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\title{Topological entropy of semi-dispersing billiards.}
% Information for first author
\author{ D. Burago}
% Address of record for the research reported here
\address{ Department of Mathematics,
The Pennsylvania State University, University Park, PA 16802 }
% Current address
%\curraddr{Department of Mathematics and Statistics,
%Case Western Reserve University, Cleveland, Ohio 43403}
\email{burago@math.psu.edu}
% \thanks will become a 1st page footnote.
\thanks{The first author is partially supported by the NSF Grant DMS
???. }
% Information for second author
\author{S. Ferleger}
\email{ferleger@math.psu.edu}
% Information for third author
\author{A. Kononenko}
\address{ Department of Mathematics,
University of Pennsylvania, Philadelphia, PA 19104-6395 }
\email{alexko@math.upenn.edu}
\begin{document}
\newfont{\Blackbb}{msbm10 scaled \magstep 0}
\newfont{\frakfurt}{eufm10 scaled \magstep 0}
\begin{abstract}
In this paper we continue to explore the applications of the connections
between singular
Riemannian geometry and billiard systems that were first
used in \cite{burago-estimate} to prove local estimates on the number of
collisions in non-degenerate semi-dispersing billiards.
In this paper we show that the topological entropy of a compact
non-degenerate semi-dispersing
billiard on any manifold of non-positive sectional curvature is
finite. Also, we prove exponential estimates on the number of periodic
points (for the first return map to the boundary) and the number of
periodic trajectories (for the billiard flow). In
Section~\ref{sec-lorentz}
we prove some estimates for the topological entropy of Lorentz gas.
\end{abstract}
\maketitle
\section{Summary of Results.}
\label{sec-summary}
The results of this paper rely on the connection between the
singular Riemannian geometry and semi-dispersing
billiard systems. Namely,
for every billiard trajectory one can construct a singular Riemannian
space such that the trajectory corresponds to a geodesic in this space.
Moreover, the Alexsandrov
curvature of this space is not bigger then the curvature of the
original billiard manifold. Thus, if we start with a billiard on a
manifold
of non-positive curvature, then the corresponding space also has
non-positive
curvature. The proof of the finiteness of entropy for
semi-dispersing
billiards is based on a singular analog of the following well known
for regular manifolds of non-positive curvature statement: if two
geodesics
in a simply connected manifold of non-positive curvature have
``close'' end
points, then they are ``close'' to each other everywhere, and not
only
on the manifold but also in its tangent bundle.
It is interesting to notice that our proof fails immediately if
there are any regions of positive curvature inside the billiard.
Moreover,
we strongly suspect that if we allow even arbitrarily small portions
of positive curvature, then it is possible to construct examples of
semi-dispersing billiards with infinite topological entropy.
Let us proceed with the rigorous formulations of our results.
Let $M$ be an arbitrary Riemannian manifold of non-positive bounded
sectional curvature without boundary. Consider a collection of $n$
geodesically convex compact subsets ({\bf walls}) $B_i \subset M,
i=1,\ldots,n,$ of $M$,
such that their boundaries are $C^1$ submanifolds of codimension one.
Let
$B=M\backslash(\bigcup_{i=1}^n{Int(B_i)}),$ where $Int(B_i)$ denotes the
interior of the set $B_i.$ The set $B \in M$ will be called a billiard
table. A
semi-dispersing billiard flow $\{ T^t \}_{t=-\infty}^{\infty} $ acts on
a
certain subset $\tilde{TB}$ of full Liouville measure of the unit
tangent bundle
to $B.$ To be more precise,
$\tilde{TB}$ consists of such points $(x,v) \in TM,$ $x \in M,$ $v \in
T_x M,$
that $x \in B,$ vector $v$ is directed ``strictly inside of $B,$'' and
the
orbit of $(x,v)$ is defined for all $t \in (-\infty, \infty)$
(see, for example, \cite{dynamicalsystems-2} for the rigorous
definitions). The projections of the orbits of that flow to $B$ are
called the
billiard trajectories and correspond to free motions of particles inside
$B.$
Namely, the particle moves inside the set $B$ with unit speed along a
geodesic
until it reaches one of the sets $B_i$ ({\bf collision}) where it
reflects
according to the law ``the angle of incidence is equal to the angle of
reflection.''
The purpose of this paper is to establish
the finiteness of topological entropy
for a large class of semi-dispersing billiards, namely for
non-degenerate semi-dispersing billiards, i.e., billiards on tables
that satisfy a certain geometric non-degeneracy
condition (see below). It was shown in \cite{burago-estimate} that this
non-degeneracy condition
implies the existence of local uniform estimates on the number of
collisions.
Estimates like that play an important role in various
questions about billiards. For example, they appear as conditions for
Sinai-Chernov's formulas for metric entropy of semi-dispersing billiards
\cite{chernov-entropy}, \cite{sinai-entropy}.
Note that
Sinai-Chernov's formulas imply the finiteness of the metric entropy
of non-degenerate semi-dispersing billiards in $\Bbb{R}^n$ or
$\Bbb{T}^n$ with respect to the Liouville
measure. However, little is known about the topological entropy of
general semi-dispersing billiards. Most of the results known to the
authors of this paper are proven only for two-dimensional
semi-dispersing billiards (the connection between the topological
entropy
and the number of periodic points \cite{chernov-topentropy}
and the results of \cite{katokstrelcyn}). The only result
about the topological entropy of billiards of arbitrary dimension,
that we are aware of, is the fact that the topological
entropy of polygonal and polyhedral billiards is zero (proved for the
two-dimensional case in \cite{katok-points}, the proof of general
case is also outlined in \cite{katok-points}, the rigorous proof
can be found in
\cite{gutkin}; see also \cite{chernov-entropy} for the similar result
about metric entropy).
In this paper we prove that the topological entropy of
compact non-degenerate semi-dispersing
billiards is finite. Moreover, our results are true not only for
billiards in $\Bbb{R}^n$ or
$\Bbb{T}^n$ but for billiards on
any manifolds of non-positive sectional curvature.
In \cite{stojanov} exponential estimates
on the number of periodic points for the first return map to the
boundary,
for
billiards in $\Bbb{R}^k,$
and the number of periodic trajectories for the flow,
for
non-degenerate billiards in $\Bbb{R}^2,$ were proven. In
Section~\ref{sec-points} we prove the analogs of those
results for billiards
on arbitrary manifolds of non-positive curvature.
In Section~\ref{sec-lorentz}
we prove some estimates for the topological entropy of Lorentz gas.
In particular we prove the existence of a limit of topological entropy
of the Lorentz gas flow when the radius of the scatterer approaches
zero.
The following non-degeneracy condition for semi-dispersing billiards was
introduced in \cite{burago-estimate}.
\begin{df}
A billiard table $B$ is {\bf non-degenerate} in a subset $U\subset M$
(with constant $C>0$),
if for any
$I \subset \{1,\ldots,n \}$ and for any $y \in (U \bigcap B )\backslash
(\bigcap_{j \in I} B_j),$
$$ \max_{k \in I} \frac{dist(y,B_k)}{dist(y, \bigcap_{j \in I} B_j)}
\geq C,$$
whenever $\bigcap_{j \in I} B_j $ is non-empty.
\end{df}
Roughly speaking, it means that if a point is $d$-close to all the walls
from
$I$ then it is $d/C$-close to their intersection.
We will say that $B$ is {\bf non-degenerate} if there exist $\delta>0$
and $C>0$
such that $B$ is non-degenerate, with constant $C,$ in any
$\delta$-ball.
