Content-Type: multipart/mixed; boundary="-------------0712211540525" This is a multi-part message in MIME format. ---------------0712211540525 Content-Type: text/plain; name="07-313.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="07-313.comments" MSC codes: 37D50, 37D25, 37A25 ---------------0712211540525 Content-Type: text/plain; name="07-313.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="07-313.keywords" hyperbolic, billiards, focusing, defocusing, invariant cones, Wojtkowski, Markarian, Donnay, Bunimovich ---------------0712211540525 Content-Type: application/x-tex; name="nearlyi-arx.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nearlyi-arx.tex" \documentclass[12pt,twoside]{article} \usepackage{amssymb,graphicx} \usepackage[small,sc]{caption2} %\usepackage{showkeys} \setlength{\topmargin}{0truecm} \setlength{\headsep}{+1truecm} \setlength{\oddsidemargin}{+.5truecm} \setlength{\evensidemargin}{+.5truecm} \setlength{\textwidth}{15truecm} \setlength{\textheight}{22truecm} %%% The above is a format that I think is ok with both US and A4 %%% paper. To maximize the page size use respectively %\input{myus} %\input{mya4} \pagestyle{myheadings} \markboth{L.~Bussolari and M.~Lenci}{Billiards with nearly flat boundaries} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{rmrk}[theorem]{Remark} %%% indexes equations within sections \makeatletter \@addtoreset{equation}{section} \makeatother \renewcommand{\theequation} {\thesection.\arabic{equation}} \newcommand{\fig}[3] { \medskip\smallskip \begin{figure}[htb] \centering \includegraphics[width=#2]{#1.eps} \begin{minipage}[t]{0.80\linewidth} \caption{#3} \protect\label{#1} \end{minipage} \end{figure} \medskip } \newenvironment{remark} {\begin{rmrk} \em} {\end{rmrk}} %%% standard macros \newcommand{\bi} {billiard} \newcommand{\fn} {function} \newcommand{\me} {measure} \newcommand{\tr} {trajector} \newcommand{\erg} {ergodic} \newcommand{\sy} {system} \newcommand{\hyp} {hyperbolic} \newcommand{\pr} {probability} \newcommand{\ra} {random} \newcommand{\dsy} {dynamical system} \renewcommand{\o} {orbit} \newcommand{\R} {\mathbb{R}} \newcommand{\C} {\mathbb{C}} \newcommand{\Q} {\mathbb{Q}} \newcommand{\Z} {\mathbb{Z}} \newcommand{\N} {\mathbb{N}} \newcommand{\qed} {\hfill {\small Q.E.D.} \par\medskip} \newcommand{\skippar} {\par\medskip} \newcommand{\ds} {\displaystyle} \newcommand{\proof} {\noindent \textsc{Proof.} } \newcommand{\proofof}[1] {\noindent \textsc{Proof of {#1}.} } \newcommand{\article}[3] {\textsc{{#1}}, {\itshape {#2}}, {{#3}}.} \newcommand{\book}[3] {\textsc{{#1}}, {\itshape {#2}}, {{#3}}.} \newcommand{\vol} {\textbf} \newcommand{\eps} {\varepsilon} \newcommand{\rset}[2] {\left\{ #1 \: \left| \: #2 \right. \! \right\} } \newcommand{\lset}[2] {\left\{ \left. \! #1 \: \right| \: #2 \right\} } \newcommand{\nota}[1] {\medskip\par {\bf nota: #1 } \medskip\par} \renewcommand{\iff} {if and only if\ } %%% macros for this paper \newcommand{\ta} {\Omega} % billiard table \newcommand{\bo} {\Gamma} % border of billiard table \newcommand{\ps} {\mathcal{M}} % phase space \newcommand{\ma} {\mathcal{F}} % phase space \renewcommand{\a} {\alpha} % angle \newcommand{\si} {\mathcal{S}} % singularity set \begin{document} \title{\textbf{Hyperbolic billiards with nearly flat \\ focusing boundaries. I}} \author{ \textsc{Luca Bussolari} \thanks{ Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ 07030, U.S.A.} $^\ddagger$ \qquad \textsc{Marco Lenci} $^*$\thanks{ Dipartimento di Matematica, Universit\`a di Bologna, P.zza di Porta S.~Donato 5, 40126 Bologna, ITALY} \thanks{E-mails: \texttt{lbussola@math.stevens.edu}, \texttt{lenci@dm.unibo.it}} } \date{December 2007} \maketitle \begin{abstract} The standard Wojtkowski--Markarian--Donnay--Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. This is due to the physical principle that is used in the proofs, the so-called defocusing mechanism of geometrical optics. In this paper we construct examples of hyperbolic billiards with a focusing boundary component of arbitrarily small curvature whose diameter is bounded by a constant independent of that curvature. Our proof employs a nonstardard cone bundle that does not solely use the familiar dispersing and defocusing mechanisms. \bigskip\noindent Mathematics Subject Classification: 37D50, 37D25, 37A25. \end{abstract} \section{Introduction} \label{sec-intro} Much has been written, in the scientific literature, about the \hyp ity of \bi s in two dimensions. So much that general principles have even been devised for the `design of \bi s with nonvanishing Lyapunov exponents'. The expression is taken from the title of the 1986 seminal paper by Wojtkowski \cite{w2}, in which he beautifully links the question of exponential instability (i.e., positivity of a Lyapunov exponent) to a few simple observations from geometrical optics. By means of the powerful \emph{invariant cone technique} \cite{w1, k, cm}, Wojtkowski gives sufficient conditions for a planar \bi\ to have nonzero Lyapunov exponents, this implying a fuller range of \hyp\ properties via the general results of Katok and Strelcyn on Pesin's theory for \dsy s with singularities \cite{ks}. Wojtkowski's conditions are rather undemanding for \emph{dispersing} and \emph{semidispersing} \bi s (i.e., \bi s in a domain $\Omega \subset \R^2$, a.k.a.\ \emph{table}, whose boundary is the finite union of smooth convex pieces, when seen from inside $\Omega$), and much more restrictive for \emph{focusing}, \emph{semifocusing} and \emph{mixed} \bi s (that is, cases when $\partial \Omega$ is made up---completely or partially, respectively---by concave pieces). (Both in the dispersing and in the focusing case, the prefix semi- means that $\partial \Omega$ has some flat parts as well.) For the latter type of \bi s, further work has been done by Markarian \cite{m1,m2}, Donnay \cite{d} and Bunimovich \cite{b3} (see \cite[Chap.~9]{cm} for an overview of the subject and \cite{de} for an interesting variation). If we call \emph{boundary component} each smooth piece of $\partial \Omega$, one of the conditions in \cite{w2} is that the inner semiosculating disc at any given point of a focusing boundary component must not intersect other components, or the semiosculating discs relative to other focusing components (\cite{m1} has a similar condition). This is required in order to implement the so-called \emph{defocusing mechanism}, which can be loosely described like this: One wants diverging beams of \tr ies to keep diverging after every collision with the boundary. But at a focusing portion of the boundary a diverging beam may be bounced back as a converging beam. A solution around this problem is to let the converging beam travel untouched for a sufficienly long time until the \tr ies focus among themselves and then start to diverge again. The defocusing mechanism is the closest extension of Sinai's original idea of extracting \hyp ity from the expanding features of dispersing boundaries \cite{s}. At least to our knowledge, it has remained unsurpassed since Bunimovich introduced it in 1974 \cite{b1}, to become very popular a few years later, when it was used to work out the famous stadium \bi\ \cite{b2}. Sticking too much to the standard principles, however, creates a problem and somehow a paradox. The condition on the semiosculating discs, and each of its later analogues, requires a table with focusing components to have a diameter of the order of the largest radius of curvature among the focusing points of the boundary. To illustrate how this may seem a paradox, consider the following example: Take a unit square and replace three of its sides with circular arcs of curvature $k_d \in (-\sqrt{2},0)$ having their endpoints in the vertices of the square. In this paper we use the convention that the curvature is positive at focusing points of the boundary and negative at dispersing points, so the arcs are convex relative to the interior of the square; the