coupling term $\\left\\langle \\varphi_{K}^{s}\\left|H\\right|\\psi_{1}^{s}\\right\\rangle $\nstrongly mixes the states $\\left|\\varphi_{K}^{s}\\right\\rangle $ and\n$\\left|\\psi_{1}^{s}\\right\\rangle $ and the two vectors $\\left|\\varphi_{K}^{s=\\pm}\\right\\rangle $\nmust be adjoined to the previous set $\\left\\{ \\left|\\psi_{i=1,2}^{s=\\pm}\\right\\rangle \\right\\} $.\nIn this subspace, the six eigenvectors of the Hamiltonian now read\n\\[\n\\left\\{ \\left|\\chi_{\\pm}^{s=\\pm}\\right\\rangle \\equiv\\frac{\\left|\\psi_{1}^{s}\\right\\rangle \\pm\\left|\\varphi_{K}^{s}\\right\\rangle }{\\sqrt{2}},\\left|\\chi_{0}^{s=\\pm}\\right\\rangle \\equiv\\left|\\psi_{2}^{s}\\right\\rangle \\right\\} \n\\]\nand the energy degeneracy is lifted. An initial state of the form\n$\\ket{\\Psi(0)} =\\sum_{k,s}c_{k}^{s}\\ket{\\psi_{k}^{s}}$\nnow has components on the six new eigenvectors, \\emph{i.e.\\@} \n$\\ket{\\Psi(0)} =\\sum_{r=\\pm,0}\\sum_{s=\\pm}d_{r}^{s}\\ket{\\chi_{r}^{s}}$\nand therefore the time-averaged state \n\\begin{align}\n\\label{rho}\n\\bar{\\rho} & =\\sum_{r=\\pm,0}\\sum_{s=\\pm}\\left|d_{r}^{s}\\right|^{2}\\left|\\chi_{r}^{s}\\right\\rangle \\left\\langle \\chi_{r}^{s}\\right|\n\\end{align}\nnow contains a highly localized component, on the atom at position\n$K$ or $\\left(N-K\\right)$. Accordingly, the probability distribution\n$\\mathcal{P}_{k}$ exhibits a strongly peaked behavior at $k=K,\\left(N-K\\right)$.\nThis localization phenomenon is in good qualitative agreement with\nwhat we observe with the full simulation: in particular, the appearance\nof the localization lines indeed happens when $\\Lambda\\approx\\frac{7}{6}$\n(see Fig.~\\ref{fig:localisationExc2}). This validates the simplified\nanalytical treatment we have just carried out which indeed seems to\nretain the main physical ingredients of the system and its evolution.\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{Fig7}\\protect\\caption{\\label{fig:EvsLambda} $E_{1},E_{2}$ and $\\varepsilon_{k}$ as functions\nof $\\Lambda$, computed by our simplified analytical treatment for\n$N=100$. The values of $\\varepsilon_{k}$ form a quasi-continuum.\nAs discussed in the text, localization peaks arise when a resonance\ntakes place,\\emph{i.e.\\@} when there exists a value $k=K$ such that\n$\\varepsilon_{K}=E_{1}$. This happens for $\\Lambda\\geq\\frac{7}{6}$\nas can be shown analytically (see Appendix~\\ref{App:Localization}) and graphically checked\non the present Figure. }\n\\end{figure}\n\n\n\\section{Discussion}\n\n\\label{sec:Comparison}\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{Fig8-eps-converted-to}\\protect\\caption{\\label{fig:Probabilities} Probability to have $\\nu$ excitations\nas a function of $\\Lambda$, with $N=100$, according to the microcanonical\npredictions (red), our numerical simulation (green) and the analytical\ntreatment (blue). }\n\\end{figure}\n\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{Fig9-eps-converted-to} \\protect\\caption{\\label{fig:exc_proba_1,5_} Spatial distribution of excitations at\n$\\Lambda=1.5$, with $N=100$, according to the microcanonical predictions\n(red), our numerical simulation (green) and the analytical model (blue).}\n\\end{figure}\n\nThis section is devoted to the comparison of the results obtained\nabove following the different approaches \nand to discussions on their differences.\n\nFig.~\\ref{fig:Probabilities} displays plots of the probability $\\mathcal{P}(\\nu)$\nto have $\\nu$ Rydberg excitations in the sample, as a function\nof $\\Lambda$ (for $0.9\\leq\\Lambda\\leq2$), calculated according to:\ni) the microcanonical hypothesis (Sec.~\\ref{sec:Thermal}); ii)\nthe full simulation of the system (Sec.~\\ref{sec:Numerics}); iii)\nthe approximate diagonalization of $H$ in a reduced $6$-dimensional\nHilbert space (Sec.~\\ref{SimpleAnalyticalTreatment}). While the schemes\nii) and iii) yield very similar results (as expected), assumption\ni) induces quite different behaviors. The same comparison can be performed\non the spatial probability distribution $\\mathcal{P}_{k}$ which is\ndisplayed on Fig.~\\ref{fig:exc_proba_1,5_}. Again, the shapes obtained\nvia schemes ii) and iii) are in very good qualitative agreement: in\nboth cases, one observes two localization peaks on a ``background\ncurve'', which coincide satisfactorily. (Note that, according to\nour calculations, excitations are more likely to be localized at the\nborders). The spatial probability distribution obtained according\nto assumption i) differs strongly: no excitation localization effect\nis observed and the background curve is far from what is observed\nin the full simulation.\n\nThe discrepancies observed above can be partly explained by the following\n``parity balance property'' established in Appendix~\\ref{App:Parity}: \nfor any eigenstate $\\ket\\psi$ of the Hamiltonian $H$ of nonzero energy, \nthe\nprojections $\\left|\\psi_{\\text{odd}}\\right\\rangle $ and \n$\\left|\\psi_{\\text{even}}\\right\\rangle $\nonto the orthogonal and plementary subspaces $\\mathcal{H}_{\\text{odd}}$\nand $\\mathcal{H}_{\\text{even}}$, respectively spanned by the states with\nan odd and even number of Rydberg excitations, have the same norm,\n\\emph{i.e.\\@} $\\left|\\psi\\right\\rangle =\\left|\\psi_{\\text{odd}}\\right\\rangle +\\left|\\psi_{\\text{even}}\\right\\rangle $\nwith $\\left\\Vert \\left|\\psi_{\\text{odd}}\\right\\rangle \\right\\Vert =\\left\\Vert \\left|\\psi_{\\text{even}}\\right\\rangle \\right\\Vert =\\frac{1}{\\sqrt2}$.\nThis property conflicts directly with the microcanonical predictions\naccording to which the probability to have $\\nu<\\nu_{\\max}$ excitations\nis negligible compared to the probability to have the maximum number\nof excitations. For example, suppose $\\nu_{\\max}=1$, the microcanonical\nensemble implies that $P(\\nu=0)=\\frac{1}{1+N}$ and $P(\\nu=1)=\\frac{N}{1+N}$.\nBy contrast, the parity balance property implies $P(\\nu=0)=P(\\nu=1)=0.5$.\nFurthermore, one can see that in Fig.~\\ref{fig:ExcProba}, each time\none of the probability curve is above $\\frac{1}{2}$, the parity balance\ncondition is therefore impossible to fulfill. In almost all cases,\nthe even\/odd parity balance property and the simple microcanonical\napproach presented in Sec.~\\ref{sec:Thermal} disagree.\n\nThe inaccuracy of the predictions deduced from the microcanonical\nassumption can also be explained by the choice of \n$\\left|\\varnothing\\right\\rangle $\nas initial state: the low connectivity of this state to the rest\nof the Hilbert space constitutes indeed a strongly limiting factor\nto the thermalization process \\cite{OML10}. In particular, the vaccum\nstate being symmetric as well as the Hamiltonian, the system remains\nin a symmetric state during its evolution. The direct application\nof the microcanonical assumption, taking into account all the states\nwhich are allowed by the Rydberg blockade, is therefore incorrect\n: for a proper use of the microcanonical hypothesis, one should actually\ntake this extra symmetry selection rule into consideration and count\nonly the accessible, \\emph{i.e.\\@} symmetric, states. Note that the\nvacuum state is the natural starting point from an experimental perspective\nto study the build-up of excitations and is therefore widely used\n\\cite{AL12,BMFALA13}.\n\nAnother choice of initial state can actually be considered. Starting\nwith a random initial state, Ates et al. \\cite{AGL12} showed that\nin the regime of strong nearest neighbor interaction ($\\Lambda>\\frac{N}{2}$),\nthe dynamics of the system is well described by the microcanonical\nensemble. In the regime studied in the present article, $\\Lambda\\ll N$,\na similar random choice of initial state leads to an essentially ``frozen\nevolution'' as seen by the following dimensionality arguments. From\nEq.~\\eqref{eq:Nnu2}, the number of states containing at most $\\nu_{\\max}-1$\nexcitations is $\\propto N^{\\nu_{\\max}-1}$ and the dimension of the\ngenerated subspace $\\mathcal{H}_{\\nu\\leq\\nu_{\\max}-1}$ is a small\nfraction $O(\\frac{1}{N})$ of the dimension of the total Hilbert space\n$\\mathcal{H}$. As $N$ increases, the Hilbert space is therefore\nessentially composed by states containing $\\nu_{\\max}$ excitations.\nFurthermore, since all eigenvectors of $H$ with non-zero eigenvalue\nfollow the parity balance property, \n\\begin{align*}\n\\text{dim}(\\mathcal{H}\\setminus\\text{ker}(H)) & \\leq2\\min\\left(\\text{dim}(\\mathcal{H}_{\\text{even}}),\\text{dim}(\\mathcal{H}_{\\text{odd}})\\right)\\\\\n & \\sim2\\text{dim}\\left(\\mathcal{H}_{\\nu=\\nu_{\\max}-1}\\right)\\\\\n & \\sim O\\left(N^{\\nu_{\\max}-1}\\right)\n\\end{align*}\nAs a consequence, the Hilbert space is mainly spanned by the states\nin $\\mathrm{ker}\\left(H\\right)$ with $\\nu_{\\max}$ excitations. \nThis can be seen, for example, on Fig.~\\ref{fig:DiffEnergy}. \nTherefore,\nthe projector on $\\text{ker}(H)$ is a ``gentle'' operator \\cite{W99} for the\nensemble of states picked uniformly at random: with high probability,\na state from this ensemble will have a large component on $\\text{ker}(H)$\nand its evolution will essentially be ``frozen'', which contradicts\nthe microcanonical predictions. \n\nConversely, if one chooses the initial\nstate in the $\\mathcal{H}_{\\nu\\leq\\nu_{\\max}-1}$ subspace, the system\nwill not explore $\\text{ker}(H)$: the dimensionality of the actual\nmicrocanonical ensemble is therefore again much less than the number\nof states allowed by the Rydberg blockade.\nNote that this initial state choice is a natural generalization of \n$\\ket{\\varnothing}$ to study the buildup of excitations.\n\n\\section{Conclusion}\n\n\\label{sec:Conclusion}\n\nIn the present article, we studied the dynamics of a 1D-Rydberg ensemble\nin the regime of at most 2 excitations. In the same conditions as\nin \\cite{AL12,BMFALA13}, we tested the validity of the microcanonical\npredictions and found it cannot be used straightforwardly to account\nfor the thermalization process which occurs in this particular regime.\nThough the observed discrepancies can be related to our specific choice\nof initial state and its particular symmetry properties, we also proved,\nby an argument involving the dimension of the kernel of the Hamiltonian,\nthat the same restriction holds for a randomly chosen initial state.\n\nFurther investigations are needed to better understand when and how\nto apply the (micro)canonical predictions. In particular, the results\npresented here all rely on the hardcore sphere assumption. Refining\nthe model and considering the full Rydberg-Rydberg interaction Hamiltonian\nEq.~\\eqref{V_dd} might actually change our conclusions and make\nthe microcanonical assumption more adapted, as shown in \\cite{LOG10}.\nIndeed, in that case, all states become, strictly speaking, allowed,\nthough more or less accessible, and the connectivity accordingly increases\nbetween states of the Hilbert space. Moreover, as suggested by our\ndiscussion, the systematic study of symmetry properties of the system\nat stake, as well as the selection rules they impose, appears to be\na crucial point in the proper application of microcanonical assumption.\n\n\n\\section*{Acknowledgments}\nWe thank Stefano Bettelli for digging up the raw data \\cite{BMFALA13data}.\nMaurice Raoult's highlighting of the simplicity of \n$2\\times2$ matrix diagonalization inspired us for\nthe simplified analytical treatment of section \\ref{SimpleAnalyticalTreatment}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe motivation of this paper is the classical family of \\defn{stack-sortable} permutations introduced by D.~Knuth in his textbook~\\cite[Sect.~2.2.1]{Knuth-TAOCP1} and characterized by the following equivalent conditions for a permutation~$\\pi \\in \\fS_n$:\n\\begin{enumerate}[(i)]\n\\item \\label{condition:stackSorting}\n$\\pi$ is sent to the identity by the stack sorting~$S$ defined inductively by~$S(\\tau n \\rho) \\eqdef S(\\tau) S(\\rho) n$.\n\\item \\label{condition:patternStack}\n$\\pi$ avoids the pattern $231$ (\\ie there is no~$p < q < r$ such that~$\\pi_r < \\pi_p < \\pi_q$).\n\\item \\label{condition:minimalLinearExtentionBinaryTree}\n$\\pi$ is minimal among all linear extensions of a binary tree on $n$ nodes (seen as a poset, where the nodes are labeled in inorder and the edges are oriented towards the leaves).\n\\item \\label{condition:alignedStack}\nFor $i < j < k$, the inversion set~$\\inv(\\pi) \\eqdef \\set{(\\pi_p, \\pi_q)}{p < q \\text{ and } \\pi_p > \\pi_q}$ of~$\\pi$ contains the inversion $(k,j)$ as soon as it contains the inversion~$(k,i)$.\n\\item \\label{condition:sortableStack}\n$\\pi$ admits a reduced expression of the form~$\\pi = c_{I_1} \\cdots c_{I_p}$ with nested subsets~$I_1 \\supseteq \\dots \\supseteq I_p$, where~$c_{\\{i_1 < \\dots < i_j\\}} \\eqdef s_{i_j} \\cdots s_{i_1}$ is a product of the simple transpositions~$s_i \\eqdef (i \\;\\; i+1)$.\n\\end{enumerate}\nIt follows from~\\eqref{condition:minimalLinearExtentionBinaryTree} that these permutations are counted by the Catalan number~$C_n \\eqdef \\frac{1}{n+1} \\binom{2n}{n}$.\n\nIn his seminal work on lattice congruences~\\cite{Reading-latticeCongruences, Reading-CambrianLattices, Reading-CoxeterSortable}, N.~Reading defined natural counterparts to conditions~\\eqref{condition:minimalLinearExtentionBinaryTree}, \\eqref{condition:alignedStack}, and~\\eqref{condition:sortableStack} above, parametrized by the choice of a Coxeter element~$c$ in a finite Coxeter group~$W$: the minimality in $c$-Cambrian classes, the $c$-alignment, and the $c$-sortability.\n(We skip the general definitions of these conditions here as we stick with the combinatorics of the symmetric group.)\nIn the situation of the symmetric group~$\\fS_n$, we can think of a Coxeter element on~$\\fS_n$ as an orientation of an $(n-1)$-path, or equivalently as a partition of~$\\{2, \\dots, n-1\\}$ into two subsets~$U$ and~$D$.\nThe Cambrian analogues of the conditions~\\eqref{condition:patternStack}, \\eqref{condition:minimalLinearExtentionBinaryTree}, \\eqref{condition:alignedStack} and~\\eqref{condition:sortableStack} above are the following equivalent conditions for a permutation~$\\pi \\in \\fS_n$:\n\\begin{enumerate}[(i')]\n\\addtocounter{enumi}{1}\n\\item \\label{condition:patternCambrian}\nFor~$i < j < k$, the permutation~$\\pi$ does not contain the subword $jki$ if~$j \\in U$ and~$kij$ if~$j \\in D$.\n\\item \\label{condition:minimalLinearExtentionCambrian}\n$\\pi$ is minimal among all linear extensions of a $c$-Cambrian tree on $n$ nodes. A $c$-Cambrian tree is an oriented tree on~$[n]$ where node~$j$ has one parent if~$j \\notin U$ and two parents if~$j \\in U$, and one child if~$j \\notin D$ and two children if~$j \\in D$, with an additional local condition at each node similar to the binary search tree condition~\\cite{ChatelPilaud}.\n\\item \\label{condition:alignedCambrian}\nFor~$i < j < k$, if~$\\inv(\\pi)$ contains~$(k,i)$, then it also contains $(k,j)$ if~$j \\in U$ and $(j,i)$ if $j \\in D$.\n\\item \\label{condition:sortableCambrian}\n$\\pi$ admits a reduced expression of the form~$\\pi = c_{I_1} \\cdots c_{I_p}$ with nested subsets~$I_1 \\supseteq I_2 \\supseteq I_p$, where~$c_I \\eqdef c_{i_1} \\cdots c_{i_{|I|}}$ denotes the subword of~$c \\eqdef c_1 \\cdots c_{n-1}$ indexed by~$I \\eqdef \\{i_1 < \\dots < i_j\\}$.\n\\end{enumerate}\nIt turns out that for any Coxeter element~$c$, the permutations satisfying these conditions are still counted by the Catalan number~$C_n$.\n\nThese Cambrian combinatorics motivated the introduction of permutree combinatorics~\\cite{PilaudPons-permutrees}.\nPermutrees generalize and interpolate between permutations, binary trees, and binary sequences, and explain the combinatorial, geometric, and algebraic similarities between them.\nThe data is now given by two subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$ that are not anymore required to form a partition of~$\\{2, \\dots, n-1\\}$ (they may intersect and may not cover all the set).\nIt was proved in~\\cite{PilaudPons-permutrees, ChatelPilaudPons} that the conditions~(\\ref{condition:patternCambrian}'), (\\ref{condition:minimalLinearExtentionCambrian}'), and~(\\ref{condition:alignedCambrian}') are still equivalent for a permutation~$\\pi \\in \\fS_n$.\nThe number of permutations satisfying these conditions is called $(U,D)$-factorial-Catalan number and admits recursive formulae interpolating between the formulae for the factorial and for the Catalan number.\n\nThe objective of this paper is to discuss characterizations of permutree minimal permutations in terms of their reduced expressions.\nIn other words, we aim at a condition playing the role of condition~(\\ref{condition:sortableCambrian}') and equivalent to conditions~(\\ref{condition:patternCambrian}'), (\\ref{condition:minimalLinearExtentionCambrian}'), and~(\\ref{condition:alignedCambrian}') for arbitrary subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$.\nWe first focus on the case where~$U = \\varnothing$ and~$D = \\{j\\}$ for some~${j \\in \\{2, \\dots, n-1\\}}$, or the opposite.\nTo characterize the permutree minimal permutations in terms of their reduced expressions in that situation, we use two automata~$\\automatonU(j)$ and $\\automatonD(j)$ defined inductively as shown in \\cref{fig:automataRecursive}.\nThe induction stops at~$\\automatonU(n)$ and~$\\automatonD(1)$, which are defined by deleting the transitions~$s_n$ and~$s_0$ respectively in \\cref{fig:automataRecursive}.\n\\cref{fig:automataComplete} presents the complete automaton $\\automatonU(j)$ after all recursion is done, and \\cref{fig:TreePartialOrientations} shows the automata~$\\automatonU(2)$, $\\automatonD(2)$, $\\automatonU(3)$, and~$\\automatonD(3)$.\nIn all these pictures the initial state is marked with ``start'', the accepting states are doubly circled, all transitions are labeled with simple transpositions~$s_i$ for~$i \\in [n-1]$, and all missing transitions are loops (we assume the reader familiar with basic automata theory, see for instance~\\cite{HopcroftUllman}).\nOur main tool is the following statement, proved in \\cref{sec:proofPatternAvoidance}.\n\n\\begin{theorem}\\label{thm:patternAvoidance}\nFix $j \\in \\{2, \\dots, n-1\\}$.\nThe following conditions are equivalent for~$\\pi \\in \\fS_n$:\n\\begin{itemize}\n\t\\item $\\pi$ admits a reduced expression accepted by the automaton~$\\automatonU(j)$ (resp.~$\\automatonD(j)$),\n\t\\item $\\pi$ contains no subword $jki$ (resp.~$kij$) with~$i < j < k$.\n\\end{itemize}\n\\end{theorem}\n\nLet us warn the reader on the fact that~$j$ is fixed in \\cref{thm:patternAvoidance}, while $i$ and~$k$ are arbitrary such that~$1 \\le i < j < k \\le n$.\nA priori, we should try all possible reduced expressions of~$\\pi$ to decide if one is accepted by the automaton~$\\automatonU(j)$ (resp.~$\\automatonD(j)$).\nHowever, we can show that if~$\\pi$ contains no subword $jki$ (resp.~$kij$) with~$i < j < k$ and has a descent~$s_\\ell$ distinct from~$s_{j-1}$ (resp.~$s_j$), then it has a reduced expression starting with~$s_\\ell$ and accepted by the automaton~$\\automatonU(j)$ (resp.~$\\automatonD(j)$).\nIn other words, there is no loss of generality in starting constructing a reduced expression for~$\\pi$ as long as we stay in the states of the top row of~$\\automatonU(j)$ (resp.~$\\automatonD(j)$).\nThis yields a simple algorithm to construct a reduced expression accepted by~$\\automatonU(j)$ (resp.~$\\automatonD(j)$).\nIt also yields natural tree structures on the permutations characterized by \\cref{thm:patternAvoidance}, which can be glanced upon in \\cref{fig:TreePartialOrientations}.\nThese algorithmic and combinatorial consequences of \\cref{thm:patternAvoidance} are explored in \\cref{sec:algorithmicCombinatorialConsequences}.\nMost results of \\cref{sec:proofPatternAvoidance,sec:algorithmicCombinatorialConsequences} are stated with respect to both automata~$\\automatonU(j)$ and $\\automatonD(j)$ but proved only for $\\automatonU(j)$ as all proofs for~$\\automatonD(j)$ are symmetric.\n\n\\begin{figure}\n\t\\centerline{\n\t\t$\\automatonU(j) \\eqdef$\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto, baseline=-1.5cm]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (q_0) {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (q_1) [below= 1.5cm of q_0] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (q_2) [below= 1.5cm of q_1] {}; \n\t\t\t\\node (q_3) [right= 2.5cm of q_0] {$\\automatonU(j+1)$};\n\t\t\t\\path[->] \n\t\t\t\t(q_0) edge node [swap] {$s_{j-1}$} (q_1)\n\t\t\t\t\t  edge node {$s_j$} (q_3)\n\t\t\t\t(q_1) edge node [swap] {$s_j$} (q_2);\n\t\t\\end{tikzpicture}\n\t\t\\qquad\\qquad\n\t\t$\\automatonD(j) \\eqdef$\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto, baseline=-1.5cm] \n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (q_0) {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (q_1) [below= 1.5cm of q_0] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (q_2) [below= 1.5cm of q_1] {}; \n\t\t\t\\node (q_3) [right= 2.5 cm of q_0] {$\\automatonD(j-1)$};\n\t\t\t\\path[->] \n\t\t\t\t(q_0) edge node [swap] {$s_{j}$} (q_1)\n\t\t\t\t\t  edge node {$s_{j-1}$} (q_3)\n\t\t\t\t(q_1) edge node [swap] {$s_{j-1}$} (q_2);\n\t\t\\end{tikzpicture}\n\t}\n\t\\caption{The automata $\\automatonU(j)$ (left) and $\\automatonD(j)$ (right) defined recursively.}\n\t\\label{fig:automataRecursive}\n\\end{figure}\n\n\\begin{figure}\n\t\\centerline{\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj-1)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj-1) [below= 1.5cm of ij-1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj) [right= 2.5cm of hj-1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 2.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj+1) [below= 1.5cm of ij+1] {};\t  \n\t\t\t%\n\t\t\t\\node (void) [right= 2.5cm of hj+1] {\\dots};\n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hn-2) [right= 2.5cm of void] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (in-2) [below= 1.5cm of hn-2] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dn-2) [below= 1.5cm of in-2] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hn-1) [right= 2.5cm of hn-2] {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (in-1) [below= 1.5cm of hn-1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj-1) edge node [swap] {$s_{j-1}$} (ij-1)\n\t\t\t\t\t   edge node {$s_j$} (hj)\n\t\t\t\t(ij-1) edge node [swap] {$s_j$} (dj-1)\n\t\t\t\t%\n\t\t\t\t(hj) edge node [swap] {$s_{j}$} (ij)\n\t\t\t\t\t edge node {$s_{j+1}$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_{j+1}$} (dj)\t\t\t\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_{j+1}$} (ij+1)\n\t\t\t\t\t   edge  node {$s_{j+2}$} (void)\n\t\t\t\t(ij+1) edge  node [swap] {$s_{j+2}$} (dj+1)\t\t\t  \n\t\t\t\t%\n\t\t\t\t(void) edge node {$s_{n-1}$} (hn-2)\n\t\t\t\t%\n\t\t\t\t(hn-2) edge node [swap] {$s_{n-2}$} (in-2)\n\t\t\t\t\t   edge node {$s_{n-1}$} (hn-1)\n\t\t\t\t(in-2) edge node [swap] {$s_{n-1}$} (dn-2)\n\t\t\t\t%\n\t\t\t\t(hn-1) edge node [swap] {$s_{n-1}$} (in-1);\n\t\t\\end{tikzpicture}\n\t\t}\n\t\\caption{The complete automaton~$\\automatonU(j)$.}\n\t\\label{fig:automataComplete}\n\\end{figure}\n\n\\enlargethispage{-.5cm}\nConsider now arbitrary subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$.\nIt follows from \\cref{thm:patternAvoidance} that a permutation is minimal in its $(U,D)$-permutree class if and only if it admits a reduced expression accepted by~$\\automatonU(j)$ for each~$j \\in U$ and by~$\\automatonD(j)$ for each~$j \\in D$.\nIn general, the reduced expressions accepted by the automata~$\\automatonU(j)$ for each~$j \\in U$ and by~$\\automatonD(j)$ for each~$j \\in D$ are distinct.\nWe prove however in \\cref{sec:intersectionsAutomata} that there is a reduced expression simultaneously accepted by all these automata when~$U$ and~$D$ are disjoint.\n\n\\begin{theorem}\\label{thm:permutreeMinimal}\nConsider two disjoint subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$.\nThe following conditions are equivalent for~$\\pi \\in \\fS_n$:\n\\begin{itemize}\n\t\\item $\\pi$ admits a reduced expression accepted by all automata~$\\automatonU(j)$ for~$j \\in U$ and~$\\automatonD(j)$ for~${j \\in D}$,\n\t\\item $\\pi$ contains no subword $jki$ if~$j \\in U$ and~$kij$ if~$j \\in D$ for any~$i < j < k$.\n\\end{itemize}\n\\end{theorem}\n\n\\cref{thm:permutreeMinimal} implies that given any permutation~$\\pi$ avoiding $jki$ if~$j \\in U$ and~$kij$ if~$j \\in D$, we can sort~$\\pi$ while preserving these avoiding conditions.\nThe resulting sorting procedures, that we call \\defn{$(U,D)$-permutree sorting}, are discussed in~\\cref{subsec:permutreeSorting}.\nFor instance, stack sorting is a $(\\{2, \\dots, n-1\\}, \\varnothing)$-permutree sorting.\n\nFinally, in the particular situation when the subsets~$U$ and~$D$ form a partition of~$\\{2, \\dots, n-1\\}$, we actually show that the reduced expression simultaneously accepted by the automata $\\automatonU(j)$ for~${j \\in U}$ and~$\\automatonD(j)$ for~$j \\in D$ is the $c$-sorting word of~$\\pi$ as defined in~\\cite{Reading-CoxeterSortable}.\nThis yields in particular an alternative proof that condition~(\\ref{condition:sortableCambrian}') characterizes the Cambrian minimal permutations.\nThis new perspective on $c$-sortability is explored in \\cref{sec:coxeterSortable}.\n\n\n\\section{Automata for reduced expressions}\\label{sec:proofPatternAvoidance}\n\n\\subsection{Reduced expressions, automata, and subword avoiding}\n\nWe start with properly fixing the few notations needed in this paper.\nWe consider the \\defn{symmetric group}~$\\fS_n$ of permutations of the set~$[n] \\eqdef \\{1,\\dots,n\\}$.\nIt is generated by the \\defn{transpositions}~$s_i \\eqdef (i \\;\\; i+1)$ for $i \\in [n-1]$ which are involutions~$s_i^2 = id$ and satisfy the commutation relations $s_i \\cdot s_j =s_j \\cdot s_i$ if $|i-j|>1$ and the braid relations~$s_i \\cdot s_{i+1} \\cdot s_i = s_{i+1} \\cdot s_i \\cdot s_{i+1}$.\nNote that we multiply permutations as usual, so that the left multiplication by~$s_i$ exchanges the entries with values~$i$ and~$i+1$, while the right multiplication by~$s_i$ exchanges the entries at positions~$i$ and~$i+1$.\nEach permutation~$\\pi$ decomposes into products of transpositions of the form~$\\pi = s_{i_1} \\cdots s_{i_k}$ with~$i_1, \\dots, i_k \\in [n-1]$.\nThe minimal number of transpositions in such a decomposition is the \\defn{length}~$\\ell(\\pi)$ of~$\\pi$ and the decompositions of length~$\\ell(\\pi)$ are the \\defn{reduced expressions} for~$\\pi$.\n\nConsider now the automata $\\automatonU(j)$ and $\\automatonD(j)$ described in the introduction, see \\cref{fig:automataRecursive,fig:automataComplete,fig:TreePartialOrientations}.\nWe call a state \\defn{healthy}, \\defn{ill}, or \\defn{dead} depending on whether it belongs to the top, middle, or bottom row of the automata.\nEach state has $n-1$ possible transitions, one for each $s_i$ for~$i \\in [n-1]$, but we only explicitly indicate the ones between different states.\nThe automata $\\automatonU(j)$ and $\\automatonD(j)$ take as entry a reduced expression~$s_{i_1} \\cdots s_{i_\\ell}$ for a permutation of~$\\fS_n$ and read it from left to right.\nWe start at the initial state (marked with ``start''), and at step~$t$ we follow the transition marked by the letter~$s_{i_t}$ if any, or stay in the current state otherwise.\nAfter~$\\ell$ steps, the reduced expression~$s_{i_1} \\cdots s_{i_\\ell}$ is declared accepted if the current state is accepting (doubly circled, healthy or ill states), and rejected otherwise (dead states).\n\nFor a fixed~$j \\in \\{2, \\dots, n-1\\}$, we say that a permutation~$\\pi$ \\defn{avoids} $jki$ (resp.~$kij$) if for any~$i < j < k$, the word~$jki$ (resp.~$kij$) does not appear as a subword of the one-line notation of~$\\pi$, or said differently if there are no positions~$p < q < r$ such that~$\\pi(r) < \\pi(p) = j < \\pi(q)$ (resp.~$\\pi(q) < \\pi(r) = j < \\pi(p)$).\nWe insist on the fact that while the value~$j$ is fixed, $i$ and~$k$ take all possible values such that~$1 \\le i < j < k \\le n$.\nThis convenient notion here should not be mixed up with the notion of pattern avoidance where~$j$ is not fixed.\nFor instance, a permutation avoids the pattern~$231$ if and only if it avoids $jki$ for all~$j \\in \\{2, \\dots, n-1\\}$.\n\n\\begin{example}\\label{exm:patternAvoidance}\n\tThe permutation~$42135$ avoids~$2ki$, $3ki$, and~$4ki$ (and therefore the pattern~$231$), but contains~$ki3$ (and therefore the pattern~$312$) because its one-line notation contains~$423$. \n\\end{example}\n\n\\subsection{Behavior under left multiplication}\n\nIn the perspective of proving \\cref{thm:patternAvoidance}, we study the two properties ``$\\pi$ admits a reduced expression accepted by~$\\automatonU(j)$ (resp.~$\\automatonD(j)$)'' and ``$\\pi$ avoids $jki$ (resp.~$kij$)''.\nIn this section, we study the behavior of these properties under left multiplication.\nWe treat separately the cases when we multiply by a permutation commuting with both~$s_{j-1}$ and~$s_j$ (\\cref{lem:vincent2}), by $s_{j-1}$ (\\cref{lem:vincent3}), and by~$s_j$ (\\cref{lem:vincent4}).\n\n\\begin{lemma}\\label{lem:vincent2}\nIf two permutations~$\\sigma, \\tau \\in \\fS_n$ are such that $\\sigma([j-1]) = [j-1]$, $\\sigma(j) = j$, and $\\sigma([n]\\ssm [j]) = [n]\\ssm [j]$ and $\\length(\\sigma \\cdot \\tau) = \\length(\\sigma) + \\length(\\tau)$, then:\n\\begin{enumerate}\n\t\\item $\\tau$ admits a reduced expression accepted by $\\automatonU(j)$ (resp.~$\\automatonD(j)$) if and only if $\\sigma \\cdot \\tau$ admits a reduced expression accepted by $\\automatonU(j)$ (resp.~$\\automatonD(j)$),\n\t\\item $\\tau$ avoids $jki$ (resp.