The following estimate on the number of collisions in non-degenerate
semi-dispersing
billiards was proven in \cite{burago-estimate}.
\begin{prop}
\label{prop-globalestimate}
For any non-degenerate
semi-dispersing billiard there exists a constant $P$ such that, for
every $t,$
every trajectory
of the billiard flow makes no more than $P(t+1)$ collisions with the
boundary in the
time interval $[0,t].$
\end{prop}
Recall that there is a standard way to introduce a distance function in
the
tangent bundle $TM$ to $M$ (sometimes this distance function is called
Sasaki
metric). We will denote this distance function by $d_{TM}(\cdot , \cdot
).$ Now
we can use the distance $d_{TM}(\cdot , \cdot )$ to define the
topological
entropy $h_{top}(f)$ of any transformation $f$ of any subset of $TM$
(for a
rigorous definition of the topological entropy of transformations of a
non-compact space see, for example, \cite{pesinpitskel}).
\begin{df}
\label{df-entropy}
The topological entropy of the time-one map $T^1$
of the billiard flow will be called
{the topological entropy of the billiard.}
\end{df}
Notice that the straightforward definition of the topological entropy
of the
billiard as the topological entropy of the whole billiard flow is
meaningless,
because, due to the discontinuity of the flow, the
topological entropy of the whole billiard flow is always infinite.
%Another type of topological entropy that makes sense for billiards is
the %topological
%entropy of the first return map $T_1$ to the boundary $N$
%of the phase space of the billiard flow.
%To be more precise, $N$ consists of those points $X=(x,v)$ in the unit
tangent %bundle
%to $B$ that satisfy the following conditions
%\begin{enumerate}
%\item $X$ belongs to the domain of definition of the billiard flow
$T^t;$
%\item $x$ belongs to the boundary of $B$ in $M.$
%\end{enumerate}
We apply methods of singular Riemannian geometry
(see \cite{alexandrov}, \cite{reshetnyak}, \cite{gromov}) to
prove some estimates on the topological entropy of
non-degenerate semi-dispersing
billiards. In particular we will prove
\begin{thm}
\label{thm-entropy}
The topological entropy of a compact non-degenerate semi-dispersing
billiard on
any manifold of non-positive sectional curvature is finite.
%The topological entropy of the first return map to the boundary of
%a non-degenerate semi-dispersing billiard with finite horizon on any
%manifold of non-positive curvature is finite.
\end{thm}
%Note that for billiards with ``infinite horizon''
%the first-return map may have infinite
%topological entropy (for example, the ``Lorentz gas,''
%\cite{chernov-topentropy}).
Let us call a point $x \in \tilde{TB}$ $\Bbb{Z}$-{\bf regular} if $T^i
(x)$
belongs to the interior of $TB$ for all $i \in \Bbb{Z}.$ For example,
almost
all points of $\tilde{TB}$ are $\Bbb{Z}$-regular with respect to the
Liouville
measure. Clearly the restriction of the time-one map $T^1$ to the set
$TB_{\Bbb{Z}}$ of $\Bbb{Z}$-regular points in $B$ is continuous, and its
topological entropy is less or equal to the topological entropy of
$T^1$ on
$B.$ Thus, Theorem~\ref{thm-entropy} together with Pesin and Pitskel
\cite{pesinpitskel} results concerning the variational principle for the
continuous maps of non-compact spaces, yields the following \begin{cor}
\label{cor-metricentropy} Metric entropy, of a compact non-degenerate
semi-dispersing billiard on any manifold of non-positive bounded
sectional
curvature, with respect to any $T^1$-invariant probability measure $\mu$
such
that $\mu (TB_{\Bbb{Z}})=1,$ is finite. In particular, metric entropy
is
finite for any measure which is invariant with respect to the whole flow
$T^t.$
\end{cor}
\section{Outline of the proof of Theorem~\ref{thm-entropy} for simply
connected $M.$}
\label{sec-outline}
% Notice, that it will be enough to prove Theorem~\ref{thm-entropy} for
the case
% when $M$ is simply connected. Indeed, if $M$ is not simply connected,
consider
% a billiard $\overline{B}
=\overline{M}\backslash(\bigcup_{i=1}^n{Int(\overline{B_i})}), $ % on
the universal cover $\overline{M}$ of $M,$ where $\overline{B_i},$
$i=1,\ldots,n$ are % the pre-images of the sets $B_i,$ $i=1,\ldots,n$
under the covering map. Then, it is easy % to see that the topological
entropy of $B$ is no bigger than the topological entropy of % %
$\overline{B}.$ Note that (due to the fact that covering map is a local
isometry) % $\overline{B}$ is non-degenerate with the same constant as
$B.$ Thus, topological entropy of $\overline{B}$ is finite provided
topological entropy of $B$ is finite.
% Therefore, from now on we will assume that $M$ is simply connected.
In order to keep the main ideas of the proof of
Theorem~\ref{thm-entropy}
more transparent we will first prove
Theorem~\ref{thm-entropy} for simply connected $M.$ In the end of
Section~\ref{sec-general}
we will show how to adapt our
arguments to the general case. Therefore, from now till the end of
Section~\ref{sec-general}, we assume $M$ to be simply connected.
Before we begin the proof, let us introduce the following
\begin{df}
We will say that two trajectories $\Gamma_1$ and $\Gamma_2$ are of the
same {\bf combinatorial class} if they collide with the same sequence of
walls.
Additionally, if $\Gamma_1$ and $\Gamma_2$ have the same length
$l \in \Bbb{R}$ and for each $t=1,2, \ldots [l]$
they experience the same number of collisions by the time $t,$ we will
say that $\Gamma_1$ and $\Gamma_2$ are of the same {\bf strict
combinatorial class}.
\end{df}
For every piece-wise smooth curve $\gamma$ in $M$ denote by
$\dot{\gamma}(t)$
the right derivative of $\gamma$ at the point $\gamma (t).$
For every $l \in \Bbb{N}$ and $\epsilon > 0$ we will construct an
$\epsilon$-net
$A^l(\epsilon) \subset {\tilde {TB}}$ for the distance $d_l(x,y) =
\max_{0 \leq i \leq l}
d_{TM}(T^ix,T^iy)$ and estimate the number of its elements.
The construction of the $\epsilon$-net $A^l(\epsilon)$ is based on the
following
lemma which will be proven in the next section.
\begin{lmm}
\label{lmm-continuity}
For every $\epsilon > 0$ there exists $\delta > 0$ such that if
$\Gamma_1, \Gamma_2 $
\begin{enumerate} \item are of the same strict combinatorial class;
\item have equal
length
$l \in \Bbb{N};$
\item $d_M(\Gamma_1(0), \Gamma_2(0))
< \delta$ and $d_M(\Gamma_1(l), \Gamma_2(l)) < \delta,$
\end{enumerate} then $d_l(
\dot{\Gamma}_1(0), \dot{\Gamma}_2(0)) \leq \epsilon.
$\end{lmm}
Let us show how to construct $A^l(\epsilon)$ using
Lemma~\ref{lmm-continuity}.