condition $|k_d| < \sqrt{2}$ ensures that each pair of adjacent arcs intersects only at the common endpoint. The resulting \bi\ is semidispersing, thus belongs to the standard class and is well-know to be uniformly \hyp, Bernoulli, and so on \cite{cm}. Now perturb the fourth side into a focusing circular arc of curvature $k_f \ll 1$. Now matter how small the perturbation, this new \bi\ will never satisfy Wojtkowski's principle and is not currently known to be \hyp, although it presumably is. This may not sound too strange. After all, certain perturbations of dispersing \bi s are known to possess elliptic islands \cite{rt, tr}. But the paradox is that the smaller the perturbation, the less adequate the standard technique; that is, the closer the \bi\ comes to be dispersing, the worse the method applies which is supposed to exploit the dispersing nature of the boundaries. Up until $k_f=0$, at which point everything suddenly, and abruptly, works again to the fullest power of the theory of \hyp\ \bi s. \skippar Here we address this problem and, although we cannot yet prove that the perturbed square \bi\ is \hyp, we devise a couple of models that make clear what the difficulties are in extending the current methodology. These \bi s, which are modifications of the example just discussed, are depicted in Figs.~\ref{fig-t1intro} and \ref{fig-t2intro}. They are indeed two families of \bi s, as we are interested in the case when the curvature of the focusing boundary goes to zero. We define an invariant cone bundle that exploits the fact that the focusing component is nearly flat, and thus almost always acts as a semidispersing boundary. \fig{fig-t1intro} {12cm} {The main \bi\ table} In any event, we are able to answer the following questions in the affirmative: \begin{enumerate} \item Can one design a \bi\ whose \hyp ity is proved via a set of invariant cones that does not use exclusively the dispersing/defocusing mechanism for beams of \tr ies? \item \label{pt2} Can one construct a family of \hyp\ \bi\ tables with a (nonvanishing) focusing component whose maximum curvature approaches zero, and such that the area of the table is bounded above? \item \label{pt3} Can one require the diameter to be bounded above as well? \item Are these \bi s \erg? (This will be proved in \cite{bl}.) \end{enumerate} \fig{fig-t2intro} {6.2cm} {A modification of the main \bi\ table} Points \ref{pt2} and \ref{pt3} show, independently of the method utilized, that one can go beyond the apparent implication `almost flat focusing boundaries imply very large tables'. \skippar This is the plan of the paper: In Section \ref{sec-prel} we review the basic definitions of \bi\ dynamics. In Section \ref{sec-cones} we present and adapt Wojtkowsky's theory of invariant cones derived from geometrical optics. In Section \ref{sec-hyp} we define the first of our models and choose suitable cones to prove its \hyp ity. In Section \ref{sec-conf} we show that the \bi\ introduced before can be chosen with a bounded area, and finally we present a second model which has a bounded diameter as well. \bigskip \noindent \textbf{Acknowledgments.} \ We would like to thank Gianluigi Del Magno for an instructive discussion on the subject. M.L.\ acknowledges partial support from NSF Grant DMS-0405439. \section{Preliminaries} \label{sec-prel} A planar \bi\ is the \dsy\ generated by the flow of a point particle that moves inertially inside a closed region $\ta \subset \R^2$ and collides elastically at the boundary; the latter is assumed to have an infinite mass. This implies that the \tr y of the particle, near the collision point, verifies the well-known \emph{Fresnel law}: the angle of incidence equals the angle of reflection. The region $\ta$ is called the \emph{\bi\ table}. We denote $\bo = \partial \ta$ and assume that $\bo$ is piecewise smooth (at least $C^3$). Let $(q(t),u(t))$ represent the position and the velocity of the particle at time $t$. It is an easy consequence of the conservation of energy that $\|u(t)\| =$ constant. Therefore, by a rescaling of time, one can always reconduct to the situation where $\|u\| = 1$, which we assume throughout the paper. The product $\ta \times S^1$ is the natural phase space of the \bi\ flow, with a couple of extra specifications: First, if $q \in \bo$ and $u$ points outwardly, then $(q,u)$ is identified with $(q,u')$, where $u'$ is the outgoing (i.e., inward) velocity of a collision at $q$ with incoming velocity $u$. Second, if $q$ is in a corner, the flow is not defined. The \bi\ flow preserves the Lebesgue \me\ on $\ta \times S^1$, as it can be verified directly or by applying the Liouville Theorem to this nonsmooth Hamiltonian \sy. Now let $\ps \subset \ta \times S^1$ be the set of all pairs $(q,u)$ with $q \in \bo$ and $u$ pointing inside the table. These pairs are sometimes called \emph{line elements} \cite{s} and $\ps$ is evidently a global cross section for the flow. The corresponding Poincar\'e map $\ma: \ps \longrightarrow \ps$ is called the \emph{\bi\ map} and acts as follows: if $q' \in \bo$ is the first collision point of the flow-\tr y with initial conditions $(q,u)$, and $u'$ is the postcollisional velocity there, then $\ma(q,u) = (q',u')$. $\ma (q,u)$ is undefined when $q'$ is a vertex of $\bo$, and is discontinuous at tangential collisions, i.e., when $u'$ is tangent to $\bo$ in $q'$. For the sake of simplicity, those latter line elements are removed as well from the domain of $\ma$. The set of all removed $(q,u)$ is denoted $\si_1$, or $\si_1^+$. We identify $\ps$ with the rectangle $[0,L] \times [-\pi/2,\pi/2]$, where $L$ is the perimeter of $\ta$: each $(q,u)$ is identified with the pair $(s,\a)$, where $s$ is the arclength coordinate of $q$ (relative to a fixed choice of the origin $s=0$ and oriented counterclockwise) and $\alpha$ is the angle (oriented clockwise) between $u$ and the inner normal to $\bo$ in $q$. The Lebesgue \me\ on $\ta \times S^1$ induces an $\ma$-invariant \me\ $\mu$ on $\ps$ which, in the above coordinates, is described by $d\mu(s,\a) = c \, \cos\a \, ds d\a$. The constant $c$ is customarily chosen so that $\mu$ is a probability \me. Let us indicate with $\si_0$ the set of all pairs $(s,\a) \in \ps$ where $s$ corresponds to a vertex of $\bo$ or $\a = \pm \pi/2$. The set $\si_1 = \si_1^+$ introduced earlier is morally given by ``$\si_1^+ := \ma^{-1} \si_0$''. For historical reasons, this is usually called the \emph{singularity set} of $\ma$, even though the differential of $\ma$ is singular only at line elements resulting in tangential hits. Analogously, for $n>1$, $\si_n^+ := \si_1^+ \cup \ma^{-1} \si_1^+ \cup \cdots \cup \ma^{-n+1} \si_1^+$ is the set where $\ma^n$ is not defined, which is called the singularity set of $\ma^n$. We also introduce ``$\si_1^- := \ma \si_0$'' and, for $n>1$, $\si_n^- := \si_1^- \cup \ma \si_1^- \cup \cdots \cup \ma^{n-1} \si_1^-$. These are the singularity sets for the powers of the inverse map $\ma^{-1}$. Lastly, $\si_\infty^+ := \bigcup_{n=1}^{\infty} \si_n^+$, $\si_\infty^- := \bigcup_{n=1}^{\infty} \si_n^-$, and $\si := \si_\infty^+ \cup \si_\infty^-$. Each $\si_n^\pm$ is the union of smooth curves whose endpoints lie either on another such curve or on the \emph{generalized boundary} of $\ps = [0,L] \times [-\pi/2,\pi/2]$, which is defined as the boundary of $\ps$ plus all the vertical segments $s=s_i$, where $s_i$ is the boundary coordinate of a vertex of $\bo$. If $L < \infty$, the number of vertices is finite, and the curvature of $\ta$ is bounded, then $\si_n^\pm$ comprises only a finite number of smooth curves. Under the above assumptions, $\ma$ is a piecewise differentiable map with singularities, of the type studied by Katok and Strelcyn in \cite{ks}. Among their results is a suitable version of the Oseledec Theorem which guarantees, for a.e.