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\begin{document}
\title{Sinai billiards under small external forces}
\author{N.I. Chernov\thanks{Partially supported by NSF grant DMS-9732728.}
\\ {\small Department of Mathematics}\\
{\small University of Alabama at Birmingham}\\
{\small Birmingham, AL 35294}\\
{\small E-mail: chernov@math.uab.edu}
%Fax: 1-(205)-934-9025
}
\date{\today}
\maketitle
\begin{abstract}
Consider a particle moving freely on the torus and colliding
elastically with some fixed convex bodies. This model is called a
periodic Lorentz gas, or a Sinai billiard. It is a Hamiltonian
system with a smooth invariant measure, whose ergodic and
statistical properties have been well investigated. Now let the
particle be subjected to a small external force. This new system
is not likely to have a smooth invariant measure. Then a
Sinai-Ruelle-Bowen (SRB) measure describes the evolution of
typical phase trajectories. We find general sufficient conditions
on the external force under which the SRB measure
for the collision map exists, is
unique, and enjoys good ergodic and statistical properties,
including Bernoulliness and an exponential decay of correlations.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
%\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{sublemma}[theorem]{Sublemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\section{Introduction}
\label{secI} \setcounter{equation}{0}
Let ${\cal B}_1,\ldots, {\cal B}_s$ be open convex domains on the
unit 2-dimensional torus $\IT^2$. Assume that $\bar{\cal
B}_i\cap\bar{\cal B}_j=\emptyset$ for $i\neq j$, and for each $i$
the boundary $\partial {\cal B}_i$ is a $C^3$ smooth closed curve
with nonvanishing curvature.
Consider a particle of unit mass moving in
$Q:=\IT^2\setminus\cup_i {\cal B}_i$ according to equations
\be
\dot{\bf q}={\bf p},\ \ \ \ \ \dot{\bf p}={\bf F}
\label{emotion}
\ee
where ${\bf q}=(x,y)$ is the position vector, ${\bf p}=(u,v)$
is the momentum (or velocity) vector, and ${\bf
F}(x,y,u,v)=(F_1,F_2)$ is a stationary force (the force is
independent of time). Upon reaching the boundary $\partial
Q=\cup_i\partial {\cal B}_i$, the particle reflects elastically,
according to the usual rule
\be
{\bf p}^+ = {\bf p}^- - 2\, ({\bf n}({\bf q})\cdot {\bf p}^-)\, {\bf n}({\bf q})
\label{p+-}
\ee
Here ${\bf q}\in\partial Q$ is the point of reflection, ${\bf
n}({\bf q})$ is the unit normal vector to $\partial Q$ pointing
inside $Q$ , and ${\bf p}^-$, ${\bf p}^+$ are the incoming and
outgoing velocity vectors, respectively.
The case ${\bf F}=0$ corresponds to the ordinary billiard dynamics
on the table $Q$. It preserves the kinetic energy $K=\frac 12
||{\bf p}||^2$, so that one can fix it, usually by setting $||{\bf
p}||=1$. Then the phase space of the system is a compact
three-dimensional manifold ${\cal M}_0:=Q\times S^1$, with
identification of incoming and outgoing velocity vectors, i.e.
${\bf p}^-$ and ${\bf p}^+$ in (\ref{p+-}), at every point of
reflection. The dynamics $\Phi^t_0$ on ${\cal M}_0$ preserves the
Liouville measure, which is simply a uniform measure on ${\cal
M}_0$.
In the study of billiards, one usually considers the following
two-dimensional cross-section of ${\cal M}_0$:
\be
M_0:=\{({\bf q},{\bf p})
\in{\cal M}_0:\, {\bf q}\in\partial Q,\, ({\bf p}\cdot {\bf n}({\bf q}))\geq 0\}
\label{M0}
\ee
which consists of all outgoing velocity vectors at reflection
points. Then the first return map $T_0:\, M_0\to M_0$ is well
defined, it is called the collision map or
billiard map. The cross-section $M_0$
can be parametrized by $(r,\varphi)$, where $r$ is the arclength
parameter along $\partial Q$ and $\varphi\in [-\pi/2,\pi/2]$ is
the angle between $\bf p$ and ${\bf n}({\bf q})$. In these
coordinates, $M_0=\partial Q\times [-\pi/2,\pi/2]$. The map
$T_0$ preserves a finite smooth measure on $M_0$, induced by
the Liouville measure on ${\cal M}_0$. It is given by
$$
d\nu_0={\rm const}\cdot \cos\varphi\, dr\,d\varphi
$$
Since each obstacle ${\cal B}_i$ is convex, it acts as a scatterer, so
that parallel bundles of trajectories diverge upon reflection, see
Fig.~1. Billiard with this property are said to be dispersing, or
Sinai billiards. The map $T_0$ and the flow $\Phi^t_0$ for dispersing
billiards are proved to be
hyperbolic (i.e., they have one positive and one negative Lyapunov
exponents), ergodic, mixing, K-mixing and Bernoulli
\cite{Si70,GO}. The map $T_0$ enjoys strong statistical
properties: exponential decay of correlations and satisfies a
central limit theorem and weak invariance principle
\cite{BSC91,Y98,Ch99b}.