Consider an arbitrary $\delta$-cover $\Delta$ of the billiard $B.$ Let
$C$ be an
arbitrary strict combinatorial class of trajectories. For each pair of
sets
$U,V \in
\Delta$ consider a billiard trajectory $\Gamma_{U,V}$ of class $C$ such
that
$\Gamma_{U,V}(0) \in U$ and $\Gamma_{U,V}(l) \in V$ (provided such a
trajectory
exists) and set $A_C^l(\epsilon) = \{\dot{\Gamma}_{U,V}(0)| {U,V \in
\Delta}
\}.$ One has $Card(A_C^l) \leq Card(\Delta)^2 \leq K$ where $K$ is a
positive
constant that depends only on the billiard $B$ and the number $\epsilon$
(clearly
it depends only on $B$ and $\delta,$ but $\delta$ is determined by
$\epsilon$).
Now remark that since our
billiard is non-degenerate, according to
Proposition~\ref{prop-globalestimate}
the number of collisions that may occur in time $l$ is not greater than
$P(l+1).$ Therefore, there is no more than $n^{P(l+1)+l}$ different
strict
combinatorial classes of trajectories that contain trajectories of
length $l.$
We take $A^l(\epsilon) = \bigcup A_C^l (\epsilon)$ where the union is
taken over all strict combinatorial classes $C.$
Clearly, $A^l(\epsilon)$ is an $\epsilon$-net with respect to the metric
$d_l$ on $TM,$ and $$ Card
(A^l(\epsilon)) \leq Kn^{P(l+1)+l} \leq Kn^{(P+1)(l+1)} $$ and,
therefore, $$
h_{top}(T^1, \epsilon) \leq \lim_{\epsilon \rightarrow 0}
\overline{\lim_{\l
\rightarrow \infty}} \frac{{\rm ln}(Card(A^l(\epsilon)))}{l} \leq $$ $$
\leq \lim_{\l
\rightarrow \infty} \frac{{\rm ln}(Kn^{(P+1)(l+1)})}{l} = (P+1){\rm
ln}(n). $$
Therefore, $h_{top}(T^1) \leq (P+1){\rm ln}(n).$
\section{Proof of Lemma~\ref{lmm-continuity} and the
general case of Theorem~\ref{thm-entropy}.}
\label{sec-general}
To prove Lemma~\ref{lmm-continuity} we apply the methods of
singular Riemannian geometry similarly to the way we did in
\cite{burago-estimate}.
First of all we have to recall the construction of a singular Riemannian
space corresponding to a given billiard trajectory $\Gamma$ which starts
at the point $X_0,$ ends at $X_{j+1}$ and has collision points $ X_1,
\ldots, X_j.$
We construct a singular Riemannian space $\bar{M}$ in the following
way:
take $j+1$ isometric copies $M_i,$ $i=0,\ldots,j,$ of $M$ and, for all
$i=0,\ldots,j-1,$ glue together $M_i$ and $M_{i+1}$ by the set $B_k,$
which
contains $X_{i+1}.$ Notice that by construction, for each
$i=0,\ldots,j-1$ there is a canonical isometric embedding $E_i:M
\rightarrow \bar{M},$ which is an
isometry between $M$ and $M_i$ and maps the subsets $B_k, k=1, \ldots,
n,$ in $M$ into the subsets $B_k$ in $M_i.$
The curve $G(\Gamma)=\bigcup_{i=0}^j E_i(X_{i}X_{i+1}) \in {\bar{M}}$ is
a
geodesic in $\bar{M}$ corresponding to the trajectory $\Gamma$ in $M$
and it
has the same length in $\bar{M}$ as $\Gamma$ in $M.$ (Here and in the
future, we
denote the piece of geodesic in $M$ connecting points $A$ and $B$ by
$AB.$)
Notice that if two trajectories have the same combinatorial
class $C$ than the singular Riemanian spaces corresponding to them are
naturally isometric. We will denote this space by $M_C.$
It follows immediately from the construction of $M_C,$ the fact that $M$
is
simply connected, and Reshetnyak's gluing theorem (\cite{1960b},
also see
Theorem 6.1 in \cite{reshetnyak})
that $M_C$ is a singular space of non-positive curvature.
Let $\Gamma_1, \Gamma_2 $ be as in Lemma~\ref{lmm-continuity}.
Consider the geodesics $G(\Gamma_1)(t)$ and $G(\Gamma_2)(t),$ where
$t$ is the time parameter along $G(\Gamma_1)$ and $G(\Gamma_2).$
Since, $M_C$ is a space of non-positive curvature the function
$D(t)=d_{M_C}(G(\Gamma_1)(t), G(\Gamma_2)(t))$ is convex (see
\cite{ballman},
Theorem 14).
Therefore, for any $\delta >0,$ the fact that $D(0)<\delta $ and
$D(l)<\delta $ implies that $D(t)<\delta $ for all $0 \leq t \leq
l.$
Notice that the distance between the points
$G(\Gamma_1)(t)$ and $G(\Gamma_2)(t)$ in $M_C$ is bigger or equal to
the distance
between the points $\Gamma_1(t)$ and $\Gamma_2(t)$ in $M.$
Thus, we immediately have the following
\begin{lmm}
\label{lmm-space}
For any $\delta >0,$ any semi-dispersing billiard $B,$ any real
number
$t_0,$ if two billiard trajectories $\Gamma_1(t)$ and $\Gamma_2(t)$
\begin{enumerate}
\item have the same combinatorial class;
\item have the same length $t_0;$
\item $d_M(\Gamma_1(0),\Gamma_2(0)) <\delta $ and
$d_M(\Gamma_1(t_0),\Gamma_2(t_0)) <\delta ,$
\end{enumerate} then
$d_M(\Gamma_1(t),\Gamma_2(t)) <\delta $ for all $0 \leq t \leq
t_0.$
\end{lmm}
Now, to finish the proof of Lemma~\ref{lmm-continuity} it will be
enough to
prove the following
\begin{lmm}
\label{lmm-one}
Let $\Gamma_1(t)$ and $\Gamma_2(t),$
$0 \leq t \leq 1$
be two trajectories of the same combinatorial class C. Then for
every
$\epsilon$ there exists
$\delta_C >0 $ such that if $d_M(\Gamma_1(0),\Gamma_2(0)) <\delta_C
$ and
$d_M(\Gamma_1(1),\Gamma_2(1)) <\delta_C $ then
$d_{TM}(\dot{\Gamma}_1(0), \dot{\Gamma}_2(0)) < \epsilon.$
\end{lmm}
Suppose that Lemma~\ref{lmm-one} is proven. Due to
Proposition~\ref{prop-globalestimate} there exist no more than $n^{2P}$
different combinatorial classes containing a trajectory of length one.
For a fixed $\epsilon$ let $\delta$ be equal to the minimum of all
$\delta_C$
over all possible combinatorial classes $C.$
It follows from Lemma~\ref{lmm-space} that if $\Gamma_1$ and $\Gamma_2$
are as
in Lemma~\ref{lmm-continuity} then we can apply
Lemma~\ref{lmm-one} to
each of the pairs of segments of $\Gamma_1(t),$ $i \leq t \leq i+1$
and $\Gamma_2(t),$
$i \leq t \leq i+1$ for all $i=0, \ldots, l-1.$ This immediately
implies Lemma~\ref{lmm-continuity}.