~$kij$) if and only if $\\sigma \\cdot \\tau$ avoids $jki$ (resp.~$kij$).\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe deal with the two statements separately:\n\\begin{enumerate} \n\t\\item The conditions on~$\\sigma$ imply that none of its reduced expressions contain the transpositions $s_{j-1}$ or $s_j$. Therefore, while reading any reduced expression for~$\\sigma$, the automaton~$\\automatonU(j)$ stays in the initial state. The result immediately follows.\n\t\\item Since~$\\sigma$ permutes only values smaller than $j$ between themselves and values greater than $j$ between themselves, we see a subword~$jki$ with~$i < j < k$ in~$\\tau$ if and only if we see a subword~$jk'i'$ with~$i' < j < k'$ in~$\\sigma \\cdot \\tau$, where~$i' = \\sigma(i)$ and~$k' = \\sigma(k)$.\n\t\\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\\begin{example}\\label{exm:lemVincent2}\nConsider~$j \\eqdef 4$ and the permutations~$\\sigma \\eqdef 312465 = s_2 \\cdot s_1 \\cdot s_5$, $\\tau_1 \\eqdef 143256 = s_3 \\cdot s_2 \\cdot s_3$, and~$\\tau_2 \\eqdef 124536 = s_3 \\cdot s_4$.\nMultiplying we obtain~$\\sigma\\cdot\\tau_1=342165$ and~$\\sigma\\cdot\\tau_2 = 314625$. Observe that\n\\begin{enumerate}\n\t\\item $\\automatonU(4)$ accepts all reduced expressions of both~$\\tau_1$ and~$\\sigma\\cdot \\tau_1$ on its first ill state, and rejects all reduced expressions of both~$\\tau_2$ and~$\\sigma\\cdot \\tau_2$,\n\t\\item both~$\\tau_1$ and~$\\sigma\\cdot\\tau_1$ avoid~$4ki$, while both~$\\tau_2$ and~$\\sigma\\cdot\\tau_2$ contain~$4ki$.\n\\end{enumerate}\n\\end{example}\n\n\\begin{lemma}\\label{lem:vincent3}\nIf a permutation $\\tau \\in \\fS_n$ has a reduced expression starting with $s_{j-1}$ (resp.~$s_j$) and accepted by $\\automatonU(j)$ (resp.~$\\automatonD(j)$), then\n\\begin{enumerate} \n\t\\item $\\tau$ does not permute $j$ and $j+1$ (resp.~$j-1$ and $j$),\n\t\\item $\\tau$ avoids $jki$ (resp.~$kij$).\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nConsider a reduced expression $w$ starting with $s_{j-1}$ and accepted by~$\\automatonU(j)$. \nWe deal with the two statements separately:\n\\begin{enumerate} \n\t\\item Since $w$ starts with~$s_{j-1}$, the values~$j-1$ and~$j$ are reversed in~$\\tau$. If~$j$ and~$j+1$ were also reversed in~$\\tau$, we would obtain that~$j-1$ and~$j+1$ are reversed. It follows that $w$ must contain a~$s_j$ at some point after the~$s_{j-1}$. But this would lead to a dead state, contradicting the assumption that~$w$ is accepted.\n\t\\item Let $\\tau = s_{j-1}\\cdot \\rho$. Since any reduced expression of $\\rho$ cannot contain~$s_j$ (as $w$ is be accepted by $\\automatonU(j)$), we have that $\\rho([j]) = [j]$ and $\\rho([n+1] \\ssm [j]) = [n+1] \\ssm [j]$ and find that $\\rho$ contains no subword~$ki$ with $i < j < k$. Therefore, $\\tau$ avoids $jki$.\n\t\\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\\begin{example}\\label{exm:lemVincent3}\nConsider~$j \\eqdef 4$ and the permutation~$\\tau \\eqdef 413265$, whose reduced expression~$s_3 \\cdot s_5 \\cdot s_2 \\cdot s_1 \\cdot s_3$ is accepted by~$\\automatonU(4)$.\nObserve that\n\\begin{enumerate}\n\t\\item $\\tau$ indeed does not permute the values~$4$ and~$5$,\n\t\\item $\\tau$ avoids~$4ki$.\n\\end{enumerate}\n\\end{example}\n\n\\begin{lemma}\\label{lem:vincent4}\nIf a permutation~$\\tau \\in \\fS_n$ does not permute $j$ and $j+1$ (resp.~$j-1$ and~$j$), then\n\\begin{enumerate} \n\t\\item $s_j \\cdot \\tau$ (resp.~$s_{j-1} \\cdot \\tau$) admits a reduced expression accepted by $\\automatonU(j)$ (resp.~$\\automatonD(j)$) if and only if $\\tau$ admits a reduced expression accepted by $\\automatonU(j+1)$ (resp.~$\\automatonD(j-1)$),\n\t\\item $s_j \\cdot \\tau$ (resp.~$s_{j-1} \\cdot \\tau$) avoids $jki$ (resp.~$kij$) if and only if $\\tau$ avoids $(j+1)ki$ (resp.~$ki(j-1)$).\n\\end{enumerate}\n\\end{lemma}\n\n\\pagebreak\n\\begin{proof}\nWe deal with the two statements separately:\n\\begin{enumerate} \n\t\\item Suppose that $w$ is a reduced expression for $\\tau$ accepted by $\\automatonU(j+1)$. Since $\\tau$ does not permute $j$ and $j+1$, we know that $s_j \\cdot w$ is a reduced expression for~$s_j \\cdot \\tau$, and it is accepted by~$\\automatonU(j)$ by construction. Conversely assume that $s_j \\cdot \\tau$ admits a reduced expression $w$ accepted by $\\automatonU(j)$. Since $s_j \\cdot \\tau$ permutes $j$ and $j+1$, $w$ must contain a $s_j$ and cannot start by $s_{j-1}$ by \\cref{lem:vincent3}. Due to \\cref{lem:vincent2} we can also assume that $w$ starts with $s_j$. Thus the suffix is a reduced expression for $\\tau$ that is accepted by $\\automatonU(j+1)$.\n\t\\item Observe that since~$j$ and~$j+1$ are reversed in~$s_j \\cdot \\tau$ and not in~$\\tau$, the value $j+1$ cannot serve as~$k$ in a subword~$jki$ of~$s_j \\cdot \\tau$ and the value~$j$ cannot serve as~$i$ in a subword $(j+1)ki$ in~$\\tau$. The result thus immediately follow from the fact that the left multiplication by~$s_j$ only exchanges the values $j$ and $j+1$.\n\t\\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\\begin{example}\\label{exm:lemVincent4}\nConsider~$j \\eqdef 4$ and the permutations~$\\tau_1 \\eqdef 142536$ and~$\\tau_2 \\eqdef 142563$ that do not permute~$4$ and~$5$. \nMultiplying we obtain~$s_4 \\cdot \\tau_1 = 152436$ and~$s_4 \\cdot \\tau_2 = 152463$. Observe that\n\\begin{enumerate}\n\t\\item the reduced expression~$s_4 \\cdot s_3 \\cdot s_4 \\cdot s_2$ of~$s_4 \\cdot \\tau_1$ is accepted by~$\\automatonU(4)$ and the reduced expression~$s_3 \\cdot s_4 \\cdot s_2$ of~$\\tau_1$ is accepted by~$\\automatonU(5)$, while all reduced expressions of~$s_4 \\cdot \\tau_2$ are rejected by~$\\automatonU(4)$ and all reduced expressions of~$\\tau_2$ are rejected by~$\\automatonU(5)$,\n\t\\item $s_4 \\cdot \\tau_1$ avoids~$4ki$ and $\\tau_1$ avoids~$5ki$, while $s_4 \\cdot \\tau_2$ contains~$463$ and $\\tau_2$ contains~$463$.\n\\end{enumerate}\n\\end{example}\n\n\\subsection{Proof of \\cref{thm:patternAvoidance}}\n\nWith these lemmas in hand, we are now ready to show \\cref{thm:patternAvoidance} that we repeat here for convenience.\n\n\\newtheorem*{thm:patternAvoidance}{\\cref{thm:patternAvoidance}}\n\\begin{thm:patternAvoidance}\nFix $j \\in \\{2, \\dots, n-1\\}$.\nThe following conditions are equivalent for~$\\pi \\in \\fS_n$:\n\\begin{itemize}\n\t\\item $\\pi$ admits a reduced expression accepted by the automaton~$\\automatonU(j)$ (resp.~$\\automatonD(j)$),\n\t\\item $\\pi$ contains no subword $jki$ (resp.~$kij$) with~$i < j < k$.\n\\end{itemize}\n\\end{thm:patternAvoidance}\n\n\\begin{proof}[Proof of \\cref{thm:patternAvoidance}]\nWe work by induction on the length of the permutations.\nAssume that a permutation $\\pi$ admits a reduced expression accepted by $\\automatonU(j)$. Let $s_i$ be the first letter of this reduced expression and let $\\tau$ be such that $\\pi = s_i \\cdot \\tau$. We distinguish three cases:\\begin{itemize}\n\t\\item if $i = j-1$, then $\\pi$ avoids $jki$ by \\cref{lem:vincent3}\\,(2).\n\t\\item if $i = j$, then $\\tau$ admits a reduced expression accepted by $\\automatonU(j+1)$ by \\cref{lem:vincent4}\\,(1). We obtain by induction that $\\tau$ avoids $(j+1)ki$. Thus $\\pi = s_j \\cdot \\tau$ avoids $jki$ by \\cref{lem:vincent4}\\,(2).\n\t\\item otherwise, $\\tau$ admits a reduced expression accepted by $\\automatonU(j)$ by \\cref{lem:vincent2}\\,(1), so that $\\tau$ avoids $jki$ by induction. Thus $\\pi = s_i \\cdot \\tau$ avoids $jki$ by \\cref{lem:vincent2}\\,(2).\n\\end{itemize}\nIn all three cases, we proved that $\\pi$ avoids $jki$.\n\nAssume now that a permutation $\\pi$ avoids $jki$. Here, we have to be careful because not all reduced expressions for $\\pi$ will be accepted by $\\automatonU(j)$ \\apriori. So we have to construct a good reduced expression for~$\\pi$. We distinguish two cases:\n\\begin{itemize}\n\t\\item Assume first that there is $m > j$ such that $\\pi$ reverses $j$ and $m$, and pick $m$ minimal for this property. It follows that $\\pi$ reverses $\\ell$ and $m$ for all $\\ell$ in $\\{j,\\dots,m-1\\}$. In other words, $\\pi$ admits a reduced expression starting by the cyclic permutation $(j, j+1, ..., m) = s_{m-1} \\cdot s_{m-2} \\cdots s_{j+1} \\cdot s_j$. Define $\\sigma = s_{m-1} \\cdot s_{m-2}\\cdots s_{j+1}$ and $\\tau$ such that $\\pi = \\sigma \\cdot s_j \\cdot \\tau$ and so that this expression is reduced. By Lemmas \\ref{lem:vincent2}\\,(2) and \\ref{lem:vincent4}\\,(2), $\\tau$ avoids $(j+1)ki$. By induction, we obtain that it admits a reduced expression accepted by $\\automatonU(j+1)$. By Lemmas \\ref{lem:vincent2}\\,(1) and \\ref{lem:vincent4}\\,(1), we conclude that $\\pi$ admits a reduced expression accepted by $\\automatonU(j)$.\n\t\\item Assume now that $j$ appears before all $m > j$ in $\\pi$. Consider any reduced expression for $\\pi$. If this expression is accepted by $\\automatonU(j)$, we are done. Otherwise, it first uses $s_{j-1}$ (otherwise, $j$ and some $m > j$ would be exchanged) and then $s_j$. Call $i$ and $k$ the two elements that are exchanged when the reduced expression first uses $s_j$. We have $i < j < k$ and $jki$ in $\\pi$ (because $j$ and $k$ are not exchanged in $\\pi$, and $i$ and $k$ are already exchanged so they will remain exchanged in $\\pi$), a contradiction.\n\\end{itemize}\nIn both cases, we proved that $\\pi$ admits a reduced expression accepted by $\\automatonU(j)$.\n\\end{proof}\n\n\n\\pagebreak\n\\section{Structure of accepted reduced expressions}\\label{sec:algorithmicCombinatorialConsequences}\n\nIn this section, we explore some additional properties of the set of reduced expressions accepted by the automata~$\\automatonU(j)$ and~$\\automatonD(j)$ and derive relevant algorithmic and combinatorial consequences.\n\n\\subsection{The set of accepted reduced expressions}\\label{subsec:principles}\n\nObserve that a given permutation~$\\pi$ may admit both accepted and rejected reduced expressions.\nFor instance, the (non-simple) transposition~${(j-1 \\;\\; j+1)}$ has reduced expressions~$s_j \\cdot s_{j-1} \\cdot s_j$ accepted by~$\\automatonU(j)$ and~$s_{j-1} \\cdot s_j \\cdot s_{j-1}$ rejected by~$\\automatonU(j)$.\nHowever, \\cref{prop:prefixesReducedExpressions,prop:algorithm,prop:sameStateAcceptedReducedExpressions} below show that the set of accepted reduced expressions satisfies the following three principles:\n\\begin{itemize}\n\t\\item \\textbf{Who can do more can do less!} --- The set of accepted reduced words is closed by~prefix.\n\t\\item \\textbf{When health goes, everything goes!} --- If $\\pi$ admits an accepted reduced expression, then $\\pi$ admits an accepted reduced expression starting with any descent that remains in the healthy states.\n\t\\item \\textbf{All roads lead to Rome!} --- All accepted reduced expressions for~$\\pi$ end at the same~state.\n\\end{itemize}\n\n\\begin{proposition}\\label{prop:prefixesReducedExpressions}\nThe set of reduced words accepted by~$\\automatonU(j)$ (resp.~$\\automatonD(j)$) is closed by prefix.\n\\end{proposition}\n\n\\begin{proof}\nThis immediately follows from the fact that the set of reduced words is closed by prefix, and that the set of accepting states of~$\\automatonU(j)$ is connected and contains the initial state.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:algorithm}\nLet $\\ell \\in [n-1]$ distinct from $j-1$ (resp.~$j$).\nA permutation $\\pi \\in \\fS_n$ that avoids $jki$ (resp.~$kij$) and reverses $\\ell$ and $\\ell+1$ admits a reduced expression starting with $s_\\ell$ and accepted by~$\\automatonU(j)$ (resp.~$\\automatonD(j)$).\n\\end{proposition}\n\n\\begin{proof}\nSince $\\pi$ reverses $\\ell$ and $\\ell+1$, it admits a reduced expression of the form~$\\pi = s_\\ell \\cdot \\tau$. Now consider two cases depending on the value of $\\ell$:\n\\begin{itemize}\n\t\\item If $\\ell = j$, then $\\tau$ avoids $(j+1)ki$ by \\cref{lem:vincent4}\\,(2), thus $\\tau$ has a reduced expression accepted by $\\automatonU(j+1)$ by \\cref{thm:patternAvoidance}, and we conclude by \\cref{lem:vincent4}\\,(1).\n\t\\item Otherwise, $\\ell$ is neither $j-1$ nor $j$, so that $\\tau$ avoids $jki$ by \\cref{lem:vincent2}\\,(2), thus $\\tau$ has a reduced expression accepted by $\\automatonU(j)$ by \\cref{thm:patternAvoidance}, and we conclude by \\cref{lem:vincent2}\\,(1).\n\t\\qedhere\n\\end{itemize}\n\\end{proof}\n\n\\begin{proposition}\\label{prop:sameStateAcceptedReducedExpressions}\nGiven a permutation $\\pi \\in \\fS_n$, all the reduced expressions for~$\\pi$ accepted by $\\automatonU(j)$ (resp.~$\\automatonD(j)$) end at the same state.\n\\end{proposition}\n\nTo prove \\cref{prop:sameStateAcceptedReducedExpressions}, it would be enough to check that any two reduced words accepted by~$\\automatonU(j)$ that differ by a single commutation or a single braid relation indeed end at the same state.\nHowever, we prefer to prove instead the following stronger but more technical version of \\cref{prop:sameStateAcceptedReducedExpressions}.\n\n\\begin{proposition}\\label{prop:sameStateAcceptedReducedExpressionsRefined}\nFor a permutation~$\\pi \\in \\fS_n$, let~$\\ninv^j(\\pi) = |\\set{(j,i)}{i < j \\text{ and } \\pi^{-1}(i) > \\pi^{-1}(j)}|$ and~$\\ninv_j(\\pi) = |\\set{(k,j)}{j < k \\text{ and } \\pi^{-1}(j) > \\pi^{-1}(k)}|$.\n\\begin{enumerate}[(i)]\n\t\\item if $\\ninv^j(\\pi) = 0$, then all reduced expressions for~$\\pi$ end at the same healthy state of~$\\automatonU(j)$,\n\t\\item if~$\\ninv_j(\\pi) = 0$, then all reduced expressions for~$\\pi$ end at the same state of~$\\automatonU(j)$, which might be healthy if $\\pi$ avoids~$ji$, ill if~$\\pi$ contains~$ji$ but avoids~$jki$, or dead if~$\\pi$ contains~$jki$,\n\t\\item if $\\ninv^j(\\pi) \\ne 0 \\ne \\ninv_j(\\pi)$, all accepted reduced expressions for~$\\pi$ end at the same ill state of~$\\automatonU(j)$ while the rejected reduced expressions may end at distinct dead states of~$\\automatonU(j)$.\n\\end{enumerate}\nMoreover, all reduced expressions for~$\\pi$ accepted by~$\\automatonU(j)$ end in the $(\\ninv_j(\\pi)+1)$st column of~$\\automatonU(j)$.\nA similar statement holds for~$\\automatonD(j)$ by exchanging $\\ninv_j(\\pi)$ and~$\\ninv^j(\\pi)$.\n\\end{proposition}\n\n\\begin{proof}\nThe proof works by induction on the length of~$\\pi$.\nConsider an arbitrary reduced expression~$w$ for~$\\pi$, starting with a transposition~$s_\\ell$, and write~$w = s_\\ell \\cdot w'$ and~$\\pi = s_\\ell \\cdot \\tau$.\nObserve~that:\n\\begin{itemize}\n\t\\item if $\\ell \\notin \\{j-1, j\\}$, then~$s_\\ell$ loops in~$\\automatonU(j)$, $\\ninv^j(\\pi) = \\ninv^j(\\tau)$ and $\\ninv_j(\\pi) = \\ninv_j(\\tau)$,\n\t\\item if~$\\ell = j$, then~$s_j$ goes to~$\\automatonU(j+1)$, $\\ninv^j(\\pi) = \\ninv^{j+1}(\\tau)$ and~${\\ninv_j(\\pi) = \\ninv_{j+1}(\\tau) + 1}$,\n\t\\item if~$\\ell = j-1$, then~$s_{j-1}$ goes to the first ill state of~$\\automatonU(j)$, $\\ninv^j(\\pi) = \\ninv^{j+1}(\\tau) + 1$ and~$\\ninv_j(\\pi) = \\ninv_{j+1}(\\tau)$.\n\\end{itemize}\nBy induction, we obtain that the reduced expression~$w'$ for~$\\tau$ ends as predicted in the statement.\nThe previous observations ensure that the reduced expression~$w$ for~$\\pi$ also does.\n\\end{proof}\n\n\\begin{example}\\label{exm:sameStateAcceptedReducedExpressionsRefined}\nWe present an example of each case:\n\\begin{enumerate}[(i)]\n\t\\item For~$\\pi \\eqdef 4312$, we have that~$\\ninv^2(\\pi)=0$ and all of its~$5$ reduced expressions end at the third healthy state of $\\automatonU(2)$.\n\t\\item For~$\\pi \\eqdef 32145$ (resp.~$\\pi \\eqdef 43215$, resp.~$\\pi \\eqdef 43251$), we have~$\\ninv_4(\\pi) = 0$ and all its~$2$ (resp.~$16$, resp.~$35$) reduced expressions end at the first healthy (resp.~ill, resp.~dead) state~of~$\\automatonU(4)$.\n\t\\item For~$\\pi \\eqdef 4321$, we have~$\\ninv^2(\\pi) = |\\{(2,1)\\}| = 1$ and $\\ninv_2 = |\\{(3,2),(4,2)\\}| = 2$. Among the~$16$ reduced expressions of~$\\pi$, the automaton~$\\automatonU(2)$ accepts~$7$ at its third ill state, rejects~$7$ at its first dead state, and rejects the other~$2$ at its second dead state.\n\\end{enumerate}\n\\end{example}\n\n\\subsection{Finding accepted reduced expressions}\n\n\\cref{prop:algorithm} has a strong algorithmic consequence.\nImagine we want to test whether a permutation~$\\pi \\in \\fS_n$ is minimal in its permutree class for~$U = \\{j\\}$ and~$D = \\varnothing$.\nOf course, the quickest way is to check for all~$i < j < k$ whether~$\\pi$ contains the subword~$jki$.\nBut since this interpretation will be lost beyond type~$A$, let us impose the use of reduced expressions for~$\\pi$ to make this test.\nWhile it would be \\apriori{} necessary to check all reduced expressions on the automaton~$\\automatonU(j)$, \\cref{prop:algorithm} enables us to construct without loss of generality a candidate reduced expression for~$\\pi$ and we will just neeed to check that this one is accepted by~$\\automatonU(j)$.\nSomewhat dually, one can also construct a reduced word accepted by~$\\automatonU(j)$ that is a reduced expression for~$\\pi$ if and only if~$\\pi$ avoids~$jki$.\nThis is done in the following algorithm, that we call \\defn{$(\\{j\\}, \\varnothing)$-permutree sorting}.\nThe reader is invited to write down the symmetric $(\\varnothing, \\{j\\})$-permutree sorting.\nWe will discuss further permutree sorting in \\cref{subsec:permutreeSorting}.\n\n\\bigskip\n\\IncMargin{1em}\n\\SetKwFor{Repeat}{repeat}{}{}\n\\SetKwIF{If}{ElseIf}{Else}{if}{then}{else if}{else}{}\n\\DontPrintSemicolon\n\\begin{algorithm}[H]\n\t\\KwData{a permutation $\\pi \\in \\fS_n$ and an integer~$j \\in [n]$}\n\t\\KwResult{a reduced word accepted by~$\\automatonU(j)$, candidate reduced expression for~$\\pi$}\n\t$w \\eqdef \\varepsilon$ \\;\n\t\\Repeat{}{\n\t\t\\If{$\\exists \\; \\ell \\ne j-1$ such that $\\ell$ and $\\ell+1$ are reversed in~$\\pi$}{\n\t\t\t$\\pi \\eqdef s_\\ell \\cdot \\pi$, \\quad $w \\eqdef w \\cdot s_\\ell$ \\;\n\t\t\t\\lIf{$\\ell = j$}{\n\t\t\t\t$j \\eqdef j+1$\n\t\t\t}\n\t\t}\n\t}\n\t\\If{$j-1$ and $j$ are reversed in~$\\pi$}{\n\t\t$\\pi \\eqdef s_{j-1} \\cdot \\pi$, \\quad $w \\eqdef w \\cdot s_{j-1}$ \\;\n\t\t$w \\eqdef w \\cdot w' \\cdot w''$ where~$w'$ sorts~$\\pi_{[j]}$ and $w''$ sorts $\\pi_{[n] \\ssm [j]}$ \\;\n\t}\n\t\\Return $w$\n\t\\caption{$(\\{j\\}, \\varnothing)$-permutree sorting}\n\t\\label{algo:shortcutsUj}\n\\end{algorithm}\n\\bigskip\n\n\\begin{example}\\label{exm:algo1}\n\tLet us present the $({2},\\varnothing)$-permutree sorting algorithm in action for the permutations~$\\pi_1 \\eqdef 3421$ and~$\\pi_2 \\eqdef 4231$. The steps of the algorithm are presented in \\cref{tab:algo1}. Each row contains the states of the permutation~$\\pi$ and of the word~$w$ and the current values of~$j$ and~$\\ell$ in use at each step. Notice that for~$\\pi_1 \\eqdef 3421$ the algorithm ends with the identity, which coincides with the fact that~$\\pi_1 \\eqdef 3421$ avoids~$2ki$. In contrast, for~$\\pi_2 \\eqdef 4231$ the algorithm ends with the permutation~$1243$, meaning that~$\\pi_2$ is not~$(\\{2\\},\\varnothing)$-sortable, which coincides with the fact that~$\\pi_2 \\eqdef 4231$ contains~$2ki$. \n\n\t\\begin{table}[h!]\n\t\t\\begin{tabular}[t]{l|l|c|c}\n\t\t\t$\\pi_1$ & $w_1$ & $j_1$ & $\\ell_1$ \\\\\n\t\t\t\\hline\n\t\t\t$3421$ & $\\varepsilon$ & $2$ & $2$ \\\\\n\t\t\t$2431$ & $s_2$ & $3$ & $1$ \\\\\n\t\t\t$1432$ & $s_2 \\cdot s_1$ & $3$ & $3$ \\\\\n\t\t\t$1342$ & $s_2 \\cdot s_1 \\cdot s_3$ & $4$ & $2$ \\\\\n\t\t\t$1243$ & $s_2 \\cdot s_1 \\cdot s_3 \\cdot s_2$ & $4$ & $3$ \\\\\n\t\t\t$1234$ & $s_2 \\cdot s_1 \\cdot s_3 \\cdot s_2 \\cdot s_3$ & $4$ &\n\t\t\\end{tabular}\n\t\t\\qquad\\qquad\n\t\t\\begin{tabular}[t]{l|l|c|c}\n\t\t\t$\\pi_2$ & $w_2$ & $j_2$ & $\\ell_2$ \\\\\n\t\t\t\\hline\n\t\t\t$4231$ & $\\varepsilon$ & $2$ & $3$ \\\\\n\t\t\t$3241$ & $s_3$ & $2$ & $2$ \\\\\n\t\t\t$2341$ & $s_3 \\cdot s_2$ & $3$ & $1$ \\\\\n\t\t\t$1342$ & $s_3 \\cdot s_2 \\cdot s_1$ & $3$ & $2$ \\\\\n\t\t\t$1243$ & $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_2$ & $3$ &\n\t\t\\end{tabular}\n\t\\caption{The~$(\\{2\\},\\varnothing)$-permutree sorting of~$\\pi_1 \\eqdef 3421$ and~$\\pi_2 \\eqdef 4231$.}\n\t\\label{tab:algo1}\n\t\\end{table}\n\\end{example}               \n\n\\begin{corollary}\\label{coro:algorithm}\nFor any permutation~$\\pi$ and~$j \\in \\{2, \\dots, n-1\\}$, \\cref{algo:shortcutsUj} returns a reduced~word~$w$ accepted by $\\automatonU(j)$ with the property that~$w$ is a reduced expression for~$\\pi$ if and only if~$\\pi$ avoids~$jki$.\n\\end{corollary}\n\n\\begin{proof}\nThis algorithm constructs a candidate reduced word for $\\pi$ while following the automaton~$\\automatonU(j)$ and prioritizing healthy states over ill states.\nLines 2 to 5 start by all possible transitions~$s_\\ell$ that remain in healthy states, updating~$j$ to~$j+1$ when~$\\ell = j$ according to \\cref{lem:vincent4} (if condition at line 5).\nWhen we have exhausted all these transitions, if we need to go to an ill state (if condition at line 6) applying~$s_{j-1}$, then we are not anymore allowed to use~$s_j$ and we obtain a candidate reduced word by sorting independently the first $j$ positions of~$\\pi$ with a reduced expression in~$\\{s_1, \\dots, s_{j-1}\\}^*$ and the last $[n] \\ssm [j]$ positions of~$\\pi$ with a reduced expression in~$\\{s_{j+1}, \\dots, s_{n-1}\\}^*$.\nThe resulting reduced word~$w$ is clearly accepted by~$\\automatonU(j)$ because we never allow the transition from an ill state to the corresponding dead state.\nIf~$w$ is a reduced expression for~$\\pi$, then $\\pi$ avoids~$jki$ by \\cref{thm:patternAvoidance}.\nConversely, if~$\\pi$ avoids~$jki$, then $w$ must be a reduced expression for~$\\pi$ since the choice to start with~$s_\\ell$ is valid in lines 2 to 5 by \\cref{prop:algorithm} and forced in lines 6 to 8 (since all reduced expressions of~$\\pi$ then start by~$s_\\ell$).\n\\end{proof}\n\n\\begin{remark}\nWe really wrote \\cref{algo:shortcutsUj} as a sorting algorithm.\nIt first tries to sort the permutation~$\\pi \\in \\fS_n$ while avoiding to swap~$j-1$ and~$j$ for a certain token~$j$ (and changing the token when swapping~$j$ and~$j+1$).\nOnce it is forced to swap~$j-1$ and~$j$, it tries to sort the permutation~$\\pi$ while avoiding to swap any value of~$[j]$ with a value of~$[n] \\ssm [j]$.\nIf we were only interested in deciding whether the permutation~$\\pi$ is $(\\{j\\}, \\varnothing)$-sortable, then we could stop and accept the permutation as soon as we reach $j = n$, and we could just check at line 8 of the algorithm whether~$\\pi([j]) = [j]$ and~$\\pi([n] \\ssm [j]) = [n] \\ssm [j]$.\n\\end{remark}\n\n\\subsection{Generating trees on accepted reduced expressions}\\label{subsec:trees}\n\n\\cref{prop:prefixesReducedExpressions,prop:sameStateAcceptedReducedExpressions} also have a relevant consequence, more combinatorial this time.\nNamely, they naturally define generating trees for the $(\\{j\\}, \\varnothing)$-permutree minimal permutations, following certain special reduced expressions for them.\nTo construct these trees, pick an arbitrary priority order~$\\prec$ on~$\\{s_1, \\dots, s_{n-1}\\}$.\nFor a $(\\{j\\}, \\varnothing)$-permutree minimal permutation~$\\pi \\in \\fS_n$, denote by~$\\pi(\\{j\\}, \\varnothing, \\prec)$ the $\\prec$-lexicographic minimal reduced expression for~$\\pi$ that is accepted by~$\\automatonU(j)$.\nDenote by~$\\lexmin(n, \\{j\\}, \\varnothing, \\prec)$ the set of reduced words of the form~$\\pi(\\{j\\}, \\varnothing, \\prec)$ for all $(\\{j\\}, \\varnothing)$-permutree minimal permutations~$\\pi \\in \\fS_n$.\nThe following statement is an analogue of \\cref{prop:prefixesReducedExpressions}.\n\n\\begin{proposition}\\label{prop:generatingTrees}\nThe set $\\lexmin(n, \\{j\\}, \\varnothing, \\prec)$ is closed by prefix.\n\\end{proposition}\n\n\\begin{proof}\nConsider a reduced expression~$w = u \\cdot v$ where~$u$ is not in~$\\lexmin(n, \\{j\\}, \\varnothing, \\prec)$.\nIf~$u$ is not accepted by~$\\automatonU(j)$, neither is~$w$ by \\cref{prop:prefixesReducedExpressions}.\nOtherwise, there exists a reduced expression~$u'$ representing the same permutation as~$u$, accepted by~$\\automatonU(j)$ and $\\prec$-lexicographic smaller than~$u$.\nBy \\cref{prop:sameStateAcceptedReducedExpressions}, the reduced expressions~$u$ and~$u'$ end at the same state of~$\\automatonU(j)$.\nTherefore, if~$w = u \\cdot v$ is accepted by~$\\automatonU(j)$, so is~$u' \\cdot v$.\nSince~$u' \\cdot v$ is $\\prec$-lexicographically smaller than~$u \\cdot v$ and represents the same permutation, this ensures that~$w$ is not in~$\\lexmin(n, \\{j\\}, \\varnothing, \\prec)$.\n\\end{proof}\n\n\\cref{prop:generatingTrees} yields a natural generating tree for $\\lexmin(n, \\{j\\}, \\varnothing, \\prec)$ where the parent of a reduced word~$w$ is obtained by deleting its last letter.\nReplacing each reduced expression by the corresponding permutation, this provides a generating tree for the $(\\{j\\}, \\varnothing)$-permutree minimal permutations of~$\\fS_n$.\nOf course there is a similar generating tree for the $(\\varnothing, \\{j\\})$-permutree minimal permutations of~$\\fS_n$.\n\\cref{fig:TreePartialOrientations} presents these generating trees for~$n = 4$ and~$j = 2, 3$, with the priority order~$s_1 \\prec s_2 \\prec s_3$.\nIt is natural to draw these trees on top of the Hasse diagram of the right weak order on permutations, defined by inclusion of inversion sets.\nIn other words, the cover relations in weak order correspond to the swap of the values at two consecutive positions in a permutation, \\ie to a right multiplication by a simple transposition.\nThe edges of the trees corresponding to the right multiplications by $s_1$, $s_2$ and $s_3$ are colored by blue, red, and green respectively.