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Let us now assume that the configuration of scatterers has {\it
finite horizon} meaning that the the free motion of the billiard
particle in $Q$ is uniformly bounded (by a constant $L>0$). In
other words, any straight line of length $L$ on the torus
intersects one of the ${\cal B}_i$. Under this condition, in
addition to all the cited properties, the flow $\Phi^t_0$
satisfies a central limit theorem and weak invariance principle
\cite{BSC91}, hence the billiard particle satisfies a diffusion
equation \cite{BSC91}. It is very likely that the flow $\Phi^t_0$
enjoys exponential decay of correlations as well, but this is not
proven yet. The assumption of finite horizon seems to be necessary
for the above properties, because in billiards without horizon the
moving particle exhibits superdiffusive (ballistic) behavior
\cite{Bl}, and the correlations seem to decay very slowly, as
const$\cdot t^{-1}$, see, e.g., \cite{FM84,FM88}.
Very little is known in the general case ${\bf F}\neq 0$, though.
Clearly, a large force can change the dynamics dramatically, so
that the properties of the dynamics will be determined by the
character of $\bf F$ in (\ref{emotion}) more than by the
scattering effect of collisions with obstacles. Hence, the
dynamics can be of quite generic nature. It is presently
understood, due to the KAM theory, that generic mechanical (even
Hamiltonian) systems are not completely hyperbolic or ergodic --
typically, chaotic regions in the phase space coexist with
elliptic islands of stability. So, we have to restrict ourselves
to small forces that will not overcome the scattering effect of
collisions with obstacles. Thus we will keep the dynamics close
enough to the original billiard. Then we can hope that many
properties of the system with force will be ``inherited'' from the
original billiard model. It is now clear that the assumption on
finite horizon will be necessary -- without it the effect of even
a small force $\bf F$ may accumulate to a dangerous level during
long runs between collisions.
The system (\ref{emotion})-(\ref{p+-}) with a force ${\bf F}\neq
0$ may be hyperbolic but is likely to admit no smooth invariant
measure. Then the evolution of typical phase trajectories is
governed by the so called Sinai-Ruelle-Bowen (SRB) measures. Those
measures are characterized by smooth conditional distributions on
unstable manifolds. The SRB measures are the only physically
observable measures, they are called nonequilibrium steady states
in the language of statistical mechanics. We refer the reader to
\cite{GC,Ru99,Y99} for more discussion on SRB measures and their
role in hyperbolic dynamics and physics.
There is a remarkable example of the system
(\ref{emotion})-(\ref{p+-}) well studied in the literature. Let
$\bf F$ be a small constant electric force, possibly combined with
a small magnetic force, with a Gaussian thermostat added, see
(\ref{EBalpha}) below. An SRB measure was constructed and its
strong ergodic and statistical properties were mathematically
proved \cite{CELS,CELSa} for this particular model. Certain
transport laws of physics were then rigorously derived, including
Ohm's law, Einstein relation, Green-Kubo formula, etc. Similar
models are now getting more and more popular in physics.
The purpose of this paper is to find general classes of forces
$\bf F$ for which the system (\ref{emotion})-(\ref{p+-}) has an
SRB measure with good ergodic and statistical properties. In fact,
we try to consider the forces as general as possible, assuming
only what seems to be necessary.
First, we will assume that the force $\bf F$ is small, as
we said above. The only other major assumption we need is an
additional integral of motion. If none exists, then the phase
space of the system is a four-dimensional noncompact manifold
$Q\times \IR^2$. Then we would face two almost hopeless problems.
First, we can only ensure that two nonzero Lyapunov exponents exist
-- they are inherited from the original billiard, but the other two
may be zero or arbitrary small. This makes the system, essentially,
only partially hyperbolic with little chance for any good ergodic or
statistical properties. To make things worse, the noncompactness
of the phase space makes the existence of physically interesting
invariant measures very unlikely. In fact, without a proper
temperature control (thermostatting), the system will usually heat
up ($||{\bf p}||\to\infty$) or cool down ($||{\bf p}||\to 0$),
which effectively rules out interesting invariant measures.
Hence, we will assume that the dynamics preserves a smooth
function ${\cal E}({\bf q},{\bf p})$, an integral of motion, and
its level surface is a compact 3-D manifold. We now turn to
exact assumptions on the force $\bf F$ in our model.
\section{The model and main results}
\label{secMMR} \setcounter{equation}{0}
Here we state our assumptions on the force $\bf F$.\medskip
\noindent {\bf Assumption A (additional integral)}. A smooth
function ${\cal E}({\bf q},{\bf p})$ is preserved by the dynamics
$\Phi^t$ defined by (\ref{emotion})-(\ref{p+-}). Its level
surface, ${\cal M}:=\{ {\cal E}({\bf q},{\bf p})={\rm const}\}$ is
a compact 3-D manifold. Two extra assumptions are made for
convenience:\\ (A1) $||{\bf p}||\neq 0$ on $\cal M$,\\ (A2) for
each ${\bf q}\in Q$ and ${\bf p}\in S^1$ the ray $\{({\bf q},s{\bf
p}),\, s>0\}$ intersects the manifold $\cal M$ in exactly one
point.\medskip
Under the assumtions (A1)-(A2), $\cal M$ can be parametrized by
$(x,y,\theta)$, where $(x,y)={\bf q}\in Q$ and $0\leq \theta<2\pi$
is a cyclic coordinate, the angle between $\bf p$ and the positive
$x$-axis. The dynamics (\ref{emotion})-(\ref{p+-}) restricted to
$\cal M$ is a flow that we denote by $\Phi^t$. In the coordinates
$(x,y,\theta)$ the equations of motion (\ref{emotion}) can be
rewritten as
\be
\dot{x}=p\cos\theta,\ \ \ \ \dot{y}=p\sin\theta,\ \ \ \ \dot{\theta}=ph
\label{h}
\ee
where
$$
p=||{\bf p}||>0\ \ \ \ {\rm and}\ \ \ \
h=(-F_1\sin\theta+F_2\cos\theta)/p^2
$$
It is also useful to note that
\be
\dot{p}=F_1\cos\theta+F_2\sin\theta
\label{dotp}
\ee
Both $h=h(x,y,\theta)$ and $p=p(x,y,\theta)$ are assumed to be
$C^2$ smooth functions on $\cal M$. Due to our assumption (A1), we
have
\be
00$. The
time between collisions is uniformly bounded below as well, by
$t_{\min}=L_{\min}/p_{\max}$.