Let us prove Lemma~\ref{lmm-one}.
\begin{proof}
Let us introduce some notation. Let $x \in M,$ and let $T$ be a linear
transformation of $T_xM.$ Let $v$ be any vector, tangent to $M,$ not
necessary
at point $x.$ By $T(v)$ we will denote the vector, obtained by the
following
procedure: first we translate $v$ parallelly along a geodesic to $x$
(since
$M$ is assumed to be simply connected there is a unique geodesic joining
any two
points of $M$), and then
apply the transformation $T$ at $x.$ Notice that if $T=Id(x),$
the identity map
in $T_xM,$ then $T(v)$ is the result of the parallel translation of $v$
to $x.$
For an arbitrary trajectory $\Gamma$ of length one and the
combinatorial class $C$ we will use
expanded notation $\Gamma((t_1, \gamma_1),\ldots,(t_m, \gamma_m))$ where
each
pair $(t_k, \gamma_k), t_k \in [0,1], \gamma_k \in M, k=1,\ldots,m,$ are
the time
and the coordinate of the $k$-th collision.
Let $\gamma_0=\Gamma(0)$
and $\Gamma_0= \dot{\Gamma}(0) $ and let $\gamma_{m+1} = \Gamma(1).$
Denote, for $k=1,\ldots,m$, $\dot{\Gamma}_k = \dot{\Gamma} (t_k),$ i.e.,
the velocity vector at the time $t_k.$
Let $S(\gamma_k)$ be the reflection in $T_{\gamma_k}M$ with respect to
the
hyperplane tangent to $\partial B_{i(k)}$ at the point $\gamma_k,$ where
$\partial B_{i(k)}$ is the boundary of the wall containing $\gamma_k.$
Then the billiard motion law for the trajectory $\Gamma$ can be written
as
\begin{equation}
\label{1}
\dot{\Gamma}_k = (Id(\gamma_k)S(\gamma_{k+1})\ldots S(\gamma_{k+l}))
\dot{\Gamma}_{k+l},
\end{equation}
for any $k,l=0,\ldots m,$ such that $k+l \leq m.$
Now, denote by $\gamma$ the uniform point-wise limit of the sequence of
trajectories $\Gamma^n ((t_1^n, \gamma_1^n),\ldots,(t_m^n,
\gamma_m^n)))$ of
combinatorial class $C.$ Let $(t_k,\gamma_k) \in [0,1] \times \partial
B_{i(k)}$ be an
accumulation point of the sequence $(t_k^n,\gamma_k^n).$ Also, let
$\gamma_0=\gamma(0), t_0=0,$ $t_{m+1}=1,$ and $\gamma_{m+1} =
\gamma(1).$
(Notice that the times
$t_k$ and the points $\gamma_k$ are defined non-uniquely, except for
$k=0$ and
$k=m+1,$ i.e., there might be more than one accumulation point for the
sequences
$(t_k^n,\gamma_k^n),$ for $k=1, \ldots, m.$)
Obviously, there exists $0 \leq k \leq m$ such that the points
$\gamma_k$ and $\gamma_{k+1}$ do not coincide. Then, denote by
$\dot{\gamma}_k$ the vector, tangent at $\gamma_k$ to the geodesic
connecting $\gamma_k$ and $\gamma_{k+1}.$ Thus $\gamma_k$ is defined
for some $0 \leq k \leq m$. Let us define
$\dot{\gamma}_k$ for all the other $k.$ Namely, we put
\begin{equation}
\label{2}
\dot{\gamma}_l = Id(\gamma_l)S(\gamma_{l+1})\ldots S(\gamma_{k})
(\dot{\gamma}_k)
\end{equation}
for all $0 \leq lk.$
%Therefore, all $\dot{\gamma}_k, k=0,\ldots,m$ are defined and billiard
motion %law for $\gamma$ holds (though, it is not necessary a billiard
trajectory %anymore).
Consider the closure $\bar{C},$ in the metric of uniform
pointwise convergence: $d_{\bar{C}}(\Gamma_1,\Gamma_2) = \max_{t \in
[0,1]} d_M
(\Gamma_1 (t),\Gamma_2 (t)),$ of the set of all the trajectories of
the
combinatorial class $C$ and of length one. (Notice that,
due to Lemma~\ref{lmm-space},
$d_{\bar{C}}(\Gamma_1,\Gamma_2)=\max \{ d_M (\Gamma_1 (0),\Gamma_2 (0)),
d_M
(\Gamma_1 (1),\Gamma_2 (1)) \}.$ We claim that if $\gamma^n \in \bar{C}$
converges to $\gamma \in C,$ then $\dot{\gamma}_0^n$ converges to
$\dot{\gamma}_0.$ Fix some choice of $(t_k, \gamma_k),$ for $k=1,
\ldots ,m.$
Let $k$ be such that $\gamma_k \neq \gamma_{k+1}.$ Let
$\tilde{\gamma}^n$ be the
part of $\gamma^n$ connecting $\gamma_k^n$ and $\gamma_{k+1}^n,$ and
$\tilde{\gamma}$ be the part of $\gamma$ connecting $\gamma_k$ and
$\gamma_{k+1}.$ Then $\tilde{\gamma}^n, n=1,2,\ldots$ and
$\tilde{\gamma}$ are
geodesic in $M,$ such that $\tilde{\gamma}^n$ converges uniformly to
$\tilde{\gamma}.$ Therefore, $\dot{\gamma}_k^n$ converges to
$\dot{\gamma}_k$
and then, due to the relations (\ref{2}) and (\ref{3}),
$\dot{\gamma}_0^n$ converges to
$\dot{\gamma}_0.$ Thus, we have established that the map $$L: \bar{C}
\rightarrow TM: \, L(\gamma)=\dot{\gamma_0}$$ is continuous. It
immediately
implies that the map $$F:\bar{C} \times \bar{C} \rightarrow \Bbb{R}:
\,F(\gamma_1,\gamma_2) = d_{TM}(L(\gamma_1),L(\gamma_2))$$ is also
continuous.
Let us introduce now the map $$\Omega: \bar{C} \rightarrow B \times B;
\,
\Omega(\gamma)=(\gamma(0),\gamma(1)),$$ which is bicontinuous and
injective
(both properties are due to the fact that $M_C$ has non-positive
curvature, see
for example,
\cite{ballman}, Theorem 14). It shows that, since $B$ is compact,
$\bar {C}$
is a compact set. On the other hand, function $F$ is identically equal
to zero
on the diagonal $\{(\gamma,\gamma)| \gamma \in \bar{C} \} \subset
\bar{C} \times
\bar{C}.$ It means that for every $\epsilon >0$ there exists $\delta_C
>0$ such
that $d_{\bar{C}}(\gamma_1,\gamma_2) \leq \delta_C$ implies
$F(\gamma_1,\gamma_2)
\leq \epsilon.$ This proves Lemma~\ref{lmm-one} and, thus, finishes the
proof of
Theorem~\ref{thm-entropy} for simply connected $M.$
\end{proof}
Now, let us show how to modify the proof to include the case when $M$
is not
simply connected.
Denote by $H(t)$ the number of different homotopy classes that can be
represented by the curves, which intersect the compact set $B$ and have
length
less or equal to $t.$
We will say that two billiard trajectories $\Gamma_1$ and $\Gamma_2$ of
length
$l \in \Bbb{R}$ are of the same {\bf homotopic combinatorial class}
(respectively, the same {\bf strict homotopic combinatorial class}) if
\begin{enumerate} \item $\Gamma_1$ and $\Gamma_2$ are of the same
combinatorial class (respectively, the same strict combinatorial
class);
\item $d_M(\Gamma_1 (0), \Gamma_2 (0)) < r_0$ and $d_M(\Gamma_1 (l),
\Gamma_2(l)) < r_0,$ where $r_0$ is the minimum of the injectivity
radius of $M$
over all points of $B;$ \item the closed curve formed by $\Gamma_1,$
$\Gamma_2$
and the two shortest geodesics connecting $\Gamma_1(0)$ with
$\Gamma_2(0),$ and
$\Gamma_1(l)$ with $\Gamma_2(l),$ is homotopically trivial.
