Content-Type: multipart/mixed; boundary="-------------0610021643437" This is a multi-part message in MIME format. ---------------0610021643437 Content-Type: text/plain; name="06-274.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-274.keywords" Lorentz process, periodic configuration of scatterers, infinite horizon, corridors, non-normal domain of attraction of the gaussian law, local limit law, recurrence, ergodicity ---------------0610021643437 Content-Type: application/x-tex; name="vhg.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="vhg.tex" \documentclass[a4paper]{article} \usepackage{amsmath,amsthm,pstricks,amssymb,graphicx,calc,amscd} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} %\newenvironment{proof}{\par Proof:\ }{\hfill\qed} \begin{document} \title{Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon} \author{Domokos Sz\'asz \and Tam\'as Varj\'u} %\thanks{Research supported by the Hungarian National Foundation for % Scientific Research grants No. T046187, NK 63066 and TS 049835, further by % Hungarian Science and Technology Foundation grant No. A-9/03} \maketitle \centerline{Dedicated to Ya. G. Sinai on the occasion of his seventieth birthday } \begin{abstract} As Bleher, \cite{Bleher} observed the free flight vector of the planar, infinite horizon, periodic Lorentz process $\{S_n | n=0, 1, 2, \dots \}$ belongs to the non-standard domain of attraction of the Gaussian law --- actually with the $\sqrt{n \log n}$ scaling. Our first aim is to establish his conjecture that, indeed, $\frac{S_n}{\sqrt{n \log n}}$ converges in distribution to the Gaussian law (a Global Limit Theorem). Here the recent method of B\'alint and Gou\"ezel, \cite{BG} helped us to essentially simplify the ideas of our earlier sketchy proof \cite{SzVi}. Moreover, we can also derive a.) the local version of the Global Limit Theorem, b.) the recurrence of the planar, infinite horizon, periodic Lorentz process, and finally c.) the ergodicity of its infinite invariant measure. -------------------------------------------------------------------------- Key words: Lorentz process, periodic configuration of scatterers, infinite horizon, corridors, non-normal domain of attraction of the gaussian law, local limit law, recurrence, ergodicity \end{abstract} \renewcommand{\thefootnote}{\fnsymbol{footnote}} \footnotetext[0]{Research supported by the Hungarian National Foundation for Scientific Research grants No. T046187, NK 63066 and TS 049835, further by Hungarian Science and Technology Foundation grant No. A-9/03} \section{Introduction} The Lorentz process is the $\mathbb Z^d-$covering of a Sinai billiard --- in other words of a dispersing billiard --- given on $\mathbb T^d = \mathbb R^d/\mathbb Z^d$. If the horizon is finite, i.~e.\ the free flight vector $\psi(x)$ is bounded, then the Lorentz process is a most instructive model of the Brownian motion. \begin{figure}[htbp] \centering \setlength{\unitlength}{1cm} \begin{picture}(6.11,4.11) \psset{unit=\unitlength} %% \psframe(0,0)(6.11,4.11) \psline[linewidth=1pt, linecolor=red] (1.00456,0.495444) (1.43705,0.927937) (1.56702,1.43997) (1.47673,1.35534) (2.07161,1.42515) (2.09287,1.64029) (2.30243,2.00556) (2.4293,1.91356) (1.49794,2.21795) (2.00001,2.24749) (1.49943,2.26688) (2.0251,2.35917) (1.96873,2.46407) (2.04449,2.39235) (1.93917,2.44371) (1.93508,2.05878) (2.03908,2.11579) (1.94911,2.04992) (2.00053,2.26629) (1.97925,2.47243) (2.08732,2.43983) (2.06761,2.58051) (2.39024,2.73682) (2.09459,2.85419) (2.19917,3.00522) (2.30061,2.49482) (2.40521,2.64645) (2.10966,2.76566) (2.44052,2.93391) (2.43634,3.08333) (2.68512,3.1041) (3.00589,3.19605) (2.45696,3.10976) (2.4837,2.99225) (2.32803,3.01249) (2.10648,2.8002) (2.40646,2.64238) (2.32627,2.48808) (2.40657,3.0551) (2.54312,3.04462) (2.64311,3.40624) (2.50835,3.01684) (2.0712,2.91257) (2.18466,3.00869) (2.37611,2.46586) (2.5029,2.48819) (2.485,2.33528) (2.72396,2.10906) (3.73487,3.39032) (3.96939,3.03542) (4.41642,2.88538) (3.48634,3.3315) (3.54648,3.45305) (3.46968,3.36933) (3.66022,3.40138) (4.07118,3.07529) (4.41307,2.6232) (4.42736,2.42619) (4.63161,2.41002) (5.02168,2.14818) (5.00174,2.00735) (5.13327,2.02893) (5.40523,1.44597) (6.00997,1.31991) (5.81084,1.39518) (5.10641,0.800682) (5.39886,0.670623) (5.25287,0.499983) (5.10366,0.682736) (5.40448,0.85108) (5.31951,1.00986) (5.35811,0.475415) (5.45564,0.542753) (5.36033,0.474337) (5.34183,1.01748) (5.43054,0.91596) (5.04462,0.956875) (5.0732,1.07325) (4.95678,1.04469) (4.91556,1.43033) (5.01714,1.34097) (4.47754,1.35356) (4.55532,1.44718) (4.49145,1.31481) (4.79756,1.10685) (5.00999,1.18004) (4.7077,1.10751) (4.45541,1.10749) (4.4943,1.00341) (4.38851,1.04188) (4.36137,0.473821) (4.45141,0.548899) (4.33315,0.485768) (4.08947,1.05835) (3.49395,1.19532) (4.07909,1.06755) (4.24248,0.499887) (4.36004,1.02552) (4.46859,0.974524) (4.40601,1.05465) \multips(.25,.25)(1,0){6}{ \psarc[linewidth=.5pt, fillstyle=solid, fillcolor=lightgray](.5,-.5){.36}{43.98}{136.02} \psarc[linewidth=.5pt, fillstyle=solid, fillcolor=lightgray](0,4){.25}{214.06}{325.94} \multips(0,0)(0,1){4}{ \pscircle[linewidth=.5pt, fillstyle=solid, fillcolor=lightgray](0,0){.25} \pscircle[linewidth=.5pt, fillstyle=solid, fillcolor=lightgray](.5,.5){.36} } } \multips(.25,.25)(0,1){4}{ \psarc[linewidth=.5pt, fillstyle=solid, fillcolor=lightgray](-.5,.5){.36}{313.98}{46.02} \psarc[linewidth=.5pt, fillstyle=solid, fillcolor=lightgray](6,0){.25}{124.06}{235.94} } \end{picture} \caption{100 collisions Lorentz trajectory segment with circular scatterers and finite horizon} \end{figure} \begin{figure}[htbp] \centering \setlength{\unitlength}{1cm} \begin{picture}(10.05,6.05) \psset{unit=\unitlength} %% \psframe(0,0)(10.05,6.05) 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(6.14731,6.02206) (6.04765,5.78744) (6.26753,5.49938) (6.45688,5.68612) (6.03641,5.83928) (6.16184,6.01606) (6.10318,5.45235) (5.9072,5.49448) (5.79016,5.0473) (5.79525,5.45343) (5.95257,4.97128) (6.46792,4.85214) (5.47829,5.3519) (5.53052,5.54548) (5.2851,5.49752) (5.02539,5.631) (6.00236,5.28425) (5.5,5.25095) (6.00163,5.22153) (5.38514,5.03967) (5.02029,4.61984) (4.90717,4.00553) (5.04345,4.10915) (4.88966,4.01551) (4.86433,4.47264) (5.00766,4.31139) (4.45864,4.11227) (4.47253,3.86406) (2.03717,1.66319) (2.09276,1.44436) (1.94155,1.51912) (2.00422,1.29573) (1.81536,1.04279) (1.7318,1.45055) (1.57969,0.996969) (1.31138,1.00765) (1.03505,0.843535) (1.48348,0.88772) (1.39353,1.0453) (1.50156,0.918159) (1.25721,1.0001) (1.00937,0.900752) (1.15684,1.018) (1.22755,0.49899) (1.20507,1.00407) (0.79372,0.0467972) \multips(.25,.25)(1,0){10}{ \psarc[linewidth=.5pt, fillstyle=solid, fillcolor=lightgray](.5,-.5){.3}{56.44}{123.56} \psarc[linewidth=.5pt, fillstyle=solid, fillcolor=lightgray](0,6){.25}{233.13}{306.87} \multips(0,0)(0,1){6}{ \pscircle[linewidth=.5pt, fillstyle=solid, fillcolor=lightgray](0,0){.25} \pscircle[linewidth=.5pt, fillstyle=solid, fillcolor=lightgray](.5,.5){.3} } } \multips(.25,.25)(0,1){6}{ \psarc[linewidth=.5pt, fillstyle=solid, fillcolor=lightgray](-.5,.5){.3}{326.44}{33.56} \psarc[linewidth=.5pt, fillstyle=solid, fillcolor=lightgray](10,0){.25}{143.13}{216.87} } \end{picture} \caption{100 collissions Lorentz trajectory-segment with circular scatterers and infinite horizon} \end{figure} We note here that though this should be true in any dimension $d \ge 2$, mathematical results only exist for $d=2$. In this paper we will also restrict our attention to this case. As suggested by the beautiful --- partially rigorous, partially heuristic --- work of Bleher \cite{Bleher}, in the infinite horizon case ($d=2$!) the asymptotic behaviour of the displacement $S_n$, taken in the moment of the $n^{\mbox{\scriptsize th}}$ reflection from a scatterer, is slightly superdiffusive and $\frac{S_n}{\sqrt{n \log n}}$ was expected to possess a limiting Gaussian distribution. The first main aim of the present work is to provide the first rigorous proof of this statement. \begin{theorem}[Global limit theorem]\label{thm:GLT} Suppose that the direction vectors of infinite, collision-free flights span the plane. Let $A$ be a bounded open set in the plane. Then \[ \mu \left( \frac{S_n} {\sqrt {n\log n}} \in A \right) \rightarrow \int_A \phi_{\Sigma}(z) dz\] where $\phi$ is a nondegenerate gaussian density with zero expectation and covariance matrix $\Sigma$. \end{theorem} A mostly geometric proof of this theorem was sketched in \cite{SzVi}. However, before we had completed our paper with the technical proof of Theorem \ref{thm:GLT}, there appeared a much interesting work of B\'alint and Gou\"ezel, \cite {BG}: for the stadium billiard they gave a quite analytic proof of a global limit theorem which also uses the $\sqrt{n \log n}$ scaling. The coincidence of scalings is explained by the analogous behaviour of long free flights in our model (i.~e.\ in corridors of the Lorentz process) and that of the quasi integrable trajectories between the linear sides of the stadium billiard. The arguments of \cite {BG} also helped us to simplify our approach substantially at essentially three points: \begin{enumerate} \item once one has a tower construction \`a la Young, their Lemma 3.5 (cf.\ our Theorem \ref{theorem:integral}) provides a general, concise condition for the validity of a non-standard Gaussian limit law; \item they reduce the "tower-sums" to a more tractable, still dominant part; \item for describing excursions from the tail they use a delicate result of Chernov, \cite{Ch99} (see in our Lemma \ref{lemma:chernov}). \end{enumerate} It is a question, though, whether Chernov's result, a beautiful but quite strong tool, is indeed, necessary in this proof. A well-known spectacular property of discrete approximations of the Brownian motion, like of simple symmetric random walks, is P\'olya's classical theorem, claiming recurrence in the planar case. For the diffusive, finite horizon case of the Lorentz process this had been proved by various authors in \cite{Sch}, \cite{Conze}, \cite{SzVf}. (As a matter of fact, the problem was raised by Sinai in 1979, after he, with Bunimovich, had proved its convergence to the Brownian motion. In 1985 --- as a partial solution --- the first named author jointly with A. Kr\'amli could settle a weaker property: quasi-recurrence of the planar, finite horizon Lorentz process, cf.\ \cite{KSz1}.) The arguments of \cite{SzVf} relied on a local version, LCLT of the central limit theorem (CLT). Our second main aim here is to first deduce a local limit theorem (Theorem \ref{thm:LLT}, a LLT), a local version of Theorem \ref{thm:GLT} and then by using it to establish the recurrence of the planar Lorentz process with an infinite horizon (Theorem \ref{thm:Rec}). \begin{theorem}[Local limit theorem]\label{thm:LLT} Suppose that the direction vectors of infinite, collision-free flights span the plane. Let $k_n\in\mathbb{Z}^2$ such that $\frac {k_n} {\sqrt{n\log n}}\rightarrow k$. Then for the discretised position vector $S_n^{\kappa}$ the following holds: \[ n \log n \enskip \mu (S_n^\kappa = k_n) \rightarrow \phi_{\Sigma}(k)\] where $\phi_{\Sigma}$ is a nondegenerate gaussian density with zero expectation from Theorem \ref{thm:GLT}. \end{theorem} \begin{theorem}[Recurrence]\label{thm:Rec} \[ \mu (\exists n_k\rightarrow \infty \quad S_{n_k}^\kappa = 0) = 1\] \end{theorem} It is worth mentioning that, for the planar Lorentz process with a finite horizon but with a finite modification of a periodic scatterer configuration, the problem of recurrence and of ergodicity was studied by Lenci \cite{L03}. The recurrence itself also has an additional interesting conclusion. Let us note first that for the Lorentz process strong stochastic properties, like correlation decay, limit laws, etc.\ could only be obtained in the case of a periodic configuration of scatterers for then its factor is a Sinai billiard. For this same case, however, it is an interesting question whether the Lorentz dynamics is ergodic without this factorization as well (N.~B.\ in this case the invariant measure is infinite!). The combination of Theorem \ref{thm:Rec} with an old result of Sim\'anyi, \cite{Sim 89} also provides the answer: \begin{theorem}\label{thm:erg} The planar Lorentz dynamics with a periodic configuration of scatterers is ergodic. \end{theorem} The paper is organized as follows. Section 2 is devoted to the description of the model and to the study of its important geometric properties. In fact, we recall the notion of corridors also discussed in Bleher's work; they are distinguished parts of the phase space in whose neighbourhood the free flight vector becomes unbounded. Here we also introduce new coordinates to make easier the study of the tail behaviour of the free flight vector and its asymptotic Markov property. In particular, our geometric probabilistic estimates are sharper than those of \cite{Bleher}. Finally Section 2 closes with a brief reminder about the non-normal domain attraction of the normal law. Section 3 is preparatory to the technique used: it contains a reminder on Young towers and on the Fourier transform of the Perron-Frobenius operator. Section 4 contains the proof of Theorem \ref{thm:GLT}, while section 5 those of Theorems \ref{thm:LLT}, \ref{thm:Rec} and \ref{thm:erg}. \section{The model and its geometry} \subsection{The model} Consider finitely many scatterers $\mathcal{O}_i$ on the 2-torus, (also called obstacles) $\mathbb{T}^2 \supset \mathcal{O}= \cup \mathcal{O}_i$ such that each of the scatterers is strictly convex with a $\mathcal{C}^3$-smooth boundary. Let $n(q)$ denote the unit normal vector of the boundary $\partial \mathcal{O}$ at the point $q$, directed outwards $\mathcal{O}$. The phase space of the system is: \[ X = \{ (q \in \partial \mathcal{O}, v\in\mathbb{R}^2) \mid |v|=1, \left< v,n(q)\right> \ge 0 \}.\] The dynamics $T:X\rightarrow X$ is uniform motion with $v$ velocity vector followed by an elastic collision ($v$ is mirrored to the tangent line at the point of impact). This system has a natural invariant measure: if we denote by $l$ the total length of $\partial \mathcal{O}$, then $d\mu= \frac1{2l}\left< v,n(q)\right>dqdv$, is an invariant probability measure, since $\int_X \left< v,n(q)\right> dqdv = 2l$. The normalizing constant $\frac1{2l}$ will be denoted by $c_\mu$. This phase space will be identified with a finite number of cylinders $\partial \mathcal{O} \times \left[-\frac{\pi}2;\frac{\pi}2 \right]$. So in this paper if \(v\) denotes a velocity of a phase point it is meant as \(v\in\left[ -\frac{\pi}2;\frac{\pi}2 \right]\). The boundary of this phase space consists of tangential collisions denoted by $S_0$. The dynamics resp.\ the inverse dynamics is non-continuous in backward resp.