\medskip
Lastly, we state our assumption on finite horizon:\medskip
\noindent{\bf Assumption C (finite horizon)}. There is an $L>0$ so
that every straight line of length $L$ on the torus $\IT^2$
crosses at least one obstacle ${\cal B}_i$.\medskip
\noindent{\bf Remark}. Under Assumptions B and C, every trajectory
of length $L_{\max}$ for the system (\ref{emotion})-(\ref{p+-}),
for some $L_{\max}>L$, must hit a scatterer ${\cal B}_i$. So, the
collision-free path of the particle is uniformly bounded by
$L_{\max}$. The time between collisions is uniformly bounded as
well, by $t_{\max}=L_{\max}/p_{\min}$.\medskip
Consider the two-dimensional cross-section of the manifold $\cal
M$:
\be
M:=\{({\bf q},{\bf p})
\in{\cal M}:\, {\bf q}\in\partial Q,\, ({\bf p}\cdot {\bf n}({\bf q}))\geq 0\}
\label{M}
\ee
which, as $M_0$ in (\ref{M0}), consists of all outgoing
velocity vectors at reflection points. Then the first return map
$T:\, M\to M$ is then well defined, we also call it collision map.
The cross-section $M$ can be
parametrized by $(r,\varphi)$, where $r$ is the arclength
parameter along $\partial Q$ and $\varphi\in [-\pi/2,\pi/2]$ is
the angle between $\bf p$ and ${\bf n}({\bf q})$. In these
coordinates, $M=\partial Q\times [-\pi/2,\pi/2]$, the same as
$M_0$ in (\ref{M0}).
We choose the orientation of the coordinates $r$ and $\varphi$ as
shown on Fig.~1 (where $r$ and $\varphi$ increase in the direction
of arrows). Also, denote by $K(r)>0$ the curvature of the
curve $\partial Q$ at the point with coordinate $r$.
There are two particularly interesting types of forces satisfying
our assumptions A and B:\medskip
\noindent {\bf Type 1 forces} (potential forces). Consider an
isotropic force ${\bf F}={\bf F}({\bf q})$ (independent of $\bf
p$) such that ${\bf F}=-\nabla U$, where $U=U({\bf q})$ is a
(small) potential function. Note that $U(x,y)$ must be a smooth
function on the torus, so it is necessarily a periodic function in
$x$ and $y$. These forces preserve the total energy $T=\frac 12
||{\bf p}||^2 +U({\bf q})$. In this case we can set $T=1/2$, so
that $||{\bf p}||^2=1-2U({\bf q})\approx 1$, assuming $U({\bf q})$
be small.\medskip
\noindent{\bf Remark}. Type 1 forces preserve the Lebesgue measure
$dx\, dy\, d\theta$ on the manifold $\cal M$, since the divergence
of the vector field (\ref{h}) vanishes. This follows from the
equation ${\bf F}=-\nabla U$ by direct calculations. Therefore, the
collision map $T:M\to M$ also preserves a smooth measure $\nu$.
\medskip
Sinai billiards under type 1 forces have been studied by several
authors \cite{KSS,V1,V2}. They established ergodicity and Bernoulli
property for the measure $\nu$. \medskip
\noindent {\bf Type 2 forces} (isokinetic forces). Consider forces
satisfying $({\bf F}\cdot {\bf p})=0$. They preserve the kinetic
energy, i.e. $K=\frac 12 ||{\bf p}||^2=$const. In this case we can
set $||p||=1$, as in billiards. Note that the equations in
(\ref{h}) hold with $p=1$ and $|h|=||{\bf F}||$ (the sign of $h$
is determined by the direction of $\bf F$).
\medskip
\noindent{\bf Example of type 2 forces: thermostatting}. Let $\bf
F$ be an arbitrary force. One can modify the equations
(\ref{emotion}) so that the kinetic energy will be preserved:
\be
\dot{\bf q}={\bf p},\ \ \ \ \ \ \dot{\bf p}={\bf F}-\alpha{\bf p}
\ \ \ \ {\rm where}\ \ \ \
\alpha=({\bf F}\cdot{\bf p})/({\bf p}\cdot{\bf p})
\label{Gtherm}
\ee
It is easy to verify that $||{\bf p}||=\,$const. The added
term $\alpha{\bf p}$ is called a Gaussian thermostat, it satisfies
the Gaussian principle of least constraint. Also, $\alpha$ is
called the Gaussian friction coefficient.\medskip
\noindent{\bf Example: electric and magnetic fields}. A well
studied example of a force of type 2 is the following, see
\cite{CELS}
\be
{\bf F}({\bf q},{\bf p}) = {\bf E} +
[{\bf B}\times {\bf p}] - \alpha {\bf p}
\label{EBalpha}
\ee
Here $\bf E$ is a small constant electric field, $\bf B$ is a
small constant magnetic field (a vector in $\IR^3$ perpendicular
to the billiard table $Q$), and $\alpha{\bf p}$ is the Gaussian
thermostat $\alpha=({\bf E}\cdot{\bf p})/({\bf p}\cdot{\bf p})$.
