\documentstyle[12pt]{article}
\textwidth6.25in
\textheight8.5in
\oddsidemargin.25in
\topmargin0in
%\renewcommand{\baselinestretch}{1.7}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\Pe{P_{\epsilon}}
\def\bib{\bibitem}
\def\S{{\bf S}}
\def\dS{S_{\delta}}
\def\qed{\vrule height 5pt width 5 pt depth 0pt}
\def\IP{\hbox{\rm I\kern -1.6pt{\rm P}}}
\def\IC{{\hbox{\rm C\kern-.58em{\raise.53ex\hbox{$\scriptscriptstyle|$}}
\kern-.55em{\raise.53ex\hbox{$\scriptscriptstyle|$}} }}}
\def\IN{\hbox{I\kern-.2em\hbox{N}}}
\def\IR{\hbox{\rm I\kern-.2em\hbox{\rm R}}}
\def\ZZ{\hbox{{\rm Z}\kern-.3em{\rm Z}}}
\def\IT{\hbox{\rm T\kern-.38em{\raise.415ex\hbox{$\scriptstyle|$}} }}
\newtheorem{theorem}{Theorem}[section]
%\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{sublemma}[theorem]{Sublemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\begin{document}
\title{Ergodicity of Billiards in Polygons with Pockets}
\author{N. Chernov and S. Troubetzkoy
\\ Department of Mathematics\\
University of Alabama at Birmingham\\
Birmingham, AL 35294, USA\\
E-mail: chernov@math.uab.edu, troubetz@math.uab.edu
}
%\email{chernov@math.uab.edu}
%\email{troubetz@math.uab.edu}
\date{\today}
\maketitle
\begin{abstract}
The billiard in a polygon is not always ergodic and never K-mixing or
Bernoulli. Here we consider billiard
tables by attaching disks to each vertex of
an arbitrary simply connected, convex
polygon. We show that the billiard on such a table
is ergodic, K-mixing and Bernoulli.
\end{abstract}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\section{Introduction}
\label{secI}
\setcounter{equation}{0}
Consider the billiard problem in a polygon.
Let $P$ be a polygon in which a particle moves freely
and bounces elastically off the boundary $\partial P$.
Assuming the
speed of the particle be unit, the phase space will
be $TP=P\times S^1$. The flow $\phi_t:\, TP\to TP$
is called the billiard flow. It preserves the Liouville
measure $d\mu=dq\times dv$, where $dq$ and $dv$ are
uniform measures on $P$ and $S^1$, respectively.
All the Lyapunov exponents of the billiard flow in any
polygon are zero, its topological entropy \cite{Ka87} and
Kolmogorov-Sinai entropy \cite{BKM,Sin} are zero as well.
The ergodic properties of the billiard flow depend on
the shape of the polygon $P$. On the one hand, billiards in
the so called rational polygons, where each angle is
a rational multiple of $\pi$, are never ergodic,
their phase space $TP$ foliates by compact invariant
surfaces \cite{ZK}. On the other hand, there is a
`topologically large', dense $G_{\delta}$, subset in
the space of all polygons consisting of those where the
billiard flow is ergodic \cite{KMS}. There are no known techniques
to determine whether the billiard in a given polygon is
ergodic, however. First explicit examples of polygons
with ergodic billiard flows were found very recently
\cite{V}. It is widely believed that billiards in polygons are
never strongly mixing, but they may be weakly mixing
\cite{Ka80,GK}. It is known that they cannot be
K-mixing or Bernoulli.
In order to ensure hyperbolicity (nonzero Lyapunov exponents)
and better ergodic and mixing properties, one has to perturb
the polygonal shape of the table by putting in bumps or pockets.
Here we study one class of such perturbations.
Let $P$ be a convex simply connected polygon. Assume that at every
vertex of $P$ a small pocket is attached to the table. The pockets
are bounded by circular arcs that terminate
on the sides adjacent to the vertex, see Fig.~1.