\ forward images of this set. We will denote $S_i=T^iS_0,\ i \in \mathbb Z$. The \emph{planar Lorentz process} is the natural $\mathbb{Z}^2$ cover of the above-described toric billiard. More precisely: consider $\Pi:\mathbb{R}^2 \to \mathbb{T}^2$ the factorization by $\mathbb{Z}^2$. Its fundamental domain $D$ is a square (semi-open, semi-closed) in $\mathbb{R}^2$, so $\mathbb{R}^2 = \cup_{z \in \mathbb{Z}^2} (D+z)$, where $D+z$ is the translated fundamental domain. We lift the obstacles to the plane (i.~e.\ we take $\tilde{\mathcal{O}}=\Pi^{-1}\mathcal{O}$), and define the phase space $\tilde X$, and the dynamics $\tilde T$ exactly the same way as above. The \emph{free flight function} $\tilde\psi: \tilde X \to \mathbb{R}^2$ is defined as follows: $\tilde \psi(\tilde x)=\tilde q(\tilde T\tilde x)-\tilde q (\tilde x)$. The \emph{discrete free flight function} $\tilde \kappa : \tilde X \to \mathbb{Z}^2$ is defined as follows: $ \tilde \kappa (\tilde x) = \iota (\tilde T \tilde x) - \iota (\tilde x)$, where $ \iota (\tilde x) = z $ if $ \tilde x \in D+z$. Observe finally that $\tilde\psi$ and $\tilde \kappa$ are invariant under the $\mathbb{Z}^2$ action, so there are $\psi$ and $\kappa$ functions defined on $X$, such that $\tilde\psi= \Pi^*\psi$ and $\tilde \kappa = \Pi^* \kappa$. Actually for our purposes it will be more convenient to choose the fundamental domain in such a way that $\partial \tilde{\mathcal{O}} \cap \partial D=\emptyset$. In this way $\kappa$ will be continuous. \begin{definition} The system is said to have \emph{finite horizon} if the free flight function is bounded. Otherwise the system is said to have \emph{infinite horizon}. \end{definition} \subsection{The corridors, their geometry and the tail of the free flight} \label{sec:geom} From now on we will be only considering the \emph{infinite horizon case}. It had been observed in the physical literature (cf.\ \cite{FM84}, \cite{BD85}, \cite{Bun85} and \cite{ZGNR86}) that, in this case, the anomalous diffusive behaviour is intimately related to the unboundedness of the free flight vector. More precisely it is related to its tail behaviour that is to the behavior of the free flight vector in the neighborhood of collision-free orbits. This motivated Bleher \cite{Bleher} to introduce corridors and to study probabilistic geometric estimates of the tail behavior of free flight vectors in these corridors. Our method, however, requires more delicate estimates (the results are also sharper), and therefore we are starting to build up our language. \begin{figure}[htbp] \centering \setlength{\unitlength}{\textwidth} \begin{picture}(.935,.08) \psset{unit=\textwidth} \multips(.1,.115)(.2,0){5}{\psarc(0,0){.05}{225}{315}} \multips(.05,-.035)(.2,0){5}{\psarc(0,0){.05}{45}{135}} \psline (.075,.0083)(.875,.0717) % \psframe(0,0)(.935,.08) \psline{<->}(0,.015)(0,.065) \psline{<-|}(.1,.065)(.05,.015) \uput[0](.075,.04){\(\tilde q_1 - \tilde q_0\)} \uput{2pt}[0](0,.04){$d$} \end{picture} \caption{Free-flight crossing a corridor, and some geometric constants} \label{fig:intcorr} \end{figure} In the infinite horizon case the only reason for the unboundedness of the free flight is the presence of \emph{corridors}. These are bi-infinite strips in the billiard-table $\mathbb{R}^2 \setminus \tilde{\mathcal{O}}$. The strips are tangent to the obstacles, and their slope is necessarily rational, and, moreover, - up to $\mathbb{Z}^2$ translations - there are finitely many of them. We will suppose the - geometrically generic - condition: for each corridor, and each side of the corridor the tangent obstacles are the images of a \emph{single} scatterer under $\mathbb{Z}^2$ translations. Our results are also valid in the excluded cases, but the geometric constants, which we will calculate would have a more complicated form. \begin{figure} \centering \includegraphics[width=\textwidth/2, keepaspectratio]{corridor} \caption{A corridor, and the corresponding four phase points.} \label{fig:corr-phase} \end{figure} For such a corridor there are four corresponding points in the phase-space, as shown on figure \ref{fig:corr-phase}. These points are on the boundary of $X$, and are fixed by the dynamics. (Without the previous condition these would be only periodic points.) Outside of the neigbourhood of these points the free-flight is bounded. Let us fix such a fixed point on the boundary as $x_0=(q_0;v_0)$, where $v_0$ is either $\pi/2$ or $-\pi/2$. Let us denote by $\mathcal{O}_0$ the obstacle on which $q_0$ is placed. The free flight $\psi(x_0)$ is a lattice vector $\psi(x_0)=\kappa(x_0)$, since $Tx_0=x_0$. Denote $\kappa(x_0)=w_0$, and the curvature of $\mathcal{O}_0$ at $q_0$ by $\xi_0$. Denote the considered (small enough) neighbourhood of $x_0$ by $U_0$. In \(U_0\) (means close enough to \(x_0\)) two types of nonsingular collisions can happen. First when the moving particle is ``crossing'' the corridor (see figure \ref{fig:intcorr}). In this case the free-flight is long (actually this is the only case), the closer the phase point lies to \(x_0\) the longer the free-flight can be. The next collision happens on the ``other side'' of the corridor. To make it precise let us denote by \(x_1=(q_1;v_1)\) the phase point which corresponds to the same corridor as \(x_0\), but \(v_1=-v_0\) and \(q_1\neq q_0\) (see figure \ref{fig:corr-phase}). Deonte by \(\vec Q_0\) the planar lattice vector \(\iota (\tilde q_1)-\iota(\tilde q_0)\), where the lifting is such that \(\tilde q_1-\tilde q_0\) ``crosses'' the corridor (see figure \ref{fig:intcorr}). We also need to define the ``width'' of the corridor \(d\) which is the length of that component of \(\tilde q_1-\tilde q_0\) which is perpendicular to \(w_0\). \begin{figure}[htbp] \centering \setlength{\unitlength}{\textwidth} \begin{picture}(.92,.08) \psset{unit=\textwidth} \multips(.085,.115)(.2,0){5}{\psarc(0,0){.05}{225}{315}} \multips(.035,-.035)(.2,0){5}{\psarc(0,0){.05}{45}{135}} \psline (.06,.0083)(.225,.015)(.86,.0717) \uput[90] (.06,.0083){$T^{-1}x$} \uput[90](.225,.015){$x$} %\psframe(0,0)(.92,.08) \end{picture} \caption{If phase point $T^{-1}x$ is such that $\kappa(x)=w_0$, then if $x\in U_0$ the free flight starting in $q(x)$ ``crosses'' the corridor.} \label{fig:neighbour} \end{figure} The other type of nonsingular trajectory in \(U_0\) is when the next collision is on the ``same side'' of the corridor i.~e.\ \(\kappa=w_0\). The phase point \(Tx\) is then again close to \(x_0\), but this time it is of the first type, so consecutive ``same side'' collisions cannot happen. This can be seen on figure \ref{fig:neighbour}. \begin{proposition}\label{prop:levelmeasure} Let \(U_0 \subset X\) be a sufficiently small neighbourhood of \(x_0\) then \[\mu\{x\in U_0 \mid \kappa(x)=N w_0+ \vec Q_0\}\sim c_{\mu} d^2 |w_0|^{-1} N^{-3}\] The range of $\kappa$ in $U_0$ is $\vec Q_0+w_0\mathbb{Z}^+ $ with possibly finitely many exceptions. \end{proposition} We postpone the sketchy proof for the first fact to section \ref{sec:new_coord}. The second statement was essentially proved in the above text. \subsection{The singularity structure} \begin{figure} \centering \includegraphics[width=\textwidth,keepaspectratio]{singular} \caption{Singularity structure and singular trajectories near $x_0$} \label{fig:sing} \end{figure} We are going to describe the singularity structure, and the type of singular trajectories in $U_0$. The importance of this is that singularities bound the sets for which we want to derive measure estimates. On figure \ref{fig:sing} we also plot some trajectory-segments, the configuration component of the corresponding phase point is denoted by a small tick perpendicular to the trajectory. There is a singularity curve from $S_{-1}$ which is a preimage of tangential collisions, denoted by thick line in figure \ref{fig:sing}, starting from $x_0$. This consists of phase points where the next collision will be tangential on $\mathcal{O}_0+w_0$. This is called the ``main'' singularity. There are singularity curves printed on the left of this one, starting on the boundary of the phase space and ending on the main singalarity. These consist of phase points where the free flight ``crosses'' the corridor, and the next collision is tangential. Therefore this is also a part of $S_{-1}$. A level-set of the disretized free-flight function $\kappa$ consists of a curvilinear rectangle bounded by two neighbouring curves from this singularity family, the edge of the phase space and the main singularity. On the right of the main curve there are some curves from $S_{-2}$. These also start on the boundary and end on the main curve, but unlike the previous ones, these curves have zero angle with the main line. These consist of phase points, for which the first collision is on $\mathcal{O}_0+w_0$, and the next one is tangential after crossing the corridor. On the right of the main singularity the next collision for any phase point occurs on $\mathcal{O}_0+w_0$, therefore $\kappa$ is constant $w_0$ in this half of $U_0$. These two families of singularity curves have an infinite number of pieces accumulating in $x_0$. The closer the curve is to $x_0$ the further the tangential collision occurs, after the moving particle has crossed the corridor. All these singularity curves (including the main one) have $-\xi_0$ as slope on the boundary of the phase space. The last singularity curve we want to describe is from $S_{1}$. It consists of phase points, where the previous collision was tangential on $\mathcal{O}_0-w_0$. This is the image of the half of $\partial X\cap U_0$, namely that half where $\kappa=w_0$ (on the right of $x_0$ on figure \ref{fig:sing}). Consequently (since $x_0$ is fixed) this curve starts at $x_0$. This is drawn with a dotted line on figure \ref{fig:sing}. This curve has slope 2 on the boundary. The sign of the second derivatives of all the singularity curves can be read from the picture. \subsection{New coordinates, and the joint distribution of $\kappa$, $\kappa\circ T^{-1}$} \label{sec:new_coord} In this subsection some proofs will be omitted, some will be sketchy or require further estimates. However, since the missing parts rely on simple but tedious geometrical calculations and the application of these results in later sections does not need sharp estimates, we intended to keep this section not too long. \begin{figure} \noindent \setlength{\unitlength}{0.98\textwidth} \begin{picture}(1,.15) \psset{unit=\textwidth} % \psframe(0,0)(1,.15) \psarc(0,-.9){1}{65}{90} \psline (0,.1)(1,.1) % T^{-1}x -> x trajectory segment % \psline{->}(.2688,.09908)(.3188,.04782) \psline{->}(.3188,.04782)(0.40395,0.0605925) \psline(0.40395,0.0605925)(1,.15) % i=(.6812,.10218) 0.08515 0.0127725 \psline{<-|}(.6667,.11)(0,.11) \uput{.15}[4](.67,.1){$\alpha$} \uput[240](.3188,.04782){$x$} \uput[90](.3333,.11){$zw_0$} \uput[270](.9,.1){trajectory of $x_0$} \uput[270](.15,.07){$\mathcal{O}_0$} \end{picture} \caption{The new coordinates $z,\alpha$.}\label{fig:coord} \end{figure} Instead of using the original $(q_0,v_0)$ coordinates we are going to introdue new $(\alpha,z)$ coordinates in $U_0$. The \emph{new coordinates} $z$ and $\alpha$ are shown on figure \ref{fig:coord} (these coordinates are also different from those of \cite{Bleher}). During the free-flight, bouncing off the scatterer $\mathcal{O}_0$, the trajectory of $x$ crosses the trajectory of $x_0$. This crossing point is therefore $q_0+zw_0$ for some $z\in\mathbb{R}$. The reader should convince himself that the coordinate $z$ is, in fact, a periodic one with period $1$. The other coordinate $\alpha$ is the angle $\measuredangle (w_0, \psi(x))$. The reason for this change is that these coordinates are more suitable for computations in relation with the free-flight since they are more intrinsically related to the geometry of the model, especially asymptotically (when the free flight goes to infinity). For example, the free flight has the asymptotic form \(|\kappa|\sim\frac{d}{\alpha}\) where \(d\) is the 'width' of the corridor (cf.\ fig.\ \ref{fig:intcorr}). Also, the invariant measure is asymptotically equal to $c_{\mu}|\alpha| |w_0| dz d\alpha$. We note that the crossing point (which was the base of this coordinatization) does not exist when the next collision occurs on $\mathcal{O}_0+w_0$. So these new coordinates map only the half of $U_0$ to the $(z,\alpha)$ plane. Namely that half which is drawn on the left of the main singularity on figure \ref{fig:sing}. We will denote this part by $U'_0$ (remember that $x_0$ and $U_0$ are fixed). However, this restriction does not influence the study of asymptotics, since we miss only the $w_0$-level set of $\kappa$ inside $U_0$. \begin{proof}[Proof of Proposition \ref{prop:levelmeasure}] The level-set of \(\kappa\) is a curvilinear rectangle in the \((z,\alpha)\) plane (cf.