\end{enumerate}
(Notice, that unlike the relation of being from the same combinatorial
class
(strict combinatorial class) the relation of being from the same
homotopic combinatorial class
(strict homotopic combinatorial class) is not an equivalency relation
of the
set of trajectories, because it does not posses the transitivity
property.)
Now, Lemma~\ref{lmm-space} is true if we substitute in its statement
``homotopic combinatorial class'' instead of ``combinatorial class'' and
add the
condition that $\delta < r_0.$ The proof
is essentially the same. We consider the universal cover $\tilde{M}$ of
$M$ and
the billiard $\tilde{B}$ in the pre-image of $B$ under the covering
map. Notice
that the set $\tilde{B}$ in $\tilde{M}$ is bounded by all the pre-images
of the
sets $B_i$ in $M,$ and each connected component of the pre-image is
considered
as a separate wall of the billiard $\tilde{B}.$ The condition that
$\Gamma_1$
and $\Gamma_2$ have the same homotopic combinatorial class guarantees
that if
their
lifts, $\tilde{\Gamma}_1$ and $\tilde{\Gamma}_2,$ in $\tilde{B}$ are
such that
$d_{\tilde{M}} (\tilde{\Gamma}_1 (0) ,\tilde{\Gamma}_2 (0)) < \delta$
then
$\tilde{\Gamma}_1$ and $\tilde{\Gamma}_2$ have the
same combinatorial class in $\tilde{B}$ and
$d_{\tilde{M}} (\tilde{\Gamma}_1 (t_0) ,\tilde{\Gamma}_2 (t_0)) <
\delta.$
Then, exactly
as before, we construct the singular space $M_{\tilde{C}}$ using
the billiard $\tilde{B}$ and the combinatorial class $\tilde{C}$ of
$\tilde{\Gamma}_1$ and $\tilde{\Gamma}_2$ in $\tilde{B},$ and show that
$d_{\tilde{M}} (\tilde{\Gamma}_1 (t) ,\tilde{\Gamma}_2 (t)) < \delta$
for
all $t \in [0,t_0].$ This immediately implies that
$d_{M} (\Gamma_1 (t) ,\Gamma_2 (t)) < \delta$ for
all $t \in [0,t_0].$
Lemma~\ref{lmm-one} is true if we substitute in its statement
``homotopic
combinatorial class'' instead of ``combinatorial class.'' The proof is
again
very similar to the proof of Lemma~\ref{lmm-one} for the simply
connected case.
Let us outline it. Let compact connected set $B' \subset \tilde{B}$ be
such
that $B'$ covers $B,$ that is, every point of $B$ has a pre-image in
$B'$ and,
moreover every geodesic connecting two points of $B$ has a lift that is
contained in
$B'.$ Let $\tilde{C}'$ be an arbitrary combinatorial class in
$\tilde{B}$ that contains pre-images of some trajectories of class $C$
in $B,$
and such that those pre-images start at some points of $B'.$
Consider the set $C'$ of all trajectories of $\tilde{B}$
that have length one, start at some point of $B',$ and belong to the
class $\tilde{C}'.$
Consider the closure
$\bar{C}'$ of $C'$ in the metric of the uniform convergence for the
curves on
$\tilde{M}.$ The functions $L'$ and $F',$ on the sets $\bar{C}'$ and
$\bar{C}'
\times \bar{C}',$ are defined exactly as the functions $L$ and $F$ were
defined
for the sets $\bar{C}$ and $\bar{C} \times \bar{C}.$
The map $\Omega'$ is defined similar to the map $\Omega$ and maps
$\bar{C}'$
into a compact set $B' \times B'(1),$ where $B'(1)$ is the set of
points in
$\tilde{M}$ which are at the distance less or equal to one from the set
$B'.$
This way, we establish the compactness of $\bar{C}'$ and, thus,
the uniform continuity of $F'.$
It means that for every $ \epsilon >0$ there exists $\delta >0$ such
that $d_{\bar{C}'}(\gamma_1,\gamma_2) \leq \delta_{\tilde{C}}$ implies
$F'(\gamma_1,\gamma_2)
\leq \epsilon.$
Choose, $\delta'_C= \min \delta_{\tilde{C}},$ where the minimum is
taken over
all the classes $\tilde{C}$ in $\tilde{B}$ that contain the pre-images
of the
trajectories of length one of class $C$ in $B.$ And, let $\delta_C =
\min \{ \delta'_C , r_0 \}.$
Then, if
$\Gamma_1$
and $\Gamma_2$ have the same homotopic combinatorial class, are such
that
$d_{M} (\Gamma_1 (0) ,\Gamma_2 (0))
< \delta_C,$ $d_{M} (\Gamma_1 (1) ,\Gamma_2 (1))
< \delta_C,$
and if their
lifts, $\tilde{\Gamma}_1$ and $\tilde{\Gamma}_2,$ in $\tilde{B}$ are
chosen
so that $d_{\tilde{M}} (\tilde{\Gamma}_1 (0) ,\tilde{\Gamma}_2 (0))
< \delta_C,$ then $\tilde{\Gamma}_1$ and $\tilde{\Gamma}_2$
have the
same combinatorial class in $\tilde{B}$ and satisfy the condition
$d_{\tilde{M}} (\tilde{\Gamma}_1 (1), \tilde{\Gamma}_2(1)) < \delta_C.$
Due to the
choice of $\delta_C$ we see that it implies the statement of the
non-simply connected
version of Lemma~\ref{lmm-one}.
Exactly as before the modified versions of Lemmas ~\ref{lmm-space}
and
~\ref{lmm-one} imply Lemma~\ref{lmm-continuity} with ``strict
combinatorial
class'' being changed to ``strict homotopic combinatorial class.''
Now, the modified version of Lemma~\ref{lmm-continuity} can be used to
construct
an $\epsilon$-net $A^l (\epsilon).$
The construction is exactly the same as the construction of $A^l
(\epsilon)$
in Section~\ref{sec-outline}, except that instead of picking for each
pair of
$U$ and $V$ a single trajectory $\Gamma_{U,V}$ we pick as many
trajectories
$\Gamma^i_{U,V}$ as we can in order to satisfy the following conditions
\begin{enumerate}
\item all the trajectories $\Gamma^i_{U,V}$ satisfy the conditions
$\Gamma^i_{U,V}(0) \in U$ and $\Gamma^i_{U,V}(l) \in V;$
\item all the trajectories $\Gamma^i_{U,V}$ have the
same strict combinatorial class $C;$
\item no two trajectories among $\Gamma^i_{U,V}$ have the same
strict
homotopic combinatorial class;
\item every trajectory $\Gamma$ of the strict combinatorial class $C$
that
satisfies the conditions
$\Gamma (0) \in U$ and $\Gamma (l) \in V$ has the same strict
homotopic
combinatorial class with at least one of the trajectories
$\Gamma^i_{U,V}.$
\end{enumerate}
Clearly, for each pair of $U$ and $V$ one can choose at most
$H(2l + 2 \delta)$ different trajectories $\Gamma^i_{U,V}.$
This way, for each strict combinatorial class $C$ we construct
a set $A_C^l (\epsilon) $ that consists of no more than $K' H(2l + 2
\delta)$
different points (here, as before, $K'$ is a constant
that depends only on $B$ and $\epsilon$). The union $A^l (\epsilon) =
\bigcup A_C^l (\epsilon) $
over all the strict combinatorial classes of trajectories of length $l$
is an
$\epsilon$-net with respect to the metric $d_l$ on $TM.$
Thus,
$$
h_{top}(T^1, \epsilon) \leq \lim_{\epsilon \rightarrow 0}
\overline{\lim_{\l
\rightarrow \infty}} \frac{{\rm ln}(Card(A^l(\epsilon)))}{l} \leq
\lim_{\l
\rightarrow \infty} \frac{{\rm ln}(K'n^{(P+1)(l+1)})H(2l+2\delta) }{l}
=$$
$$ (P+1){\rm ln}(n) + \overline{\lim_{\l
\rightarrow \infty}} \frac{{\rm ln} (H(2l+2\delta)) }{l}=
(P+1){\rm ln}(n) + 2 \overline{\lim_{\l
\rightarrow \infty}} \frac{{\rm ln} (H(l))}{l}. $$
%Therefore, $h_{top}(T^1) \leq (P+1){\rm ln}(n).$
\section{Estimates on the number of periodic points and trajectories.}
\label{sec-points}
Here we will use our methods to prove some results about
periodic points and trajectories of semi-dispersing billiards.