\ $(s,\a) =: x \in \ps$: \begin{enumerate} \item A decomposition of the tangent space $T_x \ps$ into $E_x^+ \oplus E_x^-$. These one-di\-men\-sion\-al spaces are dynamics-invariant in the sense that $(D \ma)_x E_x^\pm = E_{\ma x}^\pm$, where $(D \ma)_x$ denotes the differential of $\ma$ at $x$. \item The existence of the Lyapunov exponents $\lambda_\pm(x)$, defined as \begin{equation} \lambda_\pm(x) := \lim_{n \to +\infty} \frac1n \log \| (D \ma^n)_x v_\pm \|, \end{equation} with $v_\pm \in E_x^\pm$. Since $\mu$ is absolutely continuous w.r.t.\ the Lebesgue \me\ on $\ps$, then $\lambda_+ (x) = -\lambda_- (x)$. We adopt the convention that $\lambda_+ (x) \ge 0$. \end{enumerate} The \dsy\ is \hyp, by definition, if $\lambda_+ (x) > 0$ almost everywhere. If the \sy\ is \erg\ too, then $\lambda_+ (x) =$ constant $=: \lambda_+$. \section{Geometrical optics and cone bundles} \label{sec-cones} In this section, which liberally draws from \cite{w2}, we recall the basic tenets of the invariant cone technique for the \hyp ity of planar \bi s (cf.\ also \cite{lw}), and prove a couple of results that are specifically designed for our \sy s. Given $x\in \ps$ and two linearly independent vectors $v_1, v_2 \in T_x \ps$, we define the \emph{cone with boundaries $v_1, v_2$} as the set \begin{equation} \label{cone} C(x) := \rset{av_1 + bv_2} {a,b \in \R,\ ab \ge 0 }. \end{equation} If $C(x)$ is defined at every, or almost every, $x \in \ps$ and the dependence on $x$ is measurable, we speak of $C \subset T\ps$ as a measurable cone bundle. A measurable cone bundle $C$ is said to be: \begin{itemize} \item \emph{invariant}, if $(D\ma)_x C(x) \subseteq C(\ma x)$ for $\mu$-a.e.\ $x$; \item \emph{strictly invariant}, if $(D\ma)_x C(x) \subset C(\ma x)$ for $\mu$-a.e.\ $x$; \item \emph{eventually strictly invariant}, if it is invariant and, for $\mu$-a.e.\ $x$, there exists $n(x) \in \Z^+$ such that $(D \ma^{n(x)})_x C(x) \subset C(\ma^{n(x)} x)$. \end{itemize} The next theorem was proved in \cite{w1}. \begin{theorem} \label{thm-conhyp} Given a \bi\ map $\ma$ as described above, if there exists an eventually strictly invariant measurable cone bundle, then the Lyapunov exponent $\lambda_+(x)$ is positive for $\mu$-a.e.\ $x \in \ps$. \end{theorem} In \cite{w2} Wojtkowski reduces the invariance of a cone bundle to a problem of geometrical optics concerning the behavior of a family (a \emph{beam}) of nearby \tr ies. We present the main ideas. To a tangent vector $v \in T_x \ps$ in phase space is naturally associated a differentiable curve $\varphi: (-\eps, \eps) \longrightarrow \ps$ such that $\varphi(0) = x$ and $\varphi'(0) = v$. By construction, $\sigma \mapsto \varphi(\sigma)$ is uniquely determined in linear approximation around $0$. Using the representation of $\ps$ as a subset of $\ta \times S^1$, and the notation $\varphi(\sigma) = (q(\sigma), u(\sigma)) \in \ta \times S^1$, we construct the family of lines, or \emph{rays}, $l^+ (\sigma) := \lset{q(\sigma) + r u(\sigma)} {r \in \R}$. Also, denoting by $u^-(\sigma)$ the outward-pointing, precollisional vector of $u(\sigma)$ at $q(\sigma) \in \bo$, we define $l^- (\sigma) := \lset{q(\sigma) + r u^- (\sigma)} {r \in \R}$. In first approximation, that is, when $\eps \to 0^+$, the now infinitesimal beam of rays \emph{focuses} in a point, which means that all rays, up to adjustments of order $\eps$ in $(q(\sigma), u(\sigma))$, have a common intersection. We consider the case too where the common intersection is at infinity. This \emph{focal point} is clearly a \fn\ of $v$ only: it is denoted $F^+(v)$ for the family $\{ l^+ (\sigma) \}$ and $F^-(v)$ for the family $\{ l^- (\sigma) \}$. Let us call $f^\pm (v)$ the signed distances, along $l^\pm(0)$, between $F^\pm(v)$ and $q_0 = q(0)$ ($l^\pm(\sigma)$ has the orientation induced by the parameter $r \in \R$, that is, outward for $l^- (\sigma)$ and inward for $l^+ (\sigma)$, relative to $\ta$). In the remainder, we will omit the dependence of $v$ from all the notation whenever there is no ambiguity. Indicated by $(ds, d\a)$ the components of $0 \ne v \in T \ps_{(s_0, \a_0)}$ in the natural basis $\{ \partial / \partial s,\partial / \partial\a \}$, one has \begin{equation} f^{\pm} = \left\{ \begin{array}{lll} \ds \frac{\cos \a_0} {\pm k(s_0) - \frac{d\a}{ds}}, && \mbox{if } ds \ne 0; \vspace{6pt} \\ 0, && \mbox{if } ds = 0. \end{array} \right. \label{fpm} \end{equation} Here $k(s)$ denotes the curvature of $\bo$ at the point of coordinate $s$ (as specified in the introduction, the curvature is taken positive at focusing points of the boundary, and negative at dispersing points). The formula (\ref{fpm}) is derived, e.g., in \cite{w2}. It is easy to see that $f^\pm$ are projective coordinates of $T_x \ps$. Hence any cone of the type (\ref{cone}) can be described by a closed interval in the coordinate $f^+ \in \overline{\R}$, where $\overline{\R} := \R \cup \{ \infty \}$ is the compactification of $\R$. Henceforth, for simplicity, we will drop the subscripts from the coordinates $(s_0, \a_0)$ of the collision pair. Also, we will use the imprecise terminology `the point $s \in \bo$' to mean `the point in $\bo$ of coordinate $s$'. The next lemma is known in optics as the mirror equation \cite{w2, cm}. \begin{lemma} For an infinitesimal beam of \tr ies colliding around the point $s \in \bo$ with reflection angles around $\a$, \begin{displaymath} -\frac1{f^{-}} + \frac1{f^{+}} = \frac{2k(s)} {\cos \a}. \end{displaymath} \label{lem-mirror-eq} \end{lemma} We now present a visual description of the cone $C(x) = C(s,\a)$ on the configuration plane containing $\ta$. For $s \in \bo$ and $\beta>0$, denote by $D_\beta (s)$ the closed disc of radius $1 / |\beta k(s)|$ tangent to $\bo$ in $s$ on the internal side of $\ta$. Analogously, for $\beta<0$, let $D_\beta (s)$ be the closed disc of radius $1 / |\beta k(s)|$ tangent to $\bo$ in $s$ on the external side of $\ta$. Consider also the two closed halfplanes delimited by $t(s)$, the tangent line to $\bo$ in $s$: let $D_{0+}(s)$ denote the internal halfplane, relative to $\ta$, and $D_{0-}(s)$ the external one. See Fig.~\ref{fig-dbetas}. The interior of $D_\beta (s)$ is indicated with $D_\beta^\circ (s)$. \fig{fig-dbetas} {8cm} {The tangent line $t(s)$ and some discs $D_\beta(s)$. The yellow part of the \tr y is the locus of the focal points $F^+$ corresponding to a certain cone.} \begin{lemma} Given a cone $C(s,\a)$ of the type \emph{(\ref{cone})}, $v \in C(s,\a)$ corresponds to $F^+(v) \in l^+(0) \cap D$, where $D \subset \R^2$ is one of the following sets: \begin{itemize} \item[(a)] $D = D_{\beta_1} (s)$; \item[(b)] $D = D_{\beta_1} (s) \setminus D_{\beta_2}^\circ (s)$, with $|\beta_1| < |\beta_2|$; \item[(c)] $D = D_{\beta_1} (s) \cup D_{\beta_2} (s)$, with $\beta_1 \ge 0$ and $\beta_2 \le 0$; \item[(d)] $D = \R^2 \setminus (D_{\beta_1}^\circ (s) \cup D_{\beta_2}^\circ (s) \cup \{s\} )$, with $\beta_1 \ge 0$ and $\beta_2 \le 0$. \end{itemize} Moreover, \begin{displaymath} F^+(v) \in \partial D_\beta (s) \setminus \{ s \} \quad \Longleftrightarrow \quad f^+(v) = \frac{2 \cos \a} {\beta |k(s)|}. \end{displaymath} \label{lem-fplus} \end{lemma} \proof By construction $F^+ = F^+(v) \in l^+(0)$. Since $f^+$ is a coordinate on $l^+(0)$, a closed interval in the projectivized $f^+ \in \overline{\R}$ corresponds, on $l^+(0)$, to either a closed segment or a closed halfline or the union of two disjoint closed halflines. Cases \emph{(a)}-\emph{(d)} cover all possibilities. The second statement, for $\beta>0$, comes from elementary trigonometry (see Fig.