\ figure \ref{fig:rescaled}). We are going to multiply the height, the width and the density to get the measure. The width is simply 1. The height can be obtained from the formula \(\alpha\sim \frac{d}{|\kappa|}\). Writing \(\alpha'\sim\frac{d}{|\kappa|+|w_0|}\) we get \(\alpha-\alpha' \sim \frac{d|w_0|}{|\kappa|^2}\). The density is $c_{\mu}|\alpha| |w_0|$ and by substituting \(\alpha\) we get \(c_{\mu}d\frac{|w_0|}{|\kappa|}\). So the measure is \(\sim \frac{c_{\mu}d^2|w_0|^2}{|\kappa|^3}\). \end{proof} % Let $\beta_i=\measuredangle (f_i,e_x)$, where $e_x$ is % the direction of the $x$-axis, denote the angle of the % $i^{\mathrm{th}}$ corridor. By definition $\cos (\beta_i)= w_{i,1} / % |w_i|$. Further let us assign another vector to each of the % corridors. This vector $f_i$ is parallel to $w_i$, and its length is % equal to the width of the corridor (measured perpendicularly (in % $\mathbb{R}^2$) to $w_i$) $|f_i|=d_i$. Consider the quadratic form % defined on the plane by $Q_i(v) = 2 c_\mu |w_i| \left< v, f_i \right> % ^2$. The matrix of $Q_i$ is $2 c_\mu |w_i| d_i^2 \scriptsize\left( % \begin{array}{cc} % \cos^2\beta_i & \cos \beta_i \sin \beta_i \\ % \cos \beta_i \sin \beta_i & \sin^2 \beta_i % \end{array} % \right)$. The sum $\Sigma$ over all corridors of these quadratic % forms is \textbf {the covariance of the limit in Theorems % \ref{thm:GLT} and \ref{thm:LLT}}. % First of all let us observe that our most important function $\kappa_1$ is % the discretized version of $\cos \beta_i ( z + d_i / \alpha ) $ (at least in % the case when the next scattering is on the ``other side'' of the corridor). % To make the measure estimates transparent, we are going to discuss the phase % portrait of $\{x\in X \mid \kappa_1(x) =N, \measuredangle (v(x), f_i) \ll % 1\}$. \begin{figure}[h] \centering \setlength{\unitlength}{5cm} \begin{picture}(1,1) \psset{unit=5cm} \pspolygon(1,.5)(.05,.1)(0,.5)(.95,1) \pspolygon[fillstyle=hlines](.8,.416)(.7,.868)(.6,.816)(.698,.373) %% \pspolygon[fillstyle=hlines](.1,.553)(.05,.526)(.1,.121)(.151,.142) \psline [linestyle=dashed](.2,.605)(.252,.185) \uput [180](.025,.3){$S_0$} \uput [0](.226,.395){$S_1$} \uput [116.5](.475,.75){$S_{-1}$} \uput [0](.975,.75){$S_{-1}$} \uput [296.5](.525,.3){$S_{-1}$} \end{picture} \caption{Level set of $\kappa$ inside \(U_0\), and its intersection with a level set of $\kappa\circ T^{-1}$ (in the $(z, \alpha)$ coordinate plane)} \label{fig:rescaled} \end{figure} Let us explain how this $(z, \alpha)$ image on figure \ref{fig:rescaled} is related to the phase portrait on figure \ref{fig:sing} explained before. The largest curvilinear rectangle on figure \ref{fig:rescaled} is the image of a level-set of $\kappa$ under the $(z,\alpha)$ coordinate-mapping. This level set is bounded by the boundary of the phase-space on the left, two singularity curves from the first family on the top, and on the bottom, and the main singularity on the right. There is also the dashed line, which is the singularity line from $S_1$, already explained before, too. This latter line plays an important role in the joint distribution. On the left of this line $\kappa\circ T^{-1}=w_0$. On the right of this line the mapping \(T^{-1}\) takes values in the neighbourhood of another corridor-phase-point \(x_1\) (see subsection \ref{sec:geom}). We are going to use \(w_1, \xi_1, U_1, U'_1\) for the point $x_1$ in the same sense as \(w_0, \xi_0, U_0, U'_0\) have been used for the point \(x_0\). Using this notation, on the right of the line we are describing now, there lies $TU'_1$. Talking about the joint distribution in terms of our new coordinate functions, note that the \(\alpha\) coordinate function is defined in \(U'_0\). Therefore \(\alpha\circ T^{-1}\) in the domain \(U'_0\cap TU'_1\) has to be meant as applying the same coordinatization rule in \(U_1\). Since \(w_1=w_0\), the \(\alpha\) coordinate functions in \(U_0\) and \(U_1\) are comparable also as absolute angles: the observer only has to change signs. By definition the sign of $\alpha$ is positive, and $z$ is mostly positive. More precisely remember that the asymptotic form of the invariant measure does not depend on \(z\) meaning that in the \(|\kappa|\to\infty\) limit the distribution of \(z\) is uniform. Now consider the range of \(z\) in the domain \(\kappa=\vec N\): \[z_{\min}\stackrel{\mathrm{def}}{=} \min \{ z(x) \mid x\in U_0 ,\quad\kappa(x)=\vec N\} \] and respectively \[z_{\max}\stackrel{\mathrm{def}}{=} \max \{ z(x) \mid x\in U_0 ,\quad\kappa(x)=\vec N\} \] \begin{proposition}\label{prop:extz} The asymptotics of the range of \(z\) in the domain \(\kappa=\vec N\) when \(\vec N\) is in the range of \(\kappa\) (see proposition \ref{prop:levelmeasure}) and \(|\vec N|\to\infty \) is \[z_{\min}\sim 1-z_{\max}\sim \frac{d}{2|w_0|\xi_0 |\vec N|} \] \end{proposition} Easy geometrical calculations yield the collision equation: \begin{proposition}\label{prop:aoT} On $U'_0\cap TU'_1$: \[ \alpha\circ T^{-1}\sim -\alpha +2\sqrt{\alpha^2 +2\alpha z|w_0|\xi_0} \qquad (\alpha\rightarrow 0) \] \end{proposition} \begin{proposition} \label{prop:range} In \(U_0\) the following holds: \[ \min |\kappa|\circ T^{-1} \sim \sqrt{\frac {d|\kappa|}{8\xi_0 |w_0|}} \quad \text{and} \quad \max |\kappa|\circ T^{-1} \sim \frac {8\xi_0 |w_0||\kappa|^2} {d} \qquad |\kappa|\to \infty\] \end{proposition} \begin{proof} Substituting the asymptotic maximum of $z$ ($\max z \to 1$ as $\alpha\to 0$) to the collision equation and omitting non-dominant terms we get \(\max \alpha\circ T^{-1} \sim 2\sqrt{2\alpha |w_0|\xi_0}\). Since \(\alpha\sim \frac{d}{|\kappa|}\) and \(\alpha\circ T^{-1} \sim \frac{d}{|\kappa\circ T^{-1}|}\), substituting \(\alpha\) and \(\alpha\circ T^{-1}\) and then rearranging yields the first statement of the proposition. Using the time-reversion symmetry for this formula we get the second one. \end{proof} We can also compute the joint distribution of \((\kappa,\kappa\circ T^{-1})\). \begin{proposition}\label{prop:joint} \[ \mu ( \{\kappa = \vec N\} \cap \{\kappa \circ T^{-1} = \vec M\}\cap U_0) \lesssim \frac{c_{\mu} d^3|w_0|^2}{4\xi_0}\enskip \frac {|\vec N|+|\vec M|} {|\vec N|^3|\vec M|^3}\] \end{proposition} \begin{proof} For a nonempty intersection we are going to multiply the height the width and the density to get the measure. The density and the height are the same as in the proof of proposition \ref{prop:levelmeasure}. For the width consider the derivative of the collision equation \[\frac {\partial \alpha\circ T^{-1}} {\partial z}\sim \frac{2\alpha|w_0|\xi_0}{\sqrt{\alpha^2+2\alpha z|w_0|\xi_0}}\] To express the square root in terms of \(\kappa\) and \(\kappa\circ T^{-1}\) we can rearrange the collision equation, and substitute \(\alpha\) and \(\alpha\circ T^{-1}\): \[\frac{d}{|\kappa\circ T^{-1}|} + \frac{d}{|\kappa|} \sim 2 \sqrt{\alpha^2+2\alpha z|w_0|\xi_0} \] We can express the increment of \(z\) as the inverse of the derivative multiplied by the increment of \(\alpha\circ T^{-1}\). That is \[\frac{\frac{d}{|\kappa\circ T^{-1}|} + \frac{d}{|\kappa|}}{4\frac d{\kappa} |w_0|\xi_0} \enskip \frac{d|w_0|}{(\kappa\circ T^{-1})^2} \sim d\frac{\kappa + \kappa\circ T^{-1}}{4\xi_0(\kappa\circ T^{-1})^3}\] Hence the proposition. \end{proof} The measure of the set \(\{\kappa = \vec N\} \cap \{\kappa \circ T^{-1} = \vec M\}\cap U_0\) can be zero when \(\vec N\) or \(\vec M\) is not in the range of \(\kappa\) inside \(U_0\) (see proposition \ref{prop:levelmeasure}) or they fail the range inequality (we only gave the asymptotics of this in proposition \ref{prop:range} roughly \(c_3\sqrt{|\vec N|} < |\vec M| < c_4 |\vec N|^2\)). It can be also smaller than the expression given in proposition \ref{prop:joint} when the pair \((\vec N,\vec M)\) is close to the boundary of the range inequality, but we do not want to formulate the validity precisely, we just mention that in most part of the domain the inequality is sharp. It can also be checked by summing the right hand side, and getting 1 in the limit. What this essentially means is that the previous free-flight \(|\kappa\circ T^{-1}|\) is mostly in the range of \(\sqrt{|\kappa|}\). The measure of being in any other range can be estimated from above with \(|\kappa|\) powers. To formulate precisely what we will use later in subsection \ref{sec:red}: \begin{proposition}\label{prop:joint2} \[ \mu ( \{\kappa = \vec N\} \cap \{\kappa \circ T^{-1} > |\kappa|^{\frac34} \} \cap U_0) = O(|\vec N|^{-3.5})\] \end{proposition} \begin{proof} \(\sum_{M=N^{\frac34}}\frac{N+M}{N^3M^3} = O(N^{-3.5})\) \end{proof} The other level set of \(\kappa\circ T^{-1}\) which intersects \(U_0\) is the \(\kappa\circ T^{-1}=w_0\) set. \begin{proposition}\label{prop:joint3} \[ \mu ( \{\kappa = \vec N\} \cap \{\kappa \circ T^{-1} = w_0\}\cap U_0) =O (|\vec N|^{-4})\] \end{proposition} \begin{proof} As before the measure will be estimated with the product of the height, density and the width. It remained to estimate only the width. The domain is the left-hand side of the dashed line on figure \ref{fig:rescaled}. Going from right to left inside the level set in figure \ref{fig:rescaled}, the dashed line is reached exactly when \(\alpha\circ T^{-1}\) reaches its minimum. According to proposition \ref{prop:range} we have to estimate what is the value of \(z\) for which \(\alpha\circ T^{-1}\) reaches \(\mbox{const}\cdot\alpha^2\). Exact calculations based on the derivative of the mapping would give \(O(|\vec N|^{-5})\) in the right hand side of the proposition, but here it is sufficient to observe that this singularity line denoted by the dashed curve is on the left of the \(z=0\) line, where \(\alpha=\alpha\circ T^{-1}\). So the width can be estimated by \(z_{\min}\) which was given in proposition \ref{prop:extz}. \end{proof} \subsection{Non-normal domain of attraction of the normal law} \label{sec:nnda} It is well known (cf.\ \cite{Feller}) that a random variable $R$ with distribution function $\mathbb{P}_R$ belongs to the domain of attraction of a normal distribution if its characteristic function satisfies \[ \log \int e^{itu} d\mathbb{P}_R(u) = it\nu - \frac12 t^2L(1/|t|) \] for some constant $\nu\in\mathbb{R}$ and a slowly varying function $L:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ which is bounded below. The normal (or classical) domain of attraction consists of the class $L_2$, and is characterized by the boundedness of the slowly varying function $L$. In the ``non-normal'' domain of attraction the function $L$ is unbounded and is determined (up to asymptotic equivalence) by the tails of the distribution: \begin{eqnarray*} 1-\mathbb{P}_R(x)&\sim& c_1 x^{-2}l(x) \\ \mathbb{P}_R(x)&\sim& c_2 x^{-2}l(x) \quad x\rightarrow \infty \end{eqnarray*} for some constants $c_1, c_2>0, c_1+c_2=1$ and some slowly vaying function $l$, which in turn determines $L$ by \[ L(x) = \int _{-x}^x u^2 \enskip d\mathbb{P}_{R}u. \] and it follows that $l(x)=o(L(x))$. For a random variable $R$ in the non-normal domain of attraction of the normal law, the independent sum of $\mathbb{P}_R$-distributed random variables $S_n^*$ satisfies the limit theorem: $\frac {S_n^*-n\nu}{B_n} \stackrel{\mathrm{d}}{\rightarrow } \mathcal{N}(0,1)$ as $n\rightarrow \infty$ in distribution, where $B_n$ is the normalizing sequence defined by the asymptotics $\frac{nL(B_n)}{B_n^2}\rightarrow 1$. The random variable \(\kappa\) is a vector. It was shown in proposition \ref{prop:levelmeasure} that its component function belongs to the non-normal domain of attraction of the normal law, if that component is not perpendicular to all of the corridor-free-flights \(w_i\). In this case \(l(x)\) is a constant function \(l(x)\equiv \mathbf{c}\). Since the free-flight is symmetric, we have $c_1=c_2=\frac12$, and $\nu=0$. Consequently $L(x)\sim 2 \mathbf{c} \log x$, and $B_n = \sqrt{\mathbf{c} n\log n}$ is a normalizing sequence. The constant \(\mathbf{c}\) depends on which component we are looking at. If we choose \(\vec v \in \mathbb{R}^2\) a unit vector, then the constant of the \(\vec v\) component has the following expression in the terms of the geometric constants: \begin{equation} \mathbf{c}=\sum_{x\in\partial X \mid Tx=x} c_{\mu} d_x^2 \frac{\left<\psi(x), \vec v\right>^2}{2|\psi(x)|}. \label{eq:c} \end{equation} Remember that for such points \(\psi(x)=\kappa(x)\), and we used the notation \(\psi(x_0)=w_0\) for the fixed corridor we were investigating. Also note, that every term in the above sum appears exactly four times (cf.\ figure \ref{fig:corr-phase} in section \ref{sec:geom})! Since the configuration is planar, the following is true: if the corridor free-flight vectors span the plane, then every component of \(\kappa\), hence the vector itself is in the non-normal domain. If this is not the case, then one has to apply anisotropic scaling to get a nondegenerate limit distribution. Namely it should be \(\left( \begin{smallmatrix} \sqrt n & 0 \\ 0 & \sqrt {n \log n} \end{smallmatrix}\right) \) in a basis, where the first element is perpendicular to the corridor free-flights. The goal of the forthcoming arguments is to establish Bleher's hypothesis: though the (stationary) process of the free flights of our model is not an independent process, nevertheless in many respects the partial sums behave asymptotically the same way as if the variables were independent. \section{Young towers and the Fourier transform of the Perron-Frobenius operator} \subsection{Young towers} \begin{figure} \centering \includegraphics[width=\textwidth, keepaspectratio]{Young-towers_text} \caption{Young-towers, and Markov-return} \label{fig:Young} \end{figure} According to our recent understanding the most efficient way for constructing Markov partitions for billiards is to use Young towers, cf.\ \cite {Young}. We are going to introduce the main concepts without giving a full description. The presence of singularities prevent stable and unstable curves to possess a lower bound for their size in any part of the phase-space. Therefore the product structure - the key ingredient of several hyperbolic argument - can only be introduced in a complicated set. First choose an unstable curve $W$, which is short enough to ensure: a high amount of the points possesses unstable curve of this length. Then define a subset of this curve consisting of points, which remain a certain (exponentially shrinking) distance apart from $S_{-1}$ and other more technical singularities. \[ \Omega_{\infty}:=\{y \in W\mid d(T^n y,S) >\delta_1\lambda^{-n} \quad \forall n\geq 0\}\] where $\lambda$ is the hyperbolicity constant. If $\delta_1$ is chosen small enough this set has positive measure. By construction each point in $\Omega_{\infty}$ possesses a stable curve of length $\delta_1$. So far we have one unstable curve $W$, and a family of stable curves $\{\gamma^s\}$. Let us consider all the nearby unstable curves, which are long enough, and intersect all the stable curves in the previous family. These two families of curves $\{\gamma^s\}$ and $\{\gamma^u\}$ define the hyperbolic product-set $\Lambda=(\cup \gamma^u ) \cap (\cup \gamma^s)$. This set is the base of the hyperbolic Young-tower. To continue the construction of the tower we are going to focus on recurring subsets of $\Lambda$. On figure \ref{fig:Young} we can see that some parts of $\Lambda$ are mapped to $\Lambda$. However we are only interested in those retunrs, which respect the product structure. A subset of $\Lambda$ is said to be an \emph{s-subset} if it is the product of the full family of stable curves and some part of the unstable family. The notion of \emph{u-subset} is defined -mutatis mutandis- in the same way. We can see three intersections on the figure, the lower and upper ones are u-subsets. Talking about these intersections black covers gray in the unstable direction (when the reader sees black in these intersections, then on that unstable curve black covers gray). On the contrary gray covers black in the stable direction (on each stable line black can appear only where gray is already there). The inverse image of each of these two intersections is an s-subset. % Talking about the number of intersections we can investigate, that % when we restrict the inverse mapping to a stable curve in the % defining family of $\Lambda$, and on this curve to the intersection % with the u-subsets, then the inverse takes values in two distinct % stable curves in the defining family. A Markov-return is an event when some $T^n\Lambda\cap \Lambda$ is a u-subset, and it's inverse image under $T^{-n}$ is an s-subset. The possible non-Markov returns are when the intersection is not a u-subset (this is printed as the middle intersection), or when the inverse image is not an s-subset. This latter event occurs when a recurring part goes over the edge of $\Lambda$ in the stable direction. The inverse image of the Markov-recurring part is not necessarily a solid rectangle intersected with $\Lambda$. It