If ${\bf E}=0$ (thus $\alpha=0$), then the system preserves the
Lebesgue measure $dx\, dy\, d\theta$ on $\cal M$, just as does the
pure billiard dynamics $\Phi_0^t$. If ${\bf E}\neq 0$, then the
system has no absolutely continuous invariant measure, but has a
unique SRB measure with good ergodic and statistical properties.
We refer the reader to \cite{CELS} for a detailed study of the
system (\ref{EBalpha}).\medskip
Now we state the main results of this paper.
\begin{theorem}
Under Assumptions A, B, and C, the map $T:M\to M$ is a uniformly
hyperbolic map with singularities. It admits a unique SRB measure
$\nu$, which is positive on open sets, K-mixing and Bernoulli.
\label{tmmain1}
\end{theorem}
The next theorem concerns the statistical properties of the map
$T:M\to M$. Let ${\cal H}_\eta$ be the class of H\"older
continuous functions on $M$ with exponent $\eta >0$:
$$
{\cal H}_\eta=\{f:\, M\to \IR\, |\,
\exists C>0:\, |f(X)-f(Y)|\leq C\,[{\rm dist}(X,Y)]^\eta ,\ \forall X,Y\in M\}
$$
We say that $(T,\nu)$ has exponential decay of correlations for
H\"older continuous functions if for all $\eta>0$ there is
$\lambda =\lambda(\eta)\in (0,1)$ such that for all $f,g\in{\cal
H}_\eta$ and some $C=C(f,g)>0$ we have
\be
\left |
\int_M (f\circ T^n)g\, d\nu-
\int_M f\, d\nu\int_M g\, d\nu
\right |
\leq C\lambda^{|n|}
\label{EDC}
\ee
for all $n\in\ZZ$. We say that $(T,\nu)$ satisfies the central
limit theorem for H\"older continuous functions if for all
$\eta>0,f\in{\cal H}_\eta$ with $\int f\,d\nu=0$, there is
$\sigma_f\geq 0$ such that
\be
\frac{1}{\sqrt{n}}\sum_{i=0}^{n-1}f\circ T^i
\stackrel{\rm distr}{\longrightarrow} N(0,\sigma_f^2)
\label{CLT}
\ee
which means the convergence in distribution to the normal law
$N(0,\sigma_f^2)$. Furthermore, $\sigma_f=0$ iff $f$ is
cohomologous to zero, i.e. $f=g\circ T-g$ for some $g\in L^2(\nu)$
\begin{theorem}
The measure $\nu$ enjoys exponential decay of correlations and
satisfies the central limit theorem. The decay of correlations is
uniform in the force $\bf F$, i.e. the constants $\lambda$ and $C$
in (\ref{EDC}) are independent of $\bf F$.
\label{tmmain2}
\end{theorem}
We only remark that Theorem~\ref{tmmain1} easily implies that
the flow $\Phi^t:{\cal M}\to{\cal M}$ is fully hyperbolic
and has a unique SRB measure $\mu$ that is ergodic and positive
on open sets. In a forthcoming paper, we will prove that
the flow $\Phi^t$ is actually mixing and Bernoulli and
satisfies the central limit theorem.
\section{Hyperbolicity of $\Phi^t$ and $T$}
\label{secH}
\setcounter{equation}{0}
Our first goal is to prove that the flow $\Phi^t$ on $\cal M$ is
hyperbolic, i.e. it has one positive and one negative Lyapunov
exponents. The hyperbolicity is usually obtained by constructing a
family of invariant cones in the tangent space \cite{W85}. For
Sinai billiards, invariant cones have a clear geometrical
interpretation. Unstable cones correspond to divergent bundles of
trajectories, and stable cones - to convergent bundles of
trajectories. Any divergent bundle of trajectories remains
divergent upon reflections off convex obstacles, as in Fig.~1,
this easily implies the invariance of the unstable cones.
Similarly, any convergent bundle of trajectories remains convergent
in the past (as they flow backwards).
We will prove that in our dynamics
a sufficiently divergent bundle of trajectories
remains divergent in the future. Obviously, we need to consider
runs between collisions carefully and make sure that the
divergence is not lost there. We use some new techniques to
do that.
Let $P=(x,y,\theta)\in\cal M$ be an arbitrary point and
$P=(dx,dy,d\theta)\in{\cal T}_P{\cal M}$ a tangent vector at $P$.
Pick a smooth curve $P_s=(x_s,y_s,\theta_s)\subset\cal M$ tangent
to the vector $dP$ at the point $P$, i.e. assume that $P_0=P$ and
$P'_0=dP$. In the calculations below, we denote differentiation
with respect to the auxiliary parameter $s$ by primes and that
with respect to time $t$ by dots. In particular,
$\dot{P}=(\dot{x},\dot{y},\dot{\theta})=(p\cos\theta,p\sin\theta,ph)$
is the velocity vector of the flow $\Phi^t$. It is not to be
confused with the velocity vector
$(\dot{x},\dot{y})=(p\cos\theta,p\sin\theta)$ of the moving
particle on the torus, the latter will be referred to as the {\em
particle velocity}.