\n\n\\hvFloat[floatPos=p, capWidth=h, capPos=r, capAngle=90, objectAngle=90, capVPos=c, objectPos=c]{figure}\n{\n\t\\begin{tabular}{cccc}\n\t\t$\\automatonU(2)$\n\t\t&\n\t\t$\\automatonD(2)$\n\t\t&\n\t\t$\\automatonU(3)$\n\t\t&\n\t\t$\\automatonD(3)$\n\t\t\\\\[.5cm]\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj+1) [below= 1.5cm of ij+1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+2) [right= 1.5cm of hj+1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+2) [below= 1.5cm of hj+2] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_1$} (ij)\n\t\t\t\t\t   edge node {$s_2$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_2$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_2$} (ij+1)\n\t\t\t\t\t edge node {$s_3$} (hj+2)\n\t\t\t\t(ij+1) edge node [swap] {$s_3$} (dj+1)\n\t\t\t\t%\n\t\t\t\t(hj+2) edge node [swap] {$s_3$} (ij+2);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_2$} (ij)\n\t\t\t\t\t   edge node {$s_1$} (hj-1)\n\t\t\t\t(ij) edge node [swap] {$s_1$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj-1) edge node [swap] {$s_1$} (ij-1);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_2$} (ij)\n\t\t\t\t\t   edge node {$s_3$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_3$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_3$} (ij+1);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj-1) [below= 1.5cm of ij-1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-2) [right= 1.5cm of hj-1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-2) [below= 1.5cm of hj-2] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_3$} (ij)\n\t\t\t\t\t   edge node {$s_2$} (hj-1)\n\t\t\t\t(ij) edge node [swap] {$s_2$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj-1) edge node [swap] {$s_2$} (ij-1)\n\t\t\t\t\t edge node {$s_1$} (hj-2)\n\t\t\t\t(ij-1) edge node [swap] {$s_1$} (dj-1)\n\t\t\t\t%\n\t\t\t\t(hj-2) edge node [swap] {$s_1$} (ij-2);\n\t\t\\end{tikzpicture}\n\t\t\\\\[.5cm]\n    \t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n    \t\t\\node[blue](P1234) at (0,0){1234};\n    \t\t%\n    \t\t\\node[blue](P2134) at (-1,1.5){2134};\n    \t\t\\node[blue](P1324) at ( 0,1.5){1324};\n    \t\t\\node[blue](P1243) at ( 1,1.5){1243};\n    \t\t%\n    \t\t\\node(P2314) at (-2,3){2314};\n    \t\t\\node[blue](P3124) at (-1,3){3124};\n    \t\t\\node[blue](P2143) at ( 0,3){2143};\n    \t\t\\node[blue](P1342) at ( 1,3){1342};\n    \t\t\\node[blue](P1423) at ( 2,3){1423};\n    \t\t%\n    \t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n    \t\t\\node(P2341) at (-1.5,4.5){2341};\n    \t\t\\node[blue](P3142) at (-0.5,4.5){3142};\n    \t\t\\node(P2413) at ( 0.5,4.5){2413};\n    \t\t\\node[blue](P4123) at ( 1.5,4.5){4123};\n    \t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n    \t\t%\n    \t\t\\node(P3241) at (-2,6){3241};\n    \t\t\\node(P2431) at (-1,6){2431};\n    \t\t\\node[blue](P3412) at ( 0,6){3412};\n    \t\t\\node[blue](P4213) at ( 1,6){4213};\n    \t\t\\node[blue](P4132) at ( 2,6){4132};\n    \t\t%\n    \t\t\\node[blue](P3421) at (-1,7.5){3421};\n    \t\t\\node(P4231) at ( 0,7.5){4231};\n    \t\t\\node[blue](P4312) at ( 1,7.5){4312};\n    \t\t%\n    \t\t\\node[blue](P4321) at (0,9){4321};\n    \t\t%\n    \t\t%\n    \t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n    \t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n    \t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n    \t\t%\n    \t\t\\draw(P2134) -- (P2314);\n    \t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n    \t\t\\draw[line width=0.5mm,blue](P1324) -- (P3124);\n    \t\t\\draw[line width=0.5mm,green](P1324) -- (P1342);\n    \t\t\\draw(P1243) -- (P2143);\n    \t\t\\draw[line width=0.5mm,red](P1243) -- (P1423);\n    \t\t%\n    \t\t\\draw(P2314) -- (P3214);\n    \t\t\\draw(P2314) -- (P2341);\n    \t\t\\draw[line width=0.5mm,red](P3124) -- (P3214);\n    \t\t\\draw[line width=0.5mm,green](P3124) -- (P3142);\n    \t\t\\draw(P2143) -- (P2413);\n    \t\t\\draw(P1342) -- (P3142);\n    \t\t\\draw[line width=0.5mm,red](P1342) -- (P1432);\n    \t\t\\draw[line width=0.5mm,blue](P1423) -- (P4123);\n    \t\t\\draw(P1423) -- (P1432);\n    \t\t%\n    \t\t\\draw(P3214) -- (P3241);\n    \t\t\\draw(P2341) -- (P3241);\n    \t\t\\draw(P2341) -- (P2431);\n    \t\t\\draw[line width=0.5mm,red](P3142) -- (P3412);\n    \t\t\\draw(P2413) -- (P4213);\n    \t\t\\draw(P2413) -- (P2431);\n    \t\t\\draw[line width=0.5mm,red](P4123) -- (P4213);\n    \t\t\\draw(P4123) -- (P4132);\n    \t\t\\draw[line width=0.5mm,blue](P1432) -- (P4132);\n    \t\t%\n    \t\t\\draw(P3241) -- (P3421);\n    \t\t\\draw(P2431) -- (P4231);\n    \t\t\\draw[line width=0.5mm,blue](P3412) -- (P4312);\n    \t\t\\draw[line width=0.5mm,green](P3412) -- (P3421);\n    \t\t\\draw(P4213) -- (P4231);\n    \t\t\\draw(P4132) -- (P4312);\n    \t\t%\n    \t\t\\draw(P3421) -- (P4321);\n    \t\t\\draw(P4231) -- (P4321);\n    \t\t\\draw[line width=0.5mm,green](P4312) -- (P4321);\t\n    \t\\end{tikzpicture}\n    \t&\n    \t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n    \t\t\\node[blue](P1234) at (0,0){1234};\n    \t\t%\n    \t\t\\node[blue](P2134) at (-1,1.5){2134};\n    \t\t\\node[blue](P1324) at ( 0,1.5){1324};\n    \t\t\\node[blue](P1243) at ( 1,1.5){1243};\n    \t\t%\n    \t\t\\node[blue](P2314) at (-2,3){2314};\n    \t\t\\node(P3124) at (-1,3){3124};\n    \t\t\\node[blue](P2143) at ( 0,3){2143};\n    \t\t\\node[blue](P1342) at ( 1,3){1342};\n    \t\t\\node[blue](P1423) at ( 2,3){1423};\n    \t\t%\n    \t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n    \t\t\\node[blue](P2341) at (-1.5,4.5){2341};\n    \t\t\\node(P3142) at (-0.5,4.5){3142};\n    \t\t\\node[blue](P2413) at ( 0.5,4.5){2413};\n    \t\t\\node(P4123) at ( 1.5,4.5){4123};\n    \t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n    \t\t%\n    \t\t\\node[blue](P3241) at (-2,6){3241};\n    \t\t\\node[blue](P2431) at (-1,6){2431};\n    \t\t\\node(P3412) at ( 0,6){3412};\n    \t\t\\node[blue](P4213) at ( 1,6){4213};\n    \t\t\\node(P4132) at ( 2,6){4132};\n    \t\t%\n    \t\t\\node[blue](P3421) at (-1,7.5){3421};\n    \t\t\\node[blue](P4231) at ( 0,7.5){4231};\n    \t\t\\node(P4312) at ( 1,7.5){4312};\n    \t\t%\n    \t\t\\node[blue](P4321) at (0,9){4321};\n    \t\t%\n    \t\t%\n    \t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n    \t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n    \t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n    \t\t%\n    \t\t\\draw[line width=0.5mm,red](P2134) -- (P2314);\n    \t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n    \t\t\\draw(P1324) -- (P3124);\n    \t\t\\draw[line width=0.5mm,green](P1324) -- (P1342);\n    \t\t\\draw(P1243) -- (P2143);\n    \t\t\\draw[line width=0.5mm,red](P1243) -- (P1423);\n    \t\t%\n    \t\t\\draw[line width=0.5mm,blue](P2314) -- (P3214);\n    \t\t\\draw[line width=0.5mm,green](P2314) -- (P2341);\n    \t\t\\draw(P3124) -- (P3214);\n    \t\t\\draw(P3124) -- (P3142);\n    \t\t\\draw[line width=0.5mm,red](P2143) -- (P2413);\n    \t\t\\draw(P1342) -- (P3142);\n    \t\t\\draw[line width=0.5mm,red](P1342) -- (P1432);\n    \t\t\\draw(P1423) -- (P4123);\n    \t\t\\draw(P1423) -- (P1432);\n    \t\t%\n    \t\t\\draw[line width=0.5mm,green](P3214) -- (P3241);\n    \t\t\\draw(P2341) -- (P3241);\n    \t\t\\draw[line width=0.5mm,red](P2341) -- (P2431);\n    \t\t\\draw(P3142) -- (P3412);\n    \t\t\\draw[line width=0.5mm,blue](P2413) -- (P4213);\n    \t\t\\draw(P2413) -- (P2431);\n    \t\t\\draw(P4123) -- (P4213);\n    \t\t\\draw(P4123) -- (P4132);\n    \t\t\\draw(P1432) -- (P4132);\n    \t\t%\n    \t\t\\draw[line width=0.5mm,red](P3241) -- (P3421);\n    \t\t\\draw[line width=0.5mm,blue](P2431) -- (P4231);\n    \t\t\\draw(P3412) -- (P4312);\n    \t\t\\draw(P3412) -- (P3421);\n    \t\t\\draw(P4213) -- (P4231);\n    \t\t\\draw(P4132) -- (P4312);\n    \t\t%\n    \t\t\\draw[line width=0.5mm,blue](P3421) -- (P4321);\n    \t\t\\draw(P4231) -- (P4321);\n    \t\t\\draw(P4312) -- (P4321);\n    \t\\end{tikzpicture}\n\t\t&\n    \t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n\t\t\t\\node[blue](P1234) at (0,0){1234};\n\t\t\t%\n\t\t\t\\node[blue](P2134) at (-1,1.5){2134};\n\t\t\t\\node[blue](P1324) at ( 0,1.5){1324};\n\t\t\t\\node[blue](P1243) at ( 1,1.5){1243};\n\t\t\t%\n\t\t\t\\node[blue](P2314) at (-2,3){2314};\n\t\t\t\\node[blue](P3124) at (-1,3){3124};\n\t\t\t\\node[blue](P2143) at ( 0,3){2143};\n\t\t\t\\node(P1342) at ( 1,3){1342};\n\t\t\t\\node[blue](P1423) at ( 2,3){1423};\n\t\t\t%\n\t\t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n\t\t\t\\node(P2341) at (-1.5,4.5){2341};\n\t\t\t\\node(P3142) at (-0.5,4.5){3142};\n\t\t\t\\node[blue](P2413) at ( 0.5,4.5){2413};\n\t\t\t\\node[blue](P4123) at ( 1.5,4.5){4123};\n\t\t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n\t\t\t%\n\t\t\t\\node(P3241) at (-2,6){3241};\n\t\t\t\\node[blue](P2431) at (-1,6){2431};\n\t\t\t\\node(P3412) at ( 0,6){3412};\n\t\t\t\\node[blue](P4213) at ( 1,6){4213};\n\t\t\t\\node[blue](P4132) at ( 2,6){4132};\n\t\t\t%\n\t\t\t\\node(P3421) at (-1,7.5){3421};\n\t\t\t\\node[blue](P4231) at ( 0,7.5){4231};\n\t\t\t\\node[blue](P4312) at ( 1,7.5){4312};\n\t\t\t%\n\t\t\t\\node[blue](P4321) at (0,9){4321};\n\t\t\t%\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n\t\t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n\t\t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,red](P2134) -- (P2314);\n\t\t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,blue](P1324) -- (P3124);\n\t\t\t\\draw(P1324) -- (P1342);\n\t\t\t\\draw(P1243) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,red](P1243) -- (P1423);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P2314) -- (P3214);\n\t\t\t\\draw(P2314) -- (P2341);\n\t\t\t\\draw(P3124) -- (P3214);\n\t\t\t\\draw(P3124) -- (P3142);\n\t\t\t\\draw[line width=0.5mm,red](P2143) -- (P2413);\n\t\t\t\\draw(P1342) -- (P3142);\n\t\t\t\\draw(P1342) -- (P1432);\n\t\t\t\\draw[line width=0.5mm,blue](P1423) -- (P4123);\n\t\t\t\\draw[line width=0.5mm,green](P1423) -- (P1432);\n\t\t\t%\n\t\t\t\\draw(P3214) -- (P3241);\n\t\t\t\\draw(P2341) -- (P3241);\n\t\t\t\\draw(P2341) -- (P2431);\n\t\t\t\\draw(P3142) -- (P3412);\n\t\t\t\\draw[line width=0.5mm,blue](P2413) -- (P4213);\n\t\t\t\\draw[line width=0.5mm,green](P2413) -- (P2431);\n\t\t\t\\draw(P4123) -- (P4213);\n\t\t\t\\draw[line width=0.5mm,green](P4123) -- (P4132);\n\t\t\t\\draw(P1432) -- (P4132);\n\t\t\t%\n\t\t\t\\draw(P3241) -- (P3421);\n\t\t\t\\draw(P2431) -- (P4231);\n\t\t\t\\draw(P3412) -- (P4312);\n\t\t\t\\draw(P3412) -- (P3421);\n\t\t\t\\draw[line width=0.5mm,green](P4213) -- (P4231);\n\t\t\t\\draw[line width=0.5mm,red](P4132) -- (P4312);\n\t\t\t%\n\t\t\t\\draw(P3421) -- (P4321);\n\t\t\t\\draw[line width=0.5mm,red](P4231) -- (P4321);\n\t\t\t\\draw(P4312) -- (P4321);\n    \t\\end{tikzpicture}\n    \t&\n    \t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n\t\t\t\\node[blue](P1234) at (0,0){1234};\n\t\t\t%\n\t\t\t\\node[blue](P2134) at (-1,1.5){2134};\n\t\t\t\\node[blue](P1324) at ( 0,1.5){1324};\n\t\t\t\\node[blue](P1243) at ( 1,1.5){1243};\n\t\t\t%\n\t\t\t\\node[blue](P2314) at (-2,3){2314};\n\t\t\t\\node[blue](P3124) at (-1,3){3124};\n\t\t\t\\node[blue](P2143) at ( 0,3){2143};\n\t\t\t\\node[blue](P1342) at ( 1,3){1342};\n\t\t\t\\node(P1423) at ( 2,3){1423};\n\t\t\t%\n\t\t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n\t\t\t\\node[blue](P2341) at (-1.5,4.5){2341};\n\t\t\t\\node[blue](P3142) at (-0.5,4.5){3142};\n\t\t\t\\node(P2413) at ( 0.5,4.5){2413};\n\t\t\t\\node(P4123) at ( 1.5,4.5){4123};\n\t\t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n\t\t\t%\n\t\t\t\\node[blue](P3241) at (-2,6){3241};\n\t\t\t\\node[blue](P2431) at (-1,6){2431};\n\t\t\t\\node[blue](P3412) at ( 0,6){3412};\n\t\t\t\\node(P4213) at ( 1,6){4213};\n\t\t\t\\node(P4132) at ( 2,6){4132};\n\t\t\t%\n\t\t\t\\node[blue](P3421) at (-1,7.5){3421};\n\t\t\t\\node(P4231) at ( 0,7.5){4231};\n\t\t\t\\node[blue](P4312) at ( 1,7.5){4312};\n\t\t\t%\n\t\t\t\\node[blue](P4321) at (0,9){4321};\n\t\t\t%\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n\t\t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n\t\t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,red](P2134) -- (P2314);\n\t\t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,blue](P1324) -- (P3124);\n\t\t\t\\draw[line width=0.5mm,green](P1324) -- (P1342);\n\t\t\t\\draw(P1243) -- (P2143);\n\t\t\t\\draw(P1243) -- (P1423);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P2314) -- (P3214);\n\t\t\t\\draw[line width=0.5mm,green](P2314) -- (P2341);\n\t\t\t\\draw(P3124) -- (P3214);\n\t\t\t\\draw[line width=0.5mm,green](P3124) -- (P3142);\n\t\t\t\\draw(P2143) -- (P2413);\n\t\t\t\\draw(P1342) -- (P3142);\n\t\t\t\\draw[line width=0.5mm,red](P1342) -- (P1432);\n\t\t\t\\draw(P1423) -- (P4123);\n\t\t\t\\draw(P1423) -- (P1432);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,green](P3214) -- (P3241);\n\t\t\t\\draw(P2341) -- (P3241);\n\t\t\t\\draw[line width=0.5mm,red](P2341) -- (P2431);\n\t\t\t\\draw[line width=0.5mm,red](P3142) -- (P3412);\n\t\t\t\\draw(P2413) -- (P4213);\n\t\t\t\\draw(P2413) -- (P2431);\n\t\t\t\\draw(P4123) -- (P4213);\n\t\t\t\\draw(P4123) -- (P4132);\n\t\t\t\\draw(P1432) -- (P4132);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,red](P3241) -- (P3421);\n\t\t\t\\draw(P2431) -- (P4231);\n\t\t\t\\draw[line width=0.5mm,blue](P3412) -- (P4312);\n\t\t\t\\draw(P3412) -- (P3421);\n\t\t\t\\draw(P4213) -- (P4231);\n\t\t\t\\draw(P4132) -- (P4312);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P3421) -- (P4321);\n\t\t\t\\draw(P4231) -- (P4321);\n\t\t\t\\draw(P4312) -- (P4321);\n    \t\\end{tikzpicture}\n\t\\end{tabular}\n}\n{Generating trees for the $(\\{j\\}, \\varnothing)$- and $(\\varnothing, \\{j\\})$-permutree minimal permutations of~$\\fS_4$, with priority order $s_1 \\prec s_2 \\prec s_3$.}\n{fig:TreePartialOrientations}\n\n\n\\newpage\n\\section{Intersection of automata}\\label{sec:intersectionsAutomata}\n\nWe now consider arbitrary subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$.\nWe already know from~\\cite{PilaudPons-permutrees} and \\cref{thm:patternAvoidance} that the following conditions are equivalent for~$\\pi \\in \\fS_n$:\n\\begin{enumerate}[(i)]\n\t\\item the permutation $\\pi$ is minimal in its~$(U,D)$-permutree class, \n\t\\item for~$i < j < k$, the permutation $\\pi$ does not contain the subword~$jki$ if~$j \\in U$ and~$kij$ if~$j \\in D$,\n\t\\item for each~$j \\in U$ (each~$j \\in D$), there is a reduced expression for~$\\pi$ accepted by~$\\automatonU(j)$ (resp.~by~$\\automatonD(j)$).\n\\end{enumerate}\nA natural question is whether there is a reduced expression simultaneously accepted by all these automata.\nWe start with an example showing that this is not always the case.\n\n\\begin{example}\\label{exm:problemUDintersect}\nFor~$j \\in \\{2, \\dots, n-1\\}$, consider $U = \\{j\\} = D$, and~$\\pi = s_{j-1} \\cdot s_j \\cdot s_{j-1} = s_j \\cdot s_{j-1} \\cdot s_j$.\nThen, the expression~$s_{j-1} \\cdot s_j \\cdot s_{j-1}$ is accepted by $\\automatonD(j)$ but not by~$\\automatonU(j)$, while the expression $s_j \\cdot s_{j-1} \\cdot s_j$ is accepted by $\\automatonU(j)$ but not by~$\\automatonD(j)$.\n\\end{example}\n\nThis example clearly extends to all subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$ with a non-empty intersection.\nIn contrast, we will now show that this situation cannot occur when~$U$ and~$D$ are disjoint.\n\n\\subsection{Proof of \\cref{thm:permutreeMinimal}}\n\n\\cref{exm:problemUDintersect} motivates \\cref{thm:permutreeMinimal} that we repeat here for convenience.\n\n\\newtheorem*{thm:permutreeMinimal}{\\cref{thm:permutreeMinimal}}\n\\begin{thm:permutreeMinimal}\nConsider two disjoint subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$.\nThe following conditions are equivalent for~$\\pi \\in \\fS_n$:\n\\begin{itemize}\n\\item $\\pi$ admits a reduced expression accepted by all automata~$\\automatonU(j)$ for~$j \\in U$ and~$\\automatonD(j)$ for~${j \\in D}$,\n\\item $\\pi$ contains no subword $jki$ if~$j \\in U$ and~$kij$ if~$j \\in D$ for any~$i < j < k$.\n\\end{itemize}\n\\end{thm:permutreeMinimal}\n\n\\begin{proof}\nThe direct implication is immediate from \\cref{thm:patternAvoidance}.\nFor the converse implication, consider a permutation~$\\pi$ that avoids~$jki$ for~$j \\in U$ and~$kij$ for~$j \\in D$.\nConsider $\\bar U \\eqdef \\set{j \\in U}{\\ninv_j(\\pi) \\ne 0}$ and~$\\bar D \\eqdef \\set{j \\in D}{\\ninv^j(\\pi) \\ne 0}$.\nBy \\cref{prop:sameStateAcceptedReducedExpressionsRefined}\\,(ii), any reduced expression for~$\\pi$ is accepted by~$\\automatonU(j)$ for~$j \\in U \\ssm \\bar U$ and by~$\\automatonD(j)$ for~$j \\in D \\ssm \\bar D$.\nWe can therefore assume that~$\\bar U = U$ and~$\\bar D = D$ and one of them is non-empty, say~$\\bar U = U \\ne \\varnothing$.\nLet~$j_\\circ \\eqdef \\max(U)$ and~$m$ be minimal such that~$j_\\circ < m$ and~$\\pi^{-1}(j_\\circ) > \\pi^{-1}(m)$.\nBy minimality of~$m$, we obtain that~$\\pi$ contains the subword~$mj_\\circ\\ell$ for any~$j_\\circ < \\ell < m$.\nIt implies that\n\\begin{itemize}\n\\item $\\ell$ is neither in~$U$ by maximality of~$j_\\circ$, nor in~$D$ by assumption on~$\\pi$, for all~$j_\\circ < \\ell < m$,\n\\item $\\pi$ reverses~$\\ell$ and $m$ for all $\\ell$ in $\\{j_\\circ,\\dots,m-1\\}$, so that $\\pi$ admits a reduced expression of the form $\\pi = s_{m-1} \\cdots s_{j_\\circ} \\cdot \\tau$.\n\\end{itemize}\nLemmas \\ref{lem:vincent2}\\,(2) and \\ref{lem:vincent4}\\,(2) ensure that\n\\begin{itemize}\n\\item $\\tau$ avoids~$jki$ for all~$j \\in U \\ssm \\{j_\\circ\\}$ and~$kij$ for all~$j \\in D \\ssm \\{m\\}$ (because~$j_\\circ, \\dots, m-1$ are all distinct from~$j-1$ and~$j$ in these cases by the second observation above),\n\\item $\\tau$ avoids $(j_\\circ+1)ki$,\n\\item if~$m \\in D$, then~$\\tau$ avoids~$kij_\\circ$.\n\\end{itemize}\nBy induction, it implies that~$\\tau$ admits a reduced expression~$w$ simultaneously accepted by all automata~$\\automatonU(j)$ for~$j \\in U \\ssm \\{j_\\circ\\}$ and~$j = j_\\circ + 1$, and all automata~$\\automatonD(j)$ for~$j \\in D \\ssm \\{m\\}$ and~$j = j_\\circ$ if~$m \\in D$.\nBy Lemmas \\ref{lem:vincent2}\\,(1) and \\ref{lem:vincent4}\\,(1), we conclude that $s_{m-1} \\cdots s_{j_\\circ} \\cdot w$ is a reduced expression for~$\\pi$ simultaneously accepted by all $\\automatonU(j)$ for~$j \\in U$ and~$\\automatonD(j)$ for~$j \\in D$.\n\\end{proof}\n\n\\subsection{Intersection of automata}\n\n\\cref{thm:permutreeMinimal} can be rephrased in terms of intersection of automata.\nRecall that the intersection of some automata~$\\automatonA_1, \\dots, \\automatonA_p$ is the automaton~$\\automatonA = \\bigcap_{i \\in [p]} \\automatonA_i$ such that a word is accepted by~$\\automatonA$ if and only if it is accepted by all~$\\automatonA_1, \\dots, \\automatonA_p$.\nA state of the automaton~$\\automatonA$ is $p$-tuple formed by states of the automata~$\\automatonA_1, \\dots, \\automatonA_p$, and a transition~$t$ simultaneously changes all entries of the $p$-tuple corresponding to states modified by~$t$.\nSee~\\cite[p.\\,59--60]{HopcroftUllman} for details.\nWe denote by~$\\automatonP(U,D)$ the intersection of the automata~$\\automatonU(j)$ for~$j \\in U$ and~$\\automatonD(j)$ for~$j \\in D$.\nWe thus obtain the following statement.\n\n\\begin{corollary}\n\\label{coro:permutreeMinimal}\nWhen~$U$ and~$D$ are disjoint, the following conditions are equivalent for~$\\pi \\in \\fS_n$:\n\\begin{itemize}\n\t\\item $\\pi$ admits a reduced expression accepted by the automaton~$\\automatonP(U,D)$,\n\t\\item $\\pi$ contains no subword~$jki$ if~$j \\in U$ and~$kij$ if~$j \\in D$ with~$i < j < k$.\n\\end{itemize}\n\\end{corollary}\n\nWe say that a state of~$\\automatonP(U,D)$ is \\defn{healthy} (resp.~\\defn{ill}, resp.~\\defn{dead}) when the corresponding states in~$\\automatonU(j)$ for~$j \\in U$ and~$\\automatonD(j)$ for~$j \\in D$ are all healthy (resp.~contain at least one ill state, but no dead one, resp.~contains at least one dead state).\n\\cref{fig:automataProduct} illustrates the automata~$\\automatonP(\\{4\\},\\{2\\})$ when~$n = 5$ (left), $\\automatonP(\\{3\\},\\{2\\})$ for $n=4$ (middle), and~$\\automatonP(\\{2\\},\\{4\\})$ for $n=5$ (right).\nFor the first two automata, we have drawn the complete automata on top, and their skeleta on the bottom.\nHere, we call skeleton a simplification of the automaton that recognizes the same reduced words.\nIt is obtained using the fact that the word is rejected as soon as we reach a dead state, and that the automata~$\\automatonU(n)$ and~$\\automatonD(1)$ accept all reduced expressions.\nFor the last automaton, the complete intersection is too big, so we only draw the reachable healthy states.\nIn the picture, we color the transitions in red, blue, or purple depending on whether only~$\\automatonU$, only $\\automatonD$, or both~$\\automatonU$ and $\\automatonD$ change state.\n\\hvFloat[floatPos=p, capWidth=h, capPos=r, capAngle=90, objectAngle=90, capVPos=c, objectPos=c]{figure}\n{\n\\begin{tabular}{l@{\\hspace{-.8cm}}l@{\\hspace{-.8cm}}l}\n\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto, baseline=-1.5cm]\n\t\t\\node[state,initial,accepting,minimum size=0.5cm] (11) {}; \n\t\t\\node[state,accepting,,minimum size=0.5cm] (12) [right= 1.5cm of 11] {};\n\t\t\\node[state,accepting,,minimum size=0.5cm] (21) [right= 2.5cm of 12] {};\n\t\t\\node[state,accepting,,minimum size=0.5cm] (22) [right= 1.5cm of 21] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (13) [below= 1cm of 11] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (14) [right= 1.5cm of 13] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (23) [right= 2.5cm of 14] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (24) [right= 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[color=red] node [swap] {$s_3$} (25);\n\t\t\\path[->] (24) edge [color=blue] node [left=-1mm] {$s_1$} (44);\n\t\t\\path[->] (25) edge [color=blue] node [below=1mm] {$s_1$} (45);\n\t\t%\n\t\t\\path[->] (32) edge [color=blue] node [below=1mm] {$s_1$} (52) edge [color=red] node {$s_3$} (34);\n\t\t\\path[->] (33) edge [color=red] node [swap] {$s_3$} (35) edge [color=blue] node [below=1mm] {$s_1$} (53);\n\t\t\\path[->] (34) edge [color=blue] node [below=1mm] {$s_1$} (54);\n\t\t\\path[->] (35) edge [color=blue] node [below=1mm] {$s_1$} (55);\n\t\t%\n\t\t\\path[->] (41) edge [color=red] node {$s_2$} (43) edge [color=red] node [below left=-1mm] {$s_3$} (42);\n\t\t\\path[->] (42) edge [color=red] node {$s_3$} (44);\n\t\t\\path[->] (43) edge [color=red] node {$s_3$} (45);\n\t\t%\n\t\t\\path[->] (52) edge [color=red] node {$s_3$} (54);\n\t\t\\path[->] (53) edge [color=red] node {$s_3$} (55);\n\t\\end{tikzpicture}\n\t&\n\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto, baseline=-1.5cm]\n\t\t\\node[state,initial,accepting,minimum size=0.5cm] (11) {}; \n\t\t\\node[state,accepting,,minimum size=0.5cm] (12) [right= 1.5cm of 11] {};\n\t\t\\node[state,accepting,,minimum size=0.5cm] (23) [right= 1.5cm of 12] {};\n\t\t\\node[state,accepting,,minimum size=0.5cm] (24) [right= 1.5cm of 23] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (21) [below= 1.5cm of 11] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (32) [below= 1.5cm of 12] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (33) [below= 1.5cm of 23] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (34) [below= 1.5cm of 24] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (42) [below= 1.5cm of 32] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (43) [below= 1.5cm of 33] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (44) [below= 1.5cm of 34] {};\n\t\t\\path[->] (11) edge [color=red] node [swap] {$s_2$} (21) edge [color=blue] node {$s_3$} (12);\n\t\t\\path[->] (12) edge [color=violet] node {$s_2$} (23);\n\t\t\\path[->] (23) edge [color=red] node [swap] {$s_3$} (33) edge [color=blue] node {$s_1$} (24);\n\t\t\\path[->] (24) edge [color=red] node [swap] {$s_3$} (34);\n\t\t\\path[->] (21) edge [color=violet] node {$s_3$} (32);\n\t\t\\path[->] (32) edge [color=blue] node {$s_2$} (33) edge [color=red] node [swap] {$s_4$} (42);\n\t\t\\path[->] (33) edge [color=blue] node {$s_1$} (34) edge [color=red] node [swap] {$s_4$} (43);\n\t\t\\path[->] (34) edge [color=red] node [swap] {$s_4$} (44);\n\t\t\\path[->] (42) edge [color=blue] node {$s_2$} (43);\n\t\t\\path[->] (43) edge [color=blue] node {$s_1$} (44);\n\t\\end{tikzpicture}\n\t\\\\\n\t\\\\[.3cm]\n\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto, baseline=-1.5cm]\n\t\t\\node[state,initial,accepting,minimum size=0.5cm] (h1) {}; \n\t\t\\node[state,accepting,,minimum size=0.5cm] (h2) [right= 1.5cm of h1] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (h3) [right= 2.5cm of h2] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (h4) [right= 1.5cm of h3] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (i1) [below= 1cm of h1] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (i11) [right= 1cm of i1] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (i12) [below= 1cm of i11] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (i2) [right= 1.5cm of i11] {};\n\t\t\\node[state,minimum size=0.5cm] (d) [below= 2.5cm of i2] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (i3) [below= 1.1cm of h3] {};\n\t\t%\n\t\t\\path[->] (h1) edge [bend left,color=blue] node {$s_1$} (h3) edge [swap,color=red] node {$s_3$} (i1) edge [color=red] node [swap] {$s_4$} (h2) edge [color=blue] node [below left=-2mm] {$s_2$} (i11) ;\n\t\t\\path[->] (i1) edge [bend right=50,swap,color=red] node {$s_4$} (d) edge [bend left, color=blue] node {$s_1$} (i3) edge [color=blue] node [below left=-2mm] {$s_2$} (i12);\n\t\t\\path[->] (i11) edge [swap,color=red] node {$s_3$} (i12) edge [color=blue] node {$s_1$} (d) edge [color=red] node [swap] {$s_4$} (i2);\n\t\t\\path[->] (i12) edge [color=violet] node [below left=-2mm] {${\\color{blue}s_1},{\\color{red}s_4}$} (d);\n\t\t%\n\t\t\\path[->] (h2) edge [bend left,color=blue] node {$s_1$} (h4) edge [color=blue] node [below left=-2mm] {$s_2$} (i2);\n\t\t\\path[->] (i2) edge [color=blue] node {$s_1$} (d);\n\t\t%\n\t\t\\path[->] (h3) edge [color=red] node [swap] {$s_4$} (h4) edge [swap,color=red] node {$s_3$} (i3);\n\t\t\\path[->] (i3) edge [color=red] node {$s_4$} (d);\n\t\\end{tikzpicture}\n\t&\n\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto, baseline=-1.5cm]\n\t\t\\node[state,initial,accepting,minimum size=0.5cm] (h1) {}; \n\t\t\\node[state,accepting,,minimum size=0.5cm] (h2) [right= 1.5cm of h1] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (h3) [right= 2.5cm of h2] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (h4) [right= 1.5cm of h3] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (i1) [below= 1cm of h1] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (i2) [below= 1cm of h2] {};\n\t\t\\node[state,minimum size=0.5cm] (d) [below= 1cm of i2] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (i3) [below= 1cm of h3] {};\n\t\t%\n\t\t\\path[->] (h1) edge [bend left,color=blue] node {$s_1$} (h3) edge [color=violet] node [swap] {$s_2$} (i1) edge [color=red] node [swap] {$s_3$} (h2);\n\t\t\\path[->] (i1) edge [color=violet] node [below left=-2mm] {${\\color{blue}s_1},{\\color{red}s_3}$} (d);\n\t\t%\n\t\t\\path[->] (h2) edge [bend left,color=blue] node {$s_1$} (h4) edge [color=blue] node {$s_2$} (i2);\n\t\t\\path[->] (i2) edge [color=blue] node {$s_1$} (d);\n\t\t%\n\t\t\\path[->] (h3) edge [color=red] node [swap] {$s_3$} (h4) edge [swap,color=red] node {$s_2$} (i3);\n\t\t\\path[->] (i3) edge [color=red] node {$s_3$} (d);\n\t\\end{tikzpicture}\n\\end{tabular}\n}\n{The automaton~$\\automatonP(\\{4\\},\\{2\\})$ for $n = 5$ and its skeleton (left), the automaton~$\\automatonP(\\{3\\},\\{2\\})$ for $n=4$ and its skeleton (middle), and the healthy states of the automaton~$\\automatonP(\\{2\\},\\{4\\})$ for $n=5$ (right).}\n{fig:automataProduct}\n\n\\subsection{The set of accepted reduced expressions of~$\\automatonP(U,D)$}\n\nApplying the principles of \\cref{subsec:principles} to each automaton~$\\automatonU(j)$ for~$j \\in U$ and~$\\automatonD(j)$ for~$j \\in D$, we derive similar principles for the automaton~$\\automatonP(U,D)$.\nThe following statements are direct consequences of \\cref{prop:prefixesReducedExpressions,prop:algorithm,prop:sameStateAcceptedReducedExpressions}.\n\n\\begin{proposition}\\label{prop:prefixesReducedExpressionsIntersection}\nThe set of reduced words accepted by~$\\automatonP(U,D)$ is closed by prefix.\n\\end{proposition}\n\n\\begin{proposition}\\label{prop:algorithmIntersection}\nIf a permutation~$\\pi$ avoids~$jki$ for~$j \\in U$ and~$kij$ for~$j \\in D$, and admits a reduced expression starting with~$s_\\ell$ such that the transition~$s_\\ell$ leads to an healthy state of~$\\automatonP(U,D)$, then it admits a reduced expression starting with~$s_\\ell$ and accepted by~$\\automatonP(U,D)$.\n\\end{proposition}\n\n\\begin{proposition}\\label{prop:sameStateAcceptedReducedExpressionsIntersection}\nGiven a permutation $\\pi \\in \\fS_n$, all the reduced expressions for~$\\pi$ accepted by $\\automatonP(U,D)$ end at the same state.\n\\end{proposition}\n\n\\subsection{Permutree sorting}\\label{subsec:permutreeSorting}\n\nWe have seen that for any~$(U,D)$-permutree minimal permutation, the set of reduced expressions for~$\\pi$ that are accepted by~$\\automatonP(U,D)$ is non-empty (by \\cref{coro:permutreeMinimal}) and closed by prefix (by \\cref{prop:prefixesReducedExpressionsIntersection}).\nTherefore, it is possible to sort~$\\pi$ passing only through $(U,D)$-permutree minimal permutations along the way.\nThis motivates the following definition.\n\n\\begin{definition}\nAn \\defn{$(U,D)$-permutree sorting algorithm} is a sorting procedure such that\n\\begin{itemize}\n\\item applied to an $(U,D)$-permutree minimal permutation~$\\pi$, it only passes through $(U,D)$-permutree minimal permutations and arrives to the identity permutation, \n\\item it fails to sort a non $(U,D)$-permutree minimal permutation~$\\pi$.\n\\end{itemize}\n\\end{definition}\n\n\\begin{example}\nThe stack sorting algorithm is a $(\\{2, \\dots, n-1\\}, \\varnothing)$-permutree sorting algorithm.\n\\end{example}\n\n\\enlargethispage{.1cm}\nSaid differently, any procedure that looks for a reduced expression accepted by~$\\automatonP(U,D)$ gives a $(U,D)$-permutree sorting algorithm.\nFor instance, \\cref{algo:shortcutsUj} is a $(\\{j\\}, \\varnothing)$-permutree sorting algorithm.\nWe generalize it in the following algorithm, where we opted for a recursive style.\nAs in \\cref{algo:shortcutsUj}, the algorithm will read the automaton~$\\automatonP(U,D)$ without actually constructing it.\nTo virtually follow the edges of the automaton~$\\automatonP(U,D)$, we use the following two operations on our sets $U$ and $D$:\n\\[\n\\moveU(U,\\ell) = \n\\begin{cases}\nU & \\text{ if } \\ell \\notin U, \\\\\n(U \\ssm \\{ \\ell \\}) \\cup \\{ \\ell + 1 \\} & \\text{ if } \\ell \\in U,\n\\end{cases}\n\\]\n\n\\[\n\\moveD(D,\\ell) = \n\\begin{cases}\nD & \\text{ if } \\ell + 1 \\notin D, \\\\\n(D \\ssm \\{ \\ell + 1 \\}) \\cup \\{ \\ell \\} & \\text{ if } \\ell + 1 \\in D.\n\\end{cases}\n\\]\n\n\\bigskip\n\\IncMargin{1em}\n\\SetKwFor{Repeat}{repeat}{}{}\n\\SetKwIF{If}{ElseIf}{Else}{if}{then}{else if}{else}{}\n\\DontPrintSemicolon\n\\begin{algorithm}[H]\n\t\\KwData{a permutation $\\pi \\in \\fS_n$ and two disjoint subsets~$U$ and~$D$ of~$[n]$}\n\t\\KwResult{a reduced word accepted by~$\\automatonP(U,D)$, candidate reduced expression for~$\\pi$}\n\t\\If{$\\exists \\; \\ell \\in [n-1]$ such that $\\ell$ and $\\ell+1$ are reversed in~$\\pi$, and $\\ell+1 \\notin U$ and~$\\ell \\notin D$}{\n\t\t\\Return $s_\\ell \\cdot \\text{permutreeSort}(s_\\ell \\cdot \\pi, \\; \\moveU(U, \\ell), \\; \\moveD(D,\\ell))$ \\;\n\t}\n\t\\If{$\\exists \\; \\ell \\in [n-1]$ such that $\\ell$ and $\\ell+1$ are reversed in~$\\pi$,\n\t\\\\ and ($\\ell + 1 \\notin U$ or $\\pi([\\ell+1]) = [\\ell +1]$) and ($\\ell \\notin D$ or $\\pi([\\ell -1]) = [\\ell - 1]$)\n\t}{\n\t\t\\Return $s_\\ell \\cdot \\text{permutreeSort}(s_\\ell \\cdot \\pi, \\; \\moveU(U \\ssm \\{ \\ell + 1 \\}, \\ell), \\; \\moveD(D \\ssm \\{ \\ell \\},\\ell))$ \\;\n\t}\n\t\\Return $\\varepsilon$ \\;\n\t\\caption{$(U,D)$-permutree sorting}\n\t\\label{algo:permutreeSorting}\n\\end{algorithm}\n\\bigskip\n\nNote that in \\cref{algo:permutreeSorting}, we could ignore~$n$ in the list~$U$ (resp.~$1$ in the list~$D$) since~$\\automatonU(n)$ (resp.~$\\automatonD(1)$) accepts all reduced words.\nWe have decided not to do it to be coherent with our recursive definition of~$\\automatonU(j)$ and~$\\automatonD(j)$.\n\n\\begin{example}\\label{exm:algo2}\n\tWe present in \\cref{tab:algo2-3-2} the $(\\{3\\},\\{2\\})$-permutree sorting algorithm in action for the permutations~$\\pi_1 \\eqdef 3214$, $\\pi_2 \\eqdef 1324$ and~$\\pi_3 \\eqdef 1342$, and in \\cref{tab:algo2-2-4} the $(\\{2\\},\\{4 \\})$-permutree sorting algorithm in action for the permutations~$\\pi_4 \\eqdef 54213$ and~$\\pi_5 \\eqdef 15342$. \n\tThe corresponding automata~$\\automatonP(\\{3\\},\\{2\\})$ and~$\\automatonP(\\{2\\},\\{4\\})$ are represented in \\cref{fig:automataProduct}.\n\tEach row in these tables contains the states of the permutation~$\\pi$ and of the word~$w$, the current values of~$j$ and~$\\ell$ in use at each step, and the values of~$k$ for which we have to check that~$\\pi([k]) = [k]$, crossed in red when it fails. These tables show that~$\\pi_1$ and~$\\pi_2$ are $(\\{3\\},\\{2\\})$-permutree sortable while $\\pi_3$ is not, and that~$\\pi_4$ is $(\\{2\\},\\{4 \\})$-permutree sortable while $\\pi_5$ is not.\n\n\\begin{table}[h!]