can have infinitely many ``holes'' in it, as demonstrated on figure \ref{fig:Young}. The tower is built using these Markov-type returns. The basic $\Lambda$ set is divided into s-subsets according to Markov-returns, and each subset is marked by the return time $R$. In this way $R$ will be a function on $\Lambda$ which is constant on these s-subsets. Not all Markov-returns are considered, for sophisticated details please consult \cite{Young}! The tower itself \[\Delta\stackrel{\mathrm{def}}{=}\{(x,\omega): x\in\Lambda;\enskip \omega=0,1,\dots,R(x)-1\}\] and the dynamics on the tower is \[F(x,\omega)=\left\{ \begin{array}{ll} (x,\omega+1) & \text{if}\quad \omega+1\pi_X>>(X, \mu)\\ @A\bar FAA @AFAA @ATAA \\ (\bar\Delta, \bar{\mu}_\Delta)@<\pi_{\bar\Delta}<<(\Delta, \mu_\Delta)@>\pi_X>>(X, \mu) \end{CD} \end{equation} The projection to the original phase-space is not 1-1. On figure \ref{fig:Young} the intersection in the middle has at least two inverse images. One of them is in the ground floor, and the other is on the first floor. Since the return is not Markovian these point are to be considered as different points on the tower. Functions on the original phase space $X$ can be lifted to $\Delta$. Functions on $\Delta$ which are constant along stable directions can be considered as functions on $\bar \Delta$. For any function $\psi$ on $\Delta$ there exists functions $h$ and $\varphi$, such that $\varphi-\psi=h-h\circ F$, and $\varphi$ is constant along stable directions. In this equation the regularity of the functions can be examined, but we will skip the details, and only introduce distance, and function norms on the factorised tower $\bar\Delta$. Remember that the factorised tower $\bar \Delta$ has a Cantor-structure, and a Markov-partition. The Cantor-hierarchy can be redefined with the separation time $s(x,y)=\min\{k\geq 0\mid \bar F^kx \mbox{ and } \bar F^k y\mbox{ lie in different elements of the Markov-partition}\}$. With any $0<\beta<1$ the function $\beta^s$ is a metric providing the original Cantor topology. On $\bar \Delta$ Young uses two kind of norms: the $\mathcal{C}$ norm is\[ \left\| \varphi \right\|_{\mathcal{C}} \stackrel{\mathrm{def}}{=} \sup_{l,j}\left\| \varphi|_{\Delta_{l,j}} \right\|_\infty e^{-l\epsilon}\] where $\left\|\,.\,\right\|_\infty$ is the essential supremum wrt $\mu$, and the indices $(l,j)$ refer to the elements of the Markov-partition. The $\mathcal{L}$ norm is a sum of this, and the $h$-norm: \[\left\| \varphi \right\|_h \stackrel{\mathrm{def}}{=} \sup_{l,j} \left( \sup_{x,y\in\Delta_{l,j}} \frac {\left| \varphi (x)- \varphi(y) \right|} {\beta^{s(x,y)}} \right) e^{-l\epsilon};\] where the inner $\sup$ is again essential supremum wrt $\mu\times\mu$. For a H\"older function $f$ on the original billiard phase-space, we can associate a function on $\bar\Delta$ as described above, such that for any $\beta$ smaller than a certain number (computed from the original H\"older exponent of $f$) the resulting $\varphi$ has a finite $h$-norm. In these definitions the role of $\epsilon$ is the following: without $\epsilon$ the Jacobian of the mapping would be 1 except when recurring to the base of the tower. However estimates expressed in the terms of this norm see a uniform expansion. Not to do harm the mapping should be expanding, when recurring to the base, so we have to choose $\epsilon$ smaller then the Lyapunov-exponent. The Perron-Frobenius operator $P$ is defined on functions on $\bar\Delta$ with finite ${\cal L}$-norm as the adjoint of $\bar F$ wrt the measure $\bar\mu_\Delta$. \[\int f\cdot g\circ \bar F \enskip d \bar \mu_{\Delta} = \int P(f)\cdot g \enskip d \bar \mu_{\Delta} \qquad f,g\in\mathcal{L}\] It also has an algebraic form \[P(f)(x)= \sum_{y\mid \bar F y =x} \frac{f(y)}{J(y)} \qquad f\in\mathcal{L}\] where \(J\) in the denominator is the Jacobian \(\frac {\partial \bar\mu_{\Delta}^*} {\partial \bar\mu_{\Delta}}\) the Radon-Nikodym derivative of the pullback measure wrt the original measure. On the top of the tower the number of inverse images is infinite, otherwise there is a single preimage, and the Jaocbian is 1. The main issue here is that the transfer operator \(P\) possesses a spectral gap. This is the main reason for introducing this rather complicated and implicit symbolic dynamics. This gives those analytical tools in our hands which we describe now. Although \(\kappa\) is defined in the original phase-space \(X\), we will use the same notation for the function lifted to the tower. Since \(\kappa\) is locally constant it can also be considered as a function on \(\bar \Delta\). \subsection{Fourier transform of the Perron-Frobenius operator} In our case the Birkhoff sum $S_n=\sum_k \kappa \circ T^k$ is not an independent sum. Since Nagaev's 1957 work, \cite{Nag}, for concluding a limit theorem, one traditionally uses the Fourier-transform of the transfer operator. The Fourier transform operator is defined on the Young-tower, since that is the symbolic space, on which we have the full power of analytical tools built on the transfer operator: \[P_t(h) \stackrel {\mathrm{def}} {=} P (e^{i \left} h) \qquad (h\in\mathcal{L}, t\in\mathbb{R}^2)\] It has the following simple connection with the characteristic function of the Birkhoff-sum: \[\int \exp (i\left) d\mu = \int P_t^n(1) d\bar\mu_{\Delta}.\] For small values of $t$, $P_t$ can be considered as a perturbation of $P$, since \(P_0=P\). Then one, in general, proves that $P_t$ possesses a gap between the leading simple eigenvalue $\lambda_t$ and the rest of the spectrum and the gap is uniformly bounded away from zero. For large values of \(t\) one needs to know exactly for which \(t\) values will the unit circle intersect the spectrum. In our case for the continuous free flight function this occurs when \(t\in 2\pi\mathbb{Z}^2\). For that reason we swiched to the discretized function \(\kappa\), since the latter one is an integer valued vector function, and thus \(P_t=P\) when \(t\in 2\pi\mathbb{Z}^2\), so we can factorize, and consider \(t\in 2\pi \mathbb{T}^2\). This is the question of minimality which is completely described in \cite{SzVf}. Moreover, if our interest is a global limit law, then it is sufficient to derive an appropriate expansion for $\lambda_t$ for small values of $t$. Indeed, this is also contained in the statement of Theorem \ref{theorem:integral} from \cite{BG}. As a matter of fact, our main task will be to check the key condition of that theorem and thus we will actually derive a simple asymptotics $P_t^n = \lambda_t^n + O(\vartheta^n)$, for a suitable $\vartheta<1$. Consequently, the characteristic function of the dependent sum can be approximated with a power. Thus it is only the asymptotics of $\lambda_t$ that has remained to be investigated, and we expect the same behaviour what we have seen in the single term characteristic function. This will mean that the effect of dependence is negligible: both the independent and the dependent sums have the same asymptotic expansion for the Fourier transform, namely $1+\mathbf{c} n |t|^2 \log |t|$. (In this formula \(\mathbf{c}\) describes the direction-dependence, and \(|t|\) the length, remember the definition of \(\mathbf{c}\)) in \ref{eq:c}! Further, relying upon the method of \cite {SzVf}, we will also be able to handle the local version of the global limit law. \section{Proof of Theorem \ref{thm:GLT}: The asymptotics of $\lambda_t$} \label{sec:lambda_t} In the forthcoming section we are going to establish the global limit law of Theorem \ref{thm:GLT} via an asymptotic expression at $t\to 0$ of $\lambda_t$, the leading eigenvalue of the Fourier-transform of the Perron-Frobenius operator. We will rely upon ideas of \cite{BG} and of our work \cite {SzVi}. The following theorem of \cite{BG} provides a condition for a limit law in the non-standard domain of attraction of the gaussian law for models possessing a Young tower with an exponential tail bound for the height function, in general. \begin{theorem} \label{theorem:integral} (\cite{BG}, Theorem 3.5) Assume that the distribution of $g:\bar \Delta\rightarrow\mathbb{R}$ is in the nonstandard domain of attraction of the normal law. Remember that \(\omega\) was the level function of the tower. Denote the ``tower-sum'' by $G=\sum_{k=1}^\omega g\circ \bar F^{-k}$. Let $L,l$ be as in subsection \ref{sec:nnda}. Assume, moreover, that $l(x \log x)/l(x) \rightarrow 1, \quad L(x\log x)/L(x) \rightarrow1$ when $x\rightarrow \infty$. Finally, assume that there exists a real number $a \not= -1/2$ such that \begin{equation} \label{eq:integral} \int g(e^{itG}-1) = (a+o(1))itL(1/|t|). \end{equation} Write $L_1(x)=(2a+1)L(x)$, and choose a sequence $B_n \rightarrow \infty$ such that $\frac n{B_n^2} L_1(B_n) \rightarrow 1$. Then for \(\lambda_t\) the leading eigenvalue of the \(P_t(h)=P(e^{itg}h)\) Fourier-transform-operator: \[\lambda_t = 1- \frac {t^2}2 L_1(1/|t|)(1+o(1))\] and consequently for the \(S_n=\sum_{k=0}^{n-1} g\circ\bar F^k\) Birkhoff-sum: \[ \frac{S_n - n\int g}{B_n} \stackrel {\mathrm{d}} {\rightarrow} \mathcal{N}(0,1). \] \end{theorem} We will indeed show that in our case the integral condition (\ref{eq:integral}) of the above theorem holds for \(g=\kappa\) with $a=0$. (This is actually the case discussed by \cite{ad2}.) We note that this case will allow a simpler treatment than that of the stadium because here the expansion rate is much larger (though the logarithm of the free flight function in a corridor does not form a random walk as it is the case in the stadium). Our arguments for the local limit theorem will use again the asymptotics of the main eigenvalue $\lambda_t$, which is expressed for our case by the next Corollary. \begin{corollary} The Fourier transform operator of the discretized free-flight function has the following expansion for its leading eigenvalue: $\lambda_t \sim 1 + \mathbf{c} n |t|^2 \log |t|$. \end{corollary} Turn to the proof of the integral condition (\ref{eq:integral}). In our case the function $\kappa$ is locally constant, so we can pass the integral immediately to $\Delta$. Denote the ``tower-sum'' by $K=\sum_{k=1}^\omega \kappa \circ F^{-k}$. It is easy to see that the dominating terms of the integral in (\ref{eq:integral}) are those corresponding to parts of the phase space when the process is close to a singular point say \(x_0\) (discussed in \ref{sec:geom}). The estimate of this integral is based on the following fact. We have already observed that high values of \(\kappa\) are typically reached rapidly: \(\kappa\circ T^{-1}\) is in the order of \(\sqrt{|\kappa|}\). During this fast trajectory segment the tower-sum can be estimated with the last term, hence it is also of order \(\sqrt{|\kappa|}\). Since \(|e^{it\bar K}-1|\leq t \bar K\), the integral can be estimated by \(t\sum_n n\sqrt{n}\cdot \mu\{|\kappa|=n\} =O(t)\). The trajectories which do not provide fast reach have polinomially small (in \(|\kappa|\)) relative measure in the level-sets. These domains can be discarded due to the following lemma: \begin{lemma} Any part of the integration domain $A$ with measure $\mu(A\cap \{|\kappa|=N\})=O(N^{-3-\alpha})$ with any $\alpha>0$ can be thrown away: \[ \int_{\Delta} \kappa (e^{i\left}-1) = \int_{\Delta\setminus A} \kappa (e^{i\left}-1) + O(|t|) \] \label{lem:discard} \end{lemma} This is proved in proposition (4.17) of \cite{BG}. % \begin{proof} % \[ \left| \int_{A} \bar{\kappa}_1 (e^{it\bar{K}}-1) \right| \leq \int 1_A % |\bar{\kappa}_1| t |\bar{K}| \leq |t| \left( \int (1_A |\bar{\kappa}_1|)^p % \right)^{1/p} \left( \int |\bar{K}|^q \right)^{1/q}. \] By the condition % the first integral $\int (1_A |\bar{\kappa}_1|)^p$ is finite for $p < 2 + % \alpha$. We can choose such a $p>2$. Thus $q=(1-1/p)^{-1}<2$. We are going % to show that for any $1 < q<2$, $ \bar{K}$ belongs to $L^q$. Let us % introduce for $x\in\Delta$ the return function $\varphi(x) = % \min\{k\in\mathbb{N}\mid \omega(U^kx)=0\}$. Set also % $\psi(x)=\varphi(\pi_0x)$, where $\pi_0$ is the projection $U^{-\omega}$ to % the basis. \[|\bar{K}|^q = \left| \sum_{k=0}^{\omega(x)-1} % \bar{\kappa}_1(U^k\pi_0x) \right| ^q \leq \omega(x)^{q-1} % \sum_{k=0}^{\omega(x)-1} |\bar{\kappa}_1(U^k\pi_0x)| ^q. \] Changing % variables, we get \[ \int |\bar{K}|^q \leq \int |\bar{\kappa}_1(y)|^q % \sum_{k=1}^{\varphi(y)-1} \omega(U^ky)^{q-1} \leq \int % |\bar{\kappa}_1(y)|^q \psi(y)^q. \] Since the tower has exponentially small % tails, the function $\psi$ belongs to $L^{p'}$ for any $p'<\infty$. Using % the H\"older inequality with a sufficiently large $p'$, we obtain $\int % |\bar{K}|^q < \infty$. % \end{proof} Though the integration domains are in the hyperbolic Young-tower, if we identify the sets to be disregarded on the original phase-space \(X\), then by measure preservation their pullbacks will satisfy the lemma. \subsection{Reduction of the integration domain} \label{sec:red} First we are going to discard that part $A_1$ of the neighbourhood of \(x_0\) where the last step was not fast enough, i.~e.\ $ A_1 = \{x\in U_0 \mid |\kappa\circ T^{-1}| > |\kappa|^{\frac 34}\} \subset X$. We also discard the set $A_2$ of those points where \(\kappa\circ T^{-1} =w_0\), i.~e.\ \(A_2=U_0\setminus U'_0\). \begin{figure}[h] \centering \setlength{\unitlength}{5cm} \begin{picture}(1,1) \psset{unit=5cm} \pspolygon (1,.5)(.05,.1)(0,.5)(.95,1) \psline [linestyle=dashed](.2,.605)(.252,.185) \psline (.3,.658)(.357,.229) \rput (.1,.3){\(A_2\)} \rput (.27,.4){\(A_1\)} \psline[linecolor=red](.3057,.6151)(.955,.95) \psline[linecolor=red](.3114,.5722)(.96,.9) \psline[linecolor=red](.3171,.5293)(.965,.85) \psline[linecolor=red](.3228,.4864)(.97,.8) \psline[linecolor=red](.3285,.4435)(.975,.75) \psline[linecolor=red](.3342,.4006)(.98,.7) \psline[linecolor=red](.3399,.3577)(.985,.65) \psline[linecolor=red](.3456,.3148)(.99,.6) \psline[linecolor=red](.3513,.2719)(.995,.55) \end{picture} \caption{The discarded sets $A_1$ and \(A_2\) in the level set of $\kappa$ and the foliation with nearly unstable curves} \label{fig:discarded} \end{figure} We already know that the relative measure of $(A_1\cup A_2)$ inside $\{|\kappa|=N\}$ is $O(N^{-1/2})$ (cf.\ propositions \ref{prop:joint2} and \ref{prop:joint3}). % The second discarded set $\mathcal{T}_2$ will consist of those $\{ \kappa_1 % \circ T^{-1} = M \} \cap \{\kappa_1 = N\}$ strips which intersect $S_{-1}$ % on the right i.~e.