~\ref{fig-dbetas}), and it trivially extends to the case $\beta<0$ as well. \qed The reason why, in Lemma \ref{lem-fplus}, we chose such peculiar sets $D$ to cut a (projective) closed segment on $l^+(0)$, upon intersection, will be made clear by the next lemma. In particular, we will see that describing the cones in terms of the discs $D_\beta(s)$ will eliminate the dependence on $\a$ in the mirror equation of Lemma \ref{lem-mirror-eq}. \begin{lemma} For infinitesimal beam of trajectories colliding around $s \in \bo$, $F^{-} \in \partial D_\beta (s)$ \iff $F^{+} \in \partial D_{\beta'} (s)$, where \begin{displaymath} \beta' = 4 \, \mathrm{sgn}(k(s)) - \beta \end{displaymath} (with the understanding that $F^\pm \in \partial D_{0\pm}$ means $F^\pm \in \{ s, \infty \}$). \label{lem-betap} \end{lemma} \proof Let $\a$ be the angle of reflection (and thus of incidence) of the \tr y we are perturbing. Disregarding the case $F^+ = F^- = s$, we know from Lemma \ref{lem-fplus} that $F^+ \in \partial D_{\beta'} (s)$ corresponds to $f^+ = 2 \cos\a / (\beta' |k(s)|)$. Also, $F^- \in \partial D_\beta (s)$ is equivalent to $f^- = -2 \cos\a / (\beta |k(s)|)$ (the minus sign is needed because a focal point $F^-$ lying on the internal halfplane $D_{0+} (s)$ corresponds to a negative $f^-$ along $l^-(0)$, and viceversa). Direct substitution into Lemma \ref{lem-mirror-eq} yields \begin{equation} \frac{\beta |k(s)|} {2\cos \a} + \frac{\beta' |k(s)|} {2 \cos\a} = \frac{2k(s)} {\cos\a}, \end{equation} whence the assertion. \qed With the tools of Section \ref{sec-cones}, the problem of the cone invariance along a given \tr y can be reduced to the study of the focal points of one-parameter perturbations of that \tr y. We single out the information that we need for our forthcoming proofs. \begin{proposition} For an infinitesimal beam of \tr ies colliding around $s$ we have the following: If $s$ belongs to a focusing component of $\bo$, i.e., $k(s)>0$, then: \begin{eqnarray*} F^\mp \in D_4(s) & \Longleftrightarrow & F^\pm \in D_{0-}(s); \\ F^\mp \in D_2(s) \setminus D_4^\circ (s) & \Longleftrightarrow & F^\pm \in D_{0+}(s) \setminus D_2^\circ (s). \end{eqnarray*} If $s$ belongs to a dispersing component of $\bo$, i.e., $k(s)<0$, then \begin{eqnarray*} F^\mp \in D_{-4}(s) & \Longleftrightarrow & F^\pm \in D_{0+}(s); \\ F^\mp \in D_{-2}(s) \setminus D_{-4}^\circ (s) & \Longleftrightarrow & F^\pm \in D_{0-}(s) \setminus D_{-2}^\circ (s). \end{eqnarray*} Analogous equivalences hold for the interior of such cones. The situation is illustrated in Fig.~\ref{fig-p-dbeta}. \label{prop-d-beta} \end{proposition} \proof We only prove the first statement, the other ones being completely analogous. Once again, we disregard the easy case $F^+ = F^- = s$. We have $F^- \in D_4(s)$ $\Leftrightarrow$ $F^- \in \partial D_\beta(s)$, for $\beta \in [4, +\infty)$ $\Leftrightarrow$ (by Lemma \ref{lem-betap}) $F^+ \in \partial D_{\beta'} (s)$, for $\beta' \in (-\infty, 0]$ $\Leftrightarrow$ $F^- \in D_{0-} (s)$. Clearly, nothing changes if we swap $F^-$ and $F^+$. \qed \fig{fig-p-dbeta} {13.5cm} {A geometric representation of Proposition \ref{prop-d-beta}. The left picture represents the first two statements (focusing border); the right picture represents the last two statements (dispersing border). Yellow/blue sets of focal points $F^-$ are mapped into yellow/blue sets of focal points $F^+$. The dependence on $s$ in the notation has been omitted.} \section{Hyperbolicity} \label{sec-hyp} Fig.~\ref{fig-t1} shows the \bi\ table we are mainly interested in for the rest of the paper. We refer to it for the definition of the quantities $l, h > 0$. The three dispersing components of the boundary $\bo$ are circular arcs of curvature $k_d \in (-\sqrt{2},0)$. Their union is denoted $\bo_d$. The focusing component is a circular arc of curvature $k_f > 0$ and is denoted $\bo_f$. The remining, flat, part of the boundary is denoted $\bo_s$. The two rectangular portions of $\ta$ which $\bo_s$ almost delimits will be referred to as \emph{the strips}, or \emph{the corridors}, or whatever one's fancy suggests each time. \fig{fig-t1} {12.4cm} {The definition of the table $\ta$.} The geometric constants $l, h, k_f, k_d$ are chosen via the following procedure. Keep in mind that we are interested in small values of $k_f$ (see the Introduction) and $h$ (see Section \ref{sec-conf}). One starts by fixing arbitrary values of $k_d$ and $h$. Then $k_f$ is determined by a geometric condition that we presently describe, with the help of Fig.~\ref{fig-iss}. For $s' \in \bo_d$ and $s'' \in \bo_f$, consider the straight line passing through $s'$ and $s''$, and let $I(s', s'')$ be its intersection with the disc $D_{-2}(s')$. The curvature $k_f$ must be so small that \begin{equation} \label{C1} \forall s' \in \bo_d, \ \forall s'' \in \bo_f, \quad I(s', s'') \subset D_4(s''). \end{equation} Finally, $l$ is chosen such that \begin{equation} \label{C2} l \ge \frac1 {k_f} \end{equation} \fig{fig-iss} {13cm} {Condition (\ref{C1}) for two different choices of $s'$.} \begin{remark} Condition (\ref{C1}) excludes sufficient separation between the boundary components as per the standard theory of Wojtkowski, Markarian, Donnay and Bunimovich, which is summed up, e.g., in \cite[Thm.~9.19]{cm}. The hypotheses of that theorem are evidently violated as (\ref{C1}) implies in particular that $D_4(s'')$ contains large portions of $\bo_d$, for all $s \in \bo_f$. \label{rk-sep} \end{remark} We are now going to prove the \hyp ity of this \bi\ \sy\ via Theorem \ref{thm-conhyp}. However, we will not use exactly the Poincar\'e section that we have introduced in Sections \ref{sec-prel} and \ref{sec-cones}, but a similar section that neglects the hits on the flat boundary component $\bo_s$. This is standard procedure in the theory of \hyp\ \bi s as it is basic fact that the collisions against a flat boundary do not change the \hyp\ features of a beam of \tr ies. (One easy way to see this is to \emph{unfold} the \bi\ along a given \tr y: every time the material point hits a flat side we pretend that it continues its precollisional rectilinear motion, but we reflect the table around that flat side; apart from this rigid motion of the \bi\ table, nothing changes for the \tr y or any of its infinitesimal perturbations.) Let us denote $\bar{\bo} := \bo_f \cup \bo_d$. With the usual abuse of notation, whereby a point $q \in \bo$ is identified with its arclength coordinate $s$, we define $\ps := \bar{\bo} \times [-\pi/2, \pi/2]$, whose elements we call $(s,\a)$ or $x$. Clearly $\ps$ is a global cross section for the flow. Let $\ma: \ps \longrightarrow \ps$ be its first-return map. For any $x = (s, \a) \in \ps$ and $n \in \Z$, denote $x_n := (s_n, \a_n) := \ma^n x$ and let $\tau_n$ be the length of the portion of the \tr y (equivalently, the time) between the collisions at $s_n$ and $s_{n+1}$ (notice that there can be an arbitrary number of collisions against $\bo_s$ between $s_n$ and $s_{n+1}$). Also, let $k_n := k(s_n)$ indicate the curvature of $\bo$ in $s_n$. Analogously, given $v \in T_x \ps$, denote $v_n := (D\ma^n)_x v$. The infinitesimal beam of \tr ies determined by $v_n$ (and thus by $v$) around $(s_n, \a_n)$ will have pre- and postcollisional foci denoted, respectively, $F_n^- := F^-(v_n)$ and $F_n^+ := F^+(v_n)$. The corresponding signed distances along the pre- and postcollisional lines are indicated with $f_n^-$ and $f_n^+$. The following facts are obvious: \begin{eqnarray} && F_n^- = F_{n-1}^+, \\ && f_n^- = -( \tau_{n-1} - f_{n-1}^+). \end{eqnarray} For the sake of the notation, let us drop all subscripts 0 and write $k := k_0$, $F^+ := F_0^+$, and so on. For any $x \in \ps$, we introduce the following three cones in $T_x \ps$: \begin{itemize} \item $C_0 (x)$ is the set of all tangent vectors whose correspondent family of rays focuses in linear approximation inside $D_{-2}(s)$. Using the focal distance $f^+$, \begin{equation} C_0 (x) := \rset{v \in T_x \ps} {-\frac{\cos \alpha} {|k|} \le f^+(v) \le 0}. \end{equation} \item $C_1 (x)$ is the set of all tangent vectors whose correspondent family of rays focuses in linear approximation inside $D_{0-}(s)$, i.e., all the divergent families of rays. In projective terms, \begin{equation} C_1 (x) := \rset{v \in T_x \ps} {-\infty < f^+(v) \le 0}. \end{equation} \item $C_2 (x)$ is the set of all tangent vectors whose correspondent family of rays focuses in linear approximation inside $D_2 (s) \setminus D_4^\circ (s)$, i.e., \begin{equation} C_2 (x) := \rset{v \in T_x \ps} {\frac{\cos \alpha} {2|k|} \le f^+(v) \le \frac{\cos \alpha} {|k|}}. \end{equation} \end{itemize} We use the above cones to define piecewise an invariant cone bundle $C := \{ C(x) \}_x$. For each $x = (s, \a)$, the choice $C(x) := C_i (x)$ will depend on $s$, $s_{-1}$, and what happens to the \tr y between the collisions at $s_{-1}$ and $s$. \begin{itemize} \item[(A)] If \underline{$s \in \bo_d$}, set $C(x) := C_0 (x)$. \item[(B)] If \underline{$s \in \bo_f$}, there are two subcases: \begin{itemize} \item[(B.1)] If \underline{$s_{-1} \in \bo_f$}, set $C(x) := C_2 (x)$. \item[(B.2)] If \underline{$s_{-1} \in \bo_d$}, there are two further subcases, depending on whether the piece of \tr y between $s_{-1}$ and $s$ has collisions with $\bo_s$: \begin{itemize} \item[(B.2.1)] \underline{No collisions with $\bo_s$} between $s_{-1}$ and $s$: Set $C(x) := C_1 (x)$. \item[(B.2.2)] \underline{At least one collision with $\bo_s$} between $s_{-1}$ and $s$: Set $C(x) := C_2 (x)$. \end{itemize} \end{itemize} \end{itemize} Clearly $C(x)$ is a measurable \fn\ of $x$. \begin{theorem} \label{thm-hyp} The cone bundle $C$ just defined is eventually strictly invariant relative to the map $\ma$. \end{theorem} \proof We check that $v \in C(x)$ implies $v_1 \in C(x_1)$ for all the possible cases $C(x) = C_i(x)$, $C(x_1) = C_j(x_1)$ $(i,j \in \{ 0,1,2 \})$. \begin{itemize} \item[(I)] \underline{$s, s_1 \in \bo_d$}.\ In this case $C(x) = C_0(x)$, $C(x_1) = C_0(x_1)$. $v \in C_0(x)$ implies $F^+ \in D_{-2}(s)$, hence $F_1^- = F^+ \in D_{0+}^\circ (s_1)$. By Proposition \ref{prop-d-beta}, $F_1^+ \in D_{-4}^\circ (s_1) \subset D_{-2}^\circ (s_1)$. This is equivalent to $v_1 \in C_0^\circ (x_1)$---where $C^\circ(x)$ represents the interior of $C(x)$ in $T_x \ps$. We have thus proved strict invariance for this type of collision. \item[(II)] \underline{$s \in \bo_d$, $s_1 \in \bo_f$}.\ Here $C(x) = C_0(x)$ but the cone $C(x_1)$ may take two different forms. We separately check both cases. \begin{itemize} \item[(II.1)] There are no collisions with $\bo_s$ between $s$ and $s_1$. Then $C(x_1) = C_1(x_1)$. For $v \in C_0(x)$ we have, by condition (\ref{C1}), $F_1^- = F^+ \in D_4 (s_1)$. Proposition \ref{prop-d-beta} implies that $F_1^+ \in D_{0-} (s_1)$, that is, $v_1 \in C_1 (x_1)$. In this case the invariance is not necessarily strict. \item[(II.2)] There are collisions with $\bo_s$ between $s$ and $s_1$, that is, the material point enters a strip before colliding at $s_1$. In this case $C(x_1) = C_2(x_1)$. Since the material point has to travel all the way to the end of the strip and bounce back, $\tau > 2l > 2/k_f$, having used condition (\ref{C2}). For $v \in C_0(x)$, $f^+ \le 0$, hence $f_1^- = -\tau + f^+ < -1/k_f$. Equivalently, $F_1^- \in D_{0+}(s_1) \setminus D_2(s_1)$. By Proposition \ref{prop-d-beta}, $F_1^+ \in D_2^\circ (s_1) \setminus D_4 (s_1)$, i.e., $v_1 \in C_2^\circ (x_1)$. \end{itemize} \item[(III)] \underline{$s \in \bo_f$, $s_1 \in \bo_d$}.\ Here $C(x_1) = C_0(x_1)$ and we have two subcases on $C(x)$. \begin{itemize} \item[(III.1)] $C(x) = C_1(x)$. In this case $v \in C(x)$ is equivalent to $f^+ \le 0$. Hence $f_1^- < 0$ and $F_1^- \in D_{0+}^\circ (s_1)$. Therefore (Proposition \ref{prop-d-beta}) $F_1^+ \in D_{-4}^\circ (s_1) \subset D_{-2}^\circ (s_1)$. Namely $v_1 \in C_0^\circ (x_1)$. \item[(III.2)] $C(x) = C_2(x)$. So $v \in C(x)$ means that $F^+ = F_1^- \in D_2 (s) \setminus D_4^\circ (s)$. We consider two possible types of \tr ies: \begin{itemize} \item[(III.2.1)] There are no collisions with $\bo_s$ between $s$ and $s_1$. By (\ref{C1}), $F_1^- \in D_{0-}^\circ (s_1) \setminus D_{-2}^\circ (s_1)$. Hence $F_1^+ \in D_{-2} (s_1)$. \item[(III.2.2)] There are collisions with $\bo_s$ between $s$ and $s_1$. As in case (II.2), $\tau > 2/k_f$ and $f^+ \le (\cos\a) / k_f < 0$. Thus, $f_1^- < 0$, that is, $F_1^- \in D_{0+}^\circ (s_1)$. Finally, $F_1^+ \in D_{-4}^\circ (s_1) \subset D_{-2}^\circ (s_1)$. \end{itemize} \end{itemize} \item[(IV)] \underline{$s, s_1 \in \bo_f$}.\ Definition (B.1) ensures that $C(x_1) = C_2(x_1)$. Let us branch out in two subcases depending on $C(x)$. \begin{itemize} \item[(IV.1)] $C(x) = C_1(x)$. As in case (III.1), $v \in C(x)$ implies that $f^+ \le 0$. Since, by construction of our cross section, there can be no collisions with $\bo_d$ in the piece of \tr y between $s$ and $s_1$, there are only two possibilities: either the particle enters and exits a strip, and thus $\tau > 2/k_f$; or that piece of \tr y is a chord of the arc $\bo_f$, and thus $\tau = 2 (\cos\a) /k_f$. In either case, $\tau > (\cos\a) /k_f$ and $f_1^- < -(\cos\a) /k_f$, which means that $F_1^+ \in D_{0+}^\circ (s_1) \setminus D_2 (s_1)$. By Proposition \ref{prop-d-beta}, $F_1^+ \in D_2^\circ (s_1) \setminus D_4 (s_1)$, that is, $v_1 \in C_2^\circ (x_1)$. \item[(IV.2)] $C(x) = C_2(x)$. The hypothesis $v \in C(x)$ reads $(\cos\a) / 2k_f \le f^+ \le (\cos\a) / k_f$. Once again, there are two further subcases: \begin{itemize} \item[(IV.2.1)] There are no collisions with $\bo_s$ between $s$ and $s_1$. In this case, cf.\ (IV.1), the \tr y between $s$ and $s_1$ is a chord of $\bo_f$ and $\tau = 2 (\cos\a) /k_f$. Therefore $f_1^- = -\tau + f^+ \le -(\cos\a) /k_f$, which implies $F_1^- \in D_{0+} (s_1) \setminus D_2^\circ (s_1)$. This yields $F_1^+ \in D_2 (s_1) \setminus D_4^\circ (s_1)$, namely $v_1 \in C_2(x_1)$. \item[(IV.2.2)] There are collisions with $\bo_s$ between $s$ and $s_1$. $f^+$ and $\tau$ are exactly as in case (III.2.2). Refining the estimate that is written there, $f_1^- < -1/k_f < -(\cos\a) / k_f$, that is, $F_1^- \in D_{0+}^\circ (s_1) \setminus D_2 (s_1)$. This gives $F_1^+ \in D_2^\circ (s_1) \setminus D_4 (s_1)$. \end{itemize} \end{itemize} \end{itemize} In order to show that $C$ is eventually strict invariant \emph{almost everywhere}, we notice that there are only three cases above in which the cone invariance is not strict, namely (II.1), (III.2.1), and (IV.2.1). In both (II.1) and (III.2.1), nonstrictness can only occur when the external endpoint of $I(s',s'')$ lies on $D_4 (s'')$ and $s = s'$, $s_1 = s''$, or viceversa---cf.\ (\ref{C1}) and Fig.~\ref{fig-iss}. It is not hard to realize that this situation can only occur for finitely many pairs $(s',s'')$ (at least when the table is optimized, see (\ref{C3}) and Fig.~\ref{fig-ho}, there are only two such pairs). As concerns (IV.2.1), we realize that there can only be a finite number of consecutive collisions of that type, because each such piece of \tr y is a chord of $\bo_f$ of constant length ($\tau = \tau_1$), but $\bo_f$ is smaller than a semicircle. \qed \section{Confining the table to a bounded region} \label{sec-conf} In the previous section the table $\ta$ was constructed starting with two values for $h$ and $k_d$, which determined an upper bound on the choice of $k_f$, via (\ref{C1}), which in turn determined a lower bound on the choice of $l$, via (\ref{C2}). The latter condition, in particular, forced the area of $\ta$ to diverge, as smaller and smaller values are chosen for $k_f$. Now we want to optimize, that is, minimize, the area of the table and to do so we change the order in which its geometric parameters are chosen. Given $k_d < 0$ and $k_f$ sufficiently small, we define the \emph{optimal height} and the \emph{optimal length} of the strips, respectively, as: \begin{eqnarray} \label{C3} && h_o := h_o (k_d, k_f) := \min \rset{h} {\forall s' \in \bo_d, \forall s'' \in \bo_f, \ I(s', s'') \subset D_4(s'')} ; \\ \label{C4} && l_o := l_o (k_f) = k_f^{-1} . \end{eqnarray} These definitions are well posed, in the sense that a table can be constructed with $h = h_o$ and $l = l_o$. We call it the \emph{optimal table} and we think of it as a \fn\ of $k_f$ ($k_d$ is considered fixed once and for all). The optimal table is \hyp\ by Theorem \ref{thm-hyp}. The next proposition shows that, as $k_f \to 0$, the area of the optimal table is bounded above. (In what follows, the notation $a \sim b$ means that $a = a(k_f)$, $b = b(k_f)$ and, as $k_f \to 0$, $|a/b|$ is bounded away from $0$ and $\infty$.) \begin{proposition} \label{prop-ho} As $k_f \to 0$, $h_o(k_f) \sim k_f$. \end{proposition} \proof Since $k_f \to 0$ and $k_d$ is fixed, we may assume that, given any $s'' \in \bo_f$, $D_4(s'')$ easily contains $D_{-2}(s')$, for all $s'$ in the upper component of $\bo_d$ (left picture in Fig.