The similar results for billiards in $\Bbb{R}^k$ (for periodic points)
and
$\Bbb{R}^2$ (for periodic trajectories) were
proven in \cite{stojanov}. The advantage of our method is that the use
of singular Riemannian geometry allows us to include the billiards
on manifolds of variable non-positive
curvature and at the same time to avoid a
variational calculation used in \cite{stojanov}.
We say that a periodic trajectory $\Gamma (t),$ $0 \leq t \leq l,$
is of class $C$ if we can choose
the starting point $\Gamma (0)$ so that $\Gamma (0) \in B_{i_1}$
and
then $\Gamma$ collides with $B_i,$ $i=i_2,\ldots,i_j,$ corresponding
to the
class $C,$ and eventually
$\Gamma (0) =\Gamma (l).$ Also, we will call a curve $\nu (t)$ a
periodic
pseudo-trajectory of class $C$ if it is a closed curve that consists of
pieces of
geodesics on $M$ that connect some point $x_1 \in B_{i_1},$ with
some point $x_2 \in B_{i_2},$ the point $x_2 \in B_{i_2}$
with
some point $x_3 \in B_{i_3},$ $\ldots,$ the point
$x_j \in B_{i_j}$ with
the point $x_{1} \in B_{i_1},$
and at each point $x_k,$ $k=1, \ldots, j,$ the tangent vector to $\nu
(t)$ changes according
to the billiard rule with respect to $B_{i_k}.$ (The difference with
the
usual trajectories is that a geodesic segment of a pseudo-trajectory
between $x_k$ and $x_{k+1}$ may intersect some of the bodies $B_i,$
$i=1, \ldots,n.$) Notice that if $\Gamma$ is any periodic trajectory
then
any periodic pseudo-trajectory close enough to $\Gamma$ is a periodic
trajectory.
Our main result on periodic trajectories is
\begin{thm}
\label{thm-periodmain}
Let $B$ be a semi-dispersing billiard on a simply connected manifold $M$
of non-positive sectional curvature. Let $C$ be some combinatorial
class
of trajectories. (Notice that, unlike in our previous results,
here we do not require $B$ to be compact or non-degenerate.)
Then, the periodic trajectories of class $C$ all have the same length
and form
a parallel family in the following sense: Any two periodic trajectories
$\Gamma_1$ and $\Gamma_2$
of class $C,$ can be joined by a continuous curve $\Gamma_t,$
$1 \leq t \leq 2,$ of periodic pseudo-trajectories of
type $C,$ so that
\begin{enumerate}
\item the surface $\Sigma_k,$ $k=1, \ldots, j,$ formed by the pieces of
trajectories
$\Gamma_t,$ $1 \leq t \leq 2,$ between the $k$-th and $(k+1)$-st
collisions
is a piece of $\Bbb{R}^2$ isometrically embedded into $M;$
\item the intersection $I_k$ of the boundary of $B_{i_k}$ with the
trajectories
from the curve $\Gamma_t,$ $1 \leq t \leq 2,$ are isometrically
embedded intervals of a straight line that connect the point
$\Gamma_1 \bigcap B_{i_k} $ with the point $ \Gamma_2 \bigcap
B_{i_k};$
\item inside of each flat surface $\Sigma_k,$ $k=1, \ldots, j,$
the pieces of trajectories from $\Gamma_t,$ $1 \leq t \leq 2,$ are
parallel to each other.
\end{enumerate}
\end{thm}
We immediately have the following
\begin{cor}
\label{cor-negative}
For $M,$ and $C$ as in Theorem~\ref{thm-periodmain},
\begin{enumerate}
\item if the curvature of $M$ is strictly negative, then $C$ contains no
more then one periodic trajectory;
\item if for some periodic trajectory $\Gamma$ of class $C$
at least one of the sets $B_i,$ $i=i_1, i_2,\ldots,i_j$
is strictly convex at the point
$\Gamma \bigcap B_{i_k},$
then $\Gamma$ is the only periodic trajectory in its class.
\end{enumerate}
\end{cor}
Let us prove Theorem~\ref{thm-periodmain}.
\begin{proof}
Let $\Gamma_1$ and $\Gamma_2$
be two periodic trajectories of class $C.$ Let $x = \Gamma_1
(0)$
and $y = \Gamma_2 (0)$ and $x' = E_j(\Gamma_1 (0))$
and $y' = E_j (\Gamma_2 (0)).$ Extend the geodesics $G(\Gamma_1)$
and
$G(\Gamma_2)$ a little beyond the points $x,$
$x'$ and $y,$ $y'$ correspondingly, to geodesics $\gamma_1$ and
$\gamma_2,$
in such a way that $\gamma_1,$ $\gamma_2$ belong to $B_{i_1},$ prior
to the points $x$ and $y,$ and to $E_j (B_{i_1}),$ after the points
$x'$ and $y'$ (i.e., so to say extend the geodesics ``into the
walls''). Let $q \in \gamma_1,$ $q \in B_{i_1},$ $q \neq x,$
and let $p \in \gamma_2,$ $p \in B_{i_1},$ $p \neq y.$
Then, since $\Gamma_1$ and $\Gamma_2$ are periodic trajectories,
$q'=E_j (q) \in \gamma_1,$ and $p'=E_j (p) \in \gamma_2$ (provided that
$q$ and $p$ are chosen close enough to the $x$ and $y,$ respectively).
Therefore,
\begin{equation}
\label{eq-angles}
\angle (qxy)= \angle (q'x'y') \,\, \textup{and} \, \, \angle (xyp)=
\angle (x'y'p').
\end{equation}
Thus, the sum of the angles of the geodesic quadrangle
$xx'y'y$ is equal to $2 \pi.$ Therefore, since $M_C$ has non-positive
curvature, the defects of the
triangles $xx'y'$ and $xy'y$ are both equal to zero. Also,
we see that
\begin{equation}
\label{eq-angles2}
\angle(xy'y) +\angle(xy'x')= \angle(yy'x').
%\,\, \textup{and} \, \, \angle(x'xy') +\angle(y'xy)= \angle(x'xy).