We call the new billiard table (the union of $P$ and the pockets)
by $P_{\varepsilon}$, thinking of $\varepsilon$ as the radius of the
pockets, even though the pockets do not have to be of the same radius.
We still denote by $\phi_t:\, TP_{\varepsilon}\to TP_{\varepsilon}$
the billiard flow.
Let $Q=\partial P_{\varepsilon}$ and $TQ=\{x=(q,v)\in TP_{\varepsilon}:
q\in Q\ {\rm and}\ v\ {\rm points}\ {\rm inside}\ P_{\varepsilon}\}$.
The flow $\phi_t$ induces the first return map $f:\, TQ\to TQ$
that is called the billiard ball map. It preserves a smooth
measure, $m$, on $TQ$. The ergodicity of the flow $(TP_{\varepsilon},
\phi_t,\mu)$ is equivalent to that of the map $(TQ,f,m)$.
The main result of the paper is the following.
\begin{theorem}
Both the flow $(TP_{\varepsilon},\phi_t,\mu)$ and the map
$(TQ,f,m)$ are hyperbolic and ergodic.
\label{tmmain}
\end{theorem}
The following is then standard \cite{Bu74,Bu79,CH,OW}:
\begin{corollary}
Both the flow $(TP_{\varepsilon},\phi_t,\mu)$ and the map
$(TQ,f,m)$ are K-mixing and Bernoulli.
\end{corollary}
{\em Remark}. We consider circular pockets because this model
is the most pictorial. Our results remain valid for small convex
pockets of more general shape described in \cite{W86,M88,D,Bu91},
as well as concave bumps, see Fig.~1. It is important that
pockets and/or bumps are attached to {\em every} vertex of the
polygon $P$. \medskip
We now describe the main difficulty in the proof of Theorem~\ref{tmmain}.
Let $N\subset TQ$ be the set of points whose
trajectories $\{\phi_tx:\, -\infty 0$ such that the return time sequence $m_i$ defined
by $f^{m_i} x \in U$ satisfies $m_{i+1} - m_i < C.$ Fix a $\delta > 0$
and consider the maximal
width strip $S$ containing $x$ together with its $\delta$-neighborhood
$\dS$, see Fig.~2. By maximality, the trajectory of $S$'s boundary points
come
arbitrarily close to some vertices of the polygon, thus some vertices fall
into $\dS.$ Since $x$ is uniformly recurrent, the left most and right
most boundary points of $S$ are also uniformly recurrent, thus vertices
fall with uniformly bounded gaps into each
of the two components of $\dS - S.$ By going to a subsequence we can assume
that $x_i \rightarrow x.$ As $i \rightarrow \infty$ we consider the
intersection of the strips $S_i$ with $\dS -S.$ Because the gaps between
the vertices that fall into $\dS - S$ are uniformly bounded, a vertex will
eventually appear in the interior of the strip $S_i$, see Fig.~3. This is a
contradiction.$\Box$
\section{Hyperbolicity}
\label{secH}
\setcounter{equation}{0}
Billiard tables whose boundary consists of straight
segments and convex circular arcs were introduced
by Bunimovich \cite{Bu74,Bu79}. He discovered the
defocusing mechanism, see below, and studied the
hyperbolic and ergodic properties of such billiards.
His results have been extended to wide classes
of billiards with other convex (focusing) components of
the boundary \cite{W86,M88,D,Bu91}.
We only recall here necessary definitions and properties. \medskip
{\bf Definition}. Let $B\subset \IR^2$ be a connected billiard table,
not a perfect disk, and the boundary $\partial B$ consist of a finite
number of straight segments and convex circular arcs, the latter
denoted by $\Gamma_1,\ldots,\Gamma_r$. Every $\Gamma_i$
is an arc of a circle, $C_i$, that bounds a disk, $D_i$.
Assume that $D_i\subset B$ for all $1\leq i\leq r$.
Such tables are called {\em Bunimovich-type billiards
with pockets}.\medskip
Let $F=\partial B$ and $f:TF\to TF$ be the billiard ball map
in $B$, see Introduction. The map $f$ is piecewise $C^{\infty}$.