~\ref{fig-iss}). For $s'$ belonging to the lateral components of $\bo_d$, it is not hard to realize that the worst-case scenario is the one depicted in Fig.~\ref{fig-ho} (or the specular situation w.r.t.\ the axis of symmetry of $\ta$): First of all, if $s''$ moves to the left and/or $s'$ moves upward, $I(s',s'')$ will move towards the interior of $D_4(s'')$, so that (\ref{C1}) is always verified. Secondly, setting $h_o$ to be the $h$ displayed there, one clearly sees that for $h \ge h_o$ (\ref{C1}) is verified, while for $h < h_o$ it is not. \fig{fig-ho} {10cm} {Finding $h_o$, cf.\ Proposition \ref{prop-ho}.} Referring to the notation of Fig.~\ref{fig-ho}, we see that $h_o = \tan \beta$ where $\beta$ is the angle between the two chords $s''P$ and $s''Q$ of $\partial D_4(s'')$. Recalling that, in a circle of radius $r$, the relation between the length $\ell$ of a chord and the angle $\theta$ it makes with the tangent to the circle at each of its endpoints is $\ell = 2r \sin \theta$, we have \begin{equation} \beta = \arcsin \left( \frac{k_f \, c}2 \right) - \arcsin \left( \frac{k_f}2 \right) \sim k_f, \quad \mbox{as } k_f \to 0. \end{equation} In the above $c$ is the length of $s''P$, for which it holds $1 < c < 2 + 2k_d^{-1}$. This ends the proof since $h_o \sim \beta$. \qed From a technical point of view, Proposition \ref{prop-ho} is a consequence of the fact that $\bo_f$ fails to act as a perturbation of a semidispersing component only for a few \tr ies, whose corresponding beams need to be defocused by visiting the long strips. As $k_f \to 0$, this phenomenon concerns fewer and fewer \tr ies, but its fix requires more and more space. Proposition \ref{prop-ho} tells us that the trade-off between the two effects balances out. If a \hyp\ \bi\ table with a flatter and flatter focusing component need not become bigger and bigger in terms of area, one might hope that it need not in terms of diameter, either. In our particular table, one would like to redesign the strips so that their area is better placed in the plane and can be included in a fixed compact region. In the remainder of the section we show that this is possible, for example by bending the strips around the bulk of the \bi\ (see Fig.~\ref{fig-t2intro}). \fig{fig-folding} {5cm} {Construction of a spiral as a union of trapezoids.} Let us describe this construction with the help of Fig.~\ref{fig-folding}. Substitute each strip of $\ta$ with a polygonal modification given by the union of $N$ adjacent right trapezoids $T_1, T_2, \ldots, T_N$, where $N$ will be specified later depending on $k_f$. $T_1$ is placed so that its shorter leg coincides with the opening towards the bulk of $\ta$: its height is then $h_1 := h \ge h_o$. The length of the shorter base is denoted $l_1$ and the two nonright angles are denoted $\pi/2 + \gamma_1$ and $\pi/2 - \gamma_1$, with $0 < \gamma_1 < \pi/2$. This causes the longer leg to measure $h_2 := h_1 / \cos \gamma_1$. The longer leg of $T_1$ is then used as the shorter leg of the next trapezoid, $T_2$, in the way depicted in Fig.~\ref{fig-folding}. The construction continues recursively, as values for $l_i$, $\gamma_i$ (and therefore $h_{i+1} := h_i / \cos \gamma_i$) are generated with each new trapezoid $T_i$. We call the resulting region a \emph{polygonal spiral}, or simply \emph{spiral}. There are two of them, and they need not be equal, so we denote $N^R, h_i^R, l_i^R, \gamma_i^R$, and $N^L, h_i^L, l_i^L, \gamma_i^L$, the parameters of the right and the left spiral, respectively. These will be determined later depending on $h_o$ and $l_o$, thus ultimately on $k_f$. We will see to it that the following conditions hold: \begin{itemize} \item The spirals turn counterclockwise at each corner. \item They have no self-intersections, or intersections between them or with the bulk of $\ta$. \item For $\epsilon \in \{ R,L \}$, all angles $\gamma_i^\epsilon$ are rational multples of $\pi$. \item There exists an absolute constant $K_1$ (i.e., $K_1$ does not depend on anything, including $k_f$) such that, for $\epsilon \in \{ R,L \}$, \begin{equation} \label{cond-s1} h_{N^\epsilon}^\epsilon \le K_1 h_o. \end{equation} \item There exists an absolute constant $K_2 > 1$ such that \begin{equation} \label{cond-s2} l_o \le \sum_{i=1}^{N^\epsilon} l_i^\epsilon \le K_2 \, l_o. \end{equation} \item There exists an absolute constant $K_3$ such that, $\forall i=1, 2, \ldots, N^\epsilon$, \begin{equation} \label{cond-s3} \frac{\tan \gamma_i^\epsilon} {l_i^\epsilon} \le \frac{K_3} {h_i^\epsilon}. \end{equation} (The l.h.s.\ above is a measure of the ``curvature'' of the spiral at the $i$-th corner.) \end{itemize} Under the above conditions the area of each spiral is bounded, as $k_f \to 0$, because, dropping the superscript $\epsilon$, \begin{eqnarray} \frac12 \sum_{i=1}^N ( 2l_i + h_i \tan \gamma_i ) h_i &\le& \frac{2 + K_3}2 \sum_{i=1}^N l_i h_i \nonumber \\ &\le& \mbox{const } l_o h_N \\ &\le& \mbox{const } l_o h_o \sim 1; \nonumber \end{eqnarray} having used, in this order, (\ref{cond-s3}), (\ref{cond-s2}), and (\ref{cond-s1}). Also, defining $(\ps, \ma, \mu)$ as in Section \ref{sec-hyp}, namely, as the \dsy\ corresponding to the cross section $\ps$ of all line elements based in $\bar{\bo} = \bo_f \cup \bo_d$, we have: \begin{proposition} \label{prop-hyp-s} $\ps$ is a global cross section for the \bi\ flow and $(\ps, \ma, \mu)$ is \hyp. \end{proposition} \proof First of all, $\ma$, as the first-return map onto $\ps$, is well-defined almost everywhere (e.g., by the Poincar\'e Recurrence Theorem). To prove that $\ps$ is a global cross section, we need to show that a.a.\ \bi\ \tr ies have collisions against $\bar{\bo} = \bo_f \cup \bo_d$. This is easy if we use a well-known result from the theory of polygonal \bi s \cite{zk, bkm}: Let $P$ be the union of the two spirals plus $R$, which is the rectangle (of base 1 and height $h$) joining the open ends of the spirals. $P$ is a \emph{rational polygon}, meaning that all its angles are rational multiples of $\pi$. In a rational polygonal \bi, all but countably many values of the velocity $u \in S^1$ are \emph{minimal}, in the sense that any nonsingular flow-\tr y in configuration space (i.e., the set $\{ q(t) \}_{t \in \R}$, provided that it contains no corner of $P$), with initial velocity $u$, is dense in $P$ \cite{zk, bkm}. This implies that for a.a.\ initial conditions $(q,u)$, with $q \in P$, the \bi\ \tr y in $P$ hits the boundary of $R$, which means that the true \bi\ \tr y, relative to the table $\ta$, hits $\bar{\bo}$. As for the second assertion of Proposition \ref{prop-hyp-s}, we need the following lemma, which will be proved later. \begin{lemma} \label{lem-hyp-s} A material point that enters a spiral will travel all the way to the end of the spiral. In particular, if $\tau$ is the travel time between the last collision before entering the spiral and the first collision after exiting it (a.a.