\ which do not intersect in a Markovian manner. To % estimate the relative measure we need to know the variation of $\alpha\circ % T^{-1}$ on the singularity bounding on the right. The asymptotics of the % derivative: % \begin{multline} \label{eq:matrix} % DT^{-1}(z,\alpha)= % \begin{pmatrix} % \frac {\partial z\circ T^{-1}} {\partial z}& \frac {\partial\alpha\circ % T^{-1}} {\partial z} % \\ % \frac {\partial z\circ T^{-1}} {\partial\alpha} & \frac % {\partial\alpha\circ T^{-1}} {\partial\alpha} % \end{pmatrix} % \sim \\ % \begin{pmatrix} % -\frac {d} {(\alpha\circ T^{-1})^2} \frac {4\alpha\gamma_l} {\alpha - % \alpha\circ T^{-1}} & \frac {-4\alpha\gamma_l} {\alpha - \alpha\circ % T^{-1}} % \\ % \frac 1 {4\gamma_l} - \frac {\alpha} {4\gamma_l \alpha\circ T^{-1}} % -\frac {d} {(\alpha\circ T^{-1})^2} \frac {3\alpha + \alpha\circ T^{-1} % + 4z\gamma_l} {\alpha - \alpha\circ T^{-1}} & -\frac {3\alpha + % \alpha\circ T^{-1} + 4z\gamma_l} {\alpha - \alpha\circ T^{-1}} % \end{pmatrix} % \end{multline} % Remember that we have already excluded the $\kappa_1\circ T^{-1}=1$ % set, therefore $\alpha$ and $\alpha\circ T^{-1}$ has opposite sign. % To estimate the relative measure of $\mathcal{T}_2$, consider the % axis projections of the considered part of $S_{-1}$. Since the slope % is bounded away from zero and infinity both of the widths are of % order $N^{-2}$. From the derivative matrix one can calculate that % the variation of the non-disretized free-flight $\psi\circ T^{-1}$ on % such a set tends to zero polynomially in $N$. It follows then that % if $N$ is large there are at most two level sets of $\kappa_1\circ % T^{-1}$ which can intersect $\mathcal{T}_2$. From equations % (\ref{eq:range}) and (\ref{eq:joint}) we can see that the relative % measure of $\mathcal{T}_2$ is of order $N^{-1/2}$. Let us foliate the remaining part with curves whose direction is nearly unstable (the derivative is in the unstable cone for $T^{-1}$).% The set $\mathcal{T}_3$ will consist of the union of such % curves which intersect $S_{-1}$ either on the top or at the bottom of % the picture. From (\ref{eq:matrix}) it can be read that the unstable % slope is asymptotically $\alpha^2/d$. Deviations can also be easily % calculated. The estimate of the relative measure is based on the % observation that by using the -also measure preserving- % time-reversing transformation $\mathcal{T}_3$ can be covered by % images of $\mathcal{T}_1$ and $\mathcal{T}_2$ of the same level set % -but opposite $\alpha$ sign, so swapping ``upper'' and ``lower'' % side-, for which we already have the polynomial estimate. The third discarded set $A_3$ will consist of points, for which the backward $Z\log N$-step trajectory meets the $\{|\kappa| > N^{\frac 45}\}$ set, where \(Z\) is a large number to be chosen later. To estimate the relative measure of $A_3$ we prove the following lemma: \begin{lemma}\label{lemma:chernov} There exists a constant $C$ such that, for any large enough integer $Z$ it is true: for any large enough $N$, given any unstable curve (for the mapping $T^{-1}$) in the set $\{|\kappa| = N\} \setminus \bigcup_{i=1,2} A_i$, the points for which $|\kappa|$ increases above $N^{4/5}$ within $Z\log{N}$ iterations of $T^{-1}$ occupy a subset whose relative measure is less than $CN^{-1/11}$ in that curve. \end{lemma} \begin{proof} The proof is a suitable modification of Lemma 4.14 of \cite{BG}. Note first that in the set $\{|\kappa| = N\} \setminus A_1$ one has $|\kappa \circ T^{-1}| < N^{\frac{3}{4}}.$ The points which reach $|\kappa|>N^{4/5}$ get closer to the singularity than \(\mathrm{const}\cdot N^{-8/5}\). We are going to measure the set of points of the latter type. The map $T^{-1}$ satisfies Chernov's axioms \cite{Ch99} as this was checked for our infinite horizon case by Chernov in section 8 of the same article. Using exactly the same ideas as in \cite{BG} we choose $\delta=(Z[\mathcal{D},\mathcal{D},0]n)^{-1/\sigma}$ and apply Theorem 3.1 in \cite {Ch99}. We obtain a decreasing sequence $W_0^1\supset W_1^1\supset \cdots\supset W_{[Z\log{N}]}^1$ of subsets of the given LUM such that \begin{multline*} \forall c>0,\forall 0\leq p \leq Z\log{N} \\ \mathrm{Leb}\{ x\in W_p^1 \mid \mathrm{dist} (T^{-p}x, S_0) \leq c N^{-8/5} \} \leq CcN^{-8/5} \end{multline*} (by equation (3.3) in \cite{Ch99}), and \[ \forall 0\leq p \leq Z\log{N}, \quad \mathrm{Leb} (W_p^1\setminus W_{p+1}^1) \leq \frac CN \mathrm{Leb} (\mathrm{LUM}) \] (By (iv), (3.5) in \cite {Ch99} and our choice of $\delta$). The second estimate is itself a relative measure estimate. For the first one we observe that $N^{-8/5}=N^{-1/10}N^{-6/4} \leq N^{-1/10} \mathrm{const}\cdot\mathrm{Leb}(\mathrm{LUM})$. This gives a measure at most $C\log{N}\enskip N^{-1/10}$, hence the lemma. \end{proof} % So far we have defined the set $Y$, and showed that the relative measure of % its complement in the set $\{\kappa_1 = N\}$ is less than $CN^{-1/11}$. We % also know that on $Y$ backward trajectories does not reach the $\{\kappa_1 > % N^{4/5}\}$ set for $K\log N$ iterations. % Next we are going to show that the polynomial estimate of the relative % measure of the $\cup\mathcal{T}_i$ thrown-away set ensures that this set is % really negligible. In fact, the estimates of Propositions 4.12 and 4.13 boil % down to the following lemma: The last discarded set will be defined on the tower: We are going to throw away that part $A_4$ of the integration domain, which is too high on the tower $\omega>Z\log|\kappa|$. Since the tower has exponentially small tails, if $Z$ was chosen large enough, then the discarded set has measure $\mu_{\Delta}(A_4\cap \{|\kappa|=N\})=O(N^{-4})$, and thus by lemma \ref{lem:discard} the integral can be restricted to its complement. Now it remained to estimate the integral on the non-discarded set. \begin{lemma} \[ \int_{\pi_X^{-1}(X\setminus\bigcup_{i=1}^3 A_i)\setminus A_4} \kappa (e^{i\left}-1) = O(|t|)\] \end{lemma} \begin{proof} Consider the integral on the left. By the definition of the discarded sets we have: \begin{multline*} \left| \int_{\pi_X^{-1}(X\setminus\bigcup_{i=1}^3 A_i)\setminus A_4} \kappa (e^{i\left}-1) \right| \leq \\ \leq |t| \int_{\pi_X^{-1}(X\setminus\bigcup_{i=1}^3 A_i)\setminus A_4} |\kappa| |K| \leq C |t| \sum_n \mu\{|\kappa| = n\} n \log n \; n^{4/5} \leq C|t| \end{multline*} \end{proof} \section{Proof of Theorems \ref{thm:LLT}, \ref{thm:Rec} and \ref{thm:erg}} The proof of the local limit theorems (Theorems \ref{thm:LLT2} and \ref{thm:joint} below) and that of the recurrence is -mutatis mutandis- the same as for the finite horizon case \cite{SzVf}. Here we are going to state the theorems only, and comment just the beginning, where the different norms used by the different approaches (\cite{BG} and \cite{SzVf}) could lead to a confusion. \begin{theorem}[Nagaev-type theorem, modification of \cite{SzVf} Theorem 3.3]\label{thm:nag} There are constants $\epsilon>0$, $K>0$ and $\theta<1$ and a function $\rho:(-\epsilon, \epsilon)\to \mathcal{L}$ such that \begin{equation} \label{eq:nag} \left\| P_t^n h - \lambda_t^n \rho_t \int_{\bar\Delta} h d\bar\mu_\Delta \right\|_{L^1} \leq K \theta^n \left\| h \right\|_{\mathcal{L}} \quad \forall |t|<\epsilon, n\ge 1, h\in\mathcal{L} \end{equation} and \[ \|\rho_t-1\|_{L^1}\to 0 \enskip (t\to 0), \enskip \lambda_t = 1 + (1+o(1)) \mathbf{c} |t|^2 \log |t| \] \end{theorem} The statement in \cite{SzVf} was stronger in the sense of norms, but inequality (\ref{eq:nag}) is completely enough for the proof of our local limit theorems. The proof of Theorem \ref{thm:nag} can also be straightforwardly derived from considerations in \cite{BG}, namely the proof of their theorem 3.5. \begin{theorem}[Local limit theorem]\label{thm:LLT2} Suppose that the corridor free flights \(\{\kappa(x) \mid x\in\partial X,\enskip Tx=x\}\) span the plane. Let $k_n\in\mathbb{Z}^2$ be such that $\frac{k_n}{\sqrt{n\log n}}\rightarrow k \in \mathbb{R}^2$. Then \[ n\log n \enskip \mu\{S_n = k_n\} \rightarrow \varphi(k)\] where $\varphi$ is a non-degenerate normal density function with zero expectation and covariance matrix \[\sum_{x\in\partial X\mid Tx=x} \frac{c_{\mu} d_x^2} {2|\kappa(x)|} \begin{pmatrix} \kappa_1^2(x) & \kappa_1(x)\kappa_2(x) \\ \kappa_1(x)\kappa_2(x) & \kappa_2^2(x) \end{pmatrix} \] where \(\kappa=(\kappa_1,\kappa_2)\) is the notation for the component functions. \end{theorem} The proof of recurrence follows a version of the Borel-Cantelli lemma due to Lamperti. See \cite{SzVf} for the statement, conditions and the details of application. For this part of the Borel-Cantelli lemma one needs some kind of independence. Lamperti's condition can be fulfilled when one proves an asymptotic independence statement: \begin{theorem}\label{thm:joint} Let $j_n\in \mathbb{Z}^2$ be such that $\frac{j_n} {\sqrt{n\log n}}\rightarrow j\in\mathbb{R}^2$, and $k_n\in \mathbb{Z}^2$ be such that $\frac{k_n} {\sqrt{n\log n}}\rightarrow k\in\mathbb{R}^2$. If the corridor free flights span the plane, then \[ \lim_{m,n-m\rightarrow\infty} m\log m \enskip (n-m) \log (n-m) \enskip \mu\{S_m = j_m, S_n = j_m + k_{n-m} \} = \varphi(j) \varphi(k). \] \end{theorem} The proof consists of repeating twice the same integral transformations as in the proof of the local limit theorem (again see the proof of Theorem 4.2 in \cite{SzVf} for details; we note, however, that the claim of Theorem 4.2 in \cite{SzVf} contains a misprint, in fact it ought to be analogous with that of Theorem \ref{thm:joint}). \begin{theorem} The invariant (infinite) measure of the Lorentz process $d\tilde\mu = \left< v,n(q) \right> dqdv$ (where $dq$ is the arclength measure on the boundary of infinitely many scatterers) is ergodic i.~e.\ for any invariant set $A$ either $\tilde\mu(A)=0$ or $\tilde\mu(\tilde X \setminus A)=0$. \end{theorem} This follows from recurrence and \cite{Sim 89}, where the author showed the equivalence between recurrence and the ergodicity of the infinite measure (see also \cite{Pene}). \smallskip \noindent{\bf Acknowledgement.} The authors express their sincere thanks to P\'eter B\'alint and Sebastien Gou\"ezel for making the first versions of their paper \cite{BG} available in statu nascendi. \begin{thebibliography}{BChS 91} \bibitem[AD 01a]{Aa} Jon Aaronson and Manfred Denker. 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closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 18.800985 663.91986 moveto 135.79433 663.91986 lineto 135.79433 664.30749 lineto 18.800985 664.30749 lineto 18.800985 663.91986 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 18.800985 578.64233 moveto 137.5737 578.64233 lineto 137.5737 579.02996 lineto 18.800985 579.02996 lineto 18.800985 578.64233 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 57.947067 567.40118 moveto 58.836751 567.40118 lineto 58.836751 667.40846 lineto 57.947067 667.40846 lineto 57.947067 567.40118 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 87.306595 567.78882 moveto 87.751436 567.78882 lineto 87.751436 667.40847 lineto 87.306595 667.40847 lineto 87.306595 567.78882 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 82.8582 567.78882 moveto 83.747883 567.78882 lineto 83.747883 665.85796 lineto 82.8582 665.85796 lineto 82.8582 567.78882 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -82.5606 603.06268 moveto 242.63355 603.06268 lineto 242.63355 603.83793 lineto -82.5606 603.83793 lineto -82.5606 603.06268 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 18.800985 624.38208 moveto 138.01854 624.38208 lineto 138.01854 625.15733 lineto 18.800985 625.15733 lineto 18.800985 624.38208 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -81.041801 618.18005 moveto 241.92817 618.18005 lineto 241.92817 618.9553 lineto -81.041801 618.9553 lineto -81.041801 618.18005 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -81.540359 620.50586 moveto 241.26534 620.50586 lineto 241.26534 620.89348 lineto -81.540359 620.89348 lineto -81.540359 620.50586 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 19.023409 622.05634 moveto 136.90644 622.05634 lineto 136.90644 622.44396 lineto 19.023409 622.44396 lineto 19.023409 622.05634 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 84.859993 567.40118 moveto 85.749676 567.40118 lineto 85.749676 666.63321 lineto 84.859993 666.63321 lineto 84.859993 567.40118 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 80.856407 567.40118 moveto 81.301249 567.40118 lineto 81.301249 667.40846 lineto 80.856407 667.40846 lineto 80.856407 567.40118 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 93.756821 567.78882 moveto 94.201662 567.78882 lineto 94.201662 667.7961 lineto 93.756821 667.7961 lineto 93.756821 567.78882 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 103.09849 568.17645 moveto 103.98817 568.17645 lineto 103.98817 665.85798 lineto 103.09849 665.85798 lineto 103.09849 568.17645 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 123.11634 568.17645 moveto 124.00602 568.17645 lineto 124.00602 668.57137 lineto 123.11634 668.57137 lineto 123.11634 568.17645 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 18.578562 601.51221 moveto 135.57191 601.51221 lineto 135.57191 601.89983 lineto 18.578562 601.89983 lineto 18.578562 601.51221 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 39.930973 568.56409 moveto 40.820656 568.56409 lineto 40.820656 666.63323 lineto 39.930973 666.63323 lineto 39.930973 568.56409 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 42.15517 567.40118 moveto 42.600012 567.40118 lineto 42.600012 666.24558 lineto 42.15517 666.24558 lineto 42.15517 567.40118 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 73.73893 567.78882 moveto 74.628613 567.78882 lineto 74.628613 666.2456 lineto 73.73893 666.2456 lineto 73.73893 567.78882 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -80.925011 591.43396 moveto 242.44276 591.43396 lineto 242.44276 591.82159 lineto -80.925011 591.82159 lineto -80.925011 591.43396 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -81.003654 604.61322 moveto 245.21973 604.61322 lineto 245.21973 605.00085 lineto -81.003654 605.00085 lineto -81.003654 604.61322 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -83.335808 606.55133 moveto 242.33479 606.55133 lineto 242.33479 607.71421 lineto -83.335808 607.71421 lineto -83.335808 606.55133 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -82.077614 610.03998 moveto 241.26089 610.03998 lineto 241.26089 610.4276 lineto -82.077614 610.4276 lineto -82.077614 610.03998 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 20.357931 639.11182 moveto 134.4598 639.11182 lineto 134.4598 640.4685 lineto 20.357931 640.4685 lineto 20.357931 639.11182 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 19.69068 642.01904 moveto 134.45981 642.01904 lineto 134.45981 642.40667 lineto 19.69068 642.40667 lineto 19.69068 642.01904 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 19.69068 636.97992 moveto 134.01497 636.97992 lineto 134.01497 637.36754 lineto 19.69068 637.36754 lineto 19.69068 636.97992 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 50.710617 568.0813 moveto 51.339718 568.0813 lineto 51.339718 666.34341 lineto 50.710617 666.34341 lineto 50.710617 568.0813 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 