Now consider $P_{st}=(x_{st},y_{st},\theta_{st}):=\Phi^tP_s$. The
points $P_{st}$ make a two-dimensional surface $\cal S$ in $\cal
M$. It is standard that
$$
D\Phi^t(dP)=P_{0t}'=\frac{d}{ds}P_{st}|_{s=0}
$$
In subsequent formulas, all the calculations will be done at
the point $P_{0t}$, where $s=0$, and for brevity we will often
drop the subscript $0t$. Note that the vectors
$P'=(x',y',\theta')$ and
\be
\dot{P}=(\dot{x},\dot{y},\dot{\theta})=(p\cos\theta,p\sin\theta,ph)
\label{dots}
\ee
are both tangent vectors to $\cal S$ at the point $P$ ($=P_{0t}$).
We introduce two quantities
\be
v=x'\cos\theta+y'\sin\theta
\ \ \ \ {\rm and}\ \ \ \
w=-x'\sin\theta+y'\cos\theta
\label{vw}
\ee
It is easy to see that $v$ is the component of the vector
$(x',y')$ parallel to the particle velocity $(\dot{x},\dot{y})$,
and $w$ is the perpendicular component of $(x',y')$. Solving
(\ref{vw}) for $x',y'$ gives
\be
x'=v\cos\theta-w\sin\theta
\ \ \ \ {\rm and}\ \ \ \
y'=v\sin\theta+w\cos\theta
\label{x'y'}
\ee
Now let
\be
\alpha=v/w\ \ \ \ {\rm and}\ \ \ \ \kappa=(\theta'-vh)/w
\label{alphakappa}
\ee
So, $\alpha$ is the cotangent of the angle between the vector
$(x',y')$ and the particle velocity $(\dot{x},\dot{y})$. To
describe $\kappa$ geometrically, consider the one parameter family
of trajectories $\{(x_{st},y_{st})\}$ on the torus,
%see Fig.~2,
where $s$ is the parameter of the family and $t$ is the internal
parameter along each trajectory. Then $\kappa$ is the curvature of
the orthogonal cross-section of this family. Furthermore,
$\kappa>0$ corresponds to divergent families,
%(as the one on Fig.~2)
$\kappa<0$ to convergent families, and $\kappa=0$ to
parallel families. Also, note that $|w|$ is the width of that
family in the direction perpendicular to the particle velocity,
per unit increment of the parameter $s$.
Now, consider two vectors
$$
U=(\cos\theta,\sin\theta,h)\ \ \ \
{\rm and}\ \ \ \ R=(-\sin\theta,\cos\theta,\kappa)
$$
Both are tangent vectors to the surface $\cal S$, as it follows from
the equations
$$
\dot{P}=pU\ \ \ \ {\rm and}\ \ \ \ P'=vU+wR
$$
The vector $U$ is obtained by taking a unit vector
$(\cos\theta,\sin\theta)$ in the direction of the particle
velocity $(\dot{x},\dot{y})$ and lifting it to a tangent vector to
the surface $\cal S$. Similarly, the vector $R$ is obtained by
taking a unit vector $(-\sin\theta,\cos\theta)$ in the
perpendicular direction and lifting it to a tangent
vector to the surface $\cal S$.
Denote by $p_U$ and $p_R$ the `scaled' directional derivatives of
the function $p$ along the vectors $U,R$, respectively, defined by
$$
p_U=p_x\cos\theta+p_y\sin\theta+p_{\theta}h,\ \ \ \ \
p_R=-p_x\sin\theta+p_y\cos\theta+p_{\theta}\kappa
$$
Similarly,
$$
h_U=h_x\cos\theta+h_y\sin\theta+h_{\theta}h,\ \ \ \ \
h_R=-h_x\sin\theta+h_y\cos\theta+h_{\theta}\kappa
$$
It is then straightforward that
\be
p'=p_Uv+p_Rw\ \ \ \ \ {\rm and}\ \ \ \ \ h'=h_Uv+h_Rw
\label{p'h'}
\ee
upon direct differentiation and substitution of (\ref{x'y'}) and
using
\be
\theta'=\kappa w+hv
\label{theta'}
\ee
It is also easy to see that
$$
\dot{p}=p_Up\ \ \ \ \ \ {\rm and}\ \ \ \ \ \ \dot{h}=h_Up
$$
\begin{lemma}
The evolution of the quantities $\kappa, w, \alpha$ is given by
the equations
\be
\dot{\kappa}/p=-\kappa^2-h^2+h_{R}
\label{kappat}
\ee
\be
\dot{w}=p\kappa w
\label{wt}
\ee
and
\be
\dot{\alpha}=-p\kappa\alpha +p_{U}\alpha+p_R+ph
\label{alphat}
\ee
\end{lemma}
{\em Proof}. First, note that
$dx'/dt=(\dot{x})'=p'\cos\theta-p\sin\theta\cdot\theta'$ and
similarly $dy'/dt=p'\sin\theta+p\cos\theta\cdot\theta'$. Also,
$d\theta'/dt=(\dot{\theta})'=p'h+ph'$. Hence
$$
\dot{v}=p'+phw\ \ \ \ {\rm and}\ \ \ \ \dot{w}=p\theta'-phv=p\kappa w
$$
Then direct differentiation of the equations (\ref{alphakappa})
and substitution of (\ref{p'h'}) completes the proof.