\n\t\\centerline{\n\t\\begin{tabular}[t]{l|l|c|c|c|c}\n\t\t$\\pi_1$ & $w_1$ & $U_1$ & $D_1$ & $\\ell_1$ & $k_1$ \\\\\n\t\t\\hline\n\t\t$3214$ & $\\varepsilon$ & $\\{3\\}$ & $\\{2\\}$ & $1$ & . \\\\\n\t\t$3124$ & $s_1$ & $\\{3\\}$ & $\\{1\\}$ & $2$ & $3$ \\\\\n\t\t$2134$ & $s_1 \\cdot s_2$ & $\\varnothing$ & $\\{1\\}$ & $1$ & $0$ \\\\\n\t\t$1234$ & $s_1 \\cdot s_2 \\cdot s_1$ & & &\n\t\\end{tabular}\n\t\\quad\n\t\\begin{tabular}[t]{l|l|c|c|c|c}\n\t\t$\\pi_2$ & $w_2$ & $U_2$ & $D_2$ & $\\ell_2$ & $k_2$ \\\\\n\t\t\\hline\n\t\t$1324$ & $\\varepsilon$ & $\\{3\\}$ & $\\{2\\}$ & $2$ & $1$, $3$ \\\\\n\t\t$1234$ & $s_2$ & & &\n\t\\end{tabular}\n\t\\quad\n\t\\begin{tabular}[t]{l|l|c|c|c|c} \n\t\t$\\pi_3$ & $w_3$ & $U_3$ & $D_3$ & $\\ell_3$ & $k_3$ \\\\\n\t\t\\hline\n\t\t$1342$ & $\\varepsilon$ & $\\{3\\}$ & $\\{2\\}$ & $2$ & $1$, \\textcolor{red}{$\\xcancel{\\textcolor{black}{3}}$}\n\t\\end{tabular}\n\t}\n\t\\caption{The~$(\\{3\\},\\{2 \\})$-permutree sorting of~$\\pi_1 \\eqdef 3214$, $\\pi_2 \\eqdef 1324$ and~$\\pi_3 \\eqdef 1342$.}\n\t\\label{tab:algo2-3-2}\n\\end{table}\n\n\\begin{table}[h!]\n\t\\centerline{\n\t\\begin{tabular}[t]{l|l|c|c|c|c}\n\t\t$\\pi_4$ & $w_4$ & $U_4$ & $D_4$ & $\\ell_4$ & $k_4$ \\\\\n\t\t\\hline\n\t\t$54213$ & $\\varepsilon$ & $\\{2\\}$ & $\\{4\\}$ & $3$ & . \\\\\n\t\t$53214$ & $s_3$ & $\\{2\\}$ & $\\{3\\}$ & $2$ & . \\\\\n\t\t$52314$ & $s_3 \\cdot s_2$ & $\\{3\\}$ & $\\{2\\}$ & $1$ & . \\\\\n\t\t$51324$ & $s_3 \\cdot s_2 \\cdot s_1$ & $\\{3\\}$ & $\\{1\\}$ & $4$ & . \\\\\n\t\t$41325$ & $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_4$ & $\\{3\\}$ & $\\{1\\}$ & $3$ & . \\\\\n\t\t$31425$ & $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_4 \\cdot s_3$ & $\\{4\\}$ & $\\{1\\}$ & $2$ & . \\\\\n\t\t$21435$ & $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_4 \\cdot s_3 \\cdot s_2$ & $\\{4\\}$ & $\\{1\\}$ & $1$ & . \\\\\n\t\t$12435$ & $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_4 \\cdot s_3 \\cdot s_2 \\cdot s_1$ & $\\{4\\}$ & $\\{1\\}$ & $3$ & $4$ \\\\\n\t\t$12345$ & $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_4 \\cdot s_3 \\cdot s_2 \\cdot s_1 \\cdot s_3$ & $\\{4\\}$ & $\\{1\\}$ &\n\t\\end{tabular}\n\t\\qquad\n\t\\begin{tabular}[t]{l|l|c|c|c|c}\n\t\t$\\pi_5$ & $w_5$ & $U_5$ & $D_5$ & $\\ell_5$ & $k_5$ \\\\\n\t\t\\hline\n\t\t$15342$ & $\\varepsilon$ & $\\{2\\}$ & $\\{4\\}$ & $2$ & . \\\\\n\t\t$15243$ & $s_2$ & $\\{3\\}$ & $\\{4\\}$ & $3$ & . \\\\\n\t\t$15234$ & $s_2 \\cdot s_3$ & $\\{4\\}$ & $\\{3\\}$ & $4$ & . \\\\\n\t\t$14235$ & $s_2 \\cdot s_3 \\cdot s_5$ & $\\{5\\}$ & $\\{3\\}$ & $3$ & \\textcolor{red}{$\\xcancel{\\textcolor{black}{2}}$}\n\t\\end{tabular}\n\t}\n\t\\caption{The~$(\\{2\\},\\{4 \\})$-permutree sorting of~$\\pi_4 \\eqdef 54213$ and~$\\pi_5 \\eqdef 15342$.}\n\t\\label{tab:algo2-2-4}\n\\end{table}\n\\end{example}\n\n\\begin{corollary}\\label{coro:algorithmIntersection}\nFor any permutation~$\\pi$ and any disjoint subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$, \\cref{algo:permutreeSorting} returns a reduced word~$w$ accepted by $\\automatonP(U,D)$ with the property that~$w$ is a reduced expression for~$\\pi$ if and only if~$\\pi$ avoids~$jki$ for~$j \\in U$ and~$kij$ for~$j \\in D$.\n\\end{corollary}\n\n\\begin{proof}\nThis algorithm constructs a candidate reduced word for $\\pi$ following the automaton~$\\automatonP(U,D)$ and prioritizing healthy states over ill states. \nIt begins by checking all possible transitions~$s_\\ell$ that keep $\\automatonP(U,D)$ in healthy states following \\cref{lem:vincent4} (if condition in line~1).\nDoing this in the intersection of automata translates to updating~$\\ell$ to~$\\ell+1$ when~$\\ell \\in U$ and $\\ell+1$ to~$\\ell$ when~$\\ell+1 \\in D$ (line~2).\nWhen we have exhausted all these transitions, we need to go to an ill state of~$\\automatonP(U,D)$, \\ie to apply a transposition that sends at least one automaton of the intersection to an ill state.\nIf there is~$\\ell+1 \\in U$ (resp.~$\\ell \\in D$) such that~$s_\\ell$ is a descent of~$\\pi$ and~$\\pi([\\ell+1]) = [\\ell+1]$ (resp.~$\\pi([\\ell-1]) = [\\ell-1]$), then any reduced expression for~$\\pi$ is accepted by the automaton~$\\automatonU(\\ell)$ (resp.~$\\automatonD(\\ell)$) by \\cref{prop:sameStateAcceptedReducedExpressionsRefined}\\,(i). \nWe can thus start with~$s_\\ell$ and forget about the automaton~$\\automatonU(\\ell)$ (resp.~$\\automatonD(\\ell)$) (lines 3, 4 and 5).\nFinally, if none of these options are possible, any reduced expression for~$\\pi$ will lead to a dead state in at least one of the automata, so that $\\pi$ is not $(U,D)$-sortable.\nWe thus return the empty reduced expression (line 6).\n\\end{proof}\n\n\\subsection{Generating trees}\n\nAs in \\cref{subsec:trees}, we can define natural generating trees for the $(U,D)$-permutree minimal permutations.\nNamely, fix an arbitrary priority order~$\\prec$ on~$\\{s_1, \\dots, s_{n-1}\\}$.\nFor an $(U,D)$-permutree minimal permutation~$\\pi$, we denote by~$\\pi(U, D, \\prec)$ the $\\prec$-lexicographic minimal reduced expression for~$\\pi$ that is accepted by~$\\automatonP(U,D)$.\nWe denote by~$\\lexmin(n, U, D, \\prec)$ the set of reduced words of the form~$\\pi(U, D, \\prec)$ for all $(U,D)$-permutree minimal permutations~$\\pi \\in \\fS_n$.\nThe same proof as that of \\cref{prop:generatingTrees} shows that $\\lexmin(n, U, D, \\prec)$ is closed by prefix.\nThis yields a natural generating tree on~$\\lexmin(n, U, D, \\prec)$ where the parent of a reduced word~$w$ is obtained by deleting its last letter.\nReplacing each reduced expression by the corresponding permutation, this provides a generating tree for the $(U,D)$-permutree minimal permutations of~$\\fS_n$.\n\\cref{fig:TreeCompleteOrientations} presents these generating trees for different values of~$U$ and~$D$.\n\n\\hvFloat[floatPos=p, capWidth=h, capPos=r, capAngle=90, objectAngle=90, capVPos=c, objectPos=c]{figure}\n{\n\t\\begin{tabular}{cccc}\n\t\t$\\automatonU(2) \\cap \\automatonU(3)$\n\t\t&\n\t\t$\\automatonU(2) \\cap \\automatonD(3)$\n\t\t&\n\t\t$\\automatonU(3) \\cap \\automatonD(2)$\n\t\t&\n\t\t$\\automatonD(2) \\cap \\automatonD(3)$\n\t\t\\\\[.5cm]\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj+1) [below= 1.5cm of ij+1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+2) [right= 1.5cm of hj+1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+2) [below= 1.5cm of hj+2] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_1$} (ij)\n\t\t\t\t\t   edge node {$s_2$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_2$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_2$} (ij+1)\n\t\t\t\t\t edge node {$s_3$} (hj+2)\n\t\t\t\t(ij+1) edge node [swap] {$s_3$} (dj+1)\n\t\t\t\t%\n\t\t\t\t(hj+2) edge node [swap] {$s_3$} (ij+2);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj+1) [below= 1.5cm of ij+1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+2) [right= 1.5cm of hj+1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+2) [below= 1.5cm of hj+2] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_1$} (ij)\n\t\t\t\t\t   edge node {$s_2$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_2$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_2$} (ij+1)\n\t\t\t\t\t edge node {$s_3$} (hj+2)\n\t\t\t\t(ij+1) edge node [swap] {$s_3$} (dj+1)\n\t\t\t\t%\n\t\t\t\t(hj+2) edge node [swap] {$s_3$} (ij+2);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_2$} (ij)\n\t\t\t\t\t   edge node {$s_3$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_3$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_3$} (ij+1);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_2$} (ij)\n\t\t\t\t\t   edge node {$s_1$} (hj-1)\n\t\t\t\t(ij) edge node [swap] {$s_1$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj-1) edge node [swap] {$s_1$} (ij-1);\n\t\t\\end{tikzpicture}\n\t\t\\\\[.5cm]\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_2$} (ij)\n\t\t\t\t\t   edge node {$s_3$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_3$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_3$} (ij+1);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj-1) [below= 1.5cm of ij-1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-2) [right= 1.5cm of hj-1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-2) [below= 1.5cm of hj-2] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_3$} (ij)\n\t\t\t\t\t   edge node {$s_2$} (hj-1)\n\t\t\t\t(ij) edge node [swap] {$s_2$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj-1) edge node [swap] {$s_2$} (ij-1)\n\t\t\t\t\t edge node {$s_1$} (hj-2)\n\t\t\t\t(ij-1) edge node [swap] {$s_1$} (dj-1)\n\t\t\t\t%\n\t\t\t\t(hj-2) edge node [swap] {$s_1$} (ij-2);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_2$} (ij)\n\t\t\t\t\t   edge node {$s_1$} (hj-1)\n\t\t\t\t(ij) edge node [swap] {$s_1$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj-1) edge node [swap] {$s_1$} (ij-1);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj-1) [below= 1.5cm of ij-1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-2) [right= 1.5cm of hj-1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-2) [below= 1.5cm of hj-2] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_3$} (ij)\n\t\t\t\t\t   edge node {$s_2$} (hj-1)\n\t\t\t\t(ij) edge node [swap] {$s_2$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj-1) edge node [swap] {$s_2$} (ij-1)\n\t\t\t\t\t edge node {$s_1$} (hj-2)\n\t\t\t\t(ij-1) edge node [swap] {$s_1$} (dj-1)\n\t\t\t\t%\n\t\t\t\t(hj-2) edge node [swap] {$s_1$} (ij-2);\n\t\t\\end{tikzpicture}\n\t\t\\\\[.5cm]\n\t\t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n\t\t\t\\node[blue](P1234) at (0,0){1234};\n\t\t\t%\n\t\t\t\\node[blue](P2134) at (-1,1.5){2134};\n\t\t\t\\node[blue](P1324) at ( 0,1.5){1324};\n\t\t\t\\node[blue](P1243) at ( 1,1.5){1243};\n\t\t\t%\n\t\t\t\\node(P2314) at (-2,3){2314};\n\t\t\t\\node[blue](P3124) at (-1,3){3124};\n\t\t\t\\node[blue](P2143) at ( 0,3){2143};\n\t\t\t\\node(P1342) at ( 1,3){1342};\n\t\t\t\\node[blue](P1423) at ( 2,3){1423};\n\t\t\t%\n\t\t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n\t\t\t\\node(P2341) at (-1.5,4.5){2341};\n\t\t\t\\node(P3142) at (-0.5,4.5){3142};\n\t\t\t\\node(P2413) at ( 0.5,4.5){2413};\n\t\t\t\\node[blue](P4123) at ( 1.5,4.5){4123};\n\t\t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n\t\t\t%\n\t\t\t\\node(P3241) at (-2,6){3241};\n\t\t\t\\node(P2431) at (-1,6){2431};\n\t\t\t\\node(P3412) at ( 0,6){3412};\n\t\t\t\\node[blue](P4213) at ( 1,6){4213};\n\t\t\t\\node[blue](P4132) at ( 2,6){4132};\n\t\t\t%\n\t\t\t\\node(P3421) at (-1,7.5){3421};\n\t\t\t\\node(P4231) at ( 0,7.5){4231};\n\t\t\t\\node[blue](P4312) at ( 1,7.5){4312};\n\t\t\t%\n\t\t\t\\node[blue](P4321) at (0,9){4321};\n\t\t\t%\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n\t\t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n\t\t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n\t\t\t%\n\t\t\t\\draw(P2134) -- (P2314);\n\t\t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,blue](P1324) -- (P3124);\n\t\t\t\\draw(P1324) -- (P1342);\n\t\t\t\\draw(P1243) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,red](P1243) -- (P1423);\n\t\t\t%\n\t\t\t\\draw(P2314) -- (P3214);\n\t\t\t\\draw(P2314) -- (P2341);\n\t\t\t\\draw[line width=0.5mm,red](P3124) -- (P3214);\n\t\t\t\\draw(P3124) -- (P3142);\n\t\t\t\\draw(P2143) -- (P2413);\n\t\t\t\\draw(P1342) -- (P3142);\n\t\t\t\\draw(P1342) -- (P1432);\n\t\t\t\\draw[line width=0.5mm,blue](P1423) -- (P4123);\n\t\t\t\\draw[line width=0.5mm,green](P1423) -- (P1432);\n\t\t\t%\n\t\t\t\\draw(P3214) -- (P3241);\n\t\t\t\\draw(P2341) -- (P3241);\n\t\t\t\\draw(P2341) -- (P2431);\n\t\t\t\\draw(P3142) -- (P3412);\n\t\t\t\\draw(P2413) -- (P4213);\n\t\t\t\\draw(P2413) -- (P2431);\n\t\t\t\\draw[line width=0.5mm,red](P4123) -- (P4213);\n\t\t\t\\draw[line width=0.5mm,green](P4123) -- (P4132);\n\t\t\t\\draw(P1432) -- (P4132);\n\t\t\t%\n\t\t\t\\draw(P3241) -- (P3421);\n\t\t\t\\draw(P2431) -- (P4231);\n\t\t\t\\draw(P3412) -- (P4312);\n\t\t\t\\draw(P3412) -- (P3421);\n\t\t\t\\draw(P4213) -- (P4231);\n\t\t\t\\draw[line width=0.5mm,red](P4132) -- (P4312);\n\t\t\t%\n\t\t\t\\draw(P3421) -- (P4321);\n\t\t\t\\draw(P4231) -- (P4321);\n\t\t\t\\draw[line width=0.5mm,green](P4312) -- (P4321);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n\t\t\t\\node[blue](P1234) at (0,0){1234};\n\t\t\t%\n\t\t\t\\node[blue](P2134) at (-1,1.5){2134};\n\t\t\t\\node[blue](P1324) at ( 0,1.5){1324};\n\t\t\t\\node[blue](P1243) at ( 1,1.5){1243};\n\t\t\t%\n\t\t\t\\node(P2314) at (-2,3){2314};\n\t\t\t\\node[blue](P3124) at (-1,3){3124};\n\t\t\t\\node[blue](P2143) at ( 0,3){2143};\n\t\t\t\\node[blue](P1342) at ( 1,3){1342};\n\t\t\t\\node(P1423) at ( 2,3){1423};\n\t\t\t%\n\t\t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n\t\t\t\\node(P2341) at (-1.5,4.5){2341};\n\t\t\t\\node[blue](P3142) at (-0.5,4.5){3142};\n\t\t\t\\node(P2413) at ( 0.5,4.5){2413};\n\t\t\t\\node(P4123) at ( 1.5,4.5){4123};\n\t\t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n\t\t\t%\n\t\t\t\\node(P3241) at (-2,6){3241};\n\t\t\t\\node(P2431) at (-1,6){2431};\n\t\t\t\\node[blue](P3412) at ( 0,6){3412};\n\t\t\t\\node(P4213) at ( 1,6){4213};\n\t\t\t\\node(P4132) at ( 2,6){4132};\n\t\t\t%\n\t\t\t\\node[blue](P3421) at (-1,7.5){3421};\n\t\t\t\\node(P4231) at ( 0,7.5){4231};\n\t\t\t\\node[blue](P4312) at ( 1,7.5){4312};\n\t\t\t%\n\t\t\t\\node[blue](P4321) at (0,9){4321};\n\t\t\t%\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n\t\t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n\t\t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n\t\t\t%\n\t\t\t\\draw(P2134) -- (P2314);\n\t\t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,blue](P1324) -- (P3124);\n\t\t\t\\draw[line width=0.5mm,green](P1324) -- (P1342);\n\t\t\t\\draw(P1243) -- (P2143);\n\t\t\t\\draw(P1243) -- (P1423);\n\t\t\t%\n\t\t\t\\draw(P2314) -- (P3214);\n\t\t\t\\draw(P2314) -- (P2341);\n\t\t\t\\draw[line width=0.5mm,red](P3124) -- (P3214);\n\t\t\t\\draw[line width=0.5mm,green](P3124) -- (P3142);\n\t\t\t\\draw(P2143) -- (P2413);\n\t\t\t\\draw(P1342) -- (P3142);\n\t\t\t\\draw[line width=0.5mm,red](P1342) -- (P1432);\n\t\t\t\\draw(P1423) -- (P4123);\n\t\t\t\\draw(P1423) -- (P1432);\n\t\t\t%\n\t\t\t\\draw(P3214) -- (P3241);\n\t\t\t\\draw(P2341) -- (P3241);\n\t\t\t\\draw(P2341) -- (P2431);\n\t\t\t\\draw[line width=0.5mm,red](P3142) -- (P3412);\n\t\t\t\\draw(P2413) -- (P4213);\n\t\t\t\\draw(P2413) -- (P2431);\n\t\t\t\\draw(P4123) -- (P4213);\n\t\t\t\\draw(P4123) -- (P4132);\n\t\t\t\\draw(P1432) -- (P4132);\n\t\t\t%\n\t\t\t\\draw(P3241) -- (P3421);\n\t\t\t\\draw(P2431) -- (P4231);\n\t\t\t\\draw[line width=0.5mm,blue](P3412) -- (P4312);\n\t\t\t\\draw[line width=0.5mm,green](P3412) -- (P3421);\n\t\t\t\\draw(P4213) -- (P4231);\n\t\t\t\\draw(P4132) -- (P4312);\n\t\t\t%\n\t\t\t\\draw(P3421) -- (P4321);\n\t\t\t\\draw(P4231) -- (P4321);\n\t\t\t\\draw[line width=0.5mm,green](P4312) -- (P4321);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n\t\t\t\\node[blue](P1234) at (0,0){1234};\n\t\t\t%\n\t\t\t\\node[blue](P2134) at (-1,1.5){2134};\n\t\t\t\\node[blue](P1324) at ( 0,1.5){1324};\n\t\t\t\\node[blue](P1243) at ( 1,1.5){1243};\n\t\t\t%\n\t\t\t\\node[blue](P2314) at (-2,3){2314};\n\t\t\t\\node(P3124) at (-1,3){3124};\n\t\t\t\\node[blue](P2143) at ( 0,3){2143};\n\t\t\t\\node(P1342) at ( 1,3){1342};\n\t\t\t\\node[blue](P1423) at ( 2,3){1423};\n\t\t\t%\n\t\t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n\t\t\t\\node(P2341) at (-1.5,4.5){2341};\n\t\t\t\\node(P3142) at (-0.5,4.5){3142};\n\t\t\t\\node[blue](P2413) at ( 0.5,4.5){2413};\n\t\t\t\\node(P4123) at ( 1.5,4.5){4123};\n\t\t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n\t\t\t%\n\t\t\t\\node(P3241) at (-2,6){3241};\n\t\t\t\\node[blue](P2431) at (-1,6){2431};\n\t\t\t\\node(P3412) at ( 0,6){3412};\n\t\t\t\\node[blue](P4213) at ( 1,6){4213};\n\t\t\t\\node(P4132) at ( 2,6){4132};\n\t\t\t%\n\t\t\t\\node(P3421) at (-1,7.5){3421};\n\t\t\t\\node[blue](P4231) at ( 0,7.5){4231};\n\t\t\t\\node(P4312) at ( 1,7.5){4312};\n\t\t\t%\n\t\t\t\\node[blue](P4321) at (0,9){4321};\n\t\t\t%\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n\t\t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n\t\t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,red](P2134) -- (P2314);\n\t\t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n\t\t\t\\draw(P1324) -- (P3124);\n\t\t\t\\draw(P1324) -- (P1342);\n\t\t\t\\draw(P1243) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,red](P1243) -- (P1423);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P2314) -- (P3214);\n\t\t\t\\draw(P2314) -- (P2341);\n\t\t\t\\draw(P3124) -- (P3214);\n\t\t\t\\draw(P3124) -- (P3142);\n\t\t\t\\draw[line width=0.5mm,red](P2143) -- (P2413);\n\t\t\t\\draw(P1342) -- (P3142);\n\t\t\t\\draw(P1342) -- (P1432);\n\t\t\t\\draw(P1423) -- (P4123);\n\t\t\t\\draw[line width=0.5mm,green](P1423) -- (P1432);\n\t\t\t%\n\t\t\t\\draw(P3214) -- (P3241);\n\t\t\t\\draw(P2341) -- (P3241);\n\t\t\t\\draw(P2341) -- (P2431);\n\t\t\t\\draw(P3142) -- (P3412);\n\t\t\t\\draw[line width=0.5mm,blue](P2413) -- (P4213);\n\t\t\t\\draw[line width=0.5mm,green](P2413) -- (P2431);\n\t\t\t\\draw(P4123) -- (P4213);\n\t\t\t\\draw(P4123) -- (P4132);\n\t\t\t\\draw(P1432) -- (P4132);\n\t\t\t%\n\t\t\t\\draw(P3241) -- (P3421);\n\t\t\t\\draw(P2431) -- (P4231);\n\t\t\t\\draw(P3412) -- (P4312);\n\t\t\t\\draw(P3412) -- (P3421);\n\t\t\t\\draw[line width=0.5mm,green](P4213) -- (P4231);\n\t\t\t\\draw(P4132) -- (P4312);\n\t\t\t%\n\t\t\t\\draw(P3421) -- (P4321);\n\t\t\t\\draw[line width=0.5mm,red](P4231) -- (P4321);\n\t\t\t\\draw(P4312) -- (P4321);\n\t\t\\end{tikzpicture}\n\t\t&\n    \t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n\t\t\t\\node[blue](P1234) at (0,0){1234};\n\t\t\t%\n\t\t\t\\node[blue](P2134) at (-1,1.5){2134};\n\t\t\t\\node[blue](P1324) at ( 0,1.5){1324};\n\t\t\t\\node[blue](P1243) at ( 1,1.5){1243};\n\t\t\t%\n\t\t\t\\node[blue](P2314) at (-2,3){2314};\n\t\t\t\\node(P3124) at (-1,3){3124};\n\t\t\t\\node[blue](P2143) at ( 0,3){2143};\n\t\t\t\\node[blue](P1342) at ( 1,3){1342};\n\t\t\t\\node(P1423) at ( 2,3){1423};\n\t\t\t%\n\t\t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n\t\t\t\\node[blue](P2341) at (-1.5,4.5){2341};\n\t\t\t\\node(P3142) at (-0.5,4.5){3142};\n\t\t\t\\node(P2413) at ( 0.5,4.5){2413};\n\t\t\t\\node(P4123) at ( 1.5,4.5){4123};\n\t\t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n\t\t\t%\n\t\t\t\\node[blue](P3241) at (-2,6){3241};\n\t\t\t\\node[blue](P2431) at (-1,6){2431};\n\t\t\t\\node(P3412) at ( 0,6){3412};\n\t\t\t\\node(P4213) at ( 1,6){4213};\n\t\t\t\\node(P4132) at ( 2,6){4132};\n\t\t\t%\n\t\t\t\\node[blue](P3421) at (-1,7.5){3421};\n\t\t\t\\node(P4231) at ( 0,7.5){4231};\n\t\t\t\\node(P4312) at ( 1,7.5){4312};\n\t\t\t%\n\t\t\t\\node[blue](P4321) at (0,9){4321};\n\t\t\t%\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n\t\t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n\t\t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,red](P2134) -- (P2314);\n\t\t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n\t\t\t\\draw(P1324) -- (P3124);\n\t\t\t\\draw[line width=0.5mm,green](P1324) -- (P1342);\n\t\t\t\\draw(P1243) -- (P2143);\n\t\t\t\\draw(P1243) -- (P1423);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P2314) -- (P3214);\n\t\t\t\\draw[line width=0.5mm,green](P2314) -- (P2341);\n\t\t\t\\draw(P3124) -- (P3214);\n\t\t\t\\draw(P3124) -- (P3142);\n\t\t\t\\draw(P2143) -- (P2413);\n\t\t\t\\draw(P1342) -- (P3142);\n\t\t\t\\draw[line width=0.5mm,red](P1342) -- (P1432);\n\t\t\t\\draw(P1423) -- (P4123);\n\t\t\t\\draw(P1423) -- (P1432);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,green](P3214) -- (P3241);\n\t\t\t\\draw(P2341) -- (P3241);\n\t\t\t\\draw[line width=0.5mm,red](P2341) -- (P2431);\n\t\t\t\\draw(P3142) -- (P3412);\n\t\t\t\\draw(P2413) -- (P4213);\n\t\t\t\\draw(P2413) -- (P2431);\n\t\t\t\\draw(P4123) -- (P4213);\n\t\t\t\\draw(P4123) -- (P4132);\n\t\t\t\\draw(P1432) -- (P4132);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,red](P3241) -- (P3421);\n\t\t\t\\draw(P2431) -- (P4231);\n\t\t\t\\draw(P3412) -- (P4312);\n\t\t\t\\draw(P3412) -- (P3421);\n\t\t\t\\draw(P4213) -- (P4231);\n\t\t\t\\draw(P4132) -- (P4312);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P3421) -- (P4321);\n\t\t\t\\draw(P4231) -- (P4321);\n\t\t\t\\draw(P4312) -- (P4321);\n    \t\\end{tikzpicture}\n\t\\end{tabular}\n}\n{Generating trees for the $(U,D)$-permutree minimal permutations of~$\\fS_4$, with priority order $s_1 \\prec s_2 \\prec s_3$.}\n{fig:TreeCompleteOrientations}\n\n\n\\section{Permutree sorting versus Coxeter sorting}\\label{sec:coxeterSortable}\n\nIn this section, we discuss the particular case when~$U$ and~$D$ form a partition of~$\\{2, \\dots, n-1\\}$.\nIn that situation, we connect the $(U,D)$-permutree sorting with the $c$-sorting of N.~Reading~\\cite{Reading-sortableElements}.\n\n\\subsection{Coxeter sorting word and Coxeter sortable permutations}\\label{subsec:csorting}\n\nWe first recall the theory of $c$-sorting developed by N.~Reading in~\\cite{Reading-sortableElements}.\nWhile it was defined in arbitrary finite Coxeter groups, we focus on the symmetric group in this presentation.\n\nWe consider a \\defn{Coxeter element}~$c$ of~$\\fS_n$, \\ie the product of all simple transpositions $\\{s_1, \\dots, s_{n-1}\\}$ in an arbitrary order.\nFor a permutation $\\pi \\in \\fS_n$, the \\defn{$c$-sorting word} $\\pi(c)$ is the lexicographically smallest reduced expression for $\\pi$ in the infinite word $c^\\infty = c \\cdot c \\cdot c \\cdot c \\cdots$.\nNote that strictly speaking, $\\pi(c)$ depends on a reduced expression for~$c$, not only on the Coxeter element~$c$.\nHere, we assume that we have chosen a reduced expression and hide this dependence.\nWe let~$I_1, \\dots, I_p$ denote the subsets of~$[n-1]$ such that~$\\pi(c) = c_{I_1} \\cdot c_{I_2} \\cdots c_{I_p}$ where~$c_I$ is the subword of~$c$ obtained by keeping only the letters~$s_i$ for~$i \\in I$.\nThe permutation $\\pi$ is \\defn{$c$-sortable} if~$I_1 \\supseteq I_2 \\supseteq \\dots \\supseteq I_p$.\nNote that this does not depend on the choice of the reduced expression~$c$, only on the Coxeter element~$c$.\n\nFor our proofs we will need some simple yet useful facts from \\cite{Reading-sortableElements} on how prefixes of words influence sortability.\n\n\\begin{lemma}\\label{lem:coxeterElementFacts}\nConsider a Coxeter element of the form~$c = s_\\ell \\cdot d$ and let $\\pi \\in \\fS_n$. Then\n\\begin{itemize}\n\t\\item if $\\pi = s_\\ell \\cdot \\tau$ with $\\ell(\\pi) = \\ell(\\tau) + 1$, then $\\pi(c) = s_\\ell \\cdot \\tau(d \\cdot s_\\ell)$,\n\t\\item otherwise, $\\pi(c) = \\pi(d \\cdot s_\\ell)$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:vincent8}\nLet~$s_i \\ne s_j$ be two letters that appear in the $c$-sorting word $\\pi(c)$ of a $c$-sortable permutation~$\\pi$. Then\n\\begin{enumerate} \n\t\\item if $s_i$ appears before $s_j$ in $c$, then $s_i$ appears before $s_j$ in $\\pi(c)$,\n\t\\item if $s_j$ does not appear in between two occurrences of $s_i$ in $\\pi(c)$, then it does not appear after these occurrences either.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe deal with the two statements separately:\n\t\\begin{enumerate} \n\t\t\\item Immediate from the definition since both $s_i$ and $s_j$ appear in $\\pi(c)$.\n\t\t\\item Since $\\pi$ is $c$-sortable, $\\pi(c)$ is formed by a succession of subwords that are nested. Thus if $s_j$ were to appear after the occurrences of $s_i$ it would have to appear between them as well.\n\t\t\\qedhere\n\t\\end{enumerate}\n\\end{proof}\n\n\\subsection{Coxeter sorting via permutree automata}\n\nA Coxeter element~$c$ of~$\\fS_n$ defines a partition ${\\{2, \\dots, n-1\\} = U_c \\sqcup D_c}$, where~$U_c$ (resp.~$D_c$) consists of the elements~$j \\in \\{2, \\dots, n-1\\}$ such that~$s_j$ appears before (resp.~after) $s_{j-1}$ in~$c$.\nFor instance, when~$c = s_2 \\cdot s_5 \\cdot s_4 \\cdot s_3 \\cdot s_1 \\cdot s_6$, we obtain~$U_c = \\{2,4,5\\}$ and~$D_c = \\{3,6\\}$.\nSaid differently, $j \\in U$ (resp.~$j \\in D$) if~$c$ is accepted by~$\\automatonU(j)$ but not by~$\\automatonD(j)$ (resp.~by~$\\automatonD(j)$ but not by~$\\automatonU(j)$).\nThe goal of this section is the following connection between the $c$-sorting of \\cref{subsec:csorting} and the $(U_c, D_c)$-permutree sorting of \\cref{sec:intersectionsAutomata}.\n\n\\pagebreak\n\\begin{theorem}\n\\label{thm:csorting}\nFor any Coxeter element~$c$ and any permutation~$\\pi$, the following assertions are equivalent:\n\\begin{enumerate}[(i)]\n\t\\item $\\pi$ is $c$-sortable,\n\t\\item the $c$-sorting word~$\\pi(c)$ is accepted by the automaton~$\\automatonP(U_c, D_c)$,\n\t\\item there exists a reduced expression for~$\\pi$ accepted by the automaton~$\\automatonP(U_c, D_c)$,\n\t\\item for each~$j \\in \\{2, \\dots, n-1\\}$, there exists a reduced expression for $\\pi$ that is accepted by the automaton~$\\automatonU(j)$ if~$j \\in U_c$ and~$\\automatonD(j)$ if~$j \\in D_c$,\n\t\\item $\\pi$ avoids $jki$ for~$j \\in U_c$ and $kij$ for~$j \\in D_c$.\n\\end{enumerate}\n\\end{theorem}\n\nThe equivalences (iii) $\\iff$ (iv) $\\iff$ (v) were already established earlier.\nHere, we aim at identifying the $c$-sorting word as a reduced expression for~$\\pi$ accepted by~$\\automatonP(U_c, D_c)$.\nWe split the proof of the equivalences (i) $\\iff$ (ii) $\\iff$ (iii) into the following few lemmas.\n\n\\begin{lemma}\\label{lem:vincent9}\nThe $c$-sorting word of a $c$-sortable permutation is recognized by~$\\automatonU(j)$ for~$j \\in U_c$ and by~$\\automatonD(j)$ for~$j \\in D_c$.\n\\end{lemma}\n\n\\begin{proof}\nConsider~$j \\in U_c$ (the proof for~$j \\in D_c$ is symmetric).\nWe distinguish two possible cases:\n\\begin{itemize}\n\t\\item If $\\pi(c)$ contains no $s_{j-1}$, then $\\pi(c)$ either remains in the first healthy state or ends in the first ill state of~$\\automatonU(j)$.\n\t\\item If $\\pi(c)$ contains $s_{j-1}$, then by \\cref{lem:vincent8}\\,(1) $s_{j-1}$ appears before $s_j$ in $\\pi(c)$ and $\\pi(c)$ leads to the second healthy state of~$\\automatonU(j)$. From here on out, notice that $\\pi(c)$ cannot end at a dead state because of \\cref{lem:vincent8}\\,(2). \n\\end{itemize}\nIn both cases, $\\pi(c)$ is accepted by~$\\automatonU(j)$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:Vincent10}\nA permutation $\\pi \\in \\fS_n$ whose $c$-sorting word $\\pi(c)$ is accepted by~$\\automatonP(U_c, D_c)$ is $c$-sortable.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $\\pi$ is not $c$-sortable. We will find an automaton that rejects $\\pi(c)$ among the automata~$\\automatonU(j)$ for~$j \\in U_c$ and~$\\automatonD(j)$ for~$j \\in D_c$.\nOnce again, we work by induction on the length of $\\pi$ and the size of~$c$.\nLet~$s_\\ell$ be the first letter of~$c$ and write~$c = s_\\ell \\cdot d$.\nSince~$s_\\ell$ appears before~$s_{\\ell-1}$ and~$s_{\\ell+1}$ in~$c$, we have~$\\ell \\in U_c$ and~$\\ell+1 \\in D_c$.\nMoreover, the letter~$s_\\ell$ yields to the next healthy state in both automata~$\\automatonU(\\ell)$ and~$\\automatonD(\\ell+1)$, and remains in the initial state for all other automata~$\\automatonU(j)$ for~$j \\in U_c \\ssm \\{\\ell\\}$ and~$\\automatonD(j)$ for~$j \\in D_c \\ssm \\{\\ell+1\\}$.\nWe now distinguish two cases, depending on whether~$\\ell$ and~$\\ell+1$ are reversed in~$\\pi$.\n\nAssume first that $\\ell$ and $\\ell+1$ are reversed in $\\pi$ and write $\\pi = s_\\ell \\cdot \\tau$.\nWe then have $\\pi(c) = s_\\ell \\cdot \\tau(d \\cdot s_\\ell)$ by \\cref{lem:coxeterElementFacts}, so that~$\\tau$ is not $d \\cdot s_\\ell$-sortable.\nBy induction hypothesis, $\\tau(d \\cdot s_\\ell)$ is rejected by one of the automata~$\\automatonU(j)$ for~$j \\in U_{d \\cdot s_\\ell}$ and~$\\automatonD(j)$ for~$j \\in D_{d \\cdot s_\\ell}$.\nSince~$U_{d \\cdot s_\\ell} = U_c \\symdif \\{\\ell, \\ell+1\\}$ and~$D_{d \\cdot s_\\ell} = D_c \\symdif \\{\\ell, \\ell+1\\}$, \\cref{lem:vincent2,lem:vincent4} ensure that $\\pi(c) = s_\\ell \\cdot \\tau(d \\cdot s_\\ell)$ is rejected by one of the automata~$\\automatonU(j)$ for~$j \\in U_c$ and~$\\automatonD(j)$ for~$j \\in D_c$.\n\nAssume now that $\\ell$ and $\\ell+1$ are not reversed in $\\pi$.\nThen~$\\pi(c)$ does not use~$s_\\ell$ and~$\\pi$ is not $d$-sortable in~$W_{\\langle s_\\ell \\rangle}$.\nBy induction hypothesis, $\\pi(c)$ is rejected by one of the automata~$\\automatonU(j)$ for~$j \\in U_d$ and~$\\automatonD(j)$ for~$j \\in D_d$.\nThis concludes the proof since~$U_d \\subseteq U_c$ and~$D_d \\subseteq D_c$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:Vincent11}\nIf a permutation~$\\pi \\in \\fS_n$ admits a reduced expression accepted by~$\\automatonP(U_c, D_c)$, then its $c$-sorting word~$\\pi(c)$ is accepted by~$\\automatonP(U_c, D_c)$.\n\\end{lemma}\n\n\\begin{proof}\nOnce agin, we work by induction on the length of $\\pi$.\nLet~$s_\\ell$ be the first letter of~$c$ and write~$c = s_\\ell \\cdot d$.\nSince~$s_\\ell$ appears before~$s_{\\ell-1}$ and~$s_{\\ell+1}$ in~$c$, we have~$\\ell \\in U_c$ and~$\\ell+1 \\in D_c$.\nMoreover, the letter~$s_\\ell$ yields to the next healthy state in both automata~$\\automatonU(\\ell)$ and~$\\automatonD(\\ell+1)$, and remains in the initial state for all other automata~$\\automatonU(j)$ for~$j \\in U_c \\ssm \\{\\ell\\}$ and~$\\automatonD(j)$ for~$j \\in D_c \\ssm \\{\\ell+1\\}$.\nWe now distinguish two cases, depending on whether~$\\ell$ and~$\\ell+1$ are reversed in~$\\pi$.\n\nAssume first that $\\ell$ and $\\ell+1$ are reversed in $\\pi$ and write $\\pi = s_\\ell \\cdot \\tau$. Using \\cref{lem:coxeterElementFacts} it suffices now to show that after $s_\\ell$, there is a reduced expression for $\\tau$ accepted by the automata. For each $j$, since the automaton $\\automatonD(j)$ or $\\automatonU(j)$ that we see accepts at least one reduced expression for $\\pi$ and $s_\\ell$ does not lead to a ill state, it also accepts a reduced expression for $\\pi$ starting with $s_\\ell$ by \\cref{prop:algorithm}. Observe moreover that:\n\\begin{itemize}\n\\item $\\tau$ admits a reduced expression accepted by~$\\automatonU(j)$ for each~$j \\notin \\{\\ell, \\ell+1\\}$ by \\cref{lem:vincent2}. This lines up with the fact that the order of $s_{j-1}$ and $s_j$ has not changed from $c = s_\\ell \\cdot d$ to~$d \\cdot s_\\ell$.