Denote by $S_-$ the singularity set for $f$, it consists of points
mapped into the corners of the billiard table $B$ (their further
iterations are not defined). Let $S_+$ be the singularity set
for $f^{-1}$. For $n\geq 1$ denote by $S_{+,n}=S_+\cup f(S_+)\cup
\cdots\cup f^{n-1}(S_+)$ the singularity set for $f^{-n}$, and
$S_{+\infty}=\cup S_{+,n}$. Likewise, put
$S_{-,n}=S_-\cup f^{-n}(S_-)\cup\cdots
\cup f^{-n+1}(S_-)$ and $S_{-\infty}=\cup S_{-,n}$. Let
$SS=S_{+\infty}\cap S_{-\infty}$ be the set of points
whose trajectories terminate (hit corners) both in the future and the past.
It is known that $S_{+,n}$ and $S_{-,n}$ are finite unions
of smooth curves \cite{Bu74,Bu79,BSC90,BSC91}.
The main defocusing property of billiards with pockets is the following.
Let $q_0\in \partial B$ and let $v_0$ be a unit inward velocity vector
attached to $q_0$. Let $\Sigma_0$ be an infinitesimal
bundle of rays leaving $\partial B$ in the vicinity of $q_0$,
containing $v_0$ on one of the rays and going into $B$. Let $\gamma$
be the orthogonal cross section of the bundle $\Sigma_0$ passing
through $q_0$, see Fig.~4, and $\chi_0$ be
the signed curvature of $\gamma$ at the point $q_0$.
The sign of $\chi_0$ is set to be positive if the bundle $\Sigma_0$
is diverging and negative if $\Sigma_0$ is converging (focusing),
as in Fig.~4.
At the time the bundle $\Sigma_0$ reaches $\partial B$ again it reflects
in $\partial B$ and a new bundle of rays, $\Sigma_1$, goes back into $B$.
Let $\tau_0$ be the travel time,
$q_1=q_0+\tau_0 v_0$ the point of reflection and $v_1$ the
reflected velocity vector at $q_1$. The new bundle $\Sigma_1$
has a certain curvature at $q_1$, call it $\chi_1$. It is an easy
consequence of the mirror equation \cite{W86} that
\be
\chi_1=-\frac{2\kappa_1}{\cos\varphi_1}+
\frac{1}{\tau_0+\frac{\textstyle 1}{\textstyle \chi_0}}
\label{mirror}
\ee
where $\varphi_1$ is the angle between the vector $v_1$
and the inward normal vector to $\partial B$ at $q_1$, and
$\kappa_1\geq 0$ is the curvature of $\partial B$ at the point $q_1$.
The bundle $\Sigma_0$ is said to be {\em unstable} (at $q_0$) if either\\
(i) the point $q_0$ lies on a straight segment in $\partial B$
and $\chi_0\geq 0$, or \\
(ii) the point $q_0$ lies on a circular arc $\Gamma_i$ of radius
$R_i$, and $\chi_0\leq -(R_i\cos\varphi_0)^{-1}$, where $\varphi_0$
is the angle between $v_0$ and the inward normal vector to $\Gamma_i$
at $q_0$.
\begin{theorem}[\cite{Bu74,Bu79}]
If $\Sigma_0$ is unstable, then so is $\Sigma_1$.
\end{theorem}
{\em Proof}: it is a direct calculation based on (\ref{mirror}). \medskip
In the language of the theory of dynamical systems \cite{W85},
unstable bundles specify an invariant family of unstable
cones, $C^u_x$, $x\in TF$, for the billiard ball map $f:TF\to TF$.
In the important case (ii) above, the unstable bundle $\Sigma_0$
focuses before it reaches the midpoint between the collisions.
After that it defocuses and becomes divergent.