$\Box$.\medskip
Let $\tau$ be the length parameter along the trajectory
$(x_{0t},y_{0t})$ on the torus, i.e. $d\tau/dt=p$. Then
the equations (\ref{kappat})-(\ref{alphat}) can be rewritten as
\be
d{\kappa}/d\tau=-\kappa^2-h^2+h_{R}
\label{kappatau}
\ee
\be
d{w}/d\tau=\kappa w
\label{wtau}
\ee
and
\be
d{\alpha}/d\tau=-\kappa\alpha +h+p_{U}\alpha/p+p_R/p
\label{alphatau}
\ee
Now consider a reflection at some $\partial {\cal
B}_i\subset\partial Q$ experienced by the family $P_{st}$. For
every $s$, the trajectory $P_{st}$ reflects in $\partial {\cal
B}_i$ at some moment of time $t=t_s$. The outgoing velocity
vectors of these trajectories (taken immediately after the
reflection) make a curve $\gamma$ in $M$, we call it the {\em
trace} of the family $P_{st}$ (on $M$). Let $\gamma$ satisfy an
equation $\varphi=\varphi(r)$ in the coordinates $r,\varphi$
introduced after (\ref{M}). Note that the curve $\gamma$ is also
parametrized by $s$, because each point corresponds to a
trajectory of the family $P_{st}$.
The quantities $\theta,\alpha,\kappa,w,v,p,h$ may change
discontinuously at the reflection. Denote by
$\theta^-,\alpha^-,\kappa^-$, etc., their values before the
reflection and by $\theta^+,\alpha^+,\kappa^+$, etc., their values
after the reflection. Actually, we have $p^+=p^-$ by
(\ref{ptheta+-}).
\begin{lemma}
The derivative $t'=dt_s/ds$ satisfies
\be
t'=\mp (w^{\pm}\tan\varphi\pm v^{\pm})/p^{\pm}
\label{t'}
\ee
The derivative $dr/ds$ on $\gamma$ satisfies
\be
dr/ds=\mp w^{\pm}/\cos\varphi
\label{drds}
\ee
The derivative of the function $\varphi=\varphi(r)$ satisfies
\be
d\varphi/dr=\mp K(r)+\kappa^{\pm}\cos\varphi\mp h^{\pm}\sin\varphi
\label{dvarphidr}
\ee
\end{lemma}
Recall that $K(r)>0$ is the curvature of the boundary $\partial
{\cal B}_i\subset\partial Q$ at the point $r\in\partial Q$ on the torus
$\IT^2$.
{\em Proof}. Let the boundary $\partial {\cal B}_i$ satisfy an equation
$G(x,y)=0$, where the function $G$ is chosen so that its gradient
vector $(G_x,G_y)$ is a normal vector to $\partial {\cal B}_i$ pointing
inside $Q$ (outside ${\cal B}_i$). Then $t_s$ satisfies the equation
$G(x_{st_s},y_{st_s})=0$. Differentiating with respect to $s$
before and after the reflection gives, respectively,
\be
[(x')^{\pm}+(\dot{x})^{\pm}t']G_x+[(y')^{\pm}+(\dot{y})^{\pm}t']G_y=0
\label{x'xd}
\ee
A simple geometric analysis shows that, in the orientation of
$\varphi$ specified after (\ref{M}), we have
\be
G_y/G_x=\tan(\theta^++\varphi)=\tan(\theta^--\varphi)
\label{GyGx}
\ee
Solving (\ref{x'xd}) for $t'$ then gives
$$
t'=-\frac{(x')^{\pm}+(y')^{\pm}\tan(\theta^{\pm}\pm\varphi)}{(\dot{x})^{\pm}
+(\dot{y})^{\pm}\tan(\theta^{\pm}\pm\varphi)}
$$
Substitution of (\ref{dots}) and (\ref{x'y'}) yields
(\ref{t'}).
Next, we have $|dr/ds|=\sqrt{(x'+\dot{x}t')^2+(y'+\dot{y}t')^2}$,
where $x',\dot{x},y',\dot{y}$ are all taken either before the
reflection or after it. Using (\ref{dots}), (\ref{x'y'}) and
(\ref{t'}) and taking into account our orientation of the
coordinate $r$ (to determine the sign of $dr/ds$) yields
(\ref{drds}).
Another simple geometric inspection shows that
$K(r)=-d(\tan^{-1}(G_x/G_y))/dr$ in our orientation of the
coordinate $r$. Therefore, using (\ref{GyGx}) gives
$$
d\varphi/dr=-K(r)-d\theta^+/dr=K(r)+d\theta^-/dr
$$
Lastly,
$$
\frac{d\theta^{\pm}}{dr}=\frac{d\theta^{\pm}}{ds}\Big/\frac{dr}{ds}=
\mp\Big [(\theta')^{\pm}+(\dot{\theta})^{\pm}t'
\Big ]\Big/[w^{\pm}/\cos\varphi]
$$
Now substituting (\ref{dots}), (\ref{theta'}), and (\ref{t'})
proves (\ref{dvarphidr}). Lemma is proved. $\Box$\medskip
\begin{lemma}
At each reflection, we have $v^+=v^-$, i.e. $v$ remains unchanged.