\end{equation}
Consider the triangles $XX'Y'$ and $XY'Y$ on $\Bbb{R}^2$, which have a
common side
$XY',$ and $|XY|=d_{M_C}(x,y),$ $|XY'|=d_{M_C}(x,y'),$
$|YY'|=d_{M_C}(y,y'),$
$|XX'|=d_{M_C}(x,x'),$ $|X'Y'|=d_{M_C}(x',y').$ Since the
defects of $xx'y'$ and $xy'y$ are both equal to zero, triangles
$xx'y'$ and $xy'y$ and triangles $XX'Y'$ and $XY'Y$ have equal
corresponding
angles. Due to equations~\ref{eq-angles} and ~\ref{eq-angles2} we see
that
$\angle (XYY') + \angle (YY'X') =\pi.$ This, together with the fact
that
$|XY|=|X'Y'|$, shows that $XYY'X'$ is a parallelogram. Therefore,
the $d_{M_C}(x,x') =d_{M_C}(y,y').$ Denote this length by $l.$
Consider geodesics $g(t)=xy,$ $g'(t)=x'y',$ $t \in [0, d_{M_C}(x,y)],$
connecting $x$ with $y,$ and
correspondingly $x'$ with $y'.$ Since $M_C$ has non-positive
curvature the function $f(t) = d_{M_C}(g(t), g' (t))$ is convex, and
since $f(0)=f(d_{M_C}(x,y))$, we see that $f(t)$ is a constant function.
Denote this constant by $l.$
Consider a ruled surface $S_1$ (correspondingly, $S_2$) formed by the
geodesics connecting $y'$ with the points of the geodesic
$xy$ (correspondingly, $x$ with the points of the geodesic
$x'y'$). $S_1$ and $S_2$ have non-positive curvature with respect to
the metric inherited from $M_C$ (the result of Aleksandrov,
\cite{1957a}; see also \cite{reshetnyak}, Theorem 9.1).
For piecewise smooth surfaces there is a well-defined concept
of integral curvature measure, which has many properties similar to the
integral curvature for smooth manifolds (see a review \cite{resh-1}),
in particular, it satisfies the Gauss-Bonnet formula (see \cite{resh-1},
Theorem 5.3.2).
Applying it to the surfaces $S_1$ and $S_2$, we conclude that their
integral
curvature is equal to zero everywhere (their defects are equal to zero,
and their
curvature measures are non-positive),
and therefore, $S_1$ is isometric
to the triangle $XYY'$, and $S_2$ is isometric to triangle $XX'Y'.$
Denote the isometries by $F_1$ and $F_2,$ correspondingly.
Let $S= S_1 \bigcup S_2,$ let $F: S \rightarrow XYY'X'$ be defined by
$F|_{S_1} =F_1,$ $F|_{S_2} =F_2.$ Being a result of gluing of
$S_1$ and $S_2$, the surface $S$ also has non-positive curvature.
Let $M \in XY$ and $M' \in X'Y'$ be such that $MX=M'X'.$
Consider $\alpha = F^{-1} (MM') \in S.$ The curve $\alpha$ connects
$m=F^{-1} (M) \in xy$ and $m'=F^{-1} (M') \in x'y', $
and its length is equal to the length of $MM'$, and thus is equal to
$l.$ Therefore, $\alpha$ is the geodesic in $M_C$ connecting $m$ and
$m'.$
Thus, we can describe $S$ in the following way:
surface $S$ is formed by the geodesics $G_t$ in $M_C$ connecting
$g_1(t)$ with $g_2(t),$ for $0
\leq t \leq d_{M_C}(x,y).$
Clearly,
$S$ is a piecewise smooth surface. Namely, it consists of the
smooth pieces $L_k=S \bigcap M_k,$ $k=1, \ldots, j,$ which are
glued together in the following way:
$L_k$ is glued with $L_{k+1}$ along their common boundary
$C_{k+1},$ where
$C_k,$ $k=1, \ldots, j,$
are the curves of intersection of $S$ with the boundaries of $E_k
(B_{i_k}).$
The integral curvature measure at the interior points of
$L_k,$ $k=1, \ldots, j,$ is equal to the smooth measure multiplied by
the Gaussian curvature of $S,$ and the integral
curvature measure at the points of $C_k$ is equal to the
length measure on $C_k$ multiplied by $(k_1 +k_2 ),$ where
$k_1$ and $k_2$ are the oriented curvatures of the curve $C_k$ in
$L_{k-1}$ and $L_k,$ respectively.
We already know that $S$ has zero integral curvature
measure
at all points. From the description above of the integral curvature
measure on $L_k$ and $C_k$ it immediately follows that each piece
$L_k,$
$k=1, \ldots, j,$ is flat inside and
$L_k$ is glued with $L_{k+1}$ along a piece $C_k$ of a straight
line.
The other pieces of the boundary of $L_k,$ $k=1, \ldots, j,$ are
all
pieces of straight lines, since they are geodesics and belong to
the boundary of a flat
surface $L_k.$
%Thus,
%$d_{M_C} (x, x') =d_{M_C} (y, y')$ and equalities (\ref{eq-angles})
%for the angles of the quadrangle $xx'y'y,$ imply that $S$ is a flat
%parallelogram isometrically embedded into $M_C.$
Rescale the parameter $t$ on the curves $g_1(t),$ $g_2(t)$
(and, thus, on the family $G_t$) so that $t$ would vary from
$1$ to $2.$
Let $\Gamma_t = E^{-1} (G_t),$ where $E^{-1}$ is the
map $M_C \rightarrow M$ such that
$E^{-1} (x)=E_k^{-1} (x),$ for $x \in M_k,$ $k=1,\ldots,j.$
%Since $S$ is isometric to a flat parallelogram, we have that
%for any $1 \leq t \leq 2,$ the quadrangle
%$g_1(t)xx' g_2(t))$ is a parallelogram as well.
We see that all
$G_t$ are periodic pseudo-trajectories of the same length.
This finishes the proof
of Theorem~\ref{thm-periodmain} (with $I_k = E_k^{-1} (C_k)$ and
$\Sigma_k = E_k^{-1} (L_k)$.) \end{proof}
Let us call two periodic trajectories equivalent if they are parallel
(in the sense explained in Theorem~\ref{thm-periodmain})
and let us call two
periodic points for the first return map to the boundary equivalent if
the
corresponding periodic billiard trajectories are equivalent.
Denote by $P_k$ ($\tilde{P}_k$), $k \in \Bbb{Z},$ the number of
(equivalence classes of) periodic points of period $k$ for the
first return map to the boundary of the billiard $B,$ and by
$P^t$ ($\tilde{P}^t$), $t \in \Bbb{R},$ the number of
(equivalence classes of) periodic trajectories of
the billiard flow of length less or equal to $t.$
Theorem~\ref{thm-periodmain} implies
\begin{cor}
\label{cor-boundary}
Let $B$ be a semi-dispersing billiard on a simply connected manifold
$M$ of
non-positive sectional curvature.
Then,
\begin{enumerate} \item if the curvature of $M$ is strictly negative
then
$$\tilde{P}_k = P_k \leq (n-1)^{k};$$
\item if all the
sets $B_i,$ $i=1,\ldots, n,$ are strictly convex then
$$\tilde{P}_k = P_k \leq (n-1)^{k};$$
\item otherwise,
$$ either \, \, \tilde{P}_k = P_k \leq (n-1)^{k} \,\, or \, \,
\tilde{P}_k \leq (n-1)^{k}, \, \, \, P_k =\infty.$$
\end{enumerate}
\end{cor}
For $M=\Bbb{R}^k,$ Corollary~\ref{cor-boundary} was proven in
\cite{stojanov}.