When it hits $\partial B$ again, at $q_1$, it already gets {\em wider}
than it was near the point $q_0$. Obviously, in the case (i),
$\Sigma_1$ is also wider than $\Sigma_0$. The expansion of the
bundle between the collisions (with respect to the width measured
in the direction perpendicular to the rays) is the main property
of unstable bundles. The factor of expansion is
$L=1+\tau_0\chi_0$ in the case (i) and $L=-1-\tau_0\chi_0$
in the case (ii), in both cases $L\geq 1$.
The width of unstable bundles specifies a metric, $\rho$,
in the unstable cones. It does not correspond to any metric on
$TF$, so we will call $\rho$ {\em a pseudometric}. Note that
it is monotone under the action of $f$, i.e. $Df$ expands
every unstable vector.
The unstable subspace $E^u_x$ for every $x\in TF$ is defined,
as usual, by $E^u_x=\cap_{n\geq 0} Df^nC^u_{f^{-n}x}$. This
subspace corresponds to the unstable bundle with the curvature
\be
\chi_0^u=\frac{1}{\tau_{-1}+\frac{\textstyle 1}
{\textstyle \frac{\textstyle{2\kappa}_0}{\textstyle{\cos\phi}_0}+
{\textstyle\frac{\textstyle 1}{{\textstyle \tau}_{-2}+
\frac{\textstyle 1}{\textstyle \frac{\textstyle{2\kappa}_{-1}}
{\textstyle{\cos\phi}_{-1}}+{\textstyle\frac{\textstyle 1}
{{\textstyle\tau}_{-3}+\cdots}}}}}}}
\label{cf}
\ee
Here the quantities $\tau_{-n}$, $\kappa_{-n}$, and $\phi_{-n}$
correspond to the point $x_{-n}=f^{-n}x$, $n\geq 1$.
This continuous fraction converges whenever $\sum_{n\geq 1}
\tau_{-n}=\infty$, i.e. whenever the past semitrajectory of
the point $x$ is defined, i.e. for all $x\notin S_{+\infty}$.
Hence, $E^u_x$ exists for all $x\in TF\setminus S_{+\infty}$.
It also depends continuously on $x$.
Denote by $L_x^u=|1+\tau_0\chi_0^u|$ be the factor of expansion of
the unstable subspace $E^u_x$ under $Df$. For $n\geq 1$,
denote by $L_x^u(n)=L^u_xL^u_{fx}\cdots L^u_{f^{n-1}x}$
the factor of expansion of $E^u_x$ under $Df^n$.
The factor $L^u_x(n)$ is bounded
away from unity only when the trajectory leaves an arc,
$\Gamma_i$, at time 0 and lands on another arc $\Gamma_j$,
$j\neq i$, at time $n$ (possibly, with some reflections
at straight sides in between). Every time this
happens we say that the trajectory experiences an essential
transition. During long series of consecutive reflections
at straight segments or at the same arc $\Gamma_i$ (i.e., without
essential transitions), the expansion of unstable vectors is weak,
so that the expansion factor $L^u_x(n)$ over the entire series
of $n$ reflections with no essential transitions grows at most
linearly in $n$, no matter how large $n$ is, cf. \cite{BSC90,BSC91}.
By reversing the time, one can similarly define stable
bundles of rays, stable cones $C^s_x$ with a pseudometric $\rho$,
stable subspaces $E^s_x$, and the expansion factors $L^s_x(n)\geq 1$
of $E^s_x$ under $Df^{-n}$, $n\geq 1$, for all $x\in TF
\setminus S_{-\infty}$. Stable and unstable cones $C^u_x$ and
$C^s_x$ never overlap but may have common boundaries.
\medskip
{\bf Definition}. A point $x\in TF$ is said to be sufficient
if there exists $A>1$ and two integers $nA$
and $L^s_{f^my}(m-n)>A$ for all $y\in V$. \medskip
{\bf Definition}. A point $x\in TF$ is said to be u-essential
if for any $A>1$ there is an $n\geq 1$, such that
$f^nx$ is defined, and a neighborhood $V$ of
the point $x$ such that $L^u_{y}(n)>A$
for all $y\in V$. Similarly, s-essential points
are defined (by replacing $L^u_y(n)$ with $L^s_y(n)$
and $f^nx$ with $f^{-n}x$). \medskip
The following immediately follows from the previous observations.
\begin{proposition}
A point $x$ is sufficient if its trajectory (whenever defined)
experiences at least one essential transition. A point $x$ is
u-essential (or s-essential) if its future (resp. past)
semitrajectory is entirely defined and
experiences an infinite number of essential transitions.