\ \tr ies eventually exit the spiral), then $\tau > 2l_o = 2 / k_f$. \end{lemma} Lemma \ref{lem-hyp-s} shows that Theorem \ref{thm-hyp} (and thus Theorem \ref{thm-conhyp}) applies to the present case as well, since its proof only requires of \tr ies visiting a strip---or a spiral---that the travel time $\tau$ be larger than $2 / k_f$. (Note that, since the spirals are two polygons, they will have no effect on the \hyp\ features of an infinitesimal beam of \tr ies, just like the two strips. The only, inconsequential, difference is that the spirals have more corners than the strips, resulting in more discontinuity lines in $\ps$.) \qed \proofof{Lemma \ref{lem-hyp-s}} The first assertion is an easy consequence of our design, since a point that enters $T_i$ through the shorter leg will necessarily exit it through the longer leg, thus entering $T_{i+1}$ through the shorter leg, and so on. As for the second assertion, clearly $\tau$ will be larger than twice the sum of the lengths of the shorter bases of the trapezoids. By (\ref{cond-s2}), this sum is bounded below by $l_o$. \qed \fig{fig-reel} {13cm} {The double spiral (right picture) ``wrapping'' around the bulk of $\ta$ (left picture). The double spiral starts when the two spirals coming out of the bulk of $\ta$ join. Its initial ray is $r_0$, its initial (total) width is $w_0$, each turn amounts to an angle $\bar{\gamma} = 2\pi / \bar{N}$, and the number of rounds is $M$. The point $A$ is the center of the double spiral.} Let us finally give the exact construction of the two spirals. First of all, we design the spirals to become adjacent after a finite number of turns, say $m^R$ turns for the right spiral and $m^L$ turns for the left spiral (left picture of Fig.~\ref{fig-reel}); $m^R$ and $m^L$ are absolute constants. We say that the two spirals have now joined in a \emph{regular double spiral}, since they will keep adjacent as they spiral outwards in the regular way shown in the right picture of Fig.~\ref{fig-reel}. More precisely, all trapezoids $T_i^R$, with $i \ge m^R$, and $T_i^L$, with $i \ge m^L$, are similar, and are defined by $\gamma_i^\epsilon = \bar{\gamma} := 2\pi / \bar{N}$, where $\bar{N}$ is an integer (depending on $h_o$) to be determined momentarily. The double spiral is also defined so that its initial ray (meaning the distance from the border of the spiral to its center $A$, see Fig.~\ref{fig-reel}) is $r_0$, an absolute constant so large that intersection with the bulk of $\ta$ is avoided. At each next corner, the ray (that is, the distance between that corner and $A$) increases by a factor $1 / \cos \bar{\gamma}$. Therefore, after the first round, the ray has become $r_{\bar{N}} := r_0 ( \cos \bar{\gamma} )^{-\bar{N}}$. Since the spiral wraps around itself tightly (i.e., leaving no area uncovered), its initial width is \begin{equation} \label{ds-10} w_0 := r_0 \left( \left(\cos \frac{2\pi} {\bar{N}} \right)^{-\bar{N}} \!\!\! - 1 \right). \end{equation} On the other hand, in the place where the left and right spirals join to start the double spiral, one sees that \begin{eqnarray} \label{ds-20} w_0 &=& h_{m^R}^R + h_{m^L}^L \nonumber \\ &=& \left( \prod_{i=1}^{m^R} \frac1 { \cos \gamma_i^R } + \prod_{i=1}^{m^L} \frac1 { \cos \gamma_i^L } \right) h \\ &=:& K_4 \, h . \nonumber \end{eqnarray} $K_4$ is an absolute constant if we prescribe that, for $i = 1, \ldots, m^\epsilon$, the angles $\gamma_i^\epsilon$ are rational multiples of $\pi$ and stay fixed while $k_f \to 0$ (this is geometrically possible, cf.\ Fig.~\ref{fig-reel}, left picture). The last two equations imply that \begin{equation} \label{ds-30} h = h_1 = \frac{r_0} {K_4} \left( \left(\cos \frac{2\pi} {\bar{N}} \right)^{-\bar{N}} \!\!\! - 1 \right). \end{equation} Given $k_f$ sufficiently small, we use (\ref{ds-30}) to define both $h$ and $\bar{N}$, keeping in mind that we want $h_o \le h \le K_1 h_o$, cf.\ (\ref{cond-s1}). We need this estimate from elementary calculus: \begin{equation} \label{ds-40} \lim_{n \to +\infty} \, \frac{n} {2\pi^2} \left( \left( \cos \frac{2\pi} {n} \right)^{-n} \!\!\! - 1 \right) = 1. \end{equation} So the r.h.s.\ of (\ref{ds-30}) decreases like $1/ \bar{N}$, as $\bar{N} \to \infty$. This ensures that, given any sufficiently small $h_o$, there exists an $\bar{N} = \bar{N}(h_o)$ such that the corresponding $h = h(h_o)$, as in (\ref{ds-30}), verifies $h_o \le h \le 2 h_o$. Since $h_o = h_o(k_f)$, we rename these two values, respectively, $\bar{N}(k_f)$ and $h(k_f)$ (abbreviated in $\bar{N}$ and $h$ when there is no risk of confusion). Clearly, as $k_f \to 0$, \begin{eqnarray} \label{ds-50} && h(k_f) \sim h_o \sim k_f ; \\ \label{ds-52} && \bar{N}(k_f) \sim h^{-1} \sim k_f^{-1} . \end{eqnarray} Together with $r_0$, $h_{m^R}^R$ (equivalently $h_{m^L}^L$) and $\bar{\gamma}$ (equivalently $\bar{N}$), the fourth and last parameter that completely determines the double spiral is $M$, which is defined as the number of complete rounds the spiral makes. (Once $M$ is determined, the total number of trapezoids in the right and left spirals is given by \begin{equation} \label{ds-60} N^\epsilon = m^\epsilon + M \bar{N}, \end{equation} for $\epsilon = R$ and $\epsilon = L$, respectively.) Choosing \begin{equation} \label{ds-70} M = M(k_f) := \left[ \frac{l_o} {2\pi r_0} \right] + 1 = \left[ \frac1 {2\pi r_0 k_f} \right] + 1 \end{equation} (where $[ \,\cdot\, ]$ is the integer part of a positive number) ensures that the first inequality of (\ref{cond-s2}) is verified, since $\sum_i l_i^\epsilon > M 2\pi r_0 > l_0$. Also, for $\epsilon \in \{ R,L \}$, \begin{equation} \label{ds-80} h_{N^\epsilon} = h_{m^\epsilon}^\epsilon (\cos \bar{\gamma})^{-M \bar{N}} \sim h_{m^\epsilon}^\epsilon \sim h_o, \end{equation} as $k_f \to 0$, because of (\ref{ds-40}) and the fact that $M \sim k_f^{-1}$ (whence $M \bar{N} \sim \bar{N}^2$). The above verifies (\ref{cond-s1}). As for the second inequality of (\ref{cond-s2}), we know that the trapezoids $T_i^\epsilon$, for $i \ge m^\epsilon$, are similar. Therefore, in the limit $k_f \to 0$, we obtain \begin{eqnarray} \sum_{i=1}^{N^\epsilon} l_i^\epsilon &\sim& \sum_{i=m^\epsilon}^{N^\epsilon} l_i^\epsilon \: = \ l_{m^\epsilon}^\epsilon \sum_{j=0}^{M \bar{N} - 1} (\cos \bar{\gamma})^{-j} \nonumber \\ &\sim& \tan \bar{\gamma} \, \frac{ (\cos \bar{\gamma})^{-M \bar{N}} -1 } { (\cos \bar{\gamma})^{-1} -1 } \nonumber \\ &\sim& \bar{N}^{-1} \frac1 { \bar{N}^{-2} } \sim \bar{N} \sim k_f^{-1} \\ &\sim& l_o, \nonumber \end{eqnarray} which proves (\ref{cond-s2}). In the above we have used (\ref{ds-52}) and the evident geometric equalities $l_{m^R}^R = r_0 \tan \bar{\gamma}$ and $l_{m^L}^L = ( r_0 + h_{m^R}^R ) \tan \bar{\gamma}$ (Fig.~\ref{fig-reel}). Finally, (\ref{cond-s3}) holds because, for all $i \ge m^\epsilon$, $l_i^\epsilon / h_i^\epsilon$ is constant, while $\gamma_i^\epsilon = \bar{\gamma} \to 0$, as $k_f \to 0$. The next and last result, whose proof is apparent, emphasizes the motivation behind the constructions of Section \ref{sec-conf}. \begin{proposition} The table $\ta = \ta(k_f)$ defined before is contained in a bounded region of the plane independent of $k_f$. \end{proposition} \footnotesize \begin{thebibliography}{BKM} \bibitem[BKM]{bkm} \article{C.~Boldrighini, M.~Keane and F.~Marchetti} {Billiards in polygons} {Ann. Probab. \vol{6} (1978), no.~4, 532--540} \bibitem[B1]{b1} \article{L.~A.~Bunimovich} {On billiards close to dispersing} {Math. 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{1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 346 moveto 0 0 lineto 856 0 lineto 856 346 lineto closepath clip newpath -200.1 472.4 translate 1 -1 scale % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index oldshow % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proc char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manual % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P1 exch def /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 10 setmiterlimit 0 slj 0 slc 0.06299 0.06299 sc % % Fig objects follow % % % here starts figure with depth 99 % Ellipse 7.500 slw n 13500 4500 855 720 0 360 DrawEllipse gs /PC [[1.00 1.00 1.00] [1.00 1.