\n\\item $\\tau$ admits a reduced expression accepted by~$\\automatonU(\\ell+1)$ and a reduced expression accepted by~$\\automatonD(\\ell)$ by \\cref{lem:vincent4}. This fits the fact that $\\ell$ now appears after $\\ell-1$ and $\\ell+1$ in $d \\cdot s_\\ell$.\n\\end{itemize}\nBy induction, we obtain that $\\tau(d \\cdot s_\\ell)$ is accepted by~$\\automatonP(U_{d \\cdot s_\\ell}, D_{d \\cdot s_\\ell})$, so that $\\pi(c) = s_\\ell \\cdot \\tau(d \\cdot s_\\ell)$ is accepted~$\\automatonP(U_c, D_c)$.\n\t\nAssume now that $\\ell$ and $\\ell+1$ are not reversed in $\\pi$. We want to show that $\\ell$ never appears in the reduced expressions for $\\pi$, ie. that $\\pi([\\ell]) = [\\ell]$ and $\\pi([n] \\ssm [\\ell]) = [n] \\ssm [\\ell]$. Otherwise, the reduced expression $w_\\ell$ accepted by $\\automatonU(\\ell)$ would see first a $s_{\\ell+1}$, and then a $s_\\ell$ before it sees any $s_{\\ell-1}$, so that we would have an inversion $k \\ell$ in $\\pi$ for some $\\ell < k$. Similarly, the reduced expression $w_{\\ell+1}$ accepted by $\\automatonD(\\ell+1)$ should see first a $s_{\\ell-1}$ and then a $s_\\ell$ before it sees any $s_{\\ell+1}$, so that we would have an inversion $(\\ell+1) i$ in $\\pi$ for some $i < \\ell+1$. Since $\\ell$ and $\\ell+1$ are not reversed, we see $k \\ell (\\ell+1) i$ which contradicts twice \\cref{thm:patternAvoidance}. We conclude that this case never happens, so that we can work in the parabolic subgroup of permutations that never use $s_\\ell$ in their reduced expressions.\n\\end{proof}\n\n\\subsection{Some negative observations}\n\nWe conclude this paper with some negative observations and warnings about the connection between $c$-sorting and $(U_c,D_c)$-permutree sorting.\nFirst, we want to underline that using $c$-sorting words to test whether a permutation avoids~$jki$ or $kij$ for a fixed~$j$ is dangerous for the following two reasons.\n\n\\begin{remark}\nEven if a permutation~$\\pi$ avoids~$jki$ (resp.~$kij$) for a given~$j$, there might be no Coxeter element~$c$ for which $\\pi$ is $c$-sortable and~$j \\in U_c$ (resp.~$j \\in D_c$).\nFor instance, the permutation $41325 \\in \\fS_5$ avoids $2ki$ and~$ki4$, but contains~$352$ and~$413$, so it is not $c$-sortable for any Coxeter element~$c$.\n\\end{remark}\n\n\\begin{remark}\nWhen a permutation~$\\pi$ is not $c$-sortable, there might exist~$j \\in U_c$ (resp.~$j \\in D_c$) for which the $c$-sorting word~$\\pi(c)$ is not accepted by~$\\automatonU(j)$ (resp.~$\\automatonD(j)$) even if~$\\pi$ avoids~$jki$ (resp.~$kij$).\nFor instance, consider~$c = s_2 \\cdot s_1 \\cdot s_3$ and~$\\pi = 4213 = s_3 \\cdot s_1 \\cdot s_2 \\cdot s_1 =  s_3 \\cdot s_2 \\cdot s_1 \\cdot s_2 =  s_1 \\cdot s_3 \\cdot s_2 \\cdot s_1$.\nThen~$2 \\in U_c$, and the $c$-sorting word~$\\pi(c) = s_1 \\cdot s_3 \\cdot s_2 \\cdot s_1$ is rejected by~$\\automatonU(2)$ while~$\\pi$ contains no~$2ki$ (and indeed~$s_3 \\cdot s_2 \\cdot s_1 \\cdot s_2$ is accepted by~$\\automatonU(2)$).\n\\end{remark}\n\nWe conclude the paper with an observation about sorting networks and permutree sorting.\n\n\\begin{remark}\nGiven a Coxeter element $c$, the word $c^\\infty$ which is used to compute $\\pi(c)$ is a \\defn{sorting network}. This means that we decide \\emph{beforehand} a list of transpositions to apply if appropriate. On the other hand, the permutree sorting given in Algorithm~\\ref{algo:permutreeSorting} is \\emph{not} a sorting network. Indeed, the order on transpositions depends on the permutation and more specifically on the state of the automaton we are at. A natural question then occurs: can we replace the permutree sorting algorithm by a sorting network? Or said differently, when~$U$ and~$D$ are disjoint but do not cover~$\\{2, \\dots, n-1\\}$, can we find a word $\\tilde{c}$ which plays the role of $c^\\infty$ in the sense that looking at $\\pi(\\tilde{c})$ would be enough to check whether $\\pi$ is accepted by $\\automatonP(U,D)$?\n\nThe answer is negative in general. A counter-example is found for $n = 5$, $U = \\{ 2\\}$, and $D=\\{ 4 \\}$. In this case one can check through computer exploration that no reduced word $\\tilde{c}$ of the maximal permutation $54321$ can be used as a sorting network. Namely, for all choices of $\\tilde{c}$, there exist a permutation $\\pi$ which is accepted by $\\automatonP(U,D)$ whereas the reduced expression $\\pi(\\tilde{c})$ is rejected. The healthy states of $\\automatonP(\\{ 2\\},\\{ 4 \\})$ are shown in Figure~\\ref{fig:automataProduct}. We see that accepted reduced expressions can start with either $s_2$ or $s_3$. For some permutations, such as $54213$ shown in Example~\\ref{exm:algo2}, all accepted reduced expressions start with $s_3$ whereas for some other permutations such as $35421$, all accepted reduced expressions start with $s_2$. This eventually leads to an empty intersection for the choice of $\\tilde{c}$.\n\nNevertheless, it seems interesting to study in which case the answer is positive. The Cambrien case with the $c$-sorting word when $U$ and~$D$ form a partition of~$\\{2, \\dots, n-1\\}$ is one example. The case where $|U| + |D| = 1$ is another one. This is the case corresponding to Theorem~\\ref{thm:patternAvoidance} where we have only one automaton. In this case, we can construct a word $\\tilde{c}$ by reading the healthy states of the automaton linearly, adding at each state the word $(s_{i_1} \\cdots s_{i_k})^k s_j$ where $s_{i_1}, \\dots, s_{i_k}$ are the looping transitions and $s_j$ is the transition going to the next healthy state. This process gives a prefix that can be extended in any way to obtain a proper sorting word $\\tilde{c}$. For example, if $U = \\{2\\}$, we obtain the prefix $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_3$ and indeed $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_3 \\cdot s_2 \\cdot s_1$ acts as a sorting network equivalent to the $(\\{2\\},\\varnothing)$-permutree sorting. This process actually seems to extend to all cases where, at each healthy state of the intersection automaton, the choices for the healthy transitions commute. For example, in the case where $n=5$, $U = \\{ 4 \\}$ and $D = \\{ 2 \\}$ as illustrated in Figure~\\ref{fig:automataProduct}, the word $s_1 \\cdot s_2 \\cdot s_4 \\cdot s_3 \\cdot s_2 \\cdot s_1 \\cdot s_4 \\cdot s_3 \\cdot s_2$ gives a proper sorting network. \n\\end{remark}\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nThe fine topology on an open set $\\Omega\\subset{\\mathbb R}^n$ was introduced\nby H.\\ Cartan in classical potential theory. It is defined as the\nsmallest topology on $\\Omega$ in which every super\\-harmonic function on\n$\\Omega$ is continuous.\nPotential theory on a finely open set, for example in ${\\mathbb R}^n$, was introduced\nand studied in the 1970's by the second named author\n\\cite{F1}. The harmonic and super\\-harmonic functions and the potentials in this\ntheory are termed finely [super]harmonic functions and fine\npotentials. Generally one distinguishes by the prefix `fine(ly)'\nnotions in fine potential theory from those in classical potential\ntheory on a usual (Euclidean) open set. Large parts of classical\npotential theory have been extended to fine potential theory.\n\nThe integral representation of positive ($=$ nonnegative) finely super\\-harmonic\nfunctions by using Choquet's method of extreme points was studied by the first\nnamed author in \\cite{El1}, where it was shown that the cone of positive\nsuper\\-harmonic functions equipped with the natural topology has a compact base.\nThis allowed the present authors\nin \\cite{EF1} to define the Martin compactification and the Martin boundary of\na fine domain $U$ in ${\\mathbb R}^n$. The Martin compactification $\\overline U$ of $U$\nwas defined by injection of $U$ in a compact base of the cone $\\cal S(U)$ of\npositive finely super\\-harmonic functions on $U$. While the Martin boundary of\na usual domain is closed and hence compact, all\nwe can say in the present setup is that the Martin boundary $\\Delta(U)$ of $U$\nis a $G_\\delta$ subset of the compact Riesz-Martin space\n$\\overline U=U\\cup\\Delta(U)$ endowed with the natural topology. Nevertheless\nwe have defined in \\cite{EF1} a suitably measurable Riesz-Martin kernel\n$K:U\\times\\overline U\\longrightarrow[0,+\\infty]$.\nEvery function $u\\in\\cal S(U)$ has an integral representation\n$u(x)=\\int_{\\overline U}K(x,Y)d\\mu(Y)$ in terms of a Radon measure $\\mu$ on\n$\\overline U$. This representation is unique if it is required that $\\mu$ be\ncarried by $U\\cup\\Delta_1(U)$, where $\\Delta_1(U)$ denotes the minimal Martin\nboundary of $U$, which likewise is a $G_\\delta$ in $\\overline U$. In this case\nof uniqueness we write $\\mu=\\mu_u$. It was shown that $u$ is a\nfine potential, resp.\\ an invariant function, if and only if $\\mu_u$ is\ncarried by $U$, resp.\\ by $\\Delta(U)$. The invariant functions, likewise\nstudied in \\cite{EF1}, generalize the positive harmonic functions in the\nclassical Riesz decomposition theorem. Finite valued invariant functions are\nthe same as positive finely harmonic functions.\n\nThere is a notion of minimal thinness of a set $E\\subset U$ at a point\n$Y\\in\\Delta_1(U)$, and an associated minimal-fine filter $\\cal F(Y)$,\nwhich allowed the authors\nin \\cite{EF1} to obtain a generalization of the classical Fatou-Na{\\\"\\i}m-Doob\ntheorem.\n\nIn a continuation \\cite{EF2} of \\cite{EF1} we studied sweeping on a subset\nof the Riesz-Martin space,\nboth relative to the natural topology and to the minimal-fine topology on\n$\\overline U$, and we showed that the two notions of sweeping are identical.\nIn the present further continuation of \\cite{EF1} and \\cite{EF2} we investigate\nthe Dirichlet problem at the Martin boundary of our given fine domain $U$\nby adapting the Perron-Wiener-Brelot (PWB) method to the present setup.\nIt is a complication that there is no Harnack convergence theorem for finely\nharmonic functions, and hence the infimum of a sequence of  upper\nPWB-functionss on $U$ may equal $-\\infty$ precisely\non some nonvoid proper finely closed subset of $U$.\nWe define resolutivity of a numerical function on $\\Delta(U)$ in a standard\nway and show that it is equivalent to a weaker, but technically supple\nconcept called quasi\\-resolutivity, which possibly has not been considered\nbefore in the literature (for the classical case where $U$ is Euclidean open).\nOur main result implies the corresponding known result for the classical case,\ncf.\\ \\cite[Theorem 1.VIII.8]{Do}.\nAt the end of Section 3 we obtain analogous results for the case where the\nupper and lower PWB-classes are defined in terms of the minimal-fine topology\non $\\overline U$ instead of the natural topology. It follows that the two\ncorresponding concepts of resolutivity are compatible. This result is\npossibly new even in the classical case.\nA further alternative, but actually equivalent, concept of resolutivity is\ndiscussed in the closing Section 4.\n\n{\\bf Notations}: If $U$ is a fine domain in $\\Omega$ we\ndenote by ${\\cal S}(U)$ the convex cone of positive finely super\\-harmonic\nfunctions on $U$ in the sense of \\cite{F1}. The convex cone of fine potentials\non $U$ (that is, the functions in ${\\cal S}(U)$ for which every finely\nsubharmonic minorant is $\\le 0$) is denoted by ${\\cal P}(U)$. The cone of\ninvariant functions on $U$ is the\northogonal band to ${\\cal P}(U)$ relative to ${\\cal S}(U)$.\nBy $G_U$ we denote the (fine) Green kernel for $U$, cf.\\ \\cite{F2}, \\cite{F4}.\nIf $A\\subset U$ and $f:A\\longrightarrow[0,+\\infty]$ one denotes by $R{}_f^A$,\nresp.\\ ${\\widehat R}{}_f^A$, the reduced function, resp.\\ the swept function,\nof $f$ on $A$ relative to $U$, cf.\\ \\cite[Section 11]{F1}. If $u\\in\\cal S(U)$\nand $A\\subset U$ we may write ${\\widehat R}{}_u^A$ for\n${\\widehat R}{}_f$ with $f:=1_Au$.\n\n\n\\section{The upper and lower PWB$^h$-classes of a function on $\\Delta(U)$}\n\\label{sec2}\n\nWe shall study the Dirichlet problem at $\\Delta(U)$ relative to a fixed\nfinely harmonic function $h>0$ on $U$. We denote by $\\mu_h$ the\nmeasure on $\\Delta(U)$ carried by $\\Delta_1(U)$ and representing $h$,\nthat is $h=\\int K(.,Y)d\\mu_h(Y)=K\\mu_h$. A function $u$ on $U$ (or on\nsome finely open subset of $U$) is said to be finely\n$h$-hyper\\-harmonic, finely $h$-super\\-harmonic, $h$-invariant, or a fine\n$h$-potential, respectively, if it has the form $u=v\/h$, where\n$v$ is finely hyper\\-harmonic, finely super\\-harmonic, invariant, or a\nfine potential, respectively.\n\nLet $f$ be a function on $\\Delta(U)$ with values in ${\\overline {\\mathbb R}}$.\nA finely $h$-hyper\\-harmonic function $u=v\/h$ on $U$ is said\nto belong to the upper PWB$^h$-class, denoted by\n${\\overline {\\cal U}}{}_f^h$, if $u$ is lower bounded and if\n$$\\underset{x\\to Y,\\,x\\in U}{\\liminf}u(x)\\ge f(Y)\n\\quad\\text{ for every }\\;Y\\in \\Delta(U).$$\nWe define\n $$\\dot H{}_f^h=\\inf\\,{\\overline{\\cal U}}{}_f^h,\n\\quad{\\overline H}{}_f^h=\\widehat{\\dot H{}_f^h}\n={\\widehat\\inf}\\,{\\overline {\\cal U}}{}_f^h\\;(\\le\\dot H{}_f^h).$$\nBoth functions $\\overline H{}_f^h$ and $\\dot H{}_f^h$ are needed here,\nunlike the classical\ncase where we have the Harnack convergence theorem and hence\n$\\overline H{}_f^h=\\dot H{}_f^h$.\nIn our setup, $\\dot H{}_f^h$\nmay be neither finely $h$-hyperharmonic nor identically $-\\infty$, but only\nnearly finely $h$-hyperharmonic on the finely open set\n$\\{\\overline H{}_f^h>-\\infty\\}$ which can be a nonvoid proper subset of $U$,\nsee Example \\ref{example2.0} below, which also shows that $\\Delta(U)$ can be\nnon-compact.\nAccording to the fundamental convergence theorem \\cite[Theorem 11.8]{F1}\n${\\overline H}{}_f^h$ is finely $h$-hyper\\-harmonic on\n$\\{\\overline H{}_f^h>-\\infty\\}$\nand ${\\overline H}{}_f^h={\\dot H}{}_f^h$ quasieverywhere (q.e.)\\! there;\nfurthermore, since ${\\overline{\\cal U}}{}_f^h$ is lower directed, there is a\ndecreasing sequence\n$(u_j)\\subset\\overline{\\cal U}{}_f^h$ such that $\\inf_ju_j=\\dot H{}_f^h$.\nClearly, $\\dot H{}_f^h$ is finely u.s.c.\\ on all of  $U$, and\n${\\overline H}{}_f^h$ is finely l.s.c.\\ there.\n\nThe lower PWB$^h$ class ${\\underline{\\cal U}}{}_f^h$\nis defined by\n${\\underline {\\cal U}}{}_f^h=-{\\overline {\\cal U}}_{-f}^h$, and we have\n$\\dot H{}_0^h=0$, hence also\n$\\overline H{}_0^h={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_0^h\n=\\underline H_0^h=0$.\nIt follows that if $f\\ge0$ then\n$\\dot H_f^h\\ge\\overline H_f^h\\ge0$ and therefore only positive functions\nof class $\\overline{\\cal U}{}_f^h$ need to be considered in the\ndefinition of $\\dot H{}_f^h$ and hence of $\\overline H_f^h$.\nMoreover, $\\dot H{}_{\\alpha f+\\beta}^h=\\alpha\\dot H{}_f^h+\\beta$ and hence\n$\\overline H_{\\alpha f+\\beta}^h=\\alpha\\overline H_f^h+\\beta$ for real\nconstants $\\alpha\\ge0$ and $\\beta$ (when $0$ times $\\pm\\infty$ is defined\nto be $0$).\n\n\\begin{example}\\label{example2.0} In $\\Omega={\\mathbb R}^n$ with the Green kernel\n$G(x,y)=|x-y|^{2-n}$, $n\\ge4$, let $\\omega\\subset\\Omega$ be a bounded H\\\"older\ndomain such that $\\omega$ is irregular with a single\nirregular boundary point $z$, cf.\\ e.g.\\ \\cite[Remark 6.6.17]{AG}. Take\n$U=\\omega\\cup\\{z\\}$. According to \\cite[Theorems 1 and 3.1]{Ai} the\nEuclidean boundary $\\partial\\omega$ of $\\omega$ is topologically contained\nin the Martin boundary $\\Delta(\\omega)$.\nIn particular, $z$ is non-isolated as a point of $\\Delta(\\omega)$.\nBut $\\Delta(U)=\\Delta(\\omega)\\setminus\\{z\\}$, where $z$ is identified\nwith $P_z$ (see \\cite[Section 3]{EF1}), and since $\\Delta(\\omega)$ is\ncompact we infer that $\\Delta(U)$ is noncompact.\nIn ${\\mathbb R}^n$ choose a sequence $(z_j)$ of points of\n$\\complement{\\overline\\omega}$ such that $|z_j-z|\\le2^{-j}$.\nThen $u:=\\sum_j2^{-j}G(.,z_j)$ is infinite at $z$, but finite and\nharmonic on $\\omega$. Furthermore, $u=\\sup_ku_k$,\nwhere $u_k:=\\sum_{j\\le k}2^{-j}G(.,z_j)$ is harmonic and bounded on\n$\\overline\\omega$ ($\\subset{\\mathbb R}^n$). It follows that $(u_k)_{|U}$\nis of class $\\underline{\\cal U}{}_f^h$, where $f:=u_{|\\Delta(U)}$.\nIn fact,\n$$\n\\underset{x\\to Y,\\,x\\in U}{\\lim}\\,u_k(x)=u_k(Y)\\le u(Y)=f(Y)\n$$\nfor $Y\\in\\Delta(U)$ (natural limit on $U\\cup\\Delta(U)$, or equivalently\nEuclidean limit on $\\omega\\cup((\\partial\\omega)\\setminus\\{z\\})$. Thus\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n\\ge(u_k)_{|U}$, and hence\n$$\n\\underline H{}_f^h(z)\\ge\n{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h(z)\n\\ge\\sup_ku_k(z)=u(z)=+\\infty.\n$$\nTo show that $\\underline H{}_f^h<+\\infty$ on $U\\setminus\\{z\\}$ ($\\cong\\omega$),\nlet $v\\in\\underline{\\cal U}{}_f^h$. Being upper bounded on the bounded open\nset $\\omega$, $v$ is subharmonic on $\\omega$ by \\cite[Theorem 9.8]{F1},\nand so is therefore $v-u$.\nFor any $Y\\in\\Delta(U)$ ($\\cong(\\partial\\omega)\\setminus\\{z\\}$) we have\n$$\n\\underset{x\\to Y,\\,x\\in\\omega}{\\limsup}\\,v(x)\\le f(Y)<+\\infty\n$$\n(also with Euclidean limit), or equivalently\n$$\n\\underset{x\\to Y,\\,x\\in\\omega}{\\limsup}\\,(v(x)-u(x))\\le0.\n$$\nSince $\\{z\\}$ is polar and $v-u\\le v$ is upper bounded, it follows by a\nboundary minimum principle that $v-u\\le0$, that is, $v\\le u$ on $\\omega$.\nBy varying $v\\in\\underline{\\cal U}{}_f^h$ we conclude that\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\\le u$\non $\\omega\\cong U\\setminus\\{z\\}$ and hence by regularization that\n$\\underline H{}_f^h\\le u<+\\infty$ on $U\\setminus\\{z\\}$.\nAltogether, $\\overline H{}_{-f}^h=-\\underline H_f^h$ equals\n$-\\infty$ at $z$, but is finite on $U\\setminus\\{z\\}$.\n\\end{example}\n\nHenceforth we fix the finely harmonic function $h>0$ on $U$, relative to\nwhich we shall study the Dirichlet problem at $\\Delta(U)$. Similarly to\nthe classical case, cf.\\ \\cite[p.\\ 108]{Do}, we pose the following definition,\ndenoting by $1_A$ the indicator function of a set $A\\subset\\Delta(U)$:\n\n\\begin{definition}\\label{def4.1}\n A subset $A$ of $\\Delta(U)$ is said to be $h$-harmonic measure null if\n  $\\overline H{}_{1_A}^h=0$.\n\\end{definition}\n\nIt will be shown in Corollary \\ref{cor6.10c} that $A$ is $h$-harmonic measure\nnull if and only if $A$ is $\\mu_h$-measurable with $\\mu_h(A)=0$.\n\n\\begin{prop}\\label{prop4.2}\n{\\rm{(a)}} Every countable union of $h$-harmonic measure null sets is $h$-harmonic\nmeasure null.\n\n{\\rm{(b)}} A set $A\\subset\\Delta(U)$ is $h$-harmonic measure null if and only\nif there is a finely $h$-super\\-harmonic function $u$ (positive if we like)\non $U$ such that $\\lim_{x\\to Y,\\,x\\in U}u(x)=+\\infty$ for every $Y\\in A$.\n\n{\\rm{(c)}} If $f:\\Delta(U)\\to[0,+\\infty]$ has $\\overline H{}_f^h=0$ then\n$\\{f>0\\}$ is $h$-harmonic measure null.\n\n{\\rm{(d)}} If $f:\\Delta(U)\\to[0,+\\infty]$ has $\\overline H{}_f^h<+\\infty$ then\n$\\{f=+\\infty\\}$ is $h$-harmonic measure null.\n\n{\\rm{(e)}} If $f,g:\\Delta(U)\\to[0,+\\infty]$ and if $f\\le g$ off some\n$h$-harmonic measure null set then $\\overline H{}_f^h\\le\\overline H{}_g^h$.\n\\end{prop}\n\n\\begin{proof} We adapt the proof in \\cite[p.\\ 108, 111]{Do} for the\nclassical case.\n\n(a) Fix a point $x_0$ of the co-polar subset\n$\\bigcap_j\\{\\dot H{}_{1_{A_j}}^h=0\\}$ of $U$. For given $\\varepsilon>0$ and\nintegers $j>0$ there are functions $u_j\\in\\overline{\\cal U}{}_{1_{A_j}}^h$ with\n$u_j(x_0)<2^{-j}\\varepsilon$. It follows that the function\n$u:=\\sum_ju_j$ is of class $\\overline{\\cal U}{}_{1_A}^h$ because\n$\\sum_j1_{A_j}\\ge1_A$ on $\\Delta(U)$. Consequently,\n$\\overline H{}_{1_A}^h(x_0)\\le\\dot H{}_{1_A}^h(x_0)\\le u(x_0)<\\varepsilon$,\nand the positive finely $h$-hyperharmonic function $\\overline H{}_{1_A}^h$\ntherefore equals $0$ at $x_0$ and so indeed everywhere on $U$.\n\n(b) If $\\overline H{}_{1_A}=0$ then $\\dot H{}_{1_A}^h=0$ q.e., so\nwe may choose $x_0\\in U$ with $\\dot H{}_{1_A}^h(x_0)=0$. For integers\n$j>0$ there exist positive finely $h$-superharmonic functions\n$u_j\\in\\overline{\\cal U}{}_{1_A}^h$ on $U$ such that $u_j(x_0)<2^{-j}\\varepsilon$.\nThe function $u:=\\sum_ju_j$ is positive and finely $h$-superharmonic on $U$\nbecause $u(x_0)<+\\infty$. Furthermore,\n $\\liminf_{x\\to Y,\\,x\\in U}u(x)=+\\infty$ for every  $Y\\in A$.\nConversely, if there exists a function $u$ as described in (b), we may arrange\nthat $u\\ge0$ after adding a constant. Then\n$\\varepsilon u\\in\\overline{\\cal U}{}_{1_A}^h$ for every $\\varepsilon>0$. It follows that\n$\\dot H{}_{1_A}^h\\le\\varepsilon u$ and by varying $\\varepsilon$ that\n$\\dot H{}_{1_A}^h=0$ off the polar set of infinities of $u$, and hence\nq.e.\\ on $U$. It follows that indeed $\\overline H{}_f^h=0$.\n\n(c) For integers $j\\ge1$ let $f_j$ denote the indicator function on $U$\nfor the set $\\{f>1\/j\\}$. Then $0=\\overline H{}_f^h\\ge\\overline H{}_{f_j}^h\/j$,\nso the sets $\\{f>1\/j\\}$ are $h$-harmonic measure null, and so is by (a) the\nunion $\\{f>0\\}$ of these sets.\n\n(d) Choose $x_0\\in U$ so that $\\dot H{}_f^h(x_0)=\\underline H{}_f^h(x_0)$\n($<+\\infty$) and $u\\in\\overline{\\cal U}{}_f^h$. Then\n$\\lim_{x\\to Y,\\,x\\in U}u(x)=+\\infty$ for every\n$Y\\in A:=\\{f=+\\infty\\}$. After adding a constant we arrange that the finely\n$h$-hyperharmonic function $u$ is positive. According to (b) it follows that\nindeed $\\overline H{}_{1_A}^h=0$.\n\n(e) Let $v\\in\\overline{\\cal U}{}_g^h$ and let $u$ be a positive\n$h$-superharmonic function on $U$ with limit $+\\infty$ at every point of\nthe $h$-harmonic measure null subset $\\{f>g\\}$ of $\\Delta(U)$. Then\n$\\dot H{}_f^h\\le v+\\varepsilon u\\in\\overline{\\cal U}{}_f^h$\nfor every $\\varepsilon>0$. Hence $\\dot H{}_f^h\\le v$ q.e., and so\n$\\overline H{}_f^h\\le v$ everywhere on $U$. By varying $v$ it follows that\n$\\overline H{}_f^h\\le\\dot H{}_g^h$ and so indeed by finely l.s.c.\\\nregularization $\\overline H{}_f^h\\le\\overline H{}_g^h$.\n\\end{proof}\n\n\\begin{prop}\\label{prop6.1}\nLet $f$ be a function on $\\Delta(U)$ with values in ${\\overline {\\mathbb R}}$.\n\n{\\rm{(a)}} $\\dot H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$\nand hence $\\overline H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$ and $\\dot H{}_f^h\n\\ge\\underline H{}_f^h$.\n\n{\\rm{(b)}} $\\overline H{}_f^h\n\\ge\\underline H{}_f^h$ on\n$\\{{\\overline H}{}_f^h>-\\infty\\}\\cup\n\\{{\\underline H}{}_f^h<+\\infty\\}$.\n\n{\\rm{(c)}} If $f$ is lower bounded then\n$\\overline H{}_f^h(x)=\\dot H{}_f^h(x)$ at any point $x\\in U$\nat which $\\dot H{}_f^h(x)<+\\infty$. If \\,$f\\ge0$ on $\\Delta(U)$ then\n${\\overline H}{}_f^h$ $(\\ge0)$ is either identically $+\\infty$ or\n$h$-invariant on $U$.\n\n\\end{prop}\n\n\\begin{proof} Clearly, ${\\overline{\\cal U}}{}_f^h$ is lower directed and\n$\\underline{\\cal U}{}_f^h$ is upper directed. The constant function $+\\infty$\nbelongs to ${\\overline{\\cal U}}{}_f^h$. If \\,$+\\infty$ is the only function\nof class ${\\overline{\\cal U}}{}_f^h$ then obviously $\\dot H{}_f^h=+\\infty$\nand hence ${\\overline H}{}_f^h=+\\infty$. In the remaining case it suffices\nto consider finely $h$-super\\-harmonic functions in the definition of\n$\\dot H{}_f^h$ and hence of \\,${\\overline H}{}_f^h$.\n\n(a) Let $u\\in\\overline{\\cal U}{}_f^h$ and $v\\in\\underline{\\cal U}{}_f^h$.\nThen $u-v$ is well defined, finely $h$-hyper\\-harmonic, and lower bounded\non $U$, and\n\\begin{eqnarray*}\\underset{x\\to Y,\\,x\\in U}{\\liminf}(u(x)-v(x))\\!\\!\\!\n&\\ge&\\!\\!\\!\\underset{x\\to Y,\\,x\\in U} {\\liminf}u(x)\n-\\underset{x\\to Y,\\,x\\in U} {\\limsup}v(x)\\\\\n&\\ge&\\!\\!\\! f(Y)-f(Y)=0\n\\end{eqnarray*}\nif $f(Y)$ is finite; otherwise $\\liminf u(x)-\\limsup v(x)=+\\infty\\ge0$,\nfor if for example $f(Y)=+\\infty$ then $\\liminf u=+\\infty$ whereas\n$\\limsup v<+\\infty$ since $v$ is upper bounded.\nBy the minimal-fine boundary minimum property given in\n\\cite[Corollary 3.13]{EF2} together with \\cite[Proposition 3.5]{EF2}\napplied to the finely super\\-harmonic function $hu-hv$\n(if $\\ne+\\infty$) it then follows that $u-v\\ge0$, and hence\n$u\\ge v$. By varying $u$ and $v$ in either order we obtain\n$\\dot H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$.\nSince\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n=\\sup\\underline{\\cal U}{}_f^h$\nis finely l.s.c.\\ it follows that\n${\\overline H}{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$,\nand similarly $\\dot H{}_f^h\n\\ge{\\underline H}{}_f^h$.\n\n(b) Consider any point $x_0$ of the finely open set\n$V=\\{\\overline H{}_f^h>-\\infty\\}$.\nSince ${\\overline H}{}_f^h$ is finely $h$-hyper\\-harmonic\nand hence finely continuous on $V$\nwe obtain by (a)\n$$\n{\\overline H}{}_f^h(x_0)\n=\\underset{x\\to x_0,\\,x\\in V}{\\fine\\lim}\\,{\\overline H}{}_f^h(x)\n\\ge\\underset{x\\to x_0,\\,x\\in V\\setminus E}{\\fine\\lim\\sup}\\,\n{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n={\\underline H}{}_f^h(x_0).\n$$\nThe case $x_0\\in\\{\\underline H{}_f^h<+\\infty\\}$ is treated similarly,\nor by replacing $f$ with $-f$.\n\n(c) The former assertion reduces easily to the case $f\\ge0$, whereby\n$\\overline{\\cal U}{}_f^h$ consists of positive functions. We may assume that\n$\\overline H{}_f^h\\not\\equiv+\\infty$, and hence\n$h\\overline{\\cal U}{}_f^h\\subset\\cal S(U)$.\nConsider the cover of $U$ by the finely open sets $V_k$ from\n\\cite[Lemma 2.1 (c)]{EF2}. Then $h\\overline{\\cal U}{}_f^h$ is a Perron family\nin the sense of \\cite[Definition 2.2]{EF2}. It therefore follows by\n\\cite[Theorem 2.3]{EF2} that indeed\n${\\overline H}{}_f^h=\\widehat{\\inf}\\,\\overline{\\cal U}{}_f^h$ is\n$h$-invariant, and that $\\overline H{}_f^h(x)=\\dot H{}_f^h(x)$ at any point\n$x\\in U$ at which $\\dot H{}_f^h(x)<+\\infty$.\n\\end{proof}\n\n\\begin{prop}\\label{prop6.2} Let $f,g$ be two functions on $\\Delta(U)$ with\nvalues in $\\overline{\\mathbb R}$.\n\n{\\rm{1.}} If $f+g$ is well defined everywhere on $\\Delta(U)$ then\nthe inequality $\\dot H{}_{f+g}^h\\le\\dot H{}_f^h+\\dot H{}_g^h$\nholds at each point of $U$ where $\\dot H{}_f^h+\\dot H{}_g^h$ is\nwell defined.\n\n{\\rm{2.}} If $(f+g)(Y)$ is defined arbitrarily at points $Y$ of\nundetermination then the inequality\n$\\overline H{}_{f+g}^h\\le\\overline H{}_f^h+\\overline H{}_g^h$\nholds everywhere on $\\{\\overline H{}_f^h,\\overline H{}_g^h>-\\infty\\}$.\n\n{\\rm{3.}} For any point $x\\in U$ we have $\\dot H{}_f^h(x)<+\\infty$\nif and only if $\\dot H{}_{f\\vee0}^h(x)<+\\infty$.\n\n{\\rm{4.}} Let $(f_j)$ be an increasing sequence of lower bounded functions\n$\\Delta(U)\\longrightarrow\\,]-\\infty,+\\infty]$. Writing $f=\\sup_jf_j$ we have\n$\\overline H{}_f^h=\\sup_j\\overline H{}_{f_j}^h$ and\n$\\dot H{}_f^h=\\sup_j\\dot H{}_{f_j}^h$.\n\\end{prop}\n\n\\begin{proof} For 1., 2., and 4.\\ we proceed much as in\n\\cite[1.VIII.7, Proof of (c), (b), and (e)]{Do}.\nFor Assertion 1., consider any two functions $u\\in\\overline{\\cal U}{}_f^h$\nand $v\\in\\overline{\\cal U}{}_g^h$.\nThen $u+v\\in\\overline{\\cal U}{}_{f+g}^h$ and hence $\\dot H{}_{f+g}^h\\le u+v$.\nBy varying $v$ it follows that $\\dot H{}_{f+g}^h\\le u+\\dot H{}_g^h$ on\n$\\{\\dot H{}_g^h>-\\infty\\}$.\nBy varying $u$ this leads to\n$\\dot H{}_{f+g}^h\\le\\dot H{}_f^h+\\dot H{}_g^h$ whereever the sum is well defined\non $\\{\\dot H{}_g^h>-\\infty\\}$.\nBy interchanging $u$ and $v$ we infer that\n$\\dot H{}_{f+g}^h\\le\\dot H{}_f^h+\\dot H{}_g^h$ altogether holds whereever the\nsum is well defined on $\\{\\dot H{}_f^h,\\dot H{}_g^h>-\\infty\\}$.\nOn the residual set $\\{\\dot H{}_f^h=-\\infty\\}\\cup\\{\\dot H{}_g^h=-\\infty\\}$\nit is easily seen that\n$\\dot H{}_{f+g}^h=-\\infty=\\dot H{}_f^h+\\dot H{}_g^h$ whereever the sum is well\ndefined.\n\nFor Assertion 2., suppose first that $f,g<+\\infty$ (and so $f+g$ is well\ndefined). In the proof of Assertion 1.\\ we had\n$\\dot H{}_{f+g}^h\\le u+\\dot H{}_g^h$ for $\\dot H{}_g^h>-\\infty$, which is\nsatisfied on $\\{\\overline H{}_g^h>-\\infty\\}$. It follows that\n$\\overline H{}_{f+g}^h\\le u+\\overline H{}_g^h$ there, and hence that\n$\\overline H{}_{f+g}^h\\le\\overline H{}_f^h+\\overline H{}_g^h$ there (whereever\nwell defined). In the general case\ndefine functions $f_0<+\\infty$, resp.\\ $g_0<+\\infty$, which equal $f$,\nresp.\\ $g$, except on the set $\\{f=+\\infty\\}$, resp.\\ $\\{g=+\\infty\\}$.\nWe may assume that these exceptional sets are $h$-harmonic measure null,\nfor if e.g.\\ $\\{f=+\\infty\\}$ is not $h$-harmonic measure null then\n$\\overline H{}_f^h\\equiv=+\\infty$ by Proposition \\ref{prop4.2} (d),\nin which case 1.\\ becomes trivial.\nIt therefore follows in view of Proposition\n\\ref{prop4.2} (a), (e) that $f+g=f_0+g_0$ off the $h$-harmonic measure\nnull set $\\{f=+\\infty\\}\\cup\\{g=+\\infty\\}$ and hence by Proposition\n\\ref{prop4.2} (e) that\n$$\n\\overline H{}_{f+g}^h=\\overline H{}_{f_0+g_0}^h\n\\le\\overline H{}_{f_0}^h+\\overline H{}_{g_0}^h\n\\le\\overline H{}_f^h+\\overline H{}_g^h$$\non the finely open set\n$\\{\\overline H{}_f^h>-\\infty\\}\\cap\\{\\overline H{}_g^h>-\\infty\\}$,\nthe second inequality because $f_0\\le f$ and $g_0\\le g$.\n\nFor Assertion 3., let $x\\in U$ be given, and suppose that\n$\\dot H{}_f^h(x)<+\\infty$.\nThere is then $u\\in\\overline{\\cal U}{}_f^h$ with $u(x)<+\\infty$, $u$ being\nfinely $h$-superharmonic $\\ge-c$ for some constant $c\\ge0$. It follows that\n$u+c\\in\\overline{\\cal U}{}_{f\\vee0}^h$ and hence\n$\\dot H{}_{f\\vee0}^h(x)\\le u(x)+c<+\\infty$. The converse follows from\n$\\dot H{}_f^h\\le\\dot H{}_{f\\vee0}^h$.