Furthermore, if the subspaces $E^u_x$ and $E^s_x$ are characteristic
subspaces with a positive and, respectively, negative
Lyapunov exponent, then essential transitions in the
entire trajectory of $x$ occur with a positive frequency.
\end{proposition}
For the class of billiard tables $P_{\varepsilon}$ we can
completely characterize the sets of points $x\in TQ$
that fail to be sufficient or essential. The future semitrajectory
$\{f^nx:\, n\geq 0\}$ of a point $x\notin S_{-\infty}$
experiences at least one essential
transition unless (i) $x\in N$, or (ii) the trajectory of $x$ is
periodic with all its reflection points lying on one arc,
$\Gamma_i$. Denote by $G\subset TQ$ the set of points
of type (ii). Obviously, it consists of a finite number of
disjoint segments in $TQ$ such that the angle
of reflection is constant on every of those segments.
Put $NG=N\cup G$. We then obtain the following.
\begin{proposition}
Every point $x\in TQ\setminus(SS\cup NG)$ is
sufficient. Every point $x\in TQ\setminus
(S_{-\infty}\cup NG)$ is u-essential.
Every point $x\in TQ\setminus
(S_{+\infty}\cup NG)$ is s-essential.
\end{proposition}
The last known fact we need is this \cite{BSC90}:
the tangent line to any smooth singularity curve
in $S_{+,n}$ lies strictly inside an unstable cone, and
the tangent line to any curve in $S_{-,n}$ lies
strictly inside a stable cone.
\section{Ergodicity}
\label{secE}
\setcounter{equation}{0}
\begin{theorem}
Let $x\in TQ\setminus (SS\cup NG)$.
Then there is a neighborhood $U(x)\subset TQ$ that
belongs (mod 0) in one ergodic component of $f$.
\label{tmloc}
\end{theorem}
{\em Proof}. This theorem is a version of the local ergodic theorem
(or `fundamental theorem') in the theory of hyperbolic
billiards. It was first developed in ref. \cite{SC87}
for gases of hard balls, then generalized in ref.
\cite{KSS90} to semi-dispersing billiards (in any
dimension) and in ref. \cite{LW} to Hamiltonian systems
with invariant cone families under certain conditions.
The most general and convenient for our purposes version
of that theorem was proved in ref. \cite{Ch91}. It
requires the verification of the following five
properties:\medskip
{\bf Property 1} (double singularities). For any $n\geq 1$
the set $S_{+,n}\cap S_{-,n}$ consists of a finite number
of isolated points.
{\bf Property 2} (thickness of neighborhoods of singularities).
For any $\delta>0$ let $U_{\delta}(S_+\cup S_-)$ be the
$\delta$-neighborhood\footnote{This must be measured
in a {\em monotone} (pseudo)metric, in which the expansion of
unstable vectors and the contraction of stable vectors
is monotone. Our pseudometric $\rho$ is exactly such.} of the set
$S_+\cup S_-$. Then $m(U_{\delta}(S_+\cup S_-))\leq{\rm const}\cdot\delta$.
{\bf Property 3} (continuity). The families of stable and unstable
subspaces $E^s_x$ and $E^u_x$ are continuous on their domains.
Furthermore,
the limit spaces $\lim_{y\to x}E^u_y$ and $\lim_{y\to x}
E^s_y$ are always transversal at every sufficient point $x$,
even if $E^u_x$ or $E^s_x$ does not exist.
{\bf Property 4} (``ansatz''). Almost every point of $S_+$ (with
respect to the Lebesgue length on it) is u-essential,
and almost every point of $S_-$ is s-essential.