Also, $w^+=-w^-$ and hence $\alpha^+=-\alpha^-$. The variable
$\kappa$ changes by the rule
\be
\kappa^+=\kappa^-+\Delta\kappa
\label{kappa+-}
\ee
where
\be
\Delta\kappa=\frac{2K(r)+(h^++h^-)\sin\varphi}{\cos\varphi}
\label{Deltakappa}
\ee
Generally, there is no relation between $h^+$ and $h^-$.
\label{lm+-}
\end{lemma}
All this directly follows from the previous lemma. $\Box$\medskip
Note that by setting $h\equiv 0$ in (\ref{Deltakappa}) we recover
the well known equation $\Delta\kappa=2K(r)/\cos\varphi$ for
billiards derived by Sinai, see e.g. \cite{Si70,BSC90}.
Observe that $\Delta\kappa$ does not depend on the family of
trajectories $P_{st}$ (i.e., on $\alpha,\kappa,w$). It only
depends on the point $(r,\varphi)\in M$. Hence it is a (smooth)
function on the cross-section $M$, we call it $\Theta(r,\varphi)$,
i.e.
$$
\Theta(r,\varphi)=\frac{2K(r)+(h^++h^-)\sin\varphi}{\cos\varphi}
$$
Note that this function has a positive lower bound,
$$
\Theta(r,\varphi)\geq\Theta_{\min}=2\min_r K(r)-2\delta_0>0
$$
but it is not bounded above (as, indeed, $\cos\varphi$ may be arbitrary
close to zero during almost `grazing' reflections).
We now consider a family of trajectories that are divergent before
some reflection, i.e. assume $\kappa^->0$. Then
$\kappa^+>\Theta_{\min}$ by (\ref{kappa+-}), so the curvature of
the family is big enough after the reflection. Denote by $L$ the
length of the trajectory on the torus between the current and the
next reflection points, and parametrize this trajectory segment by the
length parameter $\tau$, $0\leq\tau\leq L$. Recall that the free
path between consecutive reflections is uniformly bounded, hence
$L_{\min}\leq L\leq L_{\max}$.
\begin{lemma}
Let $\kappa^->0$. Then
\be
\frac{1}{(\kappa^+)^{-1}+\tau}-\delta_1
\leq\kappa_{\tau}\leq
\frac{1}{(\kappa^+)^{-1}+\tau}+\delta_1
\label{kappatau2}
\ee
for all $0<\tau0$. Hence,
\be
-(\kappa+\delta')^2
\leq d{\kappa}/d\tau\leq
-(\kappa-\delta')^2+(\delta'')^2
\label{kappatau3}
\ee
where $\delta'=(2\delta_0)^{1/2}$ and
$\delta''=(4\delta_0)^{1/2}$. The smallness of $\delta',\delta''$
and the initial bound $\kappa_0=\kappa^+\geq\Theta_{\min}>0$
allows direct integration of (\ref{kappatau3}) resulting in
$$
\frac{1}{(\kappa^++\delta')^{-1}+\tau}-\delta'
\leq\kappa_{\tau}\leq
\delta''\frac{Ae^{2\delta''\tau}+1}{Ae^{2\delta''\tau}-1}+\delta'
$$
where
$$
A=\frac{\kappa^+-\delta'+\delta''}{\kappa^+-\delta'-\delta''}
$$
(this, in particular, justifies the assumption $\kappa>0$).
Since $\delta',\delta''$ are small, one can now easily obtain
(\ref{kappatau2}) with $\delta_1=\delta'+2\delta''$.
$\Box$
\medskip\noindent{\bf Convention} (on $\delta$'s). Throughout the paper,
we denote by $\delta_i$ various small constants that depend on the
domain $Q$ and $\delta_0$ in Assumption~B so that all $\delta_i\to
0$ as $\delta_0\to 0$. Hence, all those constants are effectively
assumed to be small enough.\medskip
Denote
$$
\kappa_{\min}:=\frac{1}{\Theta_{\min}^{-1}+L_{\max}}-\delta_1
\ \ \ \ {\rm and}\ \ \ \
\kappa_{\max}^-:=\frac{1}{L_{\min}}+\delta_1
$$
\begin{corollary}
If a family of trajectories is divergent before a reflection at
time $t_0$, i.e., $\kappa_{t_0-0}>0$, then for all $t>t_0$ we have
$\kappa_t>\kappa_{\min}>0$, i.e. the curvature of the family stays
bounded away from zero. In addition, at each reflection that
occurs after the time $t_0$ we have $\kappa^-\leq\kappa_{\max}^-$,
i.e. the curvature of the family falling upon $\partial Q$ is
uniformly bounded above.
\label{crkappa1}
\end{corollary}
We call a family of trajectories $P_{st}$ {\em strongly divergent}
on a time interval $(t_1,t_2)$ if $\kappa_t\geq\kappa_{\min}$ for
all $t_1t_1$). We
emphasize the following:
\medskip\noindent{\bf ``Invariance principle''}. Any strongly
divergent family of trajectories remains strongly divergent in the
future under the flow $\Phi^t$. We note that later on some additional
restrictions on the class of strongly convergent families will be
assumed (see, e.g., the convention on $\alpha$'s below), but this
invariance principle will hold.\medskip
\noindent {\bf Remark}. We do not assume that the derivatives
$p_x,p_y,p_{\theta}$ are small, they are just bounded as the
function $p$ is smooth on a compact manifold $\cal M$. It is
important, though, that the function $p_U=\dot{p}/p=d(\ln p)/dt$
has uniformly bounded integrals along any orbit segment of the
flow:
$$
\left |\int_{t_1}^{t_2}p_U\, dt\right
|\leq{\rm const}=\ln(p_{\max}/p_{\min})<\infty
$$
for any $t_1t_1$ (where $t_1$ depends on
$\alpha_{t_0}$). Moreover, if $|\alpha_{t_0}|<\alpha_{\max}$, then
$|\alpha_t|\leq\bar{\alpha}_{\max}$ for all $t>t_0$.