\n\nAssertion 4.\\ reduces easily to the case of positive functions $f_j$.\nConsider first the case of $\\overline H$. Then\n${\\overline H}{}_f^h$ and each ${\\overline H}{}_{f_j}^h$ are\npositive and hence finely $h$-hyper\\-harmonic by Proposition \\ref{prop6.1} (c).\nThe inequality ${\\overline H}{}_f^h\\ge\\sup_j{\\overline H}{}_{f_j}^h$ is obvious,\nand we may therefore assume that the positive finely $h$-hyper\\-harmonic\nfunction $\\sup_j{\\overline H}{}_{f_j}^h$ is not identically $+\\infty$, and\ntherefore is $h$-invariant, again according to\nProposition \\ref{prop6.1} (c). Denote $E_j$ the polar subset\n$\\{\\overline H{}_{f_j}^h<\\dot H{}_{f_j}^h\\}$\nof $U$ and write $E:=\\bigcup_jE_j$ (polar). For a fixed $x\\in U\\setminus E$\nand for given $\\varepsilon>0$ choose functions $u_j\\in{\\overline{\\cal U}}{}_{f_j}^h$\nso that\n\\begin{eqnarray}u_j(x)\\!\\!\\!\n&<&\\!\\!\\!\\dot H{}_{f_j}^h(x)+2^{-j}\\varepsilon=\\overline H{}_{f_j}^h(x)+2^{-j}\\varepsilon.\n\\end{eqnarray}\nIn particular, $u_j$ is finely $h$-super\\-harmonic.\n Define a finely $h$-hyper\\-harmonic function $u$ by\n\\begin{eqnarray}u\\!\\!\\!\n&=&\\!\\!\\!\\sup_j{\\overline H}{}_{f_j}^h\n+\\sum_j(u_j-{\\overline H}{}_{f_j}^h)\n\\ge{\\overline H}{}_{f_k}^h+(u_k-{\\overline H}{}_{f_k}^h)=u_k\n\\end{eqnarray}\nfor any index $k$. Then\n$$\\liminf_{x\\to Y,\\,x\\in U}u(x)\n\\ge\\liminf_{x\\to Y,\\,x\\in U}u_k(x)\n\\ge f_k(Y)$$\nfor every $Y\\in\\Delta(U)$ and every index $k$.\nThus $u\\in{\\overline{\\cal U}}{}_f^h$ and\n${\\overline H}{}_f^h\\le{\\dot H}{}_f^h\\le u$.\nIn particular, by the former equality (2.2) and by (2.1),\n\\begin{eqnarray}{\\overline H}{}_f^h(x)\\!\\!\\!\n&\\le&\\!\\!\\!u(x)\n\\le\\sup_j{\\overline H}{}_{f_j}^h(x)+\\varepsilon,\n\\end{eqnarray}\nand hence the finely $h$-hyper\\-harmonic function ${\\overline H}{}_f^h$ is\nfinely $h$-super\\-harmonic.\nBecause $\\sup_j{\\overline H}{}_f^h$ is $h$-invariant and majorized by\n${\\overline H}{}_f^h$, the function\n${\\overline H}{}_f^h-\\sup_j {\\overline H}{}_{f_j}^h$ is finely\n$h$-super\\-harmonic $\\ge0$ on $U$ by \\cite [Lemma 2.2]{EF1},\nand $\\le\\varepsilon$ at $x$. For $\\varepsilon\\to0$ we obtain the remaining inequality\n${\\overline H}{}_f^h\\le\\sup_j {\\overline H}{}_{f_j}^h$.\n\nIn the remaining case of $\\dot H$ we have\n$\\sup_j\\dot H{}_{f_j}^h(x)\\le\\dot H{}_f^h$. For any point $x\\in U$ at which\n$\\sup_j\\dot H{}_{f_j}^h(x)<\\dot H{}_f^h$ and for any $\\varepsilon>0$ choose functions\n$u_j\\in{\\overline{\\cal U}}{}_{f_j}^h$ so that\n\\begin{eqnarray}u_j(x)\\!\\!\\!\n&<&\\!\\!\\!\\dot H{}_{f_j}^h(x)+2^{-j}\\varepsilon.\n\\end{eqnarray}\nProceed as in the above case of $\\overline H$ by defining the finely\n$h$-hyperharmonic function by (2.2), replacing throughout $\\bar H$\nby $\\dot H$. Corresponding to (2.3) we now end by\n$$\n\\dot H{}_f^h(x)\\le u(x)\\le\\sup_j{\\dot H}{}_{f_j}^h(x)+\\varepsilon,\n$$\nfrom which the remaining inequality\n$\\dot H{}_f^h(x)\\le\\sup_j{\\dot H}{}_{f_j}^h(x)$ follows for $\\varepsilon\\to0$.\n\\end{proof}\n\n\n\\section{$h$-resolutive and $h$-quasiresolutive functions}\\label{sec3}\n\n\\begin{definition}\\label{def6.9}\nA function $f$ on $\\Delta(U)$ with values in ${\\overline {\\mathbb R}}$ is said\nto be $h$-resolutive if $\\overline H{}_f^h=\\underline H{}_f^h$ on $U$\nand if this function, also denoted by $H_f^h$,\nis neither identically $+\\infty$ nor identically $-\\infty$.\n\\end{definition}\n\nIt follows that $H_f^h$ is finely $h$-harmonic on the finely open set\n$\\{\\overline H{}_f^h>-\\infty\\}\\cap\\{\\underline H{}_f^h>+\\infty\\}\n=\\{-\\infty<H_f^h<+\\infty\\}$.\n\nFor any function $f:\\Delta(U)\\longrightarrow{\\mathbb R}$ we consider the following\ntwo subsets of $U$:\n$$\nE_f^h=\\{\\overline H{}_f^h=-\\infty\\}\\cup\n\\{\\underline H{}_f^h=+\\infty\\}\\cup\n\\{\\overline H{}_f^h\\ne\\underline H{}_f^h\\},\n$$\n$$\nP_f^h=\\{\\dot H{}_f^h>\\overline H{}_f^h>-\\infty\\}\n\\cup\\{{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n<\\underline H{}_f^h<+\\infty\\},\n$$\nof which $P_f^h$ is polar.\n\n\\begin{definition}\\label{def2.3} A function $f$ on $\\Delta(U)$\nwith values in ${\\overline {\\mathbb R}}$ is said to be\n$h$-quasi\\-resolutive if $E_f^h$ is polar, or equivalently if the relations\n$\\overline H{}_f^h>-\\infty$,\n$\\underline H{}_f^h<+\\infty$, and\n$\\overline H{}_f^h=\\underline H{}_f^h$ hold quasieverywhere on $U$.\n\\end{definition}\n\nWhen $f$ is $h$-quasi\\-resolutive on $U$ the functions $\\overline H{}_f^h$\nand $\\underline H{}_f^h$ are finely $h$-hyperharmonic and finely\n$h$-hypoharmonic, respectively, off the polar set\n$\\{\\overline H{}_f^h=-\\infty\\}\\cup\\{\\underline H{}_f^h=+\\infty\\}$, and they\nare actually equal and hence finely $h$-harmonic off the smaller\npolar set $E_f^h$. We then denote\nby $H_f^h$ the common restriction of $\\overline H{}_f^h$ and\n$\\underline H{}_f^h$ to $U\\setminus E_f^h$.\nSince $\\overline H{}_f^h$ is finely $h$-hyperharmonic on $U\\setminus E_f$, and\n$\\underline H{}_f^h$ is finely $h$-hypoharmonic there, it\nfollows by Proposition \\ref{prop6.1} (a) that the equalities\n$$\n\\overline H{}_f^h=\\dot H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h=\n\\underline H{}_f^h\n$$\nhold q.e.\\ on $U\\setminus E_f^h$ and hence q.e.\\ on $U$.\n\n\\begin{lemma}\\label{lemma3.3} Every $h$-resolutive function $f$ is\n$h$-quasi\\-resolutive.\n\\end{lemma}\n\n\\begin{proof} The sets $E_+:=\\{H_f^h=+\\infty\\}$ and $E_-:=\\{H_f^h=-\\infty\\}$\nare finely closed and disjoint. For any fine component $V$ of\n$U\\setminus E_-$ such that $V\\cap E_+$ is nonpolar we have $V\\cap E_+=V$,\nthat is, $V\\subset E_+$, because $\\overline H{}_f^h$ is finely $h$-hyperharmonic\non $V$. Denote by $W$ the union of these fine components $V$, and by $W'$ the\n(countable) union of the remaining fine components $V'$ of $U\\setminus E_-$.\nThen $W\\subset E_+$ whereas the set $P:=W'\\cap E_+$ is polar along with each\n$V'\\cap E_+$. Since $E_+\\cap E_-=\\varnothing$ we obtain\n$$\nE_+=(U\\setminus E_-)\\cap E_+=(W\\Cup W')\\cap E_+\n=(W\\cap E_+)\\Cup(W'\\cap E_+)=W\\Cup P,\n$$\n$\\Cup$ denoting disjoint union.\nNow, $(U\\setminus P)\\cap E_+=E_+\\setminus P$ is finely closed relatively to\nthe nonvoid fine domain $U\\setminus P$ (cf.\\ \\cite[Theorem 12.2]{F1}), but also\nfinely open, being equal to $W$ as seen from the above display. Thus either\n$W=U\\setminus P$ or $W=\\varnothing$. But $W=U\\setminus P$ would imply $E_+=U$,\ncontradicting $H_f^h\\not\\equiv+\\infty$, and so actually $E_+=P$ (polar).\nSimilarly (or by replacing $f$ with $-f$) it is shown that $E_-$ is polar,\nand so $f$ is $h$-quasiresolutive because\n$\\overline H{}_f^h=\\underline H{}_f^h$ even holds everywhere on $U$.\n\\end{proof}\n\nIn view of Lemma \\ref{lemma3.3} an $h$-quasi\\-resolutive function\n$f$ is $h$-resolutive if and only if $E_f^h=\\varnothing$ (any polar subset\nof $\\Delta(U)$ being a proper subset). This implies that 1.\\ and 2.\\ in\nthe following proposition remain valid with `$h$-quasi\\-resolutive' replaced\nthroughout by `$h$-resolutive'.\nIt will be shown in Corollary \\ref{cor6.3b} that $h$-resolutivity and\n$h$-quasi\\-resolutivity are actually identical concepts.\n\n\\begin{prop}\\label{prop6.3} Let $f,g:\\Delta(U)\\longrightarrow\\overline{{\\mathbb R}}$\nbe $h$-quasi\\-resolutive. Then\n\n{\\rm{1.}} For $\\alpha\\in{\\mathbb R}$ we have $E_{\\alpha f}^h\\subset E_f^h$ and hence\n$\\alpha f$ is $h$-quasi\\-resolutive.\nFurthermore, $H{}_{\\alpha f}^h=\\alpha H{}_f^h$ on $U\\setminus E_f^h$.\n\n{\\rm{2.}} If $f+g$ is defined arbitrarily at points of undetermination then\n$E_{f+g}^h\\subset E_f^h\\cup E_g^h$ and hence $f+g$ is $h$-quasi\\-resolutive.\nFurthermore, $H{}_{f+g}^h=H{}_f^h+H{}_g^h$ on $U\\setminus(E_f^h\\cup E_g^h)$.\n\n{\\rm{3.}} $E_{f\\vee g}^h,E_{f\\wedge g}^h$ $\\subset E_f^h\\cup E_g^h\\cup P_f^h$ and\nhence $f\\vee g$ and $f\\wedge g$ are $h$-quasi\\-resolutive.\nIf for example $H_f^h\\vee H_g^h\\ge0$\nthen $H{}_{f\\vee g}^h=(1\/h){\\widehat R}{}_{(H_f^h\\vee H{}_g^h)h}$ on\n$U\\setminus(E_f^h\\cup E_g^h\\cup P_f^h)$.\n\\end{prop}\n\n\\begin{proof} For Assertion 1., consider separately the cases\n$\\alpha>0$, $\\alpha<0$, and $\\alpha=0$.\nFor 2.\\ and 3. we proceed as in \\cite[1.VIII.7 (d)]{Do}.\nFor 2.\\ we have\n\\begin{eqnarray}\nH_f^h+H_g^h\\ge\\overline H{}_{f+g}^h\\ge\\underline H{}_{f+g}^h\\ge H_f^h+H_g^h\n\\end{eqnarray}\non $U\\setminus(E_f^h\\cup E_g^h)$, the first inequality by 2.\\ in\nProposition \\ref{prop6.2},\nthe third inequality by replacing $f$ with $-f$ in the\nfirst inequality, and the second inequality holds by Proposition\n\\ref{prop6.1} (b) on\n$$\n\\{\\overline H_{f+g}^h>-\\infty\\}\n\\supset\\{\\overline H_f^h>-\\infty\\}\\cap\\{\\overline H_g^h>-\\infty\\}\n\\supset U\\setminus(E_f\\cup E_g).\n$$\nThus equality prevails on $U\\setminus(E_f\\cup E_g)$ (and hence q.e.\\ on $U$)\nin both of these inclusion relations. It follows that\n$$\n\\{\\overline H{}_{f+g}^h=-\\infty\\}\n\\subset\\{\\overline H{}_f^h=-\\infty\\}\\cup\\{\\overline H{}_g^h=-\\infty\\}\n\\subset E_f^h\\cup E_g^h.\n$$\nand similarly\n$\\{\\underline H_{f+g}^h=-\\infty\\}\\subset E_f^h\\cup E_g^h$. Finally, by (3.1)\nwith equality throughout,\n$$\\{\\overline H{}_{f+g}^h\\ne\\underline H{}_{f+g}^h\\}\n\\subset\\{\\overline H{}_f^h\\ne\\underline H{}_f^h\\}\n\\cup\\{\\overline H{}_g^h\\ne\\underline H{}_g^h\\}\n\\subset E_f^h\\cup E_g^h.\n$$\nAltogether, $E_{f+g}^h\\subset E_f^h\\cup E_g^h$,\nand so $f+g$ is indeed $h$-quasi\\-resolutive along with $f$ and $g$.\n\nFor the notation in the stated equation in 3., see \\cite[Definition 11.4]{F1}.\nSince $f\\wedge g=-[(-f)\\vee(-g)]$ and $f\\vee g=[(f-g)\\vee0]+g$  it follows by\n1.\\ and 2.\\ that 3.\\ reduces to $E_{f\\vee0}^h\\subset E_f\\cup P_f^h$,\nwhich implies the $h$-quasi\\-resolutivity of $f^+=f\\vee0$\nand the stated expression for $H_{f\\vee g}^h$ with $g=0$.\nFor given $x\\in U\\setminus(E_f^h\\cup P_f^h)$ and integers $j>0$\nchoose $u_j\\in\\overline{\\cal U}{}_f^h$ with\n$u_j(x)\\le\\dot H{}_f^h(x)+2^{-j}=\\overline H{}_f^h(x)+2^{-j}$.\nThe series $\\sum_{j=k}^\\infty(u_j-H_f^h)$ of\npositive finely $h$-super\\-harmonic functions on $U\\setminus(E_f^h\\cup P_f^h)$\n($u_j$ being likewise restricted to $U\\setminus(E_f^h\\cup P_f^h)$) has a positive\nfinely $h$-super\\-harmonic sum, finite at $x$.\nRecall that $H_f^h$ is  defined and finely $h$-harmonic on $U\\setminus E_f^h$\nand in particular on $U\\setminus(E_f^h\\cup P_f^h)$. Consequently,\n$H_f^h\\vee0$ is finely $h$-subharmonic (and positive)\non $U\\setminus(E_f^h\\cup P_f^h)$ and majorized there by $\\overline H{}_{f\\vee0}^h$,\nwhich is finite valued on $U\\setminus(E_f^h\\cup P_f^h)$ by 3.\\ in Proposition\n\\ref{prop6.2} because $\\overline H{}_f^h<+\\infty$ on\n$U\\setminus(E_f^h\\cup P_f^h)$ and because $\\overline H{}_f^h=\\dot H{}_f^h$\nthere. It follows by \\cite[Theorem 11.13]{F1}, applied with $f$ replaced by\n$h\\overline H{}_f^h\\vee0$ on $U$,\nwhich is finely subharmonic on $U\\setminus(E_f^h\\cup P_f^h)$,\nthat $\\frac1h{\\widehat R}{}_{h\\overline H{}_f\\vee0}^h$ (sweeping relative to $U$)\nis finely $h$-harmonic on $U\\setminus(E_f^h\\cup P_f^h)$,\nbeing majorized there by\n$\\overline H{}_f^h\\vee0\\le\\overline H{}_{f\\vee0}^h<+\\infty$.\nThe positive function\n\\begin{eqnarray}\n\\frac1h{\\widehat R}{}_{h\\overline H{}_f\\vee0}^h\n+\\sum_{j=k}^\\infty(u_j-H_f^h)\n\\end{eqnarray}\nrestricted to $U\\setminus(E_f^h\\cup P_f^h)$ is therefore finely\n$h$-super\\-harmonic.\nMoreover, this positive finely  $h$-super\\-harmonic function on\n$U\\setminus(E_f^h\\cup P_f^h)$ majorizes $u_k\\in{\\overline{\\cal U}}{}_f^h$ there\n(being $\\ge\\frac1h{\\widehat R}{}_{h\\overline H{}_f\\vee0}^h+(u_k-H_f^h)\\ge u_k$ there),\nand this majorization remains in force after extension by fine continuity\nto $U$, cf.\\ \\cite[Theorem 9.14]{F1}. Thus the extended positive function (3.2)\nbelongs to $\\overline{\\cal U}{}_{f\\vee 0}^h$. For $k\\to\\infty$ it follows that\n${\\overline H}{}_{f\\vee 0}^h\n\\le\\frac1h{\\widehat R}{}_{h\\overline H{}_{f\\vee 0}^h}$\non $U$. On the other hand, $\\underline H{}_{f\\vee0}^h$ majorizes both\n $\\underline H{}_f^h$ and $0$,\nso $\\underline H{}_{f\\vee0}^h\\ge\\frac1h{\\widehat R}{}_{h\\underline H{}_f^h\\vee0}\n=\\frac1h{\\widehat R}{}_{h\\overline H{}_f^h\\vee0}$ on $U$, the equality because\n$\\underline H{}_f^h=\\overline H{}_f^h$ on $U\\setminus(E_f^h\\cup P_f^h)$ and hence\nq.e.\\ on $U$.\nIt follows that\n$${\\overline H}{}_{f\\vee0}^h\\le\\frac1h{\\widehat R}{}_{h\\overline H{}_f^h\\vee0}\n=\\frac1h{\\widehat R}{}_{h\\underline H{}_f^h\\vee0}\n\\le\\underline H{}_{f\\vee0}^h\\le{\\overline H}{}_{f\\vee0}^h<+\\infty\n$$ because\n$h\\overline H{}_f^h\\vee0=h\\underline H{}_f^h\\vee0$\non $U\\setminus(E_f^h\\cup P_f^h)$ and hence q.e.\\ on $U$. (The last inequality\nin the above display follows by Proposition \\ref{prop6.1} (b) because\n$f\\vee0>-\\infty$.)\nSince $\\overline H{}_{f\\vee0}^h\\ge0\\ge-\\infty$ we conclude that $f\\vee0$\nindeed is $h$-resolutive, resp.\\ $h$-quasi\\-resolutive, and that\n$E_{f\\vee0}^h\\subset E_f^h\\cup P_f^h$ and\n$H_{f\\vee0}^h=\\frac1h{\\widehat R}{}_{h\\overline H{}_f^h\\vee0}$ on $E_f^h\\cup P_f^h$.\n\\end{proof}\n\nA version of Proposition \\ref{prop6.3} for $h$-resolutive\nfunctions instead of $h$-quasi\\-resolutive functions will\nof course follow when the identity of $h$-resolutivity and\n$h$-quasi\\-resolutivity has been established in Corollary \\ref{cor6.3b}.\nBefore that, we do however need the following step in that direction,\nbased on Proposition \\ref{prop6.2}.\n\n\\begin{lemma}\\label{lemma6.3c} Let $f$ be an $h$-quasi\\-resolutive function\non $\\Delta(U)$. If $f^+$ and $f^-$ are $h$-resolutive then so is $f$,\nand the function\n$H_f^h=\\overline H{}_f^h=\\underline H{}_f^h$ on $U$ is finite valued.\n\\end{lemma}\n\n\\begin{proof} According to 3.\\ in Proposition \\ref{prop6.3}, $f^+$ and $f^-$\nare $h$-quasi\\-resolutive (besides being $h$-resolutive), and the functions\n$H_{f+}^h:=\\overline H{}_{f^+}^h=\\underline H{}_{f^+}^h$\n(defined on $\\{\\overline H{}_{f_+}^h>-\\infty\\}=U$ since $f^+\\ge0$)\nand similarly $H_{f-}^h:=\\overline H{}_{f^-}^h=\\underline H{}_{f^-}^h$\nare therefore finite valued. Since $-f^-\\le f\\le f^+$ it follows that\n$-\\infty<-H_{f^-}^h\\le\\underline H{}_f^h,\\overline H{}_f^h\\le H_{f^+}^h<+\\infty$.\nApplying 2.\\ in Proposition \\ref{prop6.2} to the sums\n$f=f^++(-f)^+=f^+-f^-$ and $-f=f^--f_+$, which are well defined on $\\Delta(U)$,\nwe obtain\n$$\n\\overline H{}_f^h\\le H_{f^+}^h-H_{f^-}^h\\le\\underline H{}_f^h\n$$\non all of $U$, and hence $\\overline H{}_f^h=\\underline H{}_f^h$ there because\n$\\overline H{}_f^h\\ge\\underline H{}_f^h$ on all of $U$, again by\nProposition \\ref{prop6.1} (b) since we have seen that for example\n$\\overline H{}_f^h>-\\infty$.\n\\end{proof}\n\n\\begin{cor}\\label{cor4.8} Let $(f_j)$ be an increasing sequence of lower\nbounded $h$-resolutive, resp.\\  $h$-quasi\\-resolutive functions\n$\\Delta(U)\\longrightarrow\\,]-\\infty,+\\infty]$, and let $f=\\sup_jf_j$.\nIf $\\overline H{}_f^h\\not\\equiv+\\infty$ then $f$ is\n$h$-resolutive, resp.\\ $h$-quasi\\-resolutive.\n \\end{cor}\n\n\\begin{proof} By adding a constant to $f$ we reduce the claim to the case\n$f_j\\ge0$. For every $j$ we have\n$\\underline H{}_f^h\\ge\\underline H{}_{f_j}^h=\\overline H{}_{f_j}^h$,\nby Proposition \\ref{prop6.1} (b) because $\\overline H{}_{f_j}^h>-\\infty$. Hence\n$\\underline H{}_f^h\\ge\\sup_j\\overline H{}_{f_j}^h=\\overline H{}_f^h$\naccording to 4.\\ in Proposition \\ref{prop6.2}.\nBy definition of $\\underline H{}_f^h$ we have at any point $x_0\\in U$\n$$\n\\underline H{}_f^h(x_0)=\\underset{x\\to x_0,\\,x\\in U}{\\fine\\lim\\sup}\\,\n{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h(x)\n\\le\\underset{x\\to x_0,\\,x\\in U}{\\fine\\lim\\sup}\\,\n\\overline H{}_f^h(x)\n=\\overline H{}_f^h(x_0)\n$$\naccording to Proposition \\ref{prop6.1} (a), $\\overline H{}_f^h$ being finely\n$h$-hyper\\-harmonic by Proposition \\ref{prop6.1} (c). By Proposition\n\\ref{prop6.1} (b) we have\n$\\underline H{}_f^h\\le\\overline H{}_f^h$ on $\\{\\overline H{}_f^h>-\\infty\\}=U$,\nand we conclude that $\\underline H{}_f^h=\\overline H{}_f^h$. By hypothesis\nthis finely hyperharmonic function on $U$ is finite q.e.\\ on $U$, and in\nparticular this positive function is not identically $+\\infty$.\nConsequently, $f$ is indeed $h$-resolutive, resp.\\ $h$-quasi\\-resolutive.\n\\end{proof}\n\nRecall that $\\mu_h$ denotes the unique measure on $\\overline U$\ncarried by $\\Delta_1(U)$ and representing $h$,\nthat is, $h=K\\mu_h=\\int K(.,Y)d\\mu_h(Y)$.\n\n\\begin{prop}\\label{prop6.4} For any $\\mu_h$-measurable subset $A$\nof $\\Delta(U)$ the indicator function $1_A$ is $h$-resolutive, and\n\\begin{eqnarray}H{}_{1_A}^h\\!\\!\\!\n&=&\\!\\!\\!\\frac{1}{h}\\int_AK(.,Y)d\\mu_h(Y)=\\frac1h\\widehat R{}_h^A\n\\end{eqnarray}\non $U$.\nIn particular, the constant function $1$ on $\\Delta(U)$ is\n$h$-resolutive and $H_1^h=1$.\n\\end{prop}\n\n\\begin{proof} Because $h=K\\mu_h$ and because $\\mu_h$ is carried by\n$\\Delta_1(U)$ we have by \\cite[Theorem 3.10 and Proposition 3.9]{EF2}\n$$\n\\widehat R{}_h^A=\\widehat R{}_{K\\mu_h}^A\n=\\int_{\\Delta_1(U)}\\widehat R{}_{K(.,Y)}^Ad\\mu_h(Y)\n=\\int K(.,Y)1_A(Y)d\\mu_h(Y).\n$$\nConsider any finely $h$-hyper\\-harmonic function $u=v\/h\\ge0$ on $U$ such that\n$u\\ge1$ on some open set $W\\subset\\overline U$ with $W\\supset A$. Then\n$u\\in\\overline{\\cal U}{}_{1_A}^h$ and hence\n$u\\ge\\dot H{}_{1_A}^h\\ge\\overline H{}_{1_A}^h$.\nBy varying $W$ it follows by \\cite[Definition 2.4]{EF2} that\n$\\frac1h\\widehat R{}_h^A\\ge\\overline H{}_{1_A}^h$.\nWe have\n\\begin{eqnarray}\\frac1h\\int K(.,Y)1_A(Y)d\\mu_h(Y)\n=\\frac1h \\widehat R{}_h^A\\ge{\\overline H}{}_{1_A}^h.\n\\end{eqnarray}\nApplying this inequality to the $\\mu_h$-measurable set $\\Delta(U)\\setminus A$\nin place of $A$ we obtain\n\\begin{eqnarray}\\frac{1}{h}\\int K(.,Y)1_{\\Delta(U)\\setminus A}(Y)d\\mu_h(Y)\n\\ge {\\overline H}{}_{1_{\\Delta(U)\\setminus A}}^h.\n\\end{eqnarray}\nBy adding the left hand, resp.\\ right hand, members of (3.4) and (3.5)\nthis leads by 2.\\ in Proposition \\ref{prop6.2} to\n\\begin{eqnarray}1=\\frac1h\\int K(.,Y)d\\mu_h(Y)\n\\ge{\\overline H}{}_{1_A}^h+{\\overline H}{}_{1_{\\Delta(U)\\setminus A}}^h\n\\ge{\\overline H}{}_1^h=1 .\n\\end{eqnarray}\nThus equalities prevail throughout in (3.4), (3.5), and (3.6). It follows\naltogether that\n\\begin{eqnarray*}\\underline H{}_{1_A}^h\\!\\!\\!\n&=&-\\overline H{}_{-1_A}^h=1-\\overline H{}_{1-1_A}^h\n=1-\\overline H{}_{1_{\\Delta(U)\\setminus A}}^h\\\\\n&=&\\!\\!\\!\\overline H{}_{1_A}^h\n=\\frac1h\\widehat R{}_h^A=\\frac{1}{h}\\int_AK(.,Y)d\\mu_h(Y),\n\\end{eqnarray*}\nso that indeed $1_A$ is $h$-resolutive and (3.3) holds.\n\\end{proof}\n\nFor any function $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ we define\n$f(Y)K(x,Y)=0$ at points $(x,Y)$ where $f(Y)=0$ and $K(x,Y)=+\\infty$.\nIf $f$ is $\\mu_h$-measurable then so is $Y\\longmapsto f(Y)K(x,Y)$ for each\n$x\\in U$ because $K(x,Y)>0$ is $\\mu_h$-measurable (even l.s.c.) as a function\nof $Y\\in\\Delta(U)$ according to \\cite[Proposition 2.2 (i)]{EF1}.\n\n\\begin{prop}\\label{prop6.5} Let $f$ be a $\\mu_h$-measurable\nlower bounded function on $\\Delta(U)$. Then\n$$\n{\\overline H}{}_f^h=\\frac{1}{h}\\int f(Y)K(.,Y) d\\mu_h(Y)>-\\infty,\n$$\nand ${\\overline H}{}_f^h$ is either identically $+\\infty$ or the sum of\nan $h$-invariant function and a constant $\\le0$.\nFurthermore $f$ is $h$-quasi\\-resolutive if and only if $f$ is $h$-resolutive,\nand that holds if and only if\n$\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y)<+\\infty$\nq.e.\\ on $U$, or equivalently: everywhere on $U$.\nIn particular, every bounded $\\mu_h$-measurable function\n$f:\\Delta(U)\\longrightarrow{\\mathbb R}$ is $h$-resolutive.\n\\end{prop}\n\n\\begin{proof} Consider first the case of a positive $\\mu_h$-measurable\nfunction $f$. Then $f$ is the pointwise supremum of an\nincreasing sequence of positive $\\mu_h$-measurable step\nfunctions $f_j$ (that is, finite valued functions $f_j$ taking only finitely\nmany values, each finite and each on some $\\mu_h$-measurable set; in other\nwords: affine combinations of indicator functions of $\\mu_h$-measurable sets).\nFor any index $j$ it follows by Proposition \\ref{prop6.4} and by 1.\\ and 2.\\\nin Proposition \\ref{prop6.3} (the latter extended to finite sums\nand with `$h$-resolutive' throughout in place of\n`$h$-quasi\\-resolutive', cf.\\ the paragraph preceding Proposition\n\\ref{prop6.3}) that each $f_j$ is $h$-resolutive and that\n$$\nH_{f_j}^h=\\frac{1}{h}\\int f_j(Y)K(.,Y)d\\mu_h(Y)\n$$\non $U$, whence\n\\begin{eqnarray*}0\\!\\!\\!\n&\\le&\\!\\!\\!\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y)\\\\\n&=&\\!\\!\\!\\frac1{h}\\sup_j\\int f_j(Y)K(.,Y)d\\mu_h(Y)\n=\\sup_j H_{f_j}^h={\\overline H}{}_f^h\n\\end{eqnarray*}\nby 4.\\ in Proposition \\ref{prop6.2}.\nFor a general lower bounded $\\mu_h$-measurable function $f$ on $\\Delta(U)$\nthere is a constant $c\\ge0$ such that\n$g:=f+c\\ge0$ and hence $\\overline {}H_g^h=\\overline {}H_f^h+c\\ge0$.\nIt follows that\n$$\n{\\overline H}{}_f^h=\\frac1h\\int g(Y)K(.,Y)d\\mu_h(Y)-c\n=\\frac1h\\int f(Y)K(.,Y)d\\mu_h(Y)>-\\infty\n$$\nand hence by Proposition \\ref{prop6.1} (c) applied to $g$ that\n${\\overline H}{}_f^h=\\overline H{}_g^h-c$ has the asserted form.\n\nNext, consider a bounded $\\mu_h$-measurable function $f$ on $\\Delta(U)$.\nAs just shown, we have\n$${\\overline H}{}_f^h=\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y)$$\nand the same with $f$ replaced by $-f$, whence\n${\\overline H}{}_f^h={\\underline H}{}_f^h$,\nfinite valued because $f$ is bounded. Thus $f$ is $h$-resolutive.\nLet $c\\ge0$ be a constant such that $|f|\\le c$.\nThen ${\\overline H}{}_f^h=c-{\\underline H}{}_{c-f}^h$\nwhich is finely $h$-harmonic because $c-f\\ge0$ and so $H_{c-f}^h$ is\n$h$-invariant by Proposition \\ref{prop6.1} (c) and\nhence finely $h$-harmonic, being finite valued.\n\nReturning to a general lower bounded $\\mu_h$-measurable function $f$,\nsuppose first that $f$ is $h$-quasi\\-resolutive. Then, as shown in the first\nparagraph of the proof,\n$\\frac1h\\int f(Y)K(.,Y)d\\mu_h(Y)={\\overline H}_f^h$ is finite q.e.\\ on $U$.\nConversely, if $\\frac1h\\int f(Y)K(.,Y)d\\mu_h(Y)<+\\infty$ q.e.,\nthat is $\\overline H{}_f^h\\not\\equiv+\\infty$,\nthen Corollary \\ref{cor4.8} applies to the increasing\nsequence of bounded $\\mu_h$-measurable and hence $h$-resolutive functions\n$f\\wedge j$ converging to $f$, and we conclude that $f$ is $h$-resolutive\n(in particular $h$-quasi\\-resolutive) and hence that\n$\\frac1h\\int f(Y)K(.,Y)d\\mu_h(Y)={\\overline H}_f^h$ is finite everywhere on $U$.\n\\end{proof}\n\n\\begin{cor}\\label{cor4.10a} Let $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$\nbe $\\mu_h$-measurable. Then $f$ is $h$-resolutive\nif and only if $|f|$ is $h$-resolutive.\n\\end{cor}\n\n\\begin{proof} If $f$ is $h$-resolutive, and therefore $h$-quasiresolutive by\nLemma \\ref{lemma3.3}, then $|f|=f\\vee(-f)$ is\n$h$-quasi\\-resolutive according to 3.\\ and 1.\\ in Proposition \\ref{prop6.3}.\nSince $|f|$ is lower bounded (and $\\mu_h$-measurable) then by Proposition\n\\ref{prop6.5} $|f|$ is even $h$-resolutive and $|f|K(x,.)$ is\n$\\mu_h$-integrable for every $x\\in U$. So are therefore $f^+K(x,.)$ and\n$f^-K(x,.)$, and it follows, again by Proposition \\ref{prop6.5},\nthat $f^+$ and $f^-$ are $h$-resolutive.\nSo is therefore $f=f^+-f^-$ by Lemma \\ref{lemma6.3c}.\n\\end{proof}\n\n\\begin{prop}\\label{prop6.10b} Every $h$-quasi\\-resolutive\nfunction $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ is $\\mu_h$-measurable.\n\\end{prop}\n\n\\begin{proof}\nWe begin by proving this for $f=1_A$, the indicator function of a subset $A$ of\n$\\Delta(U)$, cf.\\ \\cite[p.\\ 113]{Do}. Clearly, $\\dot H{}_f^h$ and\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h(x)$ have their\nvalues in $[0,1]$, and hence $\\overline H{}_f^h=\\dot H{}_f^h$ and\n$\\underline H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$\naccording to Proposition \\ref{prop6.1} (c).\nSince $\\overline{\\cal U}{}_f^h$ is lower directed there is a decreasing\nsequence of functions $u_j\\in\\overline{\\cal U}{}_f^h$ such that\n$\\overline H{}_f^h(x_0)=\\inf_ju_j(x_0)$. Replacing $u_j$ by\n$u_j\\wedge1\\in\\overline{\\cal U}{}_f^h$ we arrange that $u_j\\le1$.\nDenote by $g_j$ the function defined on ${\\overline U}$ by\n$$g_j(Y)=\\underset{z\\to Y,\\,z\\in U}{\\liminf}u_j(z)$$ for any\n$Y\\in {\\overline U}.$\nClearly, $g_j$ is  l.s.c.\\ on ${\\overline U}$ and $1_A\\le g_j\\le1$\non $\\Delta(U)$. Write $f_2=\\inf_jg_j$ (restricted to $\\Delta(U)$).\nThen $f_2$ is Borel measurable and\n$1\\ge f_2\\ge f=1_A$, whence $\\overline H{}_{f_2}^h\\ge\\overline H{}_f^h$. For the\nopposite inequality note that $u_j\\in\\overline{\\cal U}{}_{f_2}^h$ because\n$g_j\\ge f_2$. Hence\n$\\overline H{}_f^h=\\inf_ju_j\\ge\\overline H{}_{f_2}^h$ with equality at $x_0$.\nFurthermore, $\\overline H{}_f^h$ is invariant according to\nProposition \\ref{prop6.1} (c), and hence\n$\\overline H{}_{f_2}^h-\\overline H{}_f^h$ is positive and finely\n$h$-superharmonic on $U$. Being $0$ at $x_0$ it is identically $0$, and so\n$\\overline H{}_{f_2}^h=\\overline H{}_f^h$. Similarly there is\na positive Borel measurable function $f_1\\le f$ such that\n$\\underline H{}_{f_1}^h=\\underline H{}_f^h$. Clearly, $0\\le f_1\\le f_2\\le1$.\nSince $f$ is $h$-quasi\\-resolutive\nwe obtain from Proposition \\ref{prop6.1} (a) q.e.\\ on $U$\n$$\nH_f^h=\\underline H{}_{f_1}^h\\le\\overline H{}_{f_1}^h\\le H_f^h\n\\le\\underline H{}_{f_2}^h\\le\\overline H{}_{f_2}^h=H_f^h,\n$$\nthus with equality q.e.\\ all through. Hence $f_1$ and $f_2$ are\n$h$-quasi\\-resolutive, and so is therefore $f_2-f_1$ by 1.\\ and 2.\\ in\nProposition \\ref{prop6.3}, which also shows that $H_{f_2-f_1}^h=H_{f_2}-H_{f_1}=0$\nq.e. Because $f_2-f_1$ is positive and Borel measurable on $\\Delta(U)$ it\nfollows by Proposition \\ref{prop6.5} that\n$\\frac1h\\int(f_2(Y)-f_1(Y))K(.,Y)d\\mu_h(Y)=0$,\nand hence $f_1=f_2$ $\\mu_h$-a.e.\nIt follows that $f=1_A$ is $\\mu_h$-measurable, and so is therefore $A$.\n\nNext we treat the case of a finite valued $h$-quasi\\-resolutive function $f$ on\n$\\Delta(U)$. Adapting the proof given in \\cite[p.\\ 115]{Do} in the classical\nsetting we consider the space $\\cal C(\\overline{\\mathbb R},{\\mathbb R})$ of continuous (hence\nbounded) functions $\\overline{\\mathbb R}\\to{\\mathbb R}$, and denote by $\\Phi$ the space of\nfunctions $\\phi\\in\\cal C(\\overline{\\mathbb R},{\\mathbb R})$ such that $\\phi\\circ f$ is\n$h$-quasi\\-resolutive.\nIn view of Proposition \\ref{prop6.3}, $\\Phi$ is a vector lattice,\nclosed under uniform convergence because $|\\phi_j-\\phi|<\\varepsilon$ implies\n$|H_{\\phi_j\\circ f}^h-\\overline H{}_{\\phi\\circ f}^h|\\le\\varepsilon$ and\n$|H_{\\phi_j\\circ f}^h-\\underline H_{\\phi\\circ f}^h|\\le\\varepsilon$ on $U\\setminus E^h_f$,\nand so $|\\overline H{}_{\\phi\\circ f}^h-\\underline H{}_{\\phi\\circ f}^h|\\le2\\varepsilon$\non $U\\setminus E^h_f$. We infer that\n$\\overline H{}_{\\phi\\circ f}^h=\\underline H{}_{\\phi\\circ f}^h$ (finite values) q.e.\\\non $U$, and so $\\phi\\circ f$ is indeed resolutive.\nFurthermore, $\\Phi$ includes the fuctions $\\phi_n:t\\longmapsto(1-|t-n|)\\vee0$ on\n${\\mathbb R}$ for integers $n\\ge1$, again by Proposition \\ref{prop6.3}. These functions\nseparate points of ${\\mathbb R}$. In fact, for distinct $s,t\\in{\\mathbb R}$, say $s<t$, take\n$n=[s]$ (that is, $n\\le s<n+1$). If also $n\\le t<n+1$ then clearly\n$\\phi_n(t)<\\phi_n(s)\\le1$, and in the remaining case $t\\ge n+1$ we have\n$\\phi_n(t)=0<\\phi_n(s)$. It therefore follows by the lattice version of the\nStone-Weierstrass theorem that $\\Phi=\\cal C(\\overline{\\mathbb R},{\\mathbb R})$.\nNext, the class $\\Psi$ of (not necessarily continuous) functions\n$\\psi:{\\mathbb R}\\longrightarrow{\\mathbb R}$ for which $\\psi\\circ f$ is $h$-quasi\\-resolutive\nis closed under bounded monotone convergence, by Corollary \\ref{cor4.8}\n(adapted to a bounded monotone convergent sequence of functions $f_j$).\nAlong with the continuous functions ${\\mathbb R}\\longmapsto{\\mathbb R})$, $\\Psi$ therefore\nincludes every bounded Borel measurable function ${\\mathbb R}\\longmapsto{\\mathbb R}$.\nIn particular, the indicator function $1_J$ of an interval $J\\subset{\\mathbb R}$\nbelongs to $\\Psi$, and hence $1_J\\circ f$ is $h$-quasi\\-resolutive.\nWe conclude by the first part of the proof that the $h$-quasi\\-resolutive\nindicator function $1_J\\circ f=1_{f^{-1}(J)}$ is $\\mu_h$-measurable.\n\nFinally, for an arbitrary $h$-quasi\\-resolutive function\n$f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$, write $A_+:=\\{f=+\\infty\\}$ and\n$A_-:=\\{f=-\\infty\\} $. By 3.\\ in Proposition \\ref{prop6.3}, $f\\vee0$ is\n$h$-quasi\\-resolutive and\n$$\n\\overline H{}_{(+\\infty)1_{A_+}}^h=\\overline H{}_{f1_{A_+}}^h\n\\le\\overline H{}_{f\\vee0}^h=\\dot H{}_{f\\vee0}^h<+\\infty\n$$\non $U\\setminus(E^h_f\\cup P_f^h)$, cf.\\ the opening of the present section.\nIt follows that $\\overline H{}_{1_{A_+}}=0$ q.e.\\ on $U$\nand hence $\\overline H{}_{(+\\infty)1_{A_+}}^h=0$ q.e.\\ according to 3.\\ in\nProposition \\ref{prop6.2}. Since $-f$ likewise is $h$-quasi\\-resolutive\nwe have $\\underline H{}_{(+\\infty)1_{A_-}}^h=0$ q.e. Furthermore,\n$0\\le\\underline H{}_{(+\\infty)1_{A_+}}\\le\\overline H{}_{(+\\infty)1_{A_+}}=0$ q.e.