{\bf Property 5} (transversality). At almost every point
$x\in S_+$ the subspace $E^s_x$ is defined and transversal to
$S_+$, and at almost every point $x\in S_-$ the
subspace $E^u_x$ is defined and transversal to $S_-$. \medskip
The property 1 follows from the last remark in the previous section.
The property 2 is based on certain direct but rather delicate
calculations, which are described in detail in Refs. \cite{BSC90,BSC91}.
The property 3 follows from the last remark in the previous
section and the fact that for any sufficient point $x\in TF$
at least one of the spaces $E^u_x$, $E^s_x$ lies strictly inside
the corresponding cone.
Next, observe that the sets $S_+\cap S_{-\infty}$ and $S_-\cap S_{+\infty}$
are countable, and $(S_+\cup S_-)\cap NG=\emptyset$. So, all the points
$x\in S_+\setminus S_{-\infty}$ are u-essential, and all the
points $x\in S_-\setminus S_{+\infty}$ are s-essential.
This proves 4 and 5.
Now the theorem proved in ref. \cite{Ch91} ensures
that every sufficient point, i.e. every point
$x\in TQ\setminus (SS\cup NG)$, has a neighborhood that
belongs (mod 0) in one ergodic component. $\Box$.\medskip
We now prove our main theorem~\ref{tmmain}. The set $SS$
is countable. The set $NG$ consists of a finite number
of disjoint parallel segments in $TQ$.
Therefore, the set $TQ\setminus (SS\cup NG)$
of points satisfying the assumptions of
Theorem~\ref{tmloc} is an
{\em arcwise connected} set of full measure. This proves
Theorem~\ref{tmmain}. \medskip
{\em Remark:} If we use Corollary \ref{c1} instead of
Theorem~\ref{t2} then we can prove a slightly weaker proposition which
is still enough to conclude the ergodicity of $f$, namely
that the set $TQ \setminus (SS\cup NG)$
has an arcwise connected subset of full measure.
\begin{thebibliography}{99}
\bibitem{BKM} C.~Boldrighini, M.~Keane, and F.~Marchetti {\it Billiards in
polygons} Ann. Prob. {\bf 6} (1978) 532--540.
\bibitem{Bu74} L.A. Bunimovich, {\em On billiards close to
dispersing}, Math. USSR Sbornik {\bf 23} (1974),
45--67.
\bibitem{Bu79} L.A. Bunimovich, {\em On the ergodic properties
of nowhere dispersing billiards}, Comm. Math. Phys.
{\bf 65} (1979), 295--312.
\bibitem{Bu91} L.A. Bunimovich, {\em On absolutely focusing
mirrors}, Lect. Notes Math., {\bf 1514}, Springer, New York, 1990.
\bibitem{BSC90}L.A. Bunimovich, Ya.G. Sinai and N.I. Chernov,
{\em Markov partitions for two-dimensional hyperbolic billiards},
Russ. Math. Surv. {\bf 45} (1991), 97--134.
\bibitem{BSC91}L.A. Bunimovich, Ya.G. Sinai and N.I. Chernov,
{\em Statistical properties of two-dimensional hyperbolic billiards},
Russ. Math. Surv. {\bf 46} (1991), 47--106.
\bibitem{Ch91}N.I. Chernov,
{\em On local ergodicity in hyperbolic systems with singularities},
Functs. Anal. Applic. {\bf 27} (1993), 51--54.
\bibitem{CH} N.I. Chernov, C. Haskell, {\em Nonuniformly
hyperbolic K-systems are Bernoulli},
Ergodic Theory and Dynamical Systems {\bf 16}
(1996), 19--44.
\bib{DGT} C.~Delman, G.~Galpein, and S.~Troubetzkoy, {\it Billiards in
a polygon with pockets}, manuscript.
\bibitem{D} V. Donnay, {\em Using integrability to produce
chaos: billiards with positive entropy}, Comm. Math. Phys.
{\bf 141} (1991), 225--257.
\bib{F} H.~Furstenberg, {\it Recurrence in ergodic theory and combinatorial
number theory,} Princeton NJ, Princeton Univ.~Press 1981.