65.809036 568.49243 moveto 66.123587 568.49243 lineto 66.123587 667.16567 lineto 65.809036 667.16567 lineto 65.809036 568.49243 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 76.189217 568.49243 moveto 76.503767 568.49243 lineto 76.503767 666.0693 lineto 76.189217 666.0693 lineto 76.189217 568.49243 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 68.562538 591.27441 moveto 69.229801 591.27441 lineto 69.229801 621.50917 lineto 68.562538 621.50917 lineto 68.562538 591.27441 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 70.763206 591.31055 moveto 71.392308 591.31055 lineto 71.392308 621.4607 lineto 70.763206 621.4607 lineto 70.763206 591.31055 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 48.036934 591.44763 moveto 48.351484 591.44763 lineto 48.351484 621.52927 lineto 48.036934 621.52927 lineto 48.036934 591.44763 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 45.205963 591.51617 moveto 45.520514 591.51617 lineto 45.520514 621.11815 lineto 45.205963 621.11815 lineto 45.205963 591.51617 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 26.962027 591.37909 moveto 27.433852 591.37909 lineto 27.433852 621.52925 lineto 26.962027 621.52925 lineto 26.962027 591.37909 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 100.40957 591.51617 moveto 100.88139 591.51617 lineto 100.88139 621.25519 lineto 100.40957 621.25519 lineto 100.40957 591.51617 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 118.81079 591.51617 moveto 119.12534 591.51617 lineto 119.12534 621.25519 lineto 118.81079 621.25519 lineto 118.81079 591.51617 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 96.949509 591.37909 moveto 97.264059 591.37909 lineto 97.264059 621.25515 lineto 96.949509 621.25515 lineto 96.949509 591.37909 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 95.981018 648.75391 moveto 101.31912 648.75391 lineto 101.31912 656.10184 lineto 95.981018 656.10184 lineto 95.981018 648.75391 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 43.934559 648.17816 moveto 49.272657 648.17816 lineto 49.272657 656.52506 lineto 43.934559 656.52506 lineto 43.934559 648.17816 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 77.96492 649.19427 moveto 79.299445 649.19427 lineto 79.299445 656.05718 lineto 77.96492 656.05718 lineto 77.96492 649.19427 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 27.512506 648.93848 moveto 27.827057 648.93848 lineto 27.827057 656.20192 lineto 27.512506 656.20192 lineto 27.512506 648.93848 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 34.118065 649.07556 moveto 35.061716 649.07556 lineto 35.061716 656.33901 lineto 34.118065 656.33901 lineto 34.118065 649.07556 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 53.620197 648.66443 moveto 54.249298 648.66443 lineto 54.249298 656.20197 lineto 53.620197 656.20197 lineto 53.620197 648.66443 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 68.954514 648.52734 moveto 69.269064 648.52734 lineto 69.269064 656.47602 lineto 68.954514 656.47602 lineto 68.954514 648.52734 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 71.156403 648.80145 moveto 71.628228 648.80145 lineto 71.628228 656.33899 lineto 71.156403 656.33899 lineto 71.156403 648.80145 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 90.579865 648.66443 moveto 90.894415 648.66443 lineto 90.894415 656.47606 lineto 90.579865 656.47606 lineto 90.579865 648.66443 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 118.02441 648.66443 moveto 118.33896 648.66443 lineto 118.33896 656.20197 lineto 118.02441 656.20197 lineto 118.02441 648.66443 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 120.06897 648.80145 moveto 120.69807 648.80145 lineto 120.69807 656.20195 lineto 120.06897 656.20195 lineto 120.06897 648.80145 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 126.83184 648.66443 moveto 127.30367 648.66443 lineto 127.30367 656.06492 lineto 126.83184 656.06492 lineto 126.83184 648.66443 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 110.94704 648.93848 moveto 111.26159 648.93848 lineto 111.26159 655.92783 lineto 110.94704 655.92783 lineto 110.94704 648.93848 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 17.745619 591.20734 moveto 18.940158 591.20734 lineto 18.940158 621.3933 lineto 17.745619 621.3933 lineto 17.745619 591.20734 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 14.759275 591.55432 moveto 15.157455 591.55432 lineto 15.157455 621.04636 lineto 14.759275 621.04636 lineto 14.759275 591.55432 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 12.768373 591.3808 moveto 13.365642 591.3808 lineto 13.365642 621.04632 lineto 12.768373 621.04632 lineto 12.768373 591.3808 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 4.8047824 591.3808 moveto 7.7228148 591.3808 lineto 7.7228148 621.2198 lineto 4.8047824 621.2198 lineto 4.8047824 591.3808 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 2.4156919 591.55432 moveto 3.2120517 591.55432 lineto 3.2120517 620.87288 lineto 2.4156919 620.87288 lineto 2.4156919 591.55432 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -1.9642819 591.3808 moveto -1.566102 591.3808 lineto -1.566102 620.69935 lineto -1.9642819 620.69935 lineto -1.9642819 591.3808 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -5.5479178 591.55432 moveto -4.3533781 591.55432 lineto -4.3533781 620.87288 lineto -5.5479178 620.87288 lineto -5.5479178 591.55432 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 0.22570501 591.55432 moveto 0.62388493 591.55432 lineto 0.62388493 620.87288 lineto 0.22570501 620.87288 lineto 0.22570501 591.55432 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 9.0852041 591.3808 moveto 9.483384 591.3808 lineto 9.483384 621.04632 lineto 9.0852041 621.04632 lineto 9.0852041 591.3808 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -42.678181 591.3808 moveto -39.890921 591.3808 lineto -39.890921 621.2198 lineto -42.678181 621.2198 lineto -42.678181 591.3808 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -37.900021 591.3808 moveto -37.501841 591.3808 lineto -37.501841 622.08721 lineto -37.900021 622.08721 lineto -37.900021 591.3808 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -32.922771 591.3808 moveto -31.529142 591.3808 lineto -31.529142 620.69935 lineto -32.922771 620.69935 lineto -32.922771 591.3808 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -35.112759 591.3808 moveto -34.913669 591.3808 lineto -34.913669 621.2198 lineto -35.112759 621.2198 lineto -35.112759 591.3808 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -28.741884 591.20734 moveto -28.343704 591.20734 lineto -28.343704 621.74027 lineto -28.741884 621.74027 lineto -28.741884 591.20734 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -22.171923 591.20734 moveto -20.778293 591.20734 lineto -20.778293 621.56679 lineto -22.171923 621.56679 lineto -22.171923 591.20734 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -13.013787 591.72778 moveto -12.814697 591.72778 lineto -12.814697 620.87285 lineto -13.013787 620.87285 lineto -13.013787 591.72778 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -15.801047 591.3808 moveto -15.402867 591.3808 lineto -15.402867 620.87283 lineto -15.801047 620.87283 lineto -15.801047 591.3808 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -18.588306 591.3808 moveto -18.190127 591.3808 lineto -18.190127 620.69935 lineto -18.588306 620.69935 lineto -18.588306 591.3808 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -8.036541 591.3808 moveto -7.837451 591.3808 lineto -7.837451 620.69935 lineto -8.036541 620.69935 lineto -8.036541 591.3808 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -11.086478 591.5152 moveto -10.664145 591.5152 lineto -10.664145 620.95617 lineto -11.086478 620.95617 lineto -11.086478 591.5152 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -9.2563639 591.5152 moveto -8.9748082 591.5152 lineto -8.9748082 620.95617 lineto -9.2563639 620.95617 lineto -9.2563639 591.5152 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -23.756483 591.5152 moveto -23.615705 591.5152 lineto -23.615705 621.07883 lineto -23.756483 621.07883 lineto -23.756483 591.5152 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -26.572042 591.14716 moveto -26.290487 591.14716 lineto -26.290487 620.95614 lineto -26.572042 620.95614 lineto -26.572042 591.14716 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -62.681568 591.02448 moveto -61.766512 591.02448 lineto -61.766512 621.14013 lineto -62.681568 621.14013 lineto -62.681568 591.02448 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -57.261623 591.14716 moveto -56.83929 591.14716 lineto -56.83929 621.01747 lineto -57.261623 621.01747 lineto -57.261623 591.14716 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -59.795612 591.57654 moveto -59.654834 591.57654 lineto -59.654834 621.14017 lineto -59.795612 621.14017 lineto -59.795612 591.57654 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -52.615948 591.33118 moveto -50.50428 591.33118 lineto -50.50428 621.14016 lineto -52.615948 621.14016 lineto -52.615948 591.33118 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -45.717838 591.33118 moveto -45.57706 591.33118 lineto -45.57706 621.14016 lineto -45.717838 621.14016 lineto -45.717838 591.33118 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -48.39262 591.45386 moveto -47.970287 591.45386 lineto -47.970287 620.89482 lineto -48.39262 620.89482 lineto -48.39262 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -54.30529 591.69922 moveto -54.164512 591.69922 lineto -54.164512 621.26286 lineto -54.30529 621.26286 lineto -54.30529 591.69922 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -66.975296 591.33118 moveto -66.834518 591.33118 lineto -66.834518 621.14016 lineto -66.975296 621.14016 lineto -66.975296 591.33118 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -70.776299 591.57654 moveto -70.494743 591.57654 lineto -70.494743 621.8762 lineto -70.776299 621.8762 lineto -70.776299 591.57654 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -74.295731 591.45386 moveto -73.591841 591.45386 lineto -73.591841 621.26284 lineto -74.295731 621.26284 lineto -74.295731 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -77.955963 591.2085 moveto -77.674407 591.2085 lineto -77.674407 621.01748 lineto -77.955963 621.01748 lineto -77.955963 591.2085 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -75.703522 591.45386 moveto -75.421966 591.45386 lineto -75.421966 620.77215 lineto -75.703522 620.77215 lineto -75.703522 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -65.004402 591.45386 moveto -64.722846 591.45386 lineto -64.722846 621.01749 lineto -65.004402 621.01749 lineto -65.004402 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -68.101517 591.69922 moveto -67.960739 591.69922 lineto -67.960739 621.01752 lineto -68.101517 621.01752 lineto -68.101517 591.69922 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 134.54822 591.45386 moveto 136.09677 591.45386 lineto 136.09677 621.14017 lineto 134.54822 621.14017 lineto 134.54822 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 138.63075 591.45386 moveto 138.77153 591.45386 lineto 138.77153 621.01749 lineto 138.63075 621.01749 lineto 138.63075 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 142.43178 591.33118 moveto 142.71333 591.33118 lineto 142.71333 621.01748 lineto 142.43178 621.01748 lineto 142.43178 591.33118 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 147.64055 591.2085 moveto 148.20366 591.2085 lineto 148.20366 621.38549 lineto 147.64055 621.38549 lineto 147.64055 591.2085 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 152.70853 591.57654 moveto 155.94227 591.57654 lineto 155.94227 620.89483 lineto 152.70853 620.89483 lineto 152.70853 591.57654 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 157.21346 591.45386 moveto 157.49501 591.45386 lineto 157.49501 620.77215 lineto 157.21346 620.77215 lineto 157.21346 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 149.89299 591.45386 moveto 150.03377 591.45386 lineto 150.03377 620.64948 lineto 149.89299 620.64948 lineto 149.89299 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 144.40266 591.5152 moveto 144.68422 591.5152 lineto 144.68422 620.64949 lineto 144.40266 620.64949 lineto 144.40266 591.5152 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 140.46089 591.57654 moveto 141.30556 591.57654 lineto 141.30556 620.77217 lineto 140.46089 620.77217 lineto 140.46089 591.57654 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 158.62125 591.57654 moveto 158.9028 591.57654 lineto 158.9028 620.89483 lineto 158.62125 620.89483 lineto 158.62125 591.57654 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 219.43726 591.57654 moveto 220.00037 591.57654 lineto 220.00037 620.89483 lineto 219.43726 620.89483 lineto 219.43726 591.57654 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 209.72357 591.33118 moveto 209.86435 591.33118 lineto 209.86435 621.14016 lineto 209.72357 621.14016 lineto 209.72357 591.33118 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 204.23325 591.33118 moveto 204.93714 591.33118 lineto 204.93714 621.14016 lineto 204.23325 621.14016 lineto 204.23325 591.33118 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 213.3838 591.45386 moveto 213.80614 591.45386 lineto 213.80614 620.89482 lineto 213.3838 620.89482 lineto 213.3838 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 216.34013 591.57654 moveto 216.48091 591.57654 lineto 216.48091 620.77217 lineto 216.34013 620.77217 lineto 216.34013 591.57654 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 196.34969 591.57654 moveto 196.63124 591.57654 lineto 196.63124 621.01751 lineto 196.34969 621.01751 lineto 196.34969 591.57654 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 198.60213 591.45386 moveto 198.88368 591.45386 lineto 198.88368 621.01749 lineto 198.60213 621.01749 lineto 198.60213 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 188.60689 591.45386 moveto 188.88844 591.45386 lineto 188.88844 620.89482 lineto 188.60689 620.89482 lineto 188.60689 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 191.42247 591.57654 moveto 191.8448 591.57654 lineto 191.8448 621.01751 lineto 191.42247 621.01751 lineto 191.42247 591.57654 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 200.85457 591.45386 moveto 201.13612 591.45386 lineto 201.13612 621.14017 lineto 200.85457 621.14017 lineto 200.85457 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 223.37904 591.45386 moveto 225.51072 591.45386 lineto 225.51072 621.50818 lineto 223.37904 621.50818 lineto 223.37904 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 205.92255 591.45386 moveto 208.31577 591.45386 lineto 208.31577 620.77215 lineto 205.92255 620.77215 lineto 205.92255 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 192.54866 591.33118 moveto 193.11177 591.33118 lineto 