\\ by\nProposition \\ref{prop6.1} (a), and similarly $\\overline H{}_{(+\\infty)1_{A_-}}=0$\nq.e. Writing $A=A_+\\cup A_-=\\{|f|=+\\infty\\}$ we infer from 2.\\ in\nProposition \\ref{prop6.3} that $1_A=1_{A_+}+1_{A_-}$ is $h$-quasi\\-resolutive,\nand hence so is\n$(+\\infty)1_A$ in view of Corollary \\ref{cor4.8}. As shown in the first\nparagraph of the proof it follows that $A$ is $\\mu_h$-measurable.\nDefine $g:\\Delta(U)\\longrightarrow{\\mathbb R}$ by $g=f$ except that $g=0$ on\n$A=\\{|f|=+\\infty\\}$. Then $g\\le f+(+\\infty)1_A$ and $f\\le g+(+\\infty)1_A$, and\nhence by Proposition \\ref{prop6.2} (recall that $\\overline H{}_{1_{A_+}}=0$\nq.e.\\ on $U$)\n$$\n\\overline H{}_g^h\\le H_f^h+\\overline H{}_{(+\\infty)1_A}^h=H_f^h\n\\le\\underline H{}_g^h+\\underline H{}_{(+\\infty)1_A}^h=\\underline H{}_g^h\n$$\nquasieverywhere, actually with equalities q.e., and hence $g$ is\n$h$-quasi\\-resolutive along with $f$.\nAs established in the beginning of the proof, $g$ is $\\mu_h$-measurable and so\nis $f$. In fact, $\\{f\\ne g\\}=A$ is $\\mu_h$-measurable with $\\mu_h(A)=0$ by the\nfollowing corollary.\n\\end{proof}\n\n\\begin{cor}\\label{cor6.10c} {\\rm{(Cf.\\ \\cite[p.\\ 114]{Do}.)}}\nA set $A\\subset\\Delta(U)$ is $h$-harmonic null\nif and only if $A$ is $\\mu_h$-measurable with $\\mu_h(A)=0$.\n\\end{cor}\n\n\\begin{proof} If $A$ is $\\mu_h$-measurable with $\\mu_h(A)=0$ then\n$\\overline H{}_{1_A}^h=\\frac1h\\int1_AK(.,Y)d\\mu_h(Y)=0$ according to Proposition\n\\ref{prop6.5}. Conversely, if $\\overline H{}_{1_A}^h=0$ then\n$0\\le\\underline H{}_{1_A}^h\\le\\overline H{}_{1_A}^h=0$ and so $1_A$ is\n$h$-resolutive. It follows by Proposition \\ref{prop6.10b} that $1_A$\nis $\\mu_h$-measurable, and again by Proposition \\ref{prop6.5} that\n$\\frac1h\\int1_AK(.,Y)d\\mu_h(Y)=\\overline H{}_{1_A}^h=0$. Since $K(x,.)>0$ is\nl.s.c.\\ by \\cite[Proposition 3.2]{EF1} we conclude that $\\mu_h(A)=0$.\n\\end{proof}\n\n\\begin{cor}\\label{cor6.3b} A function $f$ on $\\Delta(U)$ with values in\n$\\overline{\\mathbb R}$ is $h$-resolutive if and only if $f$ is $h$-quasi\\-resolutive.\n\\end{cor}\n\n\\begin{proof} The `only if' part is contained in Lemma \\ref{lemma3.3}.\nFor the `if' part, suppose that $f$ is $h$-quasi\\-resolutive and hence\n$\\mu_h$-measurable, by Proposition \\ref{prop6.10b}. If $f\\ge0$ then\n$f$ is $h$-resolutive according to Proposition \\ref{prop6.5}.\nFor arbitrary $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ this applies to\n$f^+$ and $f^-$, which are $h$-quasi\\-resolutive\naccording to 3.\\ in Proposition \\ref{prop6.3}. Consequently, $f=f^+-f^-$\nis likewise finely $h$-resolutive by 1.\\ and 2.\\ in Proposition \\ref{prop6.3}.\n\\end{proof}\n\n\\begin{theorem}\\label{thm6.8}\nA function $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ is\n$h$-resolutive if and only if the\nfunction $Y\\longmapsto f(Y)K(x,Y)$ on $\\Delta(U)$ is $\\mu_h$-integrable\nfor quasievery $x\\in U$. In the affirmative  case $Y\\longmapsto f(Y)K(x,Y)$\nis $\\mu_h$-integrable for every $x\\in U$, and we have everywhere on $U$\n$$\nH{}_f^h=\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y).\n$$\n\\end{theorem}\n\n\\begin{proof} Suppose first that $f$ is $h$-resolutive.\nBy Proposition \\ref{prop6.10b} $f$ is then $\\mu_h$-measurable.\nAccording to 3.\\ in Proposition \\ref{prop6.3} the function $|f|$ is also\n$h$-resolutive, and it follows by Proposition \\ref{prop6.5} that\n$\\overline H{}_{|f|}^h(x)=\\frac1h\\int |f(Y)|K(x,Y)d\\mu_h(Y)<+\\infty$\nfor every $x\\in U$. Conversely, suppose that $fK(x,.)$ is $\\mu_h$-integrable\nfor quasi\\-every\n$x\\in U\\setminus E$. For any $x\\in U$, $K(x,.)>0$ is l.s.c.\\ and hence\n$\\mu_h$-measurable on $\\overline U$ by \\cite[Proposition 3.2]{EF1}, and so\n$f^+$ and $f^-$ must be $\\mu_h$-measurable. By Proposition \\ref{prop6.5},\n$f^+$ and $f^-$ are therefore $h$-quasi\\-resolutive, that is $h$-resolutive\nby Corollary \\ref{cor6.3b}, and so is therefore\n$f=f^+-f^-$ by 1.\\ and 2.\\ in Proposition \\ref{prop6.3}.\n\\end{proof}\n\n\\begin{remark}\\label{remark6.12}\nIn the case where $U$ is Euclidean open it\nfollows by the Harnack convergence theorem for harmonic functions (not\nextendable to finely harmonic functions) for any numerical function\n$f$ on $\\Delta(U)$ that $\\overline H{}_f^h$ is\n$h$-hyper\\-harmonic on $U$ (in particular $>-\\infty$) and hence equal to\n$\\dot H{}_f^h$ (except if $\\overline H{}_f^h\\equiv-\\infty$). It follows\nby Proposition \\ref{prop6.1} (a) that\n$\\overline H{}_f^h=\\dot H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n=\\underline H{}_f^h$. If $f$ is resolutive then\nby Definition \\ref{def6.9} equality prevails on all of $U$.\nSumming up, Theorem \\ref{thm6.8} is a (proper) extension of\nthe corresponding classical result, cf.\\ e.g.\\ \\cite[Theorem 1.VIII.8]{Do}.\n\\end{remark}\n\nWe close this section with a brief discussion of an alternative, but\ncompatible concept of $h$-resolutivity based on the minimal-fine\n(mf) topology, cf.\\ \\cite[Definition 3.4]{EF2}. The mf-closure of $U$\nis $U\\cup\\Delta_1(U)$, and the relevant boundary functions $f$ are therefore\nnow defined only on the mf-boundary $\\Delta_1(U)$.\n\nGiven a function $f$ on $\\Delta_1(U)$ with values in ${\\overline {\\mathbb R}}$,\na finely $h$-hyper\\-harmonic function $u$ on $U$ is now said\nto belong to the upper PWB$^h$ class, denoted again by\n${\\overline {\\cal U}}{}_f^h$, if $u$ is lower bounded and if\n$$\\underset{x\\to Y,\\,x\\in U}{\\mfliminf}\\,u(x)\\ge f(Y)\n\\quad\\text{ for every }\\;Y\\in \\Delta_1(U).$$\nThis leads to new, but similarly denoted concepts\n $$\\dot H{}_f^h=\\inf{\\overline{\\cal U}}{}_f^h,\n\\quad{\\overline H}{}_f^h=\\widehat{\\dot H{}_f^h}\n={\\widehat\\inf}\\,{\\overline {\\cal U}}{}_f^h\\;(\\le\\dot H{}_f^h),$$\nand hence new concepts of $h$-quasi\\-resolutivity and $h$-resolutivity.\n\nWhen considering reduction $R_u^A$ and sweeping $\\widehat R{}_u^A$ of a\nfinely $h$-hyperharmonic function $u$ on $U$ onto a set\n$A\\subset\\overline U$ we similarly use the alternative, though actually\nequivalent mf-versions\n\\cite[Definition 3.14]{EF2}, cf.\\ \\cite[Theorem 3.16]{EF2}.\nThis occurs in the proof of Proposition \\ref{prop6.4} (after the first\ndisplay), where now $W\\subset\\overline U$ is mf-open (and contains $A$).\n\nThe changes as compared with the case of $h$-resolutivity relative to the\nnatural topology are chiefly as follows. A set $A\\subset U\\cup\\Delta_1(U)$\nis of course now said to be $h$-harmonic null if $\\overline H{}_{1_A}^h=0$\n(with the present mf-version of $\\overline H$). In Proposition\n\\ref{prop6.1} (b) we apply \\cite[Proposition 3.12]{EF2} in place of its\ncorollary. In the beginning of the proof of Proposition \\ref{prop6.10b}\nthe function $g_j$ shall now be defined at $Y\\in\\Delta_1(U)$ by\n$$\ng_j(Y)=\\underset{z\\to Y,\\,z\\in U}{\\mfliminf}\\,u_j(z).\n$$\nAnd $g_j$ is $\\mu_h$-measurable on $\\Delta_1(U)$ because $g_j$ equals\n$\\mu_h$-a.e.\\ the $\\mu_h$-measurable function defined $\\mu_h$-a.e.\\ on\n$\\Delta_1(U)$ by\n$Y\\longmapsto\\mflim_{z\\to Y,\\,z\\in U}\\,u_j(z)=\\frac{d\\mu_{u_j}}{d\\mu_h}(Y)$\naccording to the version of the\nFatou-Na{\\\"i}m-Doob theorem established in \\cite[Theorem 4.5]{EF1}. Here\n$\\mu_{u_j}$ denotes the representing measure for $u_j$, that is $K\\mu_{u_j}=u_j$,\nand $d\\mu_{u_j}\/d\\mu_h$ denotes the Radon-Nikod\\'ym derivative of the\n$\\mu_h$-continuous part of $\\mu_{u_j}$ (carried by $\\Delta_1(U)$) with respect\nto $\\mu_h$. The $\\mu_h$-measurability of $g_j$ on $\\Delta_1(U)$ thus established\nis all that is needed for the proof of the mf-version of Proposition\n\\ref{prop6.10b}, replacing mostly $\\Delta(U)$ with $\\Delta_1(U)$.\n\nThe following result is established like Theorem \\ref{thm6.8}\n\n\\begin{theorem}\\label{thm6.8a} A function\n$f:\\Delta_1(U)\\longrightarrow\\overline{\\mathbb R}$ is $h$-resolutive relative to the\nmf-topology if and only if the\nfunction $Y\\longmapsto f(Y)K(x,Y)$ on $\\Delta_1(U)$ is $\\mu_h$-integrable\nfor quasievery $x\\in U$. In the affirmative  case $Y\\longmapsto f(Y)K(x,Y)$\nis $\\mu_h$-integrable for every $x\\in U$, and we have everywhere on $U$\n$$\nH{}_f^h=\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y).\n$$\n\\end{theorem}\n\n\\begin{cor}\\label{cor6.8b} For any $h$-resolutive function\n$f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ relative to the natural topology,\nthe restriction of $f$ to $\\Delta_1(U)$ is resolutive relative to the\nmf-topology. Conversely, for any $h$-resolutive function\n$f:\\Delta_1(U)\\longrightarrow\\overline{\\mathbb R}$ relative to the mf-topology,\nthe extension of $f$ by $0$ on $\\Delta(U)\\setminus\\Delta_1(U)$ is\n$h$-resolutive relative to the natural topology.\n\\end{cor}\n\n\n\\section{Further equivalent concepts of $h$-resolutivity}\\label{sec4}\n\nWe again consider functions $f:\\Delta(U)\\longrightarrow\\overline{{\\mathbb R}}$.\nWe show that the equivalent concepts of $h$-resolutivity and\n$h$-quasiresolutivity do not alter when $\\overline H{}_f^h,\\underline H{}_f^h$\nin Definitions \\ref{def6.9}, \\ref{def2.3} are\nreplaced by $\\dot H{}_f^h$ and\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$,\nrespectively. Recall from Proposition \\ref{prop6.1} (c) that\n$\\dot H{}_f^h=\\overline H{}_f^h$ if $\\dot H{}_f^h<+\\infty$. This applies,\nin particular, to the indicator function $1_A$ of a set $A\\subset\\Delta(U)$.\nTherefore the ``dot''-version of the concept of an $h$-harmonic null set\n$A$ is identical with the version considered in Definition \\ref{def4.1}.\n\n\\begin{definition}\\label{def4.0} A function $f$ on $\\Delta(U)$ with values in\n${\\overline {\\mathbb R}}$ is said to be $h$-dot-resolutive if $\\dot H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$ on $U$\nand if this function, also denoted by $H_f^h$,\nis neither identically $+\\infty$ nor identically $-\\infty$.\n\\end{definition}\n\nFor any function $f:\\Delta(U)\\longrightarrow{\\mathbb R}$ we consider the following\nsubset of $U$:\n$$\n\\dot E{}_f^h=\\{\\dot H{}_f^h=-\\infty\\}\\cup\n\\{{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h=+\\infty\\}\n\\cup\\{\\dot H{}_f^h\\ne\n{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\\}.\n$$\n\n\\begin{lemma}\\label{lemma4.3} A function $f$ on $\\Delta(U)$ with values in\n${\\overline {\\mathbb R}}$ is $h$-quasi\\-resolutive if and only if $f$ is\n$h$-dot-quasi\\-resolutive in the sense that $\\dot E{}_f^h$ is polar, or\nequi\\-valently that the relations\n$\\dot H{}_f^h>-\\infty$,\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h<+\\infty$, and\n$\\dot H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$\nall hold quasi\\-everywhere on $U$.\n\\end{lemma}\n\n\\begin{proof} If these three relations hold q.e.\\ on $U$ then analogously\n$\\overline H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n=\\dot H{}_f^h>-\\infty$ q.e.\\ and similarly\n$\\underline H{}_f^h<+\\infty$ q.e. But $\\overline H{}_f^h=\\dot H{}_f^h>-\\infty$ q.e.\\\non $\\{\\overline H{}_f^h>-\\infty\\}$, hence also q.e.\\ on $U$. Similarly,\n$\\underline H{}_f^h<+\\infty$ q.e.\\ on $U$, and altogether\n$\\overline H{}_f^h=\\underline H{}_f^h$ q.e.\\ on $U$. Thus $f$ is\nquasi\\-resolutive. The converse is obvious.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma4.4} Every $h$-dot-resolutive function $f$ is\n$h$-resolutive, and hence $h$-quasi\\-resolutive (now also termed\n$h$-dot-quasi\\-resolutive).\n\\end{lemma}\n\n\\begin{proof} Suppose that $f$ is $h$-dot-resolutive then $f$. Then $f$ is\n$h$-resolutive, for $\\dot H{}_f^h,\\overline H{}_f^h,\\underline H{}_f^h$, and\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$\nare all equal because there is equality in the general inequalities\n$\\dot H{}_f^h\\ge\\overline H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$ and\n$\\dot H{}_f^h\\ge\\underline H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$,\ncf.\\ Proposition \\ref{prop6.1} (a). The rest follows from Lemma \\ref{lemma3.3}.\n\\end{proof}\n\nIn view of Lemma \\ref{lemma4.4}, an $h$-(dot-)quasi\\-resolutive function\nis $h$-dot-resolutive if and only if $\\dot E_f^h=\\varnothing$.\nAssertions 1.\\ and 2.\\ of Proposition \\ref{prop6.3} therefore remain valid\nwhen $E$ is replaced throughout by $\\dot E$.\nThe proof of the dot-version of eq.\\ (3.1) uses 1.\\ of Proposition\n\\ref{prop6.2} in place of 2.\\ there.\nThe following lemma is analogous to Lemma \\ref{lemma6.3c}:\n\n\\begin{lemma}\\label{lemma4.5}  Let $f$ be an $h$-quasi\\-resolutive function\non $\\Delta(U)$. If $f^+$ and $f^-$ are $h$-dot-resolutive then so is $f$,\nand the function\n$H_f^h=\\dot H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$\non $U$ is finite valued.\n\\end{lemma}\n\n\\begin{proof} Since $f$ is $h$-quasi\\-resolutive, so are $f^+,f^-$\n(besides being $h$-dot-resolutive) by 3.\\ in Proposition \\ref{prop6.3}.\nHence the functions $H_{f^{\\pm}}^h=\\dot H{}_{f^{\\pm}}^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_{f^{\\pm}}^h$\nare finite valued (co-polar subets of $U$ being non-void). From\n$-f^-\\le f\\le f^+$ it therefore follows by Proposition \\ref{prop6.1} (a) that\n$$-\\infty\n<-{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_{f^-}^h\n\\le{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n\\le\\dot H{}_f^h\\le\\dot H{}_{f^+}^h<+\\infty.$$\nApplying 1.\\ in Proposition \\ref{prop6.2} to the sums\n$f=f^++(-f)^+=f^+-f^-$ and $-f=f^--f_+$, which are both well defined on\n$\\Delta(U)$, we obtain\n$$\n\\dot H{}_f^h\\le H_{f^+}^h-H_{f^-}^h\n\\le{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n$$\non all of $U$. It follows that $\\dot H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n=H_{f^+}^h-H_{f^-}^h$ holds there, again by Proposition \\ref{prop6.1} (a).\n\\end{proof}\n\n\\begin{cor}\\label{cor4.8a} Let $(f_j)$ be an increasing sequence of lower\nbounded $h$-dot-resolutive functions\n$\\Delta(U)\\longrightarrow\\,]-\\infty,+\\infty]$, and let $f=\\sup_jf_j$.\nIf $\\dot H{}_f^h\\not\\equiv+\\infty$ then $f$ is $h$-dot-resolutive.\n \\end{cor}\n\n\\begin{proof} For every $j$ we have\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_{f_j}^h\n=\\dot H{}_{f_j}^h$, and hence\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n\\ge\\sup_j\\dot H{}_{f_j}^h=\\dot H{}_f^h$\naccording to 4.\\ in Proposition \\ref{prop6.2}.\nHere equality prevails on account of Proposition \\ref{prop6.1} (a).\nBy hypothesis, $H_f^h\\not\\equiv+\\infty$, and clearly $H_f^h>-\\infty$, so we\nconclude that $f$ indeed is $h$-dot-resolutive.\n\\end{proof}\n\n\\begin{prop}\\label{prop6.4a} For any $\\mu_h$-measurable subset $A$\nof $\\Delta(U)$ the indicator function $1_A$ is $h$-dot-resolutive and\n{\\rm{(3.3)}} holds.\nIn particular, the constant function $1$ on $\\Delta(U)$ is\n$h$-dot-resolutive and $H_1^h=1$.\n\\end{prop}\n\n\\begin{proof} Since $\\dot H{}_{1_A}^h\\le\\dot H_1^h=1<+\\infty$ it follows from\nProposition \\ref{prop6.1} (c) that $\\overline H{}_{1_A}^h=\\dot H{}_{1_A}^h$,\nand the assertions reduce to the analogous Proposition \\ref{prop6.4}.\n\\end{proof}\n\n\\begin{prop}\\label{prop6.5a} Let $f$ be a $\\mu_h$-measurable\nlower bounded function on $\\Delta(U)$. Then\n$$\n{\\dot H}{}_f^h=\\frac{1}{h}\\int f(Y)K(.,Y) d\\mu_h(Y)>-\\infty,$$\nand ${\\dot H}{}_f^h$ is either identically $+\\infty$ or the sum of\nan $h$-invariant function and a constant $\\le0$.\nFurthermore $f$ is $h$-quasi\\-resolutive if and only if $f$ is\n$h$-dot-resolutive,\nand that holds if and only if\n$\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y)<+\\infty$\nq.e.\\ on $U$, or equivalently: everywhere on $U$.\nIn particular, every bounded $\\mu_h$-measurable function\n$f:\\Delta(U)\\longrightarrow{\\mathbb R}$ is $h$-dot-resolutive.\n\\end{prop}\n\n\\begin{proof} In view of the case of $\\dot H$ in 4.\\ of Proposition\n\\ref{prop6.2} the proof of the analogous Proposition \\ref{prop6.5} carries over\n{\\it{mutatis mutandis}}.\n\\end{proof}\n\n\\begin{cor}\\label{cor6.10a} Let $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$\nbe $\\mu_h$-measurable. Then $f$ is $h$-dot-resolutive\nif and only if $|f|$ is $h$-dot-resolutive.\n\\end{cor}\n\n\\begin{proof} If $f$ is $h$-dot-resolutive, and therefore\n$h$-quasiresolutive by Lemma \\ref{lemma4.4}, then $|f|=f\\vee(-f)$ is\n$h$-quasi\\-resolutive according to 1.\\ and 3.\\ in Proposition \\ref{prop6.3}.\nNow, $|f|$ is lower bounded (and $\\mu_h$-measurable), and $|f|$ is therefore\neven $h$-dot-resolutive, by Proposition \\ref{prop6.5a}.\nConsequently, $|f|K(x,.)$ is $\\mu_h$-integrable for every\n$x\\in U$. So are therefore $f^+K(x,.)$ and $f^-K(x,.)$.\nAgain by Proposition \\ref{prop6.5a} it follows that $f^+$ and $f^-$ are\n$h$-dot-resolutive along with $f^+$ and $f^-$ (positive) by Lemma\n\\ref{prop6.5a}. So is therefore $f=f^+-f^-$ by Lemma \\ref{lemma4.5}.\n\\end{proof}\n\n\n\\begin{cor}\\label{cor4.6} A function $f$ on $\\Delta(U)$ with values in\n$\\overline{\\mathbb R}$ is $h$-dot-resolutive if and only if $f$ is\n$h$-quasi\\-resolutive.\n\\end{cor}\n\n\\begin{proof} The `only if' part is contained in Lemma \\ref{lemma4.4}.\nFor the `if' part, suppose that $f$ is $h$-quasi\\-resolutive and hence\n$\\mu_h$-measurable according to Proposition \\ref{prop6.10b}. If $f\\ge0$ then\n$f$ is $h$-dot-resolutive according to Proposition \\ref{prop6.5a}.\nFor arbitrary $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ this applies to\n$f^+$ and $f^-$, which are $h$-quasi\\-resolutive\naccording to 3.\\ in Proposition \\ref{prop6.3}. Consequently, $f=f^+-f^-$\nis likewise finely $h$-dot-resolutive by 1.\\ and 2.\\ in Proposition\n\\ref{prop6.3}.\n\\end{proof}\n\n\\begin{theorem}\\label{thm6.8c}\nA function $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ is\n$h$-dot-resolutive if and only if the\nfunction $Y\\longmapsto f(Y)K(x,Y)$ on $\\Delta(U)$ is $\\mu_h$-integrable  for\nevery $x\\in U$, or equivaletly for quasievery $x\\in U$.\nIn the affirmative case the solution of the PWB-problem\non $U$ with boundary function $f$ is\n$$H{}_f^h:=\\overline H{}_f^h=\\underline H{}_f^h=\\dot H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n=\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y).\n$$\n\\end{theorem}\n\n\\begin{proof} The proof of the analogous Theorem \\ref{thm6.8a} carries over\n{\\it{mutatis mutandis}}.\n\\end{proof}\n\nThe alternative concept of $h$-resolutivity relative to the mf-topology\ndiscussed at the end of the preceding section likewise has a similarly\nestablished compatible version based on $\\dot H{}_f^h$ and\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$\ninstead of $\\overline H{}_f^h$ and $\\underline H{}_f^h$.\n\n\n\\thebibliography{9}\n\n\\bibitem{Ai} Aikawa, H.:\\textit{Potential Analysis on non-smooth domains --\nMartin boundary and boundary Harnack principle}, Complex Analysis and\nPotential Theory, 235--253, CRM Proc. Lecture Notes 55, Amer. Math. Soc.,\nProvidence, RI, 2012.\n\n\\bibitem{Al} Alfsen, E.M.: \\textit{Compact Convex Sets and Boundary\nIntegrals}, Ergebnisse der Math., Vol. 57, Springer, Berlin, 2001.\n\n\\bibitem{AG} Armitage, D.H., Gardiner, S.J.: \\textit{Classical\nPotential Theory}, Springer, London, 2001.\n\n\\bibitem{Do} Doob, J.L.: \\textit{Classical Potential Theory and Its\nProbabilistic Counterpart}, Grundlehren Vol. 262, Springer, New York, 1984.\n\n\\bibitem{El1} El Kadiri, M.: \\textit{Sur la d\\'ecomposition de\nRiesz et la repr\\'esentation int\\'egrale des fonctions finement\nsurharmoniques}, Positivity {\\bf 4} (2000), no. 2, 105--114.\n\n\\bibitem{EF1} El Kadiri, M., Fuglede, B.: \\textit{Martin boundary of a\nfine domain and a Fatou-Naim-Doob theorem for finely super\\-harmonic\nfunctions}, Manuscript (2013).\n\n\\bibitem{EF2} El Kadiri, M., Fuglede, B.: \\textit{Sweeping at the Martin\nboundary of a fine domain}, Manuscript (2013).\n\n\\bibitem{F1} Fuglede, B.: \\textit{Finely Harmonic Functions}, Lecture Notes\nin Math. 289, Springer, Berlin, 1972.\n\n\\bibitem{F2} Fuglede, B.: \\textit{Sur la fonction de Green pour un\ndomaine fin}, Ann. Inst. Fourier \\textbf{25}, 3--4 (1975), 201--206.\n\n\\bibitem{F4} Fuglede, B.: \\textit{Integral representation of fine\n potentials}, Math. Ann. \\textbf{262} (1983), 191--214.\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCertain nonstandard Lagrangian and Hamiltonian dynamical systems \\cite{mukunda,arnold,Parra} encompass very interesting classes of nonlinear oscillators and admit fascinating dynamical properties \\cite{musie,cari,jose} such as   isochronous oscillations,  linearization, nonlocal transformations and so on \\cite{CSL:02,Nucci,cmee,ChML,Ch:02,EAAR,Chandru;jPA}.   In particular, the Li\\'enard class of oscillators appear in the study of a wide range of fields such as seismology \\cite{cartwright}, biological regulatory systems \\cite{nicolas}, in the study of a self graviting stellar gas cloud \\cite{shapiro}, optoelectronics, fluid mechanics \\cite{kalashnik}.  Many of these Li\\'enard class of equations admit limit cycle\/periodic oscillations which are used to model many physical phenomenon.  Identifying such classes of coupled Li\\'enard-type equations admitting isochronous oscillations is an interesting area of research.  Several procedures have been developed to construct and identify classes of isochronous oscillators.  In particular, Calogero \\cite{Calogero:08} and Calogero and Leyvraz \\cite{Calogero:08c,Calogero:08c1,Calogero:08c2,Calogero:08d,Partha:0a} have developed many techniques to generate isochronous oscillators.  In a recent paper a procedure to generate scalar isochronous systems recursively from a given Hamiltonian \\cite{CDL:01} and a method to construct higher dimensional isochronous nonsingular Hamiltonian systems have  been discussed \\cite{DGCL:01}.  In this paper we obtain a class of coupled Li\\'enard type oscillator equations admitting isochronous oscillations by generalizing the nonstandard Lagrangian of the scalar system to coupled systems.  In order to do so let us consider the scalar Li\\'enard-type linear\/ quadratic and mixed type (in velocities) systems.  For example the Li\\'enard equation with linear velocity term,\n\\begin{eqnarray}\n\\ddot x+F(x)\\dot x+G(x)=0,\\label{meee}\n\\end{eqnarray}\nadmits a class of interesting nonstandard type conserved Hamiltonian \\cite{CSL:02} and isochronous solutions for the choice \\cite{ChML},\n\\begin{eqnarray}\nF(x)=3kx, \\quad G(x)=k^2x^3+\\omega^{2}x.\\label{mee}\n\\end{eqnarray}\nFor the choice $\\omega=0$ the above equation reduces to the modified Emden equation which is well studied in the literature and it occurs in the study of equilibrium configurations of a spherical cloud acting under the mutual attraction of its molecules and subject to the laws of thermodynamics \\cite{Ch:02} and in the modelling of the fusion of pellets \\cite{EAAR}. For the choice $\\omega\\ne0$ the above equation exhibits isochronous behaviour, that is it admits periodic oscillations with frequency of oscillation independent of the amplitude \\cite{CSL:02}.\nThe general solution of this equation is given as \n\\begin{eqnarray}\nx(t)=\\frac{A\\sin(\\omega t+\\delta)}{1-\\frac{kA}{\\omega}\\cos(\\omega t+\\delta)}.\n\\end{eqnarray}\nThe corresponding system admits a nonstandard Hamiltonian\n\\begin{eqnarray}\nH=\\frac{1}{2}\\hat{F}(p)x^2+U(p), \\label{Hammee}\n\\end{eqnarray}\nwith \n\\begin{eqnarray}\n\\hat{F}(p)=\\omega^2\\bigg(1-\\frac{2 k}{\\omega^2}p  \\bigg), \\quad U(p)=\\frac{\\omega^{4}}{2k^2}\\bigg(\\sqrt{1-\\frac{2k}{\\omega^2}p}   -1\\bigg)^2,\\label{Hammee2}\n\\end{eqnarray}\nwhere the canonically conjugate momentum\n\\begin{eqnarray}\np= \\frac{\\omega^2}{2k}\\bigg(1-\\frac{\\omega^{4}}{(k\\dot x+k^2x^2+\\omega^2)^2}  \\bigg) .\n\\end{eqnarray}\n\nSystem (\\ref{Hammee}) is \\emph{PT} symmetric $(x\\rightarrow-x$, $t\\rightarrow-t$, $\\dot {x}\\rightarrow \\dot {x})$ and is also exactly quantizable in momentum space \\cite{chithi}.\nThe class of nonlinear oscillators containing both quadratic and linear terms in $\\dot x$ is called a mixed Li\\'enard-type equation.  It can be written in the form \n\\begin{eqnarray}\n\\ddot x+J(x)\\dot x^2+F(x)\\dot x+G(x)=0,\\label{lienard2}\n\\end{eqnarray}\nwhere $F(x)$, $G(x)$ and $J(x)$ are functions of $x$.  For example with the choice\n \\begin{eqnarray}\nJ(x)=\\frac{f_x}{f},\\quad F(x)=\\frac{(r+2)h_x}{(r+1)f}, \\quad G(x)=\\frac{hh_x}{(r+1)f^2}, \\,r\\ne-1\\label{func}\n\\end{eqnarray}\nthe system admits a nonstandard Lagrangian and Hamiltonian functions of the form\n\\begin{eqnarray}\nL=\\frac{1}{(f(x)\\dot x+h(x))^r},\\label{scalag}\n\\end{eqnarray}\nwhere $f(x)$ and $h(x)$ are arbitrary functions of $x$, $r$ is a suitably chosen real positive parameter and the Hamiltonian\n\\begin{eqnarray}\nH=\\frac{p}{f}\\bigg(\\bigg(\\frac{-r f}{p}\\bigg)^{\\frac{1}{r+1}}-h   \\bigg)-\\bigg(\\frac{-r f}{p}  \\bigg)^{\\frac{-r}{r+1}},\n\\end{eqnarray}\nwith the conjugate momentum\n\\begin{eqnarray}\np=-rf(f\\dot x+h)^{-(r+1)}.\\label{scalagp}\n\\end{eqnarray}\nIn the above, $r$ is a real positive number so chosen such that $H$ is real and in (\\ref{scalag})-(\\ref{scalagp}) only the principal branch is taken when fractional powers appear.  Note that the MEE is a special case of the above system with  $f=1$, $r=1$ and \n $h=kx^2$.  We also note here that this class of Lagrangian and Hamiltonian systems are also drawing considerable interest in connection with supersymmetry related partner systems \\cite{curtright,bijan} and $\\emph PT$-symmetric systems \\cite{chithi}.  \n Now, it is of considerable importance to extend the study of above type nonstandard Lagrangian and Hamiltonian systems to higher degrees of freedom.  Particularly we wish to identify two dimensional nonstandard Hamiltonian systems which are isochronous and in future to study their exact quantization as in the case of MEE \\cite{chithi}.  \n\nA natural way of generalising the above one dimensional nonstandard Lagrangian is to modify \n(\\ref {scalag}) to the form, \n \\begin{eqnarray}\nL=\\frac{1}{(f\\dot x+g\\dot y+h)^r}\\label{arblag}\n\\end{eqnarray}\nwhere $f=f(x,y), g=g(x,y), h=h(x,y)$.\nIn this case, however the Lagrangian turns out to be singular or degenerate \\cite{Parra} \nwhich can be verified from the vanishing the Hessian,\n\\begin{eqnarray}\n\\Delta\\equiv\\left|\n\\begin{array}{ccc}\n\\frac{\\partial^2 L}{\\partial \\dot{x}^2}&\\frac{\\partial^2 L}{\\partial \\dot{x}\\partial \\dot{y}}\\\\\n\\frac{\\partial^2 L}{\\partial \\dot{y}\\partial \\dot{x}}&\\frac{\\partial^2 L}{\\partial \\dot{y}^2}\n\\end{array}\n\\right|\n=0.\n\\end{eqnarray}\nTo obtain a nonsingular Lagrangian, a modification of the above Lagrangian with suitable terms is essential.  For this purpose, we make a simple minded extension of the above form (\\ref{arblag}) judiciously and succeed to identify non-trivial coupled nonstandard and nonsingular Lagrangian type nonlinear evolution equations.  We then show how from the associated equations of motion one can obtain coupled mixed Li\\'enard-type equations, which may be considered as the natural extension of the above single degree of freedom mixed Li\\'enard-type equation (\\ref{lienard2}).  Then the procedure is specifically illustrated for the MEE equation to obtain a two dimensional isochronous extension of the MEE equation.  \n\n\nThe plan of the paper is as follows.  In section 2, we  introduce an  arbitrary form of nonsingular Lagrangian and use it to obtain the corresponding Newton's equation of motion.  However the resultant equation of motion is not in the  form of mixed Li\\'enard-type class of oscillators.  By demanding the resultant dynamical equations are of generalized coupled mixed Li\\'enard-type equations, we deduce the functional form of the allowed nonstandard Lagrangian.  In section 3, we show that the conditions obtained from the coupled mixed Li\\'enard-type oscillators allow one to deduce another class of Li\\'enard-type oscillators which exhibits periodic and quasiperiodic motions.  Also we have shown that the corresponding Hamiltonian obtained from the coupled nonsingular Lagrangian can be transformed into a two dimensional harmonic oscillator Hamiltonian through appropriate canonical transformations.  In section 4, we have shown that a special case of Li\\'enard-type of oscillators leads to a two dimensional version of  the modified Emden equation which exhibits isochronous property.  \nThe general solution is shown to admit periodic as well as quasiperiodic solutions for suitable choices of parameters. Finally, in section 5, we present our conclusions.  \n\n\\section{Two dimensional coupled mixed Li\\'enard-type equations}\n\nWe wish to identify a coupled mixed Li\\'enard-type system which possesses a nonstandard Hamiltonian and admits isochronous solution by a suitable generalization of the scalar Lagrangian (\\ref{scalag}) of MEE.  As noted earlier, the Lagrangian (\\ref{arblag}) which is a natural generalization in two dimensions is singular.  One can overcome this problem by suitably redefining the Lagrangian in the form,  \n\n\n\\begin{eqnarray}\nL=\\sum_{i=1}^{2}\\frac{1}{(f_{i}\\dot x+g_{i}\\dot y+h_{i})^{r_{i}}}.\\label{eq7}\n\\end{eqnarray}\nwhere $f_{i}=f_{i}(x,y)$, $g_{i}=g_{i}(x,y)$, $h_{i}=h_{i}(x,y)$, $i=1,2$.  With this choice of the Lagrangian we can show that Hessian in the present case is non-zero,\n\\begin{eqnarray}\n\\Delta\\equiv\\left|\n\\begin{array}{ccc}\n\\frac{\\partial^2 L}{\\partial \\dot{x}^2}&\\frac{\\partial^2 L}{\\partial \\dot{x}\\partial \\dot{y}}\\\\\n\\frac{\\partial^2 L}{\\partial \\dot{y}\\partial \\dot{x}}&\\frac{\\partial^2 L}{\\partial \\dot{y}^2}\n\\end{array}\n\\right|\n\\ne0.\n\\end{eqnarray}\nConsequently, the Lagrangian is nonsingular.