Theorem~\ref{thm-periodmain} together with
Proposition~\ref{prop-globalestimate} implies
\begin{cor}
\label{cor-flow}
Let $B$ be a non-degenerate
semi-dispersing billiard on a simply connected manifold $M$ of bounded
non-positive sectional curvature.
(Notice that here again we do not require $B$ to be compact, but we do
require
the non-degeneracy of $B.$) Then,
\begin{enumerate} \item if the curvature of $M$ is strictly negative
then
$$ \tilde{P}^t = P^t \leq (n-1)^{P(t+1)};$$
\item if all the
sets $B_i,$ $i=1,\ldots, n,$ are strictly convex then
$$ \tilde{P}^t = P^t \leq (n-1)^{P(t+1)};$$
\item otherwise,
$$ either \, \, \tilde{P}^t = P^t \leq (n-1)^{P(t+1)} \,\, or \, \,
\tilde{P}^t \leq (n-1)^{P(t+1)}, \, P^t =\infty,$$
\end{enumerate}
where $P$ is the constant from Proposition~\ref{prop-globalestimate}.
\end{cor}
In \cite{stojanov} Corollary~\ref{cor-flow} was proven for
$M=\Bbb{R}^2.$
Remark: Notice that our estimates for
the number of periodic points as well as for
the topological entropy are applicable to
billiards in polygons or polyhedras. However, for those billiards
much finer results are known.
In fact, the topological
entropy of billiards in polygons or polyhedras is equal to zero
\cite{gutkin},
\cite{katok-points}, and the number of periodic points grows
subexponentially
\cite{katok-points}.
\section{Topological entropy of Lorentz gas.}
\label{sec-lorentz}
Lorentz gas model is a billiard on $\Bbb{T}^k=\Bbb{R}^k /\Bbb{Z}^k$
with one wall which is a
ball of radius $1/2 > r>0.$ In \cite{chernov-topentropy} it was
proven
that the first return map to the boundary of the Lorentz gas billiard
has infinite topological entropy, and that the metric entropy
of the Lorentz gas billiard with respect to the Liouville measure
converges to zero, when $r \rightarrow 0.$ In contrast to
those results,
we prove the following
\begin{thm}
\label{thm-lorentz}
Denote by $h_r (k)$ the topological entropy of the Lorentz gas billiard
described above. Then,
\begin{enumerate} \item $h_r (k)$ is finite;
\item there exist $\lim_{r \rightarrow 0} h_r (k)=h_0(k),$ and
$\infty > h_0 (k) >0;$
\item $h_0(k) \leq h_0 (k+1)$
and $h_0 (k) \geq {\rm ln} (2k -1).$
\end{enumerate}
\end{thm}
The first statement of Theorem~\ref{thm-lorentz} follows
immediately from Theorem~\ref{thm-entropy}. Moreover,
for a fixed $k,$
it is easy to see that $h_r (k)$ are uniformly bounded
over $r.$ Let us denote some upper bound by $Q(k).$
To prove the second statement let us first introduce some notations.
We denote by $B_r^{(m)}$ the ball of radius $r$ centered at the
point $(m)=(m_1, \ldots , m_k )$ on $\Bbb{R}^k,$ $m_i \in \Bbb{Z},$
$i=1, \ldots, k.$ Denote by $F_r (n,k)$ the number of different
combinatorial classes of billiard trajectories of length at most $n,$
$n \in \Bbb{R},$ in
$\tilde{B}=\Bbb{R}^k \backslash (\bigcup_{(m) \in \Bbb{Z}^k}
Int \, B_r^{(m)})$ starting at
the ball $B_r^{(0)},$ where $(0)=(0, \ldots ,0).$ Then, exactly
as in the proof of Theorem~\ref{thm-entropy}, we see that
$$h_r(k)=\overline{\lim}_{n \rightarrow \infty} \frac{\rm{ln}
F_r (n,k)}{n}.$$
A curve $\gamma$ in $\Bbb{R}^k$
that satisfies the following properties is called
an $r$-pseudo-trajectory
\begin{enumerate}
\item $\gamma $ belongs to $\tilde{B},$ i.e., $\gamma$
does not intersect the interiors of the balls $B_r^{(m)};$
\item $\gamma$ consists of several straight edges, with vertices that
belong to the balls $B_r^{(m)}.$
\end{enumerate}
In other words, $\gamma$ is ``almost a billiard trajectory,'' except
that it does not have to satisfy the ``angle of incidence is equal
to the angle of reflection'' law. Denote by $\Pi_r (n,k)$ the number
of different combinatorial classes of $r$-pseudo-trajectories
of length at most $n$ that start at $B_r^{(0)}.$ Then, clearly,
$\Pi_r (n,k) \geq F_r (n,k).$ On the other hand, the shortest
$r$-pseudo-trajectory in a given class $C$ must be a billiard
trajectory
of length at most $n$
(provided that $C$ contains any $r$-pseudo-trajectories of length at
most $n$). Thus, $\Pi_r (n,k) = F_r (n,k).$
Now, we will show that $\Pi_r (n,k)$ is ``almost a decreasing
function of $r.$'' Let $\gamma$ be an $r$-pseudo-trajectory
of length $l$ and with vertices $x_i \in B_r^{(m)_i},$ $i=0, \ldots,
p,$
where $(m)_0=(0).$ Then, since the distance between any two balls
$B_r^{(m)_i} $ and $B_r^{(m)_{i+1}} $ is at least $1-2r,$ we see that
$p \leq \frac{l}{1-2r}.$ Without loss of generality we may assume that
$r \leq 1/4.$ Then $p \leq 2l.$ For a fixed $r' < r$, let
$y_i$ be the intersection of the
boundary of the ball $B_{r'}^{(m)_i}$ with the interval of a straight
line
connecting $x_i$ and $(m)_i.$ Then the broken straight
line $\gamma'$ with vertices $y_i,$ $i=0, \ldots, p,$ is an
$r'$-pseudo-trajectory. Indeed, if $\gamma'$ intersects with
the interior of some ball $B_{r'}^{(m)},$ then $\gamma$ intersects
with the interior of the ball $B_{r}^{(m)},$ which contradicts
to the fact that $\gamma$ is an $r$-pseudo-trajectory. Moreover,
the length of $\gamma'$ is at most $l+2(r-r')p \leq
l(1+4(r-r').$ Thus,
$$
\Pi_r (n,k) \leq \Pi_{r'} ( n(1+4(r-r'),k).
$$
Thus, we see that
$$h_r (k) = \overline{\lim}_{n \rightarrow \infty} \frac{\rm{ln}
\Pi_r (n,k)}{n} \leq \overline{\lim}_{n \rightarrow \infty}
\frac{\rm{ln}
\Pi_{r'} ( n(1+4(r-r'), k) }{n} = $$
$$ (1+4(r-r')) \overline{\lim}_{n \rightarrow \infty}
\frac{\rm{ln}
\Pi_{r'} ( n(1+4(r-r'), k) }{ n(1+4(r-r')) } = (1+4(r-r')) h_{r'}
(k).$$
Thus,
\begin{equation}
\label{eq-lorentz1}
h_r (k)-h_{r'} (k) \leq 4(r-r') h_{r'} (k) \leq 4(r-r') Q (k)
, \, \, \textup{if} \, \, r'