193.11177 621.26283 lineto 192.54866 621.26283 lineto 192.54866 591.33118 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 195.22345 591.33118 moveto 195.36423 591.33118 lineto 195.36423 621.75351 lineto 195.22345 621.75351 lineto 195.22345 591.33118 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 185.65056 591.45386 moveto 185.93211 591.45386 lineto 185.93211 620.77215 lineto 185.65056 620.77215 lineto 185.65056 591.45386 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 229.78441 591.33118 moveto 230.62908 591.33118 lineto 230.62908 621.26283 lineto 229.78441 621.26283 lineto 229.78441 591.33118 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 226.54652 591.69922 moveto 226.82808 591.69922 lineto 226.82808 621.14019 lineto 226.54652 621.14019 lineto 226.54652 591.69922 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 217.53677 591.2085 moveto 218.09989 591.2085 lineto 218.09989 620.77213 lineto 217.53677 620.77213 lineto 217.53677 591.2085 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 234.00778 591.7605 moveto 234.85245 591.7605 lineto 234.85245 620.89479 lineto 234.00778 620.89479 lineto 234.00778 591.7605 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 238.37186 591.2085 moveto 238.65341 591.2085 lineto 238.65341 620.64946 lineto 238.37186 620.64946 lineto 238.37186 591.2085 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 239.92044 591.2085 moveto 241.75055 591.2085 lineto 241.75055 621.01748 lineto 239.92044 621.01748 lineto 239.92044 591.2085 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 162.84662 591.57654 moveto 166.05621 591.57654 lineto 166.05621 621.01751 lineto 162.84662 621.01751 lineto 162.84662 591.57654 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 160.52173 591.57654 moveto 160.66251 591.57654 lineto 160.66251 620.8335 lineto 160.52173 620.8335 lineto 160.52173 591.57654 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 168.68684 591.39252 moveto 169.81307 591.39252 lineto 169.81307 621.07882 lineto 168.68684 621.07882 lineto 168.68684 591.39252 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 179.38594 591.39252 moveto 179.52672 591.39252 lineto 179.52672 621.32417 lineto 179.38594 621.32417 lineto 179.38594 591.39252 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 174.45871 591.26984 moveto 175.02182 591.26984 lineto 175.02182 620.83347 lineto 174.45871 620.83347 lineto 174.45871 591.26984 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 182.76462 591.63788 moveto 183.04617 591.63788 lineto 183.04617 621.07885 lineto 182.76462 621.07885 lineto 182.76462 591.63788 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 176.99274 591.63788 moveto 177.27429 591.63788 lineto 177.27429 621.07885 lineto 176.99274 621.07885 lineto 176.99274 591.63788 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -80.771523 591.5152 moveto -79.926855 591.5152 lineto -79.926855 621.07883 lineto -80.771523 621.07883 lineto -80.771523 591.5152 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 135.50732 649.06378 moveto 136.10459 649.06378 lineto 136.10459 656.17657 lineto 135.50732 656.17657 lineto 135.50732 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 139.29002 649.06378 moveto 140.08638 649.06378 lineto 140.08638 656.35005 lineto 139.29002 656.35005 lineto 139.29002 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 142.07729 648.89032 moveto 142.27638 648.89032 lineto 142.27638 656.00311 lineto 142.07729 656.00311 lineto 142.07729 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 147.4527 648.54333 moveto 149.84178 648.54333 lineto 149.84178 656.35005 lineto 147.4527 656.35005 lineto 147.4527 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 153.22632 648.7168 moveto 153.6245 648.7168 lineto 153.6245 656.17655 lineto 153.22632 656.17655 lineto 153.22632 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 160.19449 648.89032 moveto 165.37083 648.89032 lineto 165.37083 656.17659 lineto 160.19449 656.17659 lineto 160.19449 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 158.20357 648.89032 moveto 158.60175 648.89032 lineto 158.60175 656.17659 lineto 158.20357 656.17659 lineto 158.20357 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 170.54715 648.89032 moveto 171.34351 648.89032 lineto 171.34351 656.00311 lineto 170.54715 656.00311 lineto 170.54715 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 168.35716 649.06378 moveto 168.55625 649.06378 lineto 168.55625 656.35005 lineto 168.35716 656.35005 lineto 168.35716 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 174.72803 648.89032 moveto 174.92712 648.89032 lineto 174.92712 656.17659 lineto 174.72803 656.17659 lineto 174.72803 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 178.11256 649.06378 moveto 179.50619 649.06378 lineto 179.50619 655.8296 lineto 178.11256 655.8296 lineto 178.11256 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 182.69164 649.06378 moveto 183.08982 649.06378 lineto 183.08982 656.00308 lineto 182.69164 656.00308 lineto 182.69164 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 186.07617 648.89032 moveto 186.47435 648.89032 lineto 186.47435 656.00311 lineto 186.07617 656.00311 lineto 186.07617 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 187.66888 648.7168 moveto 187.86797 648.7168 lineto 187.86797 655.82958 lineto 187.66888 655.82958 lineto 187.66888 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 191.65068 648.89032 moveto 194.23885 648.89032 lineto 194.23885 656.17659 lineto 191.65068 656.17659 lineto 191.65068 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 195.83159 648.89032 moveto 196.22977 648.89032 lineto 196.22977 656.00311 lineto 195.83159 656.00311 lineto 195.83159 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 200.41063 648.54333 moveto 202.00335 648.54333 lineto 202.00335 656.17657 lineto 200.41063 656.17657 lineto 200.41063 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 211.1615 648.89032 moveto 211.55968 648.89032 lineto 211.55968 656.17659 lineto 211.1615 656.17659 lineto 211.1615 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 207.97607 648.7168 moveto 209.56879 648.7168 lineto 209.56879 656.00306 lineto 207.97607 656.00306 lineto 207.97607 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 205.18881 649.06378 moveto 205.58699 649.06378 lineto 205.58699 656.00308 lineto 205.18881 656.00308 lineto 205.18881 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 215.54147 648.89032 moveto 215.74056 648.89032 lineto 215.74056 656.17659 lineto 215.54147 656.17659 lineto 215.54147 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 217.13419 648.89032 moveto 217.73146 648.89032 lineto 217.73146 655.82962 lineto 217.13419 655.82962 lineto 217.13419 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 220.31966 648.54333 moveto 220.51875 648.54333 lineto 220.51875 656.17657 lineto 220.31966 656.17657 lineto 220.31966 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 229.47777 648.89032 moveto 230.07504 648.89032 lineto 230.07504 655.82962 lineto 229.47777 655.82962 lineto 229.47777 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 225.69507 649.06378 moveto 225.89416 649.06378 lineto 225.89416 655.8296 lineto 225.69507 655.8296 lineto 225.69507 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 236.04773 648.54333 moveto 236.44591 648.54333 lineto 236.44591 656.00309 lineto 236.04773 656.00309 lineto 236.04773 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 242.61769 648.36987 moveto 244.60859 648.36987 lineto 244.60859 656.17659 lineto 242.61769 656.17659 lineto 242.61769 648.36987 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 232.8623 648.7168 moveto 233.45957 648.7168 lineto 233.45957 656.00306 lineto 232.8623 656.00306 lineto 232.8623 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 282.03754 648.19635 moveto 283.23208 648.19635 lineto 283.23208 656.52351 lineto 282.03754 656.52351 lineto 282.03754 648.19635 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 279.84753 648.7168 moveto 280.24571 648.7168 lineto 280.24571 656.69699 lineto 279.84753 656.69699 lineto 279.84753 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 271.48578 648.7168 moveto 272.08305 648.7168 lineto 272.08305 656.35003 lineto 271.48578 656.35003 lineto 271.48578 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 260.73489 648.63007 moveto 260.93398 648.63007 lineto 260.93398 656.35004 lineto 260.73489 656.35004 lineto 260.73489 648.63007 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 252.9704 648.89032 moveto 253.96583 648.89032 lineto 253.96583 656.17658 lineto 252.9704 656.17658 lineto 252.9704 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 249.78497 648.7168 moveto 249.98406 648.7168 lineto 249.98406 656.17655 lineto 249.78497 656.17655 lineto 249.78497 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 240.82591 648.54333 moveto 241.22409 648.54333 lineto 241.22409 656.00309 lineto 240.82591 656.00309 lineto 240.82591 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 264.91577 648.89032 moveto 268.49939 648.89032 lineto 268.49939 656.00311 lineto 264.91577 656.00311 lineto 264.91577 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 256.95221 648.7168 moveto 257.74857 648.7168 lineto 257.74857 656.00306 lineto 256.95221 656.00306 lineto 256.95221 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 246.99771 648.7168 moveto 247.1968 648.7168 lineto 247.1968 656.35003 lineto 246.99771 656.35003 lineto 246.99771 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -2.2629182 648.19635 moveto 1.3207009 648.19635 lineto 1.3207009 656.17655 lineto -2.2629182 656.17655 lineto -2.2629182 648.19635 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 5.1033998 648.89032 moveto 6.2979394 648.89032 lineto 6.2979394 655.65614 lineto 5.1033998 655.65614 lineto 5.1033998 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 7.6915746 648.54333 moveto 7.8906645 648.54333 lineto 7.8906645 655.8296 lineto 7.6915746 655.8296 lineto 7.6915746 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 10.677918 648.7168 moveto 11.076098 648.7168 lineto 11.076098 656.17655 lineto 10.677918 656.17655 lineto 10.677918 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 9.2842894 648.7168 moveto 9.4833793 648.7168 lineto 9.4833793 656.00306 lineto 9.2842894 656.00306 lineto 9.2842894 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 14.062451 648.54333 moveto 14.659721 648.54333 lineto 14.659721 656.00309 lineto 14.062451 656.00309 lineto 14.062451 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 12.668821 648.54333 moveto 13.465181 648.54333 lineto 13.465181 656.00309 lineto 12.668821 656.00309 lineto 12.668821 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 17.845152 648.54333 moveto 18.442422 648.54333 lineto 18.442422 656.00309 lineto 17.845152 656.00309 lineto 17.845152 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -5.8465347 648.54333 moveto -5.2492649 648.54333 lineto -5.2492649 656.35005 lineto -5.8465347 656.35005 lineto -5.8465347 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -13.411957 648.7168 moveto -13.212867 648.7168 lineto -13.212867 656.00306 lineto -13.411957 656.00306 lineto -13.411957 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -10.624697 647.93616 moveto -10.226517 647.93616 lineto -10.226517 656.17658 lineto -10.624697 656.17658 lineto -10.624697 647.93616 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -18.588306 648.54333 moveto -17.194677 648.54333 lineto -17.194677 656.17657 lineto -18.588306 656.17657 lineto -18.588306 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -15.801047 648.54333 moveto -15.402867 648.54333 lineto -15.402867 656.00309 lineto -15.801047 656.00309 lineto -15.801047 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -22.371008 648.89032 moveto -21.375558 648.89032 lineto -21.375558 656.17659 lineto -22.371008 656.17659 lineto -22.371008 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -19.782833 648.54333 moveto -19.583743 648.54333 lineto -19.583743 656.17657 lineto -19.782833 656.17657 lineto -19.782833 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -37.103661 648.54333 moveto -33.918221 648.54333 lineto -33.918221 656.17657 lineto -37.103661 656.17657 lineto -37.103661 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -28.542799 648.89032 moveto -27.945529 648.89032 lineto -27.945529 655.82962 lineto -28.542799 655.82962 lineto -28.542799 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -31.330059 649.06378 moveto -31.130969 649.06378 lineto -31.130969 656.00308 lineto -31.330059 656.00308 lineto -31.330059 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -25.954624 648.54333 moveto -25.158264 648.54333 lineto -25.158264 656.52353 lineto -25.954624 656.52353 lineto -25.954624 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -23.366449 648.36987 moveto -23.167359 648.36987 lineto -23.167359 655.82962 lineto -23.366449 655.82962 lineto -23.366449 648.36987 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -41.881824 648.54333 moveto -40.488195 648.54333 lineto -40.488195 656.35005 lineto -41.881824 656.35005 lineto -41.881824 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -39.094566 648.7168 moveto -38.696386 648.7168 lineto -38.696386 656.17655 lineto -39.094566 656.17655 lineto -39.094566 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -50.840878 649.06378 moveto -49.447248 649.06378 lineto -49.447248 656.00308 lineto -50.840878 656.00308 lineto -50.840878 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -56.21629 649.06378 moveto -55.61902 649.06378 lineto -55.61902 656.00308 lineto -56.21629 656.00308 lineto -56.21629 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -54.225407 648.89032 moveto -53.827227 648.89032 lineto -53.827227 655.82962 lineto -54.225407 655.82962 lineto -54.225407 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -63.184441 649.06378 moveto -62.188991 649.06378 lineto -62.188991 656.52353 lineto -63.184441 656.52353 lineto -63.184441 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -59.799927 648.89032 moveto -59.600837 648.89032 lineto -59.600837 656.00311 lineto -59.799927 656.00311 lineto -59.799927 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -92.649765 648.36987 moveto -91.654315 648.36987 lineto -91.654315 656.00311 lineto -92.649765 656.00311 lineto -92.649765 648.36987 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -73.338036 648.54333 moveto -72.541676 648.54333 lineto -72.541676 656.52353 lineto -73.338036 656.52353 lineto -73.338036 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -65.573532 648.89032 moveto -64.179902 648.89032 lineto -64.179902 656.69704 lineto -65.573532 656.69704 lineto -65.573532 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -70.152588 648.89032 moveto -69.754408 