\nFrom the above Lagrangian, the equation of motion can be obtained as  \n\\begin{eqnarray}\n&&\\ddot x= \\bigg(g_{2}(x,y)q_{1}(\\dot x,\\dot y,x,y)+g_{1}(x,y)q_{2}(\\dot x,\\dot y,x,y)  +  d_{1}(x,y)\\dot x^2\n\\nonumber\\\\\n&&\\hspace{0.5cm}+d_{2}(x,y)\\dot y^2+d_{3}(x,y)\\dot x \\dot y+ s_{1}(x,y)\\dot x+s_{2}(x,y)\\dot y+u_{1}(x,y)    \\bigg),\\label{seceq1}\\\\\n&&\\ddot y= -\\bigg(f_{2}(x,y)q_{1}(\\dot x,\\dot y,x,y)+f_{1}(x,y)q_{2}(\\dot x,\\dot y,x,y)+d_{4}(x,y)\\dot x^2 \n\\nonumber\\\\\n&&\\hspace{0.5cm}+d_{5}(x,y)\\dot y^2+d_{6}(x,y)\\dot x \\dot y+ s_{3}(x,y)\\dot x+ s_{4}(x,y)\\dot y+u_{2}(x,y)     \\bigg),\\label{seceq2}\\\\\n&&\\hspace{-1cm}\\mbox{where}\\nonumber\\\\ \n&&q_{1}=\\frac{(r_{2}+1)}{\\Delta^{2}r_{1}}\\bigg(r_{2}(f_{1}\\dot x+g_{1}\\dot y+h_{1})^{(r_{1}+2)} (f_{2}\\dot x+g_{2}\\dot y+h_{2})^{-(r_{2}+1)}\\nonumber\\\\\n&&\\hspace{0.5cm}\\times\\bigg(f_{2}h_{2y}-g_{2}h_{2x}+(f_{2y}-g_{2x})(\\dot x f_{2}+\\dot y g_{2})\\bigg)\\bigg),\\label{q1}\\\\\n&&q_{2}=\\frac{(r_{1}+1)}{\\Delta^{2}r_{2}}\\bigg(r_{1}(f_{1}\\dot x+g_{1}\\dot y+h_{1})^{-(r_{1}+1)} (f_{2}\\dot x+g_{2}\\dot y+h_{2})^{(r_{2}+2)}\\nonumber\\\\\n&&\\hspace{0.5cm}\\times\\bigg(f_{1}h_{1y}-g_{1}h_{1x}+(f_{1y}-g_{1x})(\\dot x f_{1}+\\dot y g_{1})\\bigg)\\bigg),\\label{q2}\n\\end{eqnarray}\n\\begin{eqnarray}\n&&d_{1}=\\frac{1}{\\Delta^{2}}\\bigg(f_{1}f_{2}(g_{1}(f_{2y}-g_{2x})(r_{1}+1)+g_{2}(f_{1y}-g_{1x})(r_{2}+1))\\nonumber\\\\\n&&\\hspace{0.5cm}+(r_{1}+1)(r_{2}+1)(g_{1}g_{2}(f_{2}f_{1x}+f_{1}f_{2x})-f_{2}f_{2x}g_{1}^2-f_{1}f_{1x}g_{2}^2) \\bigg),\\\\\n&&d_{2}=\\frac{1}{\\Delta^{2}}\\bigg(g_{1}^2(r_{1}+1)(g_{2}(f_{2y}-g_{2x})-f_{2}g_{2y}(r_{2}+1))+g_{2}^2(r_{2}+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(g_{1}(f_{1y}-g_{1x})-f_{1}g_{1y}(r_{1}+1))+g_{1}g_{2}(r_1+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(r_{2}+1)(f_{1}g_{2y}+f_{2}g_{1y})   \\bigg),\\\\\n&&d_{3}=\\frac{1}{\\Delta^{2}}\\bigg(g_{1}g_{2}(f_{2}(r_{2}+1)( f_{1y}(r_{1}+2)+g_{1x}r_{1})+f_{1}(r_{1}+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(f_{2y}(r_{2}+2)+g_{2x}r_{2}))-g_{1}^2f_{2}(r_{1}+1)(f_{2y}r_{2}+ g_{2x}(r_{2}+2))\\nonumber\\\\\n&&\\hspace{0.5cm}-g_{2}^2f_{1}(r_{2}+1)(f_{1y}r_{1}+ g_{1x}(r_{1}+2))  \\bigg),\\\\\n&&d_{4}=\\frac{1}{\\Delta^{2}}\\bigg(f_{1}^2(r_{1}+1)(f_{2}(f_{2y}-g_{2x})+ f_{2x}g_{2}(r_{2}+1))+f_{2}^{2}(r_{2}+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(f_{1}(f_{1y}-g_{1x})+ f_{1x}g_{1}(r_1+1))- f_{1}f_{2}(r_1+1)(r_2+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(f_{2x}g_{1}+f_{1x}g_{2}) \\bigg),\n\\end{eqnarray}\n\\begin{eqnarray}\n&&d_{5}=\\frac{1}{\\Delta^{2}}\\bigg(g_{1}g_{2}(f_{1}(f_{2y}-g_{2x})(r_1+1)+f_{2}(f_{1y}-g_{1x})(r_2+1))\\nonumber\\\\\n&&\\hspace{0.5cm}+(f_{2}g_{1}-f_{1}g_{2})(f_{2}g_{1y}-f_{1}g_{2y})(r_{1}+1)(r_{2}+1) \\bigg), \\\\\n&&d_{6}=\\frac{1}{\\Delta^{2}}\\bigg(f_{1}^2g_{2}(r_1+1)(g_{2x}r_{2}+f_{2y}(r_2+2))+f_{2}^2g_{1}(r_2+1)(g_{1x}r_{1}\\nonumber\\\\\n&&\\hspace{0.5cm}+ f_{1y}(r_1+2))-f_{1}f_{2}(g_{1}(r_1+1) (f_{2y}r_{2}+g_{2x}(r_2+2))+g_{2}(r_2+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(f_{1y}r_{1}+ g_{1x}(r_1+2)))\\bigg),\\\\\n&&s_{1}=\\frac{1}{\\Delta^{2}}\\bigg(g_{1}(r_1+1)(f_{1}(h_{2}(f_{2y}-g_{2x})+f_{2}h_{2y})-f_{2}g_{1}h_{2x}(r_2+2))\\nonumber\\\\\n&&\\hspace{0.5cm}+g_{2}(r_2+1)(f_{2}(h_{1}(f_{1y}-g_{1x})+f_{1}h_{1y})-f_{1}g_{2}h_{1x}(r_1+2))\\nonumber\\\\\n&&\\hspace{0.5cm}+ g_{1}g_{2}(r_1+1)(r_2+1) (f_{2}h_{1x}+f_{1}h_{2x})   \\bigg), \\\\\n&&s_{2}=\\frac{1}{\\Delta^{2} }\\bigg(g_{1}^2(r_1+1)(h_{2}(f_{2y}-g_{2x})-g_{2}h_{2x}-f_{2}h_{2y}(r_2+1))\\nonumber\\\\\n&&\\hspace{0.5cm}\n+g_{2}^2(r_2+1) (h_{1}(f_{1y}-g_{1x})-g_{1}h_{1x}-f_{1}h_{1y}(r_1+1))+g_{1}g_{2}\n\\nonumber\\\\\n&&\\hspace{0.5cm}\n\\times(f_{2}h_{1y}(r_1+2)(r_2+1)+f_{1}h_{2y}(r_1+1)(r_2+2))  \\bigg),\\\\\n&&s_{3}=\\frac{1}{\\Delta^{2}}\\bigg(f_{1}^2(r_1+1)(h_{2}(f_{2y}-g_{2x})+f_{2}h_{2y}+g_{2}h_{2x}(r_2+1))\\nonumber\\\\\n&&\\hspace{0.5cm}+f_{2}^2(r_2+1)(h_{1}(f_{1y}-g_{1x})+f_{1}h_{1y}+g_{1}h_{1x}(r_1+1))\n\\nonumber\\\\\n&&\\hspace{0.5cm}-f_{1}f_{2}(g_{2}h_{1x}(r_1+2)(r_2+1)\n+g_{1}h_{2x}(r_1+1)(r_2+2))  \\bigg),\\\\ \n&&s_{4}=\\frac{1}{\\Delta^{2}}\\bigg(f_{1}g_{1}(r_{1}+1)(h_{2}(f_{2y}-g_{2x})-g_{2}h_{2x})+f_{2}g_{2}(r_{2}+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(h_{1}(f_{1y}-g_{1x})-g_{1}h_{1x})-f_{1}f_{2}(r_{1}+1)(r_{2}+1)(g_{2}h_{1y}+g_{1}h_{2y})\\nonumber\\\\\n&&\\hspace{0.5cm}+f_{1}^2g_{2}h_{2y}(r_{1}+1)(r_{2}+2)+f_{2}^2g_{1}h_{1y}(r_{1}+2)(r_{2}+1) \\bigg),\\\\\n&&u_{1}=\\frac{1}{\\Delta^{2}}\\bigg[g_{1}h_{2}(f_{1}h_{2y}-g_{1}h_{2x})(r_1+1)\n+g_{2}h_{1}(f_{2}h_{1y}-g_{2}h_{1x})\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(r_2+1)\\bigg], \\\\\n&&\nu_{2}=\\frac{1}{\\Delta^{2}}\\bigg[f_{1}h_{2}(f_{1}h_{2y}-g_{1}h_{2x})(r_1+1)+f_{2}h_{1}(f_{2}h_{1y}-g_{2}h_{1x})\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(r_2+1)\\bigg], \\\\\n&&\\Delta=(f_{2}g_{1}-f_{1}g_{2})\\sqrt{(r_{1}+1)(r_{2}+1)}. \n\\end{eqnarray}\n\n\n\n\n\n\n\nThe mixed scalar Li\\'enard-type equation  (\\ref{lienard2}) has only linear and quadratic terms in $\\dot x$.  On the other hand  the coupled system of second order equations of motion (\\ref{seceq1}) and (\\ref{seceq2}), obtained from the nonsingular Lagrangian (\\ref{eq7}) is not in the class of mixed Li\\'enard-type oscillators because it contains higher\/different powers of $\\dot x$ and  $\\dot y$ than quadratic and linear powers.  By equating  these higher\/different degree coefficients to zero, analyzing them and making use of the results in the original coupled equations (\\ref{seceq1}) and (\\ref{seceq2}), we can obtain the relevant evolution equations.  For this purpose we equate the terms $q_{1}$ and $q_{2}$ in equations (\\ref{seceq1}) and (\\ref{seceq2}) to zero as they contain higher-degree terms in $\\dot x$, $\\dot y$ and their products.  Therefore we take\n\\begin{eqnarray}\nq_{1}(\\dot x,\\dot y, x ,y)=0, \\quad q_{2}(\\dot x,\\dot y, x ,y)=0. \\label{q1q2}\n\\end{eqnarray}\nSolving equation (\\ref{q1}) and (\\ref{q2}),  we get a set of partial differential equations for the variables $f_{i}$ and $g_{i}$, $(i=1,2)$. \nWe can easily see that from $q_{1}(\\dot x,\\dot y, x ,y)=0$, we get\n\\begin{eqnarray}\nf_{2y}=g_{2x},\\quad  f_{2}=\\frac{g_{2}h_{2x}}{h_{2y}},\\label{condition1}\n\\end{eqnarray}\n\nSimilarly from $q_{2}(\\dot x,\\dot y, x ,y)=0$, we get\n\\begin{eqnarray}\nf_{1y}=g_{1x},\\quad f_{1}=\\frac{g_{1}h_{1x}}{h_{1y}} , \\label{condition}\n\\end{eqnarray}\n\nOn substituting the above forms in equations (\\ref{seceq1}) and (\\ref{seceq2}), we obtain the coupled system of mixed  Li\\'enard-type class of oscillators of the form\n \n\\begin{eqnarray}\n&&\\ddot x=\\frac{-1}{\\hat{\\Delta}g_{1}g_{2}r_{12}}\\bigg[g_{1}g_{2}\\bigg((r_{1}+1)(r_{2}+1)\\bigg((f_{2x}g_{1}-f_{1x}g_{2})\\dot x^2-(g_{1y}g_{2}-g_{1}g_{2y})\\dot y^2\\nonumber\\\\\n&&\\hspace{0.7cm}-2(g_{1x}g_{2}-g_{1}g_{2x})\\dot x\\dot y\\bigg)   +(r_{1}+1)(r_{2}+2)(h_{2x}\\dot x+h_{2y}\\dot y)g_{1}\\nonumber\\\\\n&&\\hspace{0.7cm}-(r_{1}+2)(r_{2}+1)(h_{1x}\\dot x+h_{1y}\\dot y)g_{2}\\bigg)-g_{2}^2h_{1}h_{1y}(r_2+1)\\nonumber\\\\\n&&\\hspace{0.7cm}+g_{1}^2h_{2}h_{2y}(r_1+1)    \\bigg] ,  \\label{eq91}     \n\\end{eqnarray}\n\\begin{eqnarray}\n&&\\ddot y=\\frac{1}{\\hat{\\Delta}g_{1}g_{2}r_{12}h_{1y}h_{2y}}\\bigg[g_{1}g_{2}\n\\bigg((r_1+1)(r_2+1)  \\bigg((f_{2x}g_{1}h_{1x}h_{2y}-f_{1x}g_{2}h_{1y}h_{2x})\\dot x^2\\nonumber\\\\\n&&\\hspace{0.7cm}+(g_{2y}g_{1}h_{1x}h_{2y}-g_{1y}g_{2}h_{1y}h_{2x})\\dot y^2+2(g_{1}g_{2x}h_{1x}h_{2y}-g_{1x}g_{2}h_{1y}h_{2x})\\dot x\\dot y\\bigg)\n\\nonumber\\\\\n&&\\hspace{0.7cm}-(r_{1}+2)(r_{2}+1)g_{2}h_{1y}h_{2x}(h_{1x}\\dot x+h_{1y}\\dot y)\\nonumber\\\\\n&&\\hspace{0.7cm}+(r_{1}+1)(r_{2}+2)g_{1}h_{2y}h_{1x}(h_{2x}\\dot x+h_{2y}\\dot y)\\bigg)\n+g_{1}^2h_{2}h_{1x}h_{2y}^2(r_1+1)\\nonumber\\\\\n&&\\hspace{0.7cm}-g_{2}^2h_{1}h_{1y}^2h_{2x}(r_2+1)    \\bigg]. \\label{eq92}  \n\\end{eqnarray}\nThe Hamiltonian associated with (\\ref{eq91}) and (\\ref{eq92}) corresponding to the Lagrangian (\\ref{eq7}) can now be written down as \n\n\\begin{eqnarray}\n&&H=\\frac{r_{12}}{\\hat{\\Delta}}\\bigg[\\frac{g_{1}}{h_{1y}}(p_{2}h_{1x}-p_{1}h_{1y})\\bigg(h_{2}-\\bigg(\\frac{g_{1}(h_{1x}p_{2}-h_{1y}p_{1})r_{12}}{h_{1y}r_{2}\\hat{\\Delta}}  \\bigg)^\\frac{-1}{r_2+1}\\bigg)\\nonumber\\\\\n&&\\hspace{0.7cm}  +\\frac{g_{2}}{h_{2y}}(p_{1}h_{2y}-p_{2}h_{2x})\\bigg(h_{1}-\\bigg(\\frac{g_{2}(h_{2y}p_{1}-h_{2x}p_{2})r_{12}}{h_{2y}r_{1}\\hat{\\Delta}}\\bigg) ^\\frac{-1}{r_{1}+1}\\bigg)\\bigg]\\nonumber\\\\\n&&\\hspace{0.7cm}-\\bigg(\\frac{g_{1}(h_{1x}p_{2}-h_{1y}p_{1})r_{12}}{h_{1y}r_{2}\\hat{\\Delta}}\\bigg)^\\frac{r_{2}}{r_2+1}-\\bigg(\\frac{g_{2}(h_{2y}p_{1}-h_{2x}p_{2})r_{12}}{h_{2y}r_{1}\\hat{\\Delta}}\\bigg)^\\frac{r_{1}}{r_1+1},\n\\end{eqnarray}\nwhere $\\hat{\\Delta}=g_{1}g_{2}(h_{1y}h_{2x}-h_{1x}h_{2y})(h_{1y}h_{2y})^{-1}r_{12},\\,\\,r_{12}=[(r_{1}+1)(r_{2}+1)]^{\\frac{1}{2}}$.\nHere the conjugate momenta $p_{1}$ and $p_{2}$ are defined as \n\\begin{eqnarray}\n&&\\hspace{-2cm}p_{1}=L_{\\dot x}=-\\frac{f_{1}r_{1}}{(f_{1}\\dot x+g_{1}\\dot y+h_{1})^{r_{1}+1}}-\\frac{f_{2}r_{2}}{(f_{2}\\dot x+g_{2}\\dot y+h_{2})^{r_{2}+1}},\\\\\n&&\\hspace{-2cm}p_{2}=L_{\\dot y}=-\\frac{g_{1}r_{1}}{(f_{1}\\dot x+g_{1}\\dot y+h_{1})^{r_{1}+1}}-\\frac{g_{2}r_{2}}{(f_{2}\\dot x+g_{2}\\dot y+h_{2})^{r_{2}+1}}.\n\\end{eqnarray}\n\n\\section{\\bf{Reduction to a subclass exhibiting quasiperiodic motion}}\n~~~~~~~~\tIn our further analysis, for simplicity, we assume the parameters $r_{1}$ = $r_{2}$ = $1$ in the Lagrangian given by (\\ref{eq7}).  Now, let the quantities $(f_{i}\\dot x+g_{i}\\dot y)$, $i=1,2,$ be the total derivatives (when $r_{1}$ = $r_{2}$ = $1$) of certain functions $\\rho_{i}(x,y)$.  \n\\begin{eqnarray}\nf_{i}\\dot x+g_{i}\\dot y=\\frac{d}{dt}[\\rho_{i}(x,y)]=\\rho_{ix}\\dot x+\\rho_{iy}\\dot y.\\label{totder}\n\\end{eqnarray}\nFrom the above equation, we find $f_{i}=\\rho_{ix}$, $g_{i}=\\rho_{iy}$, $i=1,2$.  Substituting Eq. (\\ref{totder}) in Eq. (\\ref{eq7}) we get\n\\begin{eqnarray}\nL=\\sum_{i=1}^{2}\\frac{1}{(\\rho_{ix}\\dot x+\\rho_{iy}\\dot y+h_{i})}\\label{modlag}.\n\\end{eqnarray}\n\n\n\nSimilarly, the condition (\\ref{condition}) reduces to \n\\begin{eqnarray}\n\\rho_{ix}=\\frac{\\rho_{iy}h_{ix}}{h_{iy}}. \\label{frac}\n\\end{eqnarray}\nEquation (\\ref{frac}) can also be written as\n\\begin{eqnarray}\n\\left|\n\\begin{array}{ccc}\n\\rho_{ix}&\\rho_{iy}\\\\\nh_{ix}&h_{iy}\n\\end{array}\n\\right|=0.\n\\end{eqnarray}\nConsequently the term $h_{i}$ is functionally dependent on $\\rho_{i}$ that is $h_{i}=Q_{i}(\\rho_{i})$.\\\\\nThen the Lagrangian (\\ref{modlag}) can also be written as \n\\begin{eqnarray}\nL=\\sum_{i=1}^{2}\\frac{1}{(\\rho_{ix}\\dot x+\\rho_{iy}\\dot y+Q_{i}(\\rho_{i}))}.\\label{rearr1}\n\\end{eqnarray}\n~~~~~~~~Now the modified Emden equation is a special case of  Li\\'enard type of oscillators \\cite{CSL:02} which is obtained for specific forms of $Q_{i}(\\rho_{i})=\\rho_{i}^2+\\lambda_{i}$, where $\\lambda_{i}'s$ are constants. \\\\\nFrom the Lagrangian (\\ref{rearr1}), the corresponding equation of motion is \n\\begin{eqnarray}\n\\addtocounter{equation}{-1}\n\\label{equation12}\n\\addtocounter{equation}{1}\n&&\\ddot x=\\bigg((\\rho_{1xx}\\rho_{2y}-\\rho_{1y}\\rho_{2xx})\\dot x^2+(\\rho_{1yy}\\rho_{2y}-\\rho_{1y}\\rho_{2yy})\\dot y^2+3(\\rho_{1}\\rho_{1x}\\rho_{2y}\\nonumber\\\\\n&&\\hspace{0.5cm}-\\rho_{2}\\rho_{1y}\\rho_{2x})\\dot x+3 \\rho_{1y}\\rho_{2y}(\\rho_{1}-\\rho_{2})\\dot y+2(\\rho_{1xy}\\rho_{2y}-\\rho_{1y}\\rho_{2xy})\\dot x\\dot y\\nonumber\\\\\n&&\\hspace{0.5cm}+\\rho_{1}\\rho_{2y}(\\rho_{1}^2+\\lambda_{1})-\\rho_{2}\\rho_{1y}(\\rho_{2}^2+\\lambda_{2})\\bigg)(\\rho_{1y}\\rho_{2x}-\\rho_{1x}\\rho_{2y})^{-1},\\label{equation1}\\\\\n&&\\ddot y=\\bigg( (\\rho_{1x}\\rho_{2xx}-\\rho_{1xx}\\rho_{2x})\\dot x^{2}+(\\rho_{1x}\\rho_{2yy}-\\rho_{1yy}\\rho_{2x})\\dot y^{2}+3\\rho_{1x}\\rho_{2x}(\\rho_{2}\\nonumber\\\\\n&&\\hspace{0.5cm}-\\rho_{1})\\dot x+3(\\rho_{2}\\rho_{1x}\\rho_{2y}-\\rho_{1}\\rho_{1y}\\rho_{2x})\\dot y+2(\\rho_{1x}\\rho_{2xy}-\\rho_{2x}\\rho_{1xy})\\dot x\\dot y\\nonumber\\\\\n&&\\hspace{0.5cm}+\\rho_{2}\\rho_{1x}(\\rho_{2}^2+\\lambda_{2})-\\rho_{1}\\rho_{2x}(\\rho_{1}^2+\\lambda_{1})\\bigg)(\\rho_{1y}\\rho_{2x}-\\rho_{1x}\\rho_{2y})^{-1}.\\label{equation2}\n\\end{eqnarray}\nThe associated Hamiltonian becomes\n\\begin{eqnarray}\n&&H=\\bigg((\\rho_{2y}p_{1}-\\rho_{2x}p_{2})(\\rho_{1}^2+\\lambda_{1}) +(\\rho_{1x}p_{2}-\\rho_{1y}p_{1})(\\rho_{2}^2+\\lambda_{2})\\nonumber\\\\\n&&\\hspace{0.5cm} -2\\sqrt{(\\rho_{1y}\\rho_{2x}-\\rho_{1x}\\rho_{2y})}(\\sqrt{(\\rho_{1x}p_{2}-\\rho_{1y}p_{1})}+\\sqrt{(\\rho_{2y}p_{1}-\\rho_{2x}p_{2})}) \\bigg)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(\\rho_{1y}\\rho_{2x}-\\rho_{1x}\\rho_{2y})^{-1}.\\label{arbham}\n\\end{eqnarray}\nHere the conjugate momenta $p_{1}$ and $p_{2}$ are defined as\n\\begin{eqnarray}\n&&\\hspace{-1cm}p_{1}=L_{\\dot x}=-\\frac{\\rho_{1x}}{(\\rho_{1x}\\dot x+\\rho_{1y}\\dot y+\\rho_{1}^2+\\lambda_{1})^2}-\\frac{\\rho_{2x}}{(\\rho_{2x}\\dot x+\\rho_{2y}\\dot y+\\rho_{2}^2+\\lambda_{2})^2},\\\\\n&&\\hspace{-1cm}p_{2}=L_{\\dot y}=-\\frac{\\rho_{1y}}{(\\rho_{1x}\\dot x+\\rho_{1y}\\dot y+\\rho_{1}^2+\\lambda_{1})^2}-\\frac{\\rho_{2y}}{(\\rho_{2x}\\dot x+\\rho_{2y}\\dot y+\\rho_{2}^2+\\lambda_{2})^2}.\n\\end{eqnarray}\nThe Hamiltonian (\\ref{arbham}) is connected to the Hamiltonian of  a system of uncoupled  linear harmonic oscillators \n $\\tilde{H}=\\frac{1}{2}(P_{1}^2+P_{2}^2+\\lambda_{1}U_{1}^2+\\lambda_{2}U_{2}^2)$ through the following canonical transformation,\n\n\n\\begin{eqnarray}\n&&P_1=\\left[\\lambda_1+\\left(\\lambda_1^2-2\\lambda_1\\left[\\frac{p_1\\rho_{2y}-p_2\\rho_{2x}}{\\rho_{2y}\\rho_{1x}-\\rho_{1y}\\rho_{2x}}\\right]\\right)^{\\frac{1}{2}}\\right],\\label{xx}\\\\\n&&P_2=\\left[\\lambda_2+\\left(\\lambda_2^2-2\\lambda_2\\left[\\frac{p_2\\rho_{1x}-p_1\\rho_{1y}}{\\rho_{2y}\\rho_{1x}-\\rho_{1y}\\rho_{2x}}\\right]\\right)^{\\frac{1}{2}}\\right],\\\\\n&&U_1=-\\frac{\\rho_1}{\\lambda_1}\\left[\\lambda_1^2-2\\lambda_1\\left(\\frac{p_1\\rho_{2y}-p_2\\rho_{2x}}{\\rho_{2y}\\rho_{1x}-\\rho_{1y}\\rho_{2x}}\\right)\\right]^{\\frac{1}{2}},\\\\\n&&U_2=-\\frac{\\rho_2}{\\lambda_2}\\left[\\lambda_2^2-2\\lambda_2\\left(\\frac{p_2\\rho_{1x}-p_1\\rho_{1y}}{\\rho_{2y}\\rho_{1x}-\\rho_{1y}\\rho_{2x}}\\right)\\right]^{\\frac{1}{2}}.\\label{yy}\n\\end{eqnarray}\n\n\nThe general solution of (\\ref{equation1}) and (\\ref{equation2}) can be found after choosing the forms of $f_{1}$ and $f_{2}$ by using the relations (\\ref{xx}) -(\\ref{yy}) and the harmonic oscillator solution\n\\begin{eqnarray}\nU_{i}=A_{i}\\sin(\\omega_{i}t+\\delta_{i})\\nonumber\\\\\nP_{i}=A_{i}\\omega_{i}\\cos(\\omega_{i}t+\\delta_{i}),\\quad \\omega_i=\\sqrt{\\lambda_i}\n\\end{eqnarray}\nwhere $A_{i}$ and $\\delta_{i}$, ($i=1,2,$) are arbitrary constants. The above canonical transformations are identified by generalizing the knowledge of the canonical transformation for the scalar equation which is discussed in the appendix.\n\n\\subsection{ Nonlocal transformation}\nThe system of coupled mixed MEE equations (\\ref{equation1}) and (\\ref{equation2}) can also be related to a system of uncoupled simple harmonic oscillators,\n\\begin{eqnarray}\n\\ddot{u}+\\lambda_1u=0,\\qquad \\ddot{v}+\\lambda_2v=0,\\label{sho}\n\\end{eqnarray}\nthrough the nonlocal transformations,\n\\begin{eqnarray}\nu=\\rho_1(x,y)e^{\\int \\rho_1(x,y)dt},\\qquad v=\\rho_2(x,y)e^{\\int \\rho_2(x,y)dt}.\\label{nonlocal}\n\\end{eqnarray}\nEquations having such type of nonlocal transformations and their solutions have been studied in Ref. \\cite{nonlocal-connection}.  In addition to using the canonical transformation obtained in the previous subsection to find the solution of equation (\\ref{equation12}), one can also obtain the solution of it by solving the following set of coupled first order equations arising from (\\ref{sho}) and (\\ref{nonlocal}), \n\\begin{eqnarray}\n\\addtocounter{equation}{-1}\n\\label{coupled-eq1}\n\\addtocounter{equation}{1}\n\\dot{x}=\\frac{uv(\\rho_2^2\\rho_{1y}-\\rho_1^2\\rho_{2y})+\\dot{u}v\\rho_1\\rho_{2y}-u\\dot{v}\\rho_2\\rho_{1y}}{uv(\\rho_{1x}\\rho_{2y}-\\rho_{1y}\\rho_{2x})},\\\\\n\\dot{y}=\\frac{uv(\\rho_1^2\\rho_{2x}-\\rho_2^2\\rho_{1x})+u\\dot{v}\\rho_2\\rho_{1x}-\\dot{u}v\\rho_1\\rho_{2x}}{uv(\\rho_{1x}\\rho_{2y}-\\rho_{1y}\\rho_{2x})}\\label{coupled-eq2}.\n\\end{eqnarray}\nHere $u$ and $v$ are the solutions of the simple harmonic oscillator equations (\\ref{sho}). The difficulty in solving the equations (\\ref{coupled-eq1}) depends on the form of $\\rho_1$ and $\\rho_2$ (see \\cite{nonlocal-connection}).\n\n\\section{\\bf{A system of coupled Li\\'enard-type equations}}\n Next we can obtain a special case of Li\\'enard-type of oscillators (\\ref{rearr1}) by choosing \n\\begin{eqnarray}\n\\rho_{1}=k_{1}(\\alpha_{1}x^{m_{1}}+\\alpha_{2}y^{m_{2}}), \\quad \\rho_{2}=k_{2}(\\alpha_{3}y^{m_{3}}+\\alpha_{4}x^{m_{4}}). \n\\end{eqnarray}\nwhere $k_{i}$'s and $\\alpha_{j}$'s are arbitrary real parameters and $m_{j}$'s are positive integers and $i=1,2,$ and $j=1,2,3,4.$\n  In this case the specific form of the nonstandard and nonsingular Lagrangian with two degrees of freedom takes the form\n\\begin{eqnarray}\n&&\\hspace{-1.3cm}L=\\frac{1}{\\lambda_{1}+k_{1}^2(\\alpha_{1}x^{m_{1}}+\\alpha_{2}y^{m_{2}})^2+k_{1}(m_{1}\\alpha_{1}x^{m_{1}-1}\\dot x+m_{2}\\alpha_{2}y^{m_{2}-1}\\dot y)}\\nonumber\\\\\n&&\\hspace{-.5cm}+\\frac{1}{\\lambda_{2}+k_{2}^2(\\alpha_{3}y^{m_{3}}+\\alpha_{4}x^{m_{4}})^2+k_{2}(m_{3}\\alpha_{3}y^{m_{3}-1}\\dot y+m_{4}\\alpha_{4}x^{m_{4}-1}\\dot x)}.\\label{lagpowerm}\n\\end{eqnarray}\nThe equation of motion of the mixed Li\\'enard-type class of oscillators can be obtained from the above Lagrangian.  It is of the form\n\\label{secpowerm1&m2}\n\\begin{eqnarray}\n&&\\ddot x=\\frac{-1}{xy^2\\delta_{m}}\\bigg(\\dot x^2 y^2\\bigg[m_{1}m_{3}\\alpha_{1}\\alpha_{3}(m_{1}-1)x^{m_{1}}y^{m_{3}}-m_{2}m_{4}\\alpha_{2}\\alpha_{4}(m_{4}-1)x^{m_{4}}y^{m_{2}}  \\bigg]\\nonumber\\\\\n&&+\n\\dot y^2 x^2\\bigg[    m_{2} m_{3} \\alpha_{2}\\alpha_{3}(m_{2}-m_{3})y^{m_{2}+m_{3}}\\bigg]+ \\bigg[3k_{1}y^{m_{3}} (m_{1}\\alpha_{1}x^{m_{1}}y\\dot x+m_{2}\\alpha_{2}y^{m_{2}}x\\dot y)\\nonumber\\\\\n&& +k_{1}^2xy^{m_{3}+1}(\\alpha_{1} x^{m_{1}}+\\alpha_{2} y^{m_{2}})^2+\\lambda_{1}x y^{m_{3}+1} \\bigg]m_{3} \\alpha_{3} x y (\\alpha_{1} x^{m_{1}}+\\alpha_{2} y^{m_{2}})\\nonumber\\\\\n&&-\\bigg[3k_{2}y^{m_{2}} (m_{4}\\alpha_{4}x^{m_{4}}y\\dot x+m_{3}\\alpha_{3}y^{m_{3}}x\\dot y) +k_{2}^2xy^{m_{2}+1}(\\alpha_{4} x^{m_{4}}+\\alpha_{3} y^{m_{3}})^2\\nonumber\\\\\n&&+\\lambda_{2}x y^{m_{2}+1} \\bigg]m_{2} \\alpha_{2} x y (\\alpha_{4} x^{m_{4}}+\\alpha_{3} y^{m_{3}}) \\bigg),\\label{secpowerm1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&&\\ddot y=\\frac{1}{yx^2\\delta_{m}}\\bigg(\\dot y^2x^2\\bigg[m_{2}m_{4}\\alpha_{2}\\alpha_{4}(m_{2}-1)x^{m_{4}}y^{m_{2}} -m_{1}m_{3}\\alpha_{1}\\alpha_{3}(m_{3}-1)x^{m_{1}}y^{m_{3}}\\bigg]\\nonumber\\\\\n&&-\\dot x^2y^{2}\\bigg[m_{1}m_{4}\\alpha_{1}\\alpha_{4}(m_{4}-m_{1})x^{m_{1}+m_{4}} \\bigg] \n+ \\bigg[3k_{1} x^{m_{4}}(m_{1}\\alpha_{1}x^{m_{1}}y\\dot x+m_{2}\\alpha_{2}y^{m_{2}}x\\dot y)\\nonumber\\\\\n&& +k_{1}^2x^{m_{4}+1}y(\\alpha_{1} x^{m_{1}}+\\alpha_{2} y^{m_{2}})^2+\\lambda_{1}yx^{m_{4}+1}  \\bigg]m_{4} \\alpha_{4} x y (\\alpha_{1} x^{m_{1}}+\\alpha_{2} y^{m_{2}})\n\\nonumber\\\\\n&&-\\bigg[3k_{2}x^{m_{1}} (m_{4}\\alpha_{4}x^{m_{4}}y\\dot x+m_{3}\\alpha_{3}y^{m_{3}}x\\dot y) +k_{2}^2x^{m_{1}+1}y(\\alpha_{4} x^{m_{4}}+\\alpha_{3} y^{m_{3}})^2\\nonumber\\\\\n&&+\\lambda_{2}x^{m_{1}+1} y \\bigg]m_{1} \\alpha_{1} x y (\\alpha_{4} x^{m_{4}}+\\alpha_{3} y^{m_{3}}) \\bigg),\\label{secpowerm2}\n\\end{eqnarray}\n\nwhere $\\delta_{m}=m_{1}m_{3}\\alpha_{1}\\alpha_{3}x^{m_{1}}y^{m_{3}}-m_{2}m_{4}\\alpha_{2}\\alpha_{4}x^{m_{4}}y^{m_{2}}$. \\\\\n\n\\noindent The associated Hamiltonian becomes\n\\begin{eqnarray}\n&&\\hspace{-0.3cm}H=\\frac{1}{k_{1}k_{2}\\delta_{m}}\\bigg[k_{2}(m_{4}\\alpha_{4}p_{2}x^{m_{4}}y-m_{3}\\alpha_{3}p_{1}xy^{m_{3}})\\bigg(k_{1}^2 X_{m}^2+\\lambda_{1}  \\bigg)\\nonumber\\\\\n&&\\hspace{0.3cm}+k_{1}(m_{2}\\alpha_{2}p_{1}xy^{m_{2}}-m_{1}\\alpha_{1}p_{2}x^{m_{1}}y)\\bigg(k_{2}^2 Y_{m}^2+\\lambda_{2}  \\bigg)-2\\delta_{m}^{\\frac{1}{2}}\\bigg(k_{1}k_{2}^{\\frac{1}{2}}(m_{2}\\alpha_{2}p_{1}xy^{m_{2}}\\nonumber\\\\\n&&\\hspace{0.5cm}-m_{1}\\alpha_{1}p_{2}x^{m_{1}}y)^{\\frac{1}{2}}+k_{2}k_{1}^{\\frac{1}{2}}(m_{4}\\alpha_{4}p_{2}x^{m_{4}}y-m_{3}\\alpha_{3}p_{1}xy^{m_{3}})^{\\frac{1}{2}}  \\bigg)\\bigg].\n\\end{eqnarray}\nwhere $X_{m}=\\alpha_{1}x^{m_{1}}+\\alpha_{2}y^{m_{2}}$, $Y_{m}=\\alpha_{4}x^{m_{4}}+\\alpha_{3}y^{m_{3}}$.\\\\\n\\subsection{\\bf{Coupled Modified Emden Equation: A special case}}\n\tThe coupled system of equations (\\ref{secpowerm1}) and (\\ref{secpowerm2}) reduces to a coupled generalization of the modified Emden equation (\\ref{meee})-(\\ref{mee}) for the choice $m_{1}$ = $m_{2}$ = $m_{3}$ = $m_{4}$ = $1$ and is of the form \n\\begin{eqnarray}\n\\addtocounter{equation}{-1}\n\\addtocounter{equation}{1}\n&\\hspace{-1cm}\\ddot x=\\frac{-1}{\\hat{\\delta_{1}}}\\bigg[\\bigg(3 k_{1}(\\alpha_{1}\\dot x+\\alpha_{2}\\dot y)+k_{1}^2(\\alpha_{1}x+\\alpha_{2}y)^2 +\\lambda_{1}\\bigg)(\\alpha_{3}(\\alpha_{1}x+\\alpha_{2}y))\\nonumber\\\\\n&\\hspace{-1cm}-\\bigg(3 k_{2}(\\alpha_{3}\\dot y+\\alpha_{4}\\dot x)+k_{2}^2(\\alpha_{3}y+\\alpha_{4}x)^2 +\\lambda_{2}\\bigg)(\\alpha_{2}(\\alpha_{3}y+\\alpha_{4}x)) \\bigg],\\label{eq15}\n\\end{eqnarray}\n\\begin{eqnarray}\n&\\hspace{-1cm}\\ddot y=\\frac{1}{\\hat{\\delta_{1}}}\\bigg[\\bigg(3 k_{1}(\\alpha_{1}\\dot x+\\alpha_{2}\\dot y)+k_{1}^2(\\alpha_{1}x+\\alpha_{2}y)^2 +\\lambda_{1}\\bigg)(\\alpha_{4}(\\alpha_{1}x+\\alpha_{2}y))\\nonumber\\\\&\\hspace{-1cm}-\\bigg(3 k_{2}(\\alpha_{3}\\dot y+\\alpha_{4}\\dot x)+k_{2}^2(\\alpha_{3}y+\\alpha_{4}x)^2 +\\lambda_{2}\\bigg)(\\alpha_{1}(\\alpha_{3}y+\\alpha_{4}x)) \\bigg],\\label{eq15a}\n\\end{eqnarray}\nwhere $\\hat{\\delta_{1}}=(\\alpha_{1}\\alpha_{3}-\\alpha_{2}\\alpha_{4})$.  Note that the above system of equations is \\emph{PT} symmetric under the combined transformations, $(t\\rightarrow-t$, $x\\rightarrow-x$, $ y\\rightarrow- y)$.  There are other ways of generalizing the modified Emden equation to two dimensions, for example see Ref. \\cite{cmee}.  Even though the generalized version identified as the coupled modified Emden equation in Ref. \\cite{cmee} is isochronous, the system lacks a Hamiltonian description in order to quantize it.  However, we find that the system (\\ref{eq15}) and (\\ref{eq15a}) has the well defined Hamiltonian \n\\begin{eqnarray}\n&&\\hspace{-0.7cm}H=\\frac{1}{k_{1}k_{2}\\hat{\\delta_{1}}}\\bigg[k_{2}(p_{2}\\alpha_{4}-p_{1}\\alpha_{3})(k_{1}^2 X^2+\\lambda_{1})+k_{1}(p_{1}\\alpha_{2}-p_{2}\\alpha_{1})(k_{2}^2 Y^2+\\lambda_{2})\\nonumber\\\\\n&&\\hspace{-0.1cm}  -2\\sqrt{\\hat{\\delta_{1}}}(k_{1}k_{2}^{\\frac{1}{2}}(p_{1}\\alpha_{2}-p_{2}\\alpha_{1})^{\\frac{1}{2}}+   k_{2}k_{1}^{\\frac{1}{2}}(p_{2}\\alpha_{4}-p_{1}\\alpha_{3})^{\\frac{1}{2}}) \\bigg],\\label{eq16}\n\\end{eqnarray}\nwhere $X=(\\alpha_{1}x+\\alpha_{2}y)$ and $Y=(\\alpha_{3}y+\\alpha_{4}x)$.   Here we note that one can use a variable transformation $X=(\\alpha_{1}x+\\alpha_{2}y)$ and $Y=(\\alpha_{3}y+\\alpha_{4}x)$ in equation (\\ref{eq15}) to obtain two uncoupled modified Emden equations. \n\nIn order to find the general solution of the equations (\\ref{eq15}) and (\\ref{eq15a}) we use suitable canonical transformation to the above Hamiltonian to reduce it to a simpler form.  We find that the above Hamiltonian is connected to the Hamiltonian of a two dimensional linear harmonic oscillator through the following canonical transformation, see (\\ref{xx})-(\\ref{yy}),\t\n\n\\begin{eqnarray}\n&&P_1=\\lambda_1+\\left[\\lambda_1^2-\\frac{2\\lambda_1(\\alpha_3p_1-\\alpha_4p_2)}{k_1(\\alpha_1\\alpha_3-\\alpha_2\\alpha_4)}\\right]^{\\frac{1}{2}},\\label{eq119}\\\\\n&&P_2=\\lambda_2+\\left[\\lambda_2^2-\\frac{2\\lambda_2(\\alpha_1p_2-\\alpha_2p_1)}{k_2(\\alpha_1\\alpha_3-\\alpha_2\\alpha_4)}\\right]^{\\frac{1}{2}},\\\\\n&&U_1=-\\frac{k_1(\\alpha_1x+\\alpha_2y)}{\\lambda_1}\\left[\\lambda_1^2-\\frac{2\\lambda_1(\\alpha_3p_1-\\alpha_4p_2)}{k_1(\\alpha_1\\alpha_3-\\alpha_2\\alpha_4)}\\right]^{\\frac{1}{2}},\\\\\n&&U_2=-\\frac{k_2(\\alpha_4x+\\alpha_3y)}{\\lambda_2}\\left[\\lambda_2^2-\\frac{2\\lambda_2(\\alpha_1p_2-\\alpha_2p_1)}{k_2(\\alpha_1\\alpha_3-\\alpha_2\\alpha_4)}\\right]^{\\frac{1}{2}}\\label{eq120},\n\\end{eqnarray}\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.1\\columnwidth]{figure1}\n\\caption{(color online) Periodic oscillations with $\\omega_{1}:\\omega_{2}$ = $4:4$, $k_{1}$ = $k_{2}$ = 1,  $\\alpha_{1}$ = $\\alpha_{3}$ = $5.5$, $\\alpha_{2}$ = $\\alpha_{4}$ = $3$ (a) Time series plot (b) Projected phase portrait.  Similar plots can be given for the $y$ variable.}\n\\end{figure}\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.0\\columnwidth]{figure2}\n\\caption{(color online) Quasiperiodic oscillations with $\\omega_{1}:\\omega_{2}$ = $4$ : $\\sqrt{3}$, $k_{1}$ = $k_{2}$ = $1$,  $\\alpha_{1}$ = $\\alpha_{3}$ = $5.5$, $\\alpha_{2}$ = $\\alpha_{4}$ = $3$. (a) Time series plot (b) Projected phase portrait (c) Poincar\\'e section.  Similar plots can be given for the $y$ variable.}\n\\end{figure}\n\n\nThe general solution of equations (\\ref{eq15}) and (\\ref{eq15a}) can then be obtained by substituting the general solution of the two dimensional harmonic oscillator given by the expressions\n\\begin{eqnarray}\nU_{1}=A \\sin(\\omega_{1} t+ \\delta_{1}),\\quad U_{2}=B \\sin(\\omega_{2} t+ \\delta_{2}),\\nonumber \\\\\nP_{1}=A \\omega_{1}\\cos(\\omega_{1}t+\\delta_{1}),\\quad P_{2}=B \\omega_{2}\\cos(\\omega_{2}t+\\delta_{2}),\\label{harmq}\n\\end{eqnarray}\nwhere $\\omega_{j}=\\sqrt{\\lambda_j}$, j=1,2 into Eqs. (\\ref{eq119})-(\\ref{eq120}) and solving the resultant equations.  We obtain\n\\begin{eqnarray}\nx=\\frac{A\\alpha_{3}\\omega_{1}\\sin(\\omega_{1}t+\\delta_{1})}{k_{1}\\hat{\\delta_{1}}\\left(\\omega_{1}-A\\cos(\\omega_{1}t+\\delta_{1})\\right)}-\\frac{B\\alpha_{2}\\omega_{2}\\sin(\\omega_{2}t+\\delta_{2})}{k_{2}\\hat{\\delta_{1}}\\left(\\omega_{2}-B\\cos(\\omega_{2}t+\\delta_{2})\\right)},\n\\end{eqnarray}\n\\begin{eqnarray}\ny=-\\frac{A\\alpha_{4}\\omega_{1}\\sin(\\omega_{1}t+\\delta_{1})}{k_{1}\\hat{\\delta_{1}}\\left(\\omega_{1}-A\\cos(\\omega_{1}t+\\delta_{1})\\right)}+\\frac{B\\alpha_{1}\\omega_{2}\\sin(\\omega_{2}t+\\delta_{2})}{k_{2}\\hat{\\delta_{1}}\\left(\\omega_{2}-B\\cos(\\omega_{2}t+\\delta_{2})\\right)},\n\\end{eqnarray}\\\\\nwhere $A$,\\,$B$,\\,$\\delta_{1}$, $\\delta_{2}$ are arbitrary constants. \nThe above solution is oscillatory and is periodic and bounded for suitable choice of parameters namely $0<A<\\omega_{1}$ and $0<B<\\omega_{2}$.\nHere one can note that the frequency of oscillations is again independent of the amplitude for the two dimensional coupled modified Emden equation.  Figures 1 and 2 show two types of oscillatory behaviour, namely periodic and quasiperiodic oscillations, depending upon the ratio of frequencies $\\omega_{1}$ and $\\omega_{2}$.  Fig.1 shows  periodic oscillations for $\\omega_1$ : $\\omega_2$ = $4:4$, $k_{1}$ = $k_{2}$ = $1$, $\\alpha_{1}$ = $\\alpha_{3}$ = $5.5$, $\\alpha_{2}$ = $\\alpha_{4}$ = $3$ and Fig.2 shows quasiperiodic oscillations for $\\omega_1$ : $\\omega_2$ = $4$ : $\\sqrt{3}$, $k_{1}$ = $k_{2}$ = $1$, $\\alpha_{1}$ = $\\alpha_{3}$ = $5.5$, $\\alpha_{2}$ = $\\alpha_{4}$ = $3$.\n\n\n  \n\n\n\\section{Conclusion}\nIn this paper we have identified a class of coupled mixed Li\\'enard type nonlinear evolution equations.  This class of equation is identified by generalizing the nonstandard Lagrangian of the scalar MEE to a suitable nonsingular two dimensional Lagrangian.  Imposing the condition that the resultant Euler-Lagrange equation should be of mixed Li\\'enard type form, we have identified a specific class of equations which admits isochoronous solutions. The procedure is illustrated for the special case of  two dimensional modified Emden equation and is found to be isochronous as well as $\\emph PT$ symmetric for suitable choice of parameters, exhibiting quasiperiodic and periodic oscillations.  The above procedure can in principle be extended to higher degrees of freedom also.  The problem of quantization of system  (\\ref{eq16}) is also worth further consideration.\n\n\\section{Acknowledgements}\nThe work of ADD and ML is supported by a DST-IRHPA research project and the work of ML is also supported by a DAE Raja Ramanna Fellowship.   The work of V.K.C. is supported by the SERB-DST Fast Track scheme for young scientists under Grant No. YSS\/2014\/000175.\n\n\n\\section*{Appendix}\n\n\nLet us consider the following mixed Li\\'enard type equation,\n\\begin{eqnarray}\n\\ddot{x}+\\dot{x}^2\\frac{u_{xx}}{u_x}+3ku\\dot{x}+\nk^2\\frac{u^3}{u_x}+\\lambda\\frac{u}{u_x}=0,\n\\end{eqnarray}\nwhere $u(x)$ is an arbitrary function of $x$.  This equation is obtained by applying the transformation $y=u(x)$ in the linearizable MEE\n\\begin{eqnarray}\n\\ddot{y}+3ky\\dot{y}+k^2y^3+\\lambda y=0.\n\\end{eqnarray}\nThe resultant equation\npossess the Hamiltonian structure\n\\begin{eqnarray}\nH=-kp\\frac{u^2}{u_x}-\\frac{\\lambda p}{k u_{x}}+2\\sqrt{-\\frac{p}{k u_x}},\n\\end{eqnarray}\nwhich can be deduced from the Lagrangian (\\ref{arblag}).  This Hamiltonian can be transformed to the Hamiltonian of the simple\nharmonic oscillator equation through the canonical transformation\n\\begin{eqnarray}\nU=\\frac{u(x)\\sqrt{\\lambda(\\lambda-\\frac{2kp}{u_x})}}{\\lambda},\\qquad P=\\frac{\\lambda-\\sqrt{\\lambda(\\lambda-\\frac{2kp}{u_x})}}{k}.\n\\end{eqnarray}\nGeneralizing the above canonical transformation to two dimensions we obtain the canonical transformations (vide Eqs. (\\ref{xx})-(\\ref{yy})) for the coupled MEE.\\\\\n\\section*{References}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}