\bib{GKT} G.~Galperin, T.~Kr\"uger, and S.~Troubetzkoy,
{\it Local instability of orbits in polygonal and polyhedral billiards,}
Comm.~Math.~Phys. {\bf 169} (1995) 463--473.
\bib{Gu1} E.~Gutkin, {\it Billiard in polygons,} Physica D
{\bf 19} (1986) 311-333.
\bib{Gu2} E.~Gutkin, {\it Billiard in polygons: survey of
recent results,} J.~Stat.~Phys., {\bf 83} (1996) 7--26.
\bib{Gu3} E.~Gutkin, {\it oral communication}.
\bibitem{GK} E. Gutkin and A. Katok, {\it Weakly mixing billiards}
in {\it Holomorphic dynamics,} LNM {\bf 1345}, Springer-Verlag, Berlin,
(1989) 163--176.
\bibitem{Ka80} A. Katok, {\it Interval exchange transformations and some
special flows are not mixing}, Isr.~J.~Math.~{\bf 35} (1980) 301--310.
\bibitem{Ka87} A. Katok, {\em The growth rate for the number of
singular and periodic orbits for a polygonal billiard},
Comm. Math. Phys. {\bf 111} (1987), 151--160.
\bibitem{KMS} S. Kerckhoff, H. Masur, and J. Smillie,
{\em Ergodicity of billiard flows and quadratic differentials},
Annals of Math. {\bf 124} (1986), 293--311.
\bibitem{KSS90}A. Kr\'amli, N. Sim\'anyi, and D. Sz\'asz,
{\em A ``transversal'' fundamental theorem for semi-dispersing
billiards}, Comm. Math. Phys. {\bf 129} (1990), 535--560.
\bibitem{LW} C. Liverani, M. Wojtkowski, {\em Ergodicity
in Hamiltonian systems}, Dynamics Reported {\bf 4} (1995),
130--202.
\bibitem{M88} R. Markarian, {\em Billiards with Pesin region
of measure one}, Comm. Math. Phys. {\bf 118} (1988),
87--97.
\bibitem{OW} D. Ornstein and B. Weiss, {\em On the Bernoulli
nature of systems with some hyperbolic structure},
manuscript, 1995.
\bibitem{Sin} Ya.G~Sinai, {\em Introduction to ergodic theory}, Princeton
University Press, Princeton (1977).
\bibitem{SC87}Ya.G. Sinai and N.I. Chernov, {\em Ergodic properties of some
systems of 2-dimensional discs and 3-dimensional spheres},
Russ. Math. Surv. {\bf 42} (1987), 181--207.
\bib{T} S.~Tabachnikov, {\it Billiards,} ``Panoramas et
Syntheses'', Soc.~Math.~France (1995).
\bibitem{V} Ya.B. Vorobets,
{\em Ergodicity of billiards in polygons: explicit examples},
Uspekhi Mat. Nauk {\bf 51} (1996), 151--152.
\bibitem{W85} M.P. Wojtkowski, {\em Invariant families of cones
and Lyapunov exponents}, Ergod. Th. Dynam. Sys. {\bf 5} (1985),
145--161.
\bibitem{W86} M.P. Wojtkowski, {\em Principles for the design of
billiards with nonvanishing Lyapunov exponents}, Commun.
Math. Phys. {\bf 105} (1986), 391--414.
\bibitem{ZK} A. Zemlyakov and A. Katok, {\em Topological transitivity
of billiards in polygons}, Math. Notes {\bf 18} (1976), 760--764.
\end{thebibliography}
\newpage
\centerline{Figure captions}\bigskip
1. A triangle with two pockets and one `bump'. \\
2. The strip $S$ together with it's $\delta$-neighborhood $S_{\delta}$. \\
3. The intersection of $S_i$ with $S_{\delta}\setminus S$ is shaded. \\
4. An unstable focusing bundle $\Sigma_0$ gets wider
at the next reflection near the point $q_1$.
\end{document}
\end