648.89032 lineto -69.754408 656.35007 lineto -70.152588 656.35007 lineto -70.152588 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -67.763519 648.54333 moveto -67.564429 648.54333 lineto -67.564429 656.35005 lineto -67.763519 656.35005 lineto -67.763519 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -84.08889 648.63007 moveto -83.8898 648.63007 lineto -83.8898 656.69701 lineto -84.08889 656.69701 lineto -84.08889 648.63007 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -77.518929 648.36987 moveto -77.120749 648.36987 lineto -77.120749 656.52355 lineto -77.518929 656.52355 lineto -77.518929 648.36987 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -88.269775 648.89032 moveto -88.070685 648.89032 lineto -88.070685 656.35007 lineto -88.269775 656.35007 lineto -88.269775 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -81.699814 648.97705 moveto -80.704364 648.97705 lineto -80.704364 656.17658 lineto -81.699814 656.17658 lineto -81.699814 648.97705 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -86.677063 648.89032 moveto -86.477973 648.89032 lineto -86.477973 656.00311 lineto -86.677063 656.00311 lineto -86.677063 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -180.05025 647.84943 moveto -178.85571 647.84943 lineto -178.85571 656.35007 lineto -180.05025 656.35007 lineto -180.05025 647.84943 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -178.25844 648.36987 moveto -177.66117 648.36987 lineto -177.66117 656.17659 lineto -178.25844 656.17659 lineto -178.25844 648.36987 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -175.073 648.7168 moveto -173.48028 648.7168 lineto -173.48028 656.17655 lineto -175.073 656.17655 lineto -175.073 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -176.46663 649.06378 moveto -176.06845 649.06378 lineto -176.06845 656.00308 lineto -176.46663 656.00308 lineto -176.46663 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -169.29939 648.7168 moveto -168.90121 648.7168 lineto -168.90121 656.35003 lineto -169.29939 656.35003 lineto -169.29939 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -171.68846 648.7168 moveto -171.09119 648.7168 lineto -171.09119 655.82958 lineto -171.68846 655.82958 lineto -171.68846 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -166.31305 648.89032 moveto -166.11396 648.89032 lineto -166.11396 656.35007 lineto -166.31305 656.35007 lineto -166.31305 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -138.63954 648.89032 moveto -136.05137 648.89032 lineto -136.05137 656.00311 lineto -138.63954 656.00311 lineto -138.63954 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -144.61224 648.7168 moveto -144.41315 648.7168 lineto -144.41315 656.17655 lineto -144.61224 656.17655 lineto -144.61224 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -150.58493 648.89032 moveto -150.38584 648.89032 lineto -150.38584 656.17659 lineto -150.58493 656.17659 lineto -150.58493 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -157.15491 648.7168 moveto -156.35855 648.7168 lineto -156.35855 655.82958 lineto -157.15491 655.82958 lineto -157.15491 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -161.73398 648.7168 moveto -160.53944 648.7168 lineto -160.53944 656.00306 lineto -161.73398 656.00306 lineto -161.73398 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -159.1458 648.7168 moveto -158.15035 648.7168 lineto -158.15035 656.00306 lineto -159.1458 656.00306 lineto -159.1458 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -131.67139 649.06378 moveto -131.07412 649.06378 lineto -131.07412 656.69701 lineto -131.67139 656.69701 lineto -131.67139 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -141.52634 648.63007 moveto -141.02861 648.63007 lineto -141.02861 656.17656 lineto -141.52634 656.17656 lineto -141.52634 648.63007 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -148.59404 648.97705 moveto -148.19586 648.97705 lineto -148.19586 655.82961 lineto -148.59404 655.82961 lineto -148.59404 648.97705 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -121.51781 649.06378 moveto -121.31872 649.06378 lineto -121.31872 656.17657 lineto -121.51781 656.17657 lineto -121.51781 649.06378 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -101.70835 648.36987 moveto -101.31017 648.36987 lineto -101.31017 656.00311 lineto -101.70835 656.00311 lineto -101.70835 648.36987 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -108.87559 647.84943 moveto -108.6765 647.84943 lineto -108.6765 656.17659 lineto -108.87559 656.17659 lineto -108.87559 647.84943 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -115.84373 648.19635 moveto -114.25101 648.19635 lineto -114.25101 655.6561 lineto -115.84373 655.6561 lineto -115.84373 648.19635 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -124.80279 648.89032 moveto -124.20552 648.89032 lineto -124.20552 656.35007 lineto -124.80279 656.35007 lineto -124.80279 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -98.522903 649.32397 moveto -98.124724 649.32397 lineto -98.124724 655.82957 lineto -98.522903 655.82957 lineto -98.522903 649.32397 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -97.228828 648.7168 moveto -95.237929 648.7168 lineto -95.237929 656.35003 lineto -97.228828 656.35003 lineto -97.228828 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -105.78969 648.89032 moveto -105.39151 648.89032 lineto -105.39151 656.17659 lineto -105.78969 656.17659 lineto -105.78969 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -111.7624 648.89032 moveto -110.76695 648.89032 lineto -110.76695 656.17659 lineto -111.7624 656.17659 lineto -111.7624 648.89032 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -128.68504 648.54333 moveto -128.48595 648.54333 lineto -128.48595 656.17657 lineto -128.68504 656.17657 lineto -128.68504 648.54333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -118.73055 648.7168 moveto -118.33237 648.7168 lineto -118.33237 656.17655 lineto -118.73055 656.17655 lineto -118.73055 648.7168 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 14.535085 628.43909 moveto 15.661308 628.43909 lineto 15.661308 634.57262 lineto 14.535085 634.57262 lineto 14.535085 628.43909 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 19.039982 628.92975 moveto 19.603093 628.92975 lineto 19.603093 634.5726 lineto 19.039982 634.5726 lineto 19.039982 628.92975 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 10.874856 628.68439 moveto 11.437967 628.68439 lineto 11.437967 634.57258 lineto 10.874856 634.57258 lineto 10.874856 628.68439 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -0.66891152 628.68439 moveto 2.1466455 628.68439 lineto 2.1466455 634.57258 lineto -0.66891152 634.57258 lineto -0.66891152 628.68439 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -4.6106901 628.43909 moveto -4.0475787 628.43909 lineto -4.0475787 634.32728 lineto -4.6106901 634.32728 lineto -4.6106901 628.43909 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 6.3699765 628.92975 moveto 6.6515322 628.92975 lineto 6.6515322 634.5726 lineto 6.3699765 634.5726 lineto 6.3699765 628.92975 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 3.5544174 628.68439 moveto 4.1175288 628.68439 lineto 4.1175288 634.57258 lineto 3.5544174 634.57258 lineto 3.5544174 628.68439 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -7.7078185 628.68439 moveto -7.4262628 628.68439 lineto -7.4262628 634.32724 lineto -7.7078185 634.32724 lineto -7.7078185 628.68439 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -10.523377 628.43909 moveto -9.6787103 628.43909 lineto -9.6787103 634.57262 lineto -10.523377 634.57262 lineto -10.523377 628.43909 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -13.338918 628.43909 moveto -13.057362 628.43909 lineto -13.057362 634.08194 lineto -13.338918 634.08194 lineto -13.338918 628.43909 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -15.028256 628.19373 moveto -14.746701 628.19373 lineto -14.746701 635.06328 lineto -15.028256 635.06328 lineto -15.028256 628.19373 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -18.406935 628.19373 moveto -17.562268 628.19373 lineto -17.562268 634.32726 lineto -18.406935 634.32726 lineto -18.406935 628.19373 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -23.474934 628.19373 moveto -22.630266 628.19373 lineto -22.630266 634.81794 lineto -23.474934 634.81794 lineto -23.474934 628.19373 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -26.572042 628.43909 moveto -26.008931 628.43909 lineto -26.008931 634.57262 lineto -26.572042 634.57262 lineto -26.572042 628.43909 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -28.261383 628.43909 moveto -27.979827 628.43909 lineto -27.979827 634.81796 lineto -28.261383 634.81796 lineto -28.261383 628.43909 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -30.232271 628.19373 moveto -29.950715 628.19373 lineto -29.950715 634.32726 lineto -30.232271 634.32726 lineto -30.232271 628.19373 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -38.678947 628.43909 moveto -36.989613 628.43909 lineto -36.989613 634.81796 lineto -38.678947 634.81796 lineto -38.678947 628.43909 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -34.455601 628.68439 moveto -33.892489 628.68439 lineto -33.892489 633.83656 lineto -34.455601 633.83656 lineto -34.455601 628.68439 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -32.484711 628.68439 moveto -32.203155 628.68439 lineto -32.203155 634.32724 lineto -32.484711 634.32724 lineto -32.484711 628.68439 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 0 0 0 setrgbcolor newpath -29.809927 518.28076 moveto 114.3466 518.28076 lineto 114.3466 529.07578 lineto -29.809927 529.07578 lineto -29.809927 518.28076 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -32.624138 519.90601 moveto 116.29514 519.90601 lineto 116.29514 521.64083 lineto -32.624138 521.64083 lineto -32.624138 519.90601 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -31.230507 525.11047 moveto 116.09605 525.11047 lineto 116.09605 526.8453 lineto -31.230507 526.8453 lineto -31.230507 525.11047 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -31.230507 523.72266 moveto 116.09605 523.72266 lineto 116.09605 524.41659 lineto -31.230507 524.41659 lineto -31.230507 523.72266 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -31.628695 527.53925 moveto 115.69786 527.53925 lineto 115.69786 527.88621 lineto -31.628695 527.88621 lineto -31.628695 527.53925 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -28.045078 516.78333 moveto -26.850539 516.78333 lineto -26.850539 529.27407 lineto -28.045078 529.27407 lineto -28.045078 516.78333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -20.877827 517.47723 moveto -20.479647 517.47723 lineto -20.479647 529.62101 lineto -20.877827 529.62101 lineto -20.877827 517.47723 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -5.7470021 517.47723 moveto -5.3488222 517.47723 lineto -5.3488222 529.96798 lineto -5.7470021 529.96798 lineto -5.7470021 517.47723 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -13.710592 517.47723 moveto -12.516053 517.47723 lineto -12.516053 529.27405 lineto -13.710592 529.27405 lineto -13.710592 517.47723 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath -8.9324312 517.13031 moveto -8.5342513 517.13031 lineto -8.5342513 529.96802 lineto -8.9324312 529.96802 lineto -8.9324312 517.13031 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 8.5874834 516.78333 moveto 16.551081 516.78333 lineto 16.551081 531.0089 lineto 8.5874834 531.0089 lineto 8.5874834 516.78333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 1.8184193 517.82422 moveto 2.6147792 517.82422 lineto 2.6147792 530.31496 lineto 1.8184193 530.31496 lineto 1.8184193 517.82422 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 42.830936 517.47723 moveto 44.423656 517.47723 lineto 44.423656 530.66191 lineto 42.830936 530.66191 lineto 42.830936 517.47723 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 74.685333 516.78333 moveto 77.472593 516.78333 lineto 77.472593 530.31497 lineto 74.685333 530.31497 lineto 74.685333 516.78333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 108.92883 516.43634 moveto 110.52155 516.43634 lineto 110.52155 531.00888 lineto 108.92883 531.00888 lineto 108.92883 516.43634 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 103.35431 517.47723 moveto 103.75249 517.47723 lineto 103.75249 530.66191 lineto 103.35431 530.66191 lineto 103.35431 517.47723 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 91.80706 517.13031 moveto 93.79796 517.13031 lineto 93.79796 529.96802 lineto 91.80706 529.96802 lineto 91.80706 517.13031 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 97.779793 516.95679 moveto 99.372512 516.95679 lineto 99.372512 529.96798 lineto 97.779793 529.96798 lineto 97.779793 516.95679 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 82.648949 516.43634 moveto 83.047129 516.43634 lineto 83.047129 530.66191 lineto 82.648949 530.66191 lineto 82.648949 516.43634 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 87.825294 517.13031 moveto 89.418014 517.13031 lineto 89.418014 529.96802 lineto 87.825294 529.96802 lineto 87.825294 517.13031 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 49.600002 517.13031 moveto 51.192722 517.13031 lineto 51.192722 530.31498 lineto 49.600002 530.31498 lineto 49.600002 517.13031 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 39.247337 516.78333 moveto 40.043697 516.78333 lineto 40.043697 529.62103 lineto 39.247337 529.62103 lineto 39.247337 516.78333 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 32.080086 516.43634 moveto 32.478266 516.43634 lineto 32.478266 529.96798 lineto 32.080086 529.96798 lineto 32.080086 516.43634 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 23.320139 517.13031 moveto 24.514679 517.13031 lineto 24.514679 530.31498 lineto 23.320139 530.31498 lineto 23.320139 517.13031 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 28.496469 517.13031 moveto 28.894649 517.13031 lineto 28.894649 529.62105 lineto 28.496469 529.62105 lineto 28.496469 517.13031 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 18.940165 517.13031 moveto 19.736524 517.13031 lineto 19.736524 529.62105 lineto 18.940165 529.62105 lineto 18.940165 517.13031 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 52.38726 517.13031 moveto 52.78544 517.13031 lineto 52.78544 529.62105 lineto 52.38726 529.62105 lineto 52.38726 517.13031 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 57.56361 516.43634 moveto 58.35997 516.43634 lineto 58.35997 530.31495 lineto 57.56361 530.31495 lineto 57.56361 516.43634 lineto closepath fill grestore grestore gsave [0.912447 -0.409195 0.639867 0.768486 0 0] concat gsave 1 1 1 setrgbcolor newpath 66.721725 517.47723 moveto 67.916265 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