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 843.00 252.00] PATmp PATsp ef gr PATusp gs col6 s gr % here ends figure; % % here starts figure with depth 85 % Polyline 0 slj 0 slc 7.500 slw [15 45] 45 sd n 14260 4501 m 14881 4499 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13959 4379 m 14881 4129 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13921 4258 m 14794 3753 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13858 4139 m 14593 3406 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13759 4052 m 14298 3118 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13635 3995 m 13923 2905 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13499 3957 m 13497 2755 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13357 3957 m 13032 2757 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13221 4014 m 12570 2883 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13105 4107 m 12131 3131 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13001 4213 m 11759 3494 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 12934 4347 m 11491 3961 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 12898 4500 m 11319 4500 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 12902 4662 m 11334 5078 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 12970 4808 m 11490 5661 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13073 4925 m 11786 6214 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13197 5022 m 12242 6676 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13345 5084 m 12833 6987 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13499 5117 m 13502 7173 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13665 5117 m 14215 7173 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13827 5070 m 14916 6957 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 13981 4984 m 15570 6570 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 14101 4861 m 16113 6018 l gs col0 s gr [] 0 sd % Polyline [15 45] 45 sd n 14199 4707 m 16524 5325 l gs col0 s gr [] 0 sd % here ends figure; % % here starts figure with depth 80 % Ellipse 7.500 slw [120] 0 sd n 6234 5094 3050 2400 0 360 DrawEllipse gs col0 s gr [] 0 sd % Polyline 0 slj 0 slc [90] 0 sd n 13635 3780 m 6570 2700 l gs col0 s gr [] 0 sd % Polyline [90] 0 sd n 13860 5175 m 7200 7380 l gs col0 s gr [] 0 sd % here ends figure; % % here starts figure with depth 75 % Polyline 0 slj 0 slc 7.500 slw n 15407 4504 m 15277 4993 l 15037 5404 l 14712 5714 l 14324 5936 l 13910 6016 l 13499 6016 l gs col0 s gr % Polyline n 13497 6013 m 13127 5910 l 12789 5729 l 12523 5478 l 12343 5170 l 12257 4830 l 12257 4507 l gs col0 s gr % Polyline n 14381 4493 m 14323 4734 l 14217 4924 l 14074 5074 l 13898 5190 l 13700 5258 l 13500 5258 l gs col0 s gr % Polyline n 13497 3446 m 13763 3517 l 13994 3632 l 14189 3812 l 14316 4031 l 14381 4262 l 14381 4493 l gs col0 s gr % Polyline n 12257 4507 m 12349 4188 l 12494 3916 l 12704 3694 l 12943 3537 l 13216 3451 l 13497 3451 l gs col0 s gr % Polyline n 13501 5256 m 13313 5211 l 13139 5135 l 12989 5014 l 12877 4861 l 12803 4683 l 12803 4501 l gs col0 s gr % Polyline n 12802 4501 m 12843 4325 l 12919 4167 l 13039 4041 l 13174 3930 l 13335 3870 l 13499 3870 l gs col0 s gr % Polyline n 13499 3870 m 13660 3907 l 13807 3974 l 13915 4078 l 13990 4219 l 14031 4360 l 14031 4495 l gs col0 s gr % here ends figure; % % here starts figure with depth 70 % Polyline 0 slj 0 slc 15.000 slw n 13950 4499 m 13950 4371 l 13916 4253 l 13848 4134 l 13755 4049 l 13632 3986 l 13501 3952 l gs col0 s gr % Polyline n 13501 3952 m 13361 3952 l 13213 4011 l 13103 4100 l 12997 4206 l 12929 4342 l 12895 4490 l gs col0 s gr % Polyline n 12895 4490 m 12895 4651 l 12963 4808 l 13065 4931 l 13196 5028 l 13336 5087 l 13501 5126 l gs col0 s gr % Polyline n 13501 5126 m 13662 5126 l 13831 5071 l 13980 4982 l 14111 4854 l 14196 4706 l 14247 4500 l gs col0 s gr % Polyline n 14252 4500 m 14252 4301 l 14192 4098 l 14082 3911 l 13930 3767 l 13718 3682 l 13493 3624 l gs col0 s gr % Polyline n 13489 3624 m 13269 3640 l 13048 3716 l 12853 3856 l 12684 4034 l 12574 4254 l 12514 4492 l gs col0 s gr % Polyline n 12514 4492 m 12514 4767 l 12586 5034 l 12730 5276 l 12947 5470 l 13209 5598 l 13497 5653 l gs col0 s gr % Polyline n 13497 5653 m 13807 5648 l 14108 5564 l 14391 5390 l 14612 5153 l 14773 4865 l 14879 4500 l gs col0 s gr % Polyline n 14879 4500 m 14879 4123 l 14790 3763 l 14595 3407 l 14307 3115 l 13917 2898 l 13497 2754 l gs col0 s gr % Polyline n 13497 2754 m 13035 2754 l 12561 2886 l 12137 3136 l 11764 3496 l 11493 3962 l 11327 4496 l gs col0 s gr % Polyline n 11327 4496 m 11332 5072 l 11497 5657 l 11789 6208 l 12251 6670 l 12836 6987 l 13501 7174 l gs col0 s gr % Polyline n 13501 7174 m 14213 7174 l 14916 6958 l 15577 6573 l 16111 6022 l 16522 5323 l 16740 4496 l gs col0 s gr % Polyline n 13947 4502 m 14252 4502 l gs col0 s gr % Polyline n 14883 4502 m 16740 4494 l gs col0 s gr % here ends figure; % % here starts figure with depth 65 % Polyline 0 slj 0 slc 7.500 slw n 8775 4230 m 8775 4410 l gs col0 s gr % Polyline n 7875 4230 m 7875 4410 l gs col0 s gr % Polyline n 5670 4230 m 5670 4275 l 5670 4320 l 5670 4365 l 5670 4410 l gs col0 s gr % Polyline [90 45 15 45] 0 sd n 8910 4320 m 5535 4320 l gs col0 s gr [] 0 sd % Polyline [90 45 15 45] 0 sd n 14445 2520 m 13410 2520 l gs col0 s gr [] 0 sd % Polyline n 13500 2430 m 13500 2475 l 13500 2520 l 13500 2565 l 13500 2610 l gs col0 s gr % Polyline n 13950 2430 m 13950 2475 l 13950 2520 l 13950 2565 l 13950 2610 l gs col0 s gr % Polyline n 14265 2430 m 14265 2475 l 14265 2520 l 14265 2565 l 14265 2610 l gs col0 s gr /Times-Roman ff 285.75 scf sf 6885 4230 m gs 1 -1 sc (0) col0 sh gr /Times-Italic ff 412.75 scf sf 8190 4140 m gs 1 -1 sc (w) col0 sh gr /Times-Roman ff 285.75 scf sf 8460 4230 m gs 1 -1 sc (0) col0 sh gr /Times-Italic ff 412.75 scf sf 13545 2250 m gs 1 -1 sc (r) col0 sh gr /Times-Italic ff 412.75 scf sf 13950 2250 m gs 1 -1 sc (w) col0 sh gr /Times-Roman ff 285.75 scf sf 13680 2385 m gs 1 -1 sc (0) col0 sh gr /Times-Roman ff 285.75 scf sf 14175 2385 m gs 1 -1 sc (0) col0 sh gr /Times-Italic ff 412.75 scf sf 6705 4140 m gs 1 -1 sc (r) col0 sh gr /Times-Italic ff 412.75 scf sf 5535 4905 m gs 1 -1 sc (A) col0 sh gr /Times-Italic ff 412.75 scf sf 13365 4905 m gs 1 -1 sc (A) col0 sh gr % here ends figure; % % here starts figure with depth 60 % Polyline 0 slj 0 slc 15.000 slw n 4753 5535 m 4500 5535 l 4275 5715 l 4275 5940 l 4500 6165 l 6570 6165 l gs col0 s gr % Polyline n 4749 5400 m 4410 5400 l 4005 5715 l 4005 6165 l 4500 6660 l gs col0 s gr % Polyline n 4500 6660 m 6660 6660 l 8100 6210 l 8775 4950 l 8775 4500 l 8100 4500 l gs col0 s gr % here ends figure; % % here starts figure with depth 50 % Arc 15.000 slw 0 slc n 5647.0 2602.5 3067.5 107.0615 72.9385 arcn gs col0 s gr % Arc n 5647.0 667.5 3067.5 107.0615 72.9385 arcn gs col0 s gr % Arc n 9479.5 4500.0 3067.5 -162.9385 162.9385 arcn gs col0 s gr % Arc n 1814.5 4500.0 3067.5 17.0615 -17.0615 arcn gs col0 s gr % Ellipse 30.000 slw n 5670 4500 17 17 0 360 DrawEllipse gs col0 s gr % Ellipse n 13500 4500 17 17 0 360 DrawEllipse gs col0 s gr % Polyline 0 slj 15.000 slw n 6547 5535 m 7447 5535 l 8122 4950 l 8122 4500 l gs col0 s gr % Polyline n 6570 6165 m 7627 5850 l 8122 4950 l gs col0 s gr % Polyline n 6547 5400 m 7357 5400 l 7897 4950 l 7897 4500 l 8122 4500 l gs col0 s gr % here ends figure; $F2psEnd rs end showpage %%Trailer %EOF ---------------0712211540525 Content-Type: application/postscript; name="fig-t1.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fig-t1.eps" %!PS-Adobe-2.0 EPSF-2.0 %%Title: fig-t1.fig %%Creator: fig2dev Version 3.2 Patchlevel 5-alpha5 %%CreationDate: Tue Oct 16 18:06:58 2007 %%For: lenci@nap (Marco Lenci,,,) %%BoundingBox: 0 0 612 208 %Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 208 moveto 0 0 lineto 612 0 lineto 612 208 lineto closepath clip newpath -69.9 389.3 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 10 setmiterlimit 0 slj 0 slc 0.06299 0.06299 sc % % Fig objects follow % % % here starts figure with depth 70 /Times-Italic ff 349.25 scf sf 3555 6075 m gs 1 -1 sc (l) col0 sh gr /Symbol ff 381.00 scf sf 6210 6075 m gs 1 -1 sc (G) col0 sh gr 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