\label{lmalpha}
\end{lemma}
{\em Proof}. At every reflections, $\alpha$ simply changes sign,
i.e. $|\alpha^+|=|\alpha^-|$. Due to (\ref{alphatau}), we have
\be
d{\alpha}/d\tau=-\kappa(\alpha-p_{\theta}/p) +(p_{U}/p)\alpha+
(-p_x\sin\theta+p_y\cos\theta)/p+h
\label{alphatau2}
\ee
Note that the terms $p_{\theta}/p$, $p_U/p$, and
$(-p_x\sin\theta+p_y\cos\theta)/p+h$ are uniformly bounded. Since
$\kappa\geq\kappa_{\min}>0$, the first term in (\ref{alphatau2})
drives $\alpha$ back whenever it gets too large. The influence of
the second term, $(p_U/p)\alpha$, is uniformly bounded by the
previous remark. $\Box$\medskip
\noindent{\bf Convention} (on $\alpha$'s). In all that follows, we
will only consider strongly divergent families that satisfy
$|\alpha_t|\leq\alpha_{\max}$ for all relevant $t$. We also assume
that the ``invariance principle'' holds, as we may in view of
Lemma~\ref{lmalpha}.
%\medskip
\begin{lemma}
For any strongly divergent family of trajectories on an interval
$(t_0,\infty)$, its width $|w|$ grows exponentially in time:
$$
|w_t|=|w_{t_0}|\exp \left (\int_{t_0}^t p_u\kappa_u\,
du\right )\geq |w_{t_0}|\, e^{c(t-t_0)}
$$
where $c=p_{\min}\kappa_{\min}>0$.
\end{lemma}
We also need the invariance and exponential growth
for sufficiently convergent families of trajectories
as they flow backwards in time.
The following trick will do the job.\medskip
\noindent {\bf Time reversal principle}. There is a convenient way
to study backward dynamics $\Phi^{t}$ as $t\to-\infty$.
Consider the involution map ${\cal I}:\, (x,y,\theta)\mapsto
(x,y,\theta+\pi)$ on $\cal M$. The flow
$$
\Phi_-^t:={\cal I}\circ\Phi^{-t}\circ {\cal I}
$$
is governed by the equations
\be
\dot{x}=p_-\cos\theta,\ \ \ \ \dot{y}=p_-\sin\theta,\ \ \ \
\dot{\theta}=p_-h_-
\label{h-}
\ee
where $p_-(x,y,\theta)=p(x,y,\theta+\pi)$ and
$h_-(x,y,\theta)=-h(x,y,\theta+\pi)$.
So, equations (\ref{h-}) are similar to (\ref{h}).
The new flow $\Phi_-^t$ satisfies Assumption~A,
quite obviously, and Assumption~B, because the
function $h_-$ and its partial derivatives are the negatives of
those of $h$. Thus, all the properties of the flow $\Phi^t$ also
hold for $\Phi_-^t$.
It is clear that convergent families of trajectory and their
backward evolution correspond to divergent families of the flow
$\Phi^t_-$ and their forward evolution. Hence, all the properties
we proved and assumptions we made for divergent families have
their counterparts for convergent families. We will say that a
family is {\em strongly convergent} on an interval $(t_t,t_2)$ if
$\kappa_t\leq -\kappa_{\max,-}$ where $\kappa_{\max,-}>0$ is the
constant defined just as $\kappa_{\max}$, but for the flow
$\Phi_-$. Our convention on $\alpha$'s and the ``invariance
principle'' (under the backward dynamics) apply to strongly
convergent families, and they grow (in terms of the width $w$)
exponentially in time as $t\to -\infty$. \medskip
\noindent {\bf Remark}. In a particular case where
$$
p(x,y,\theta)=p(x,y,\theta+\pi)\ \ \ \ {\rm and}
\ \ \ \ h(x,y,\theta)=-h(x,y,\theta+\pi)
$$
the flows $\Phi^t$ and $\Phi^t_-$ coincide. Then we say that
the flow $\Phi^t$ is time reversible. Time reversibility is
quite common in many models of direct physical origin. For
example, potential forces (type 1) are always time reversible.
The model (\ref{EBalpha}) is time reversible, though, only
if ${\bf B}\neq 0$. Generally, time reversibility does not
follow from Assumptions~A and B.
\medskip
We now arrive at the first major theorem.
\begin{theorem}[Hyperbolicity]
The flow $\{\Phi^t\}$ on $\cal M$ is hyperbolic with respect to
any invariant measure, i.e. it has one positive and one negative
Lyapunov exponent almost everywhere. The unstable tangent vector
$dP^u=(dx^u,dy^u,d\theta^u)$ at a point $P\in{\cal M}$
corresponds to a family of
trajectories that is strongly divergent at all times
($-\infty1$ such that for every nonzero unstable
vector $V=(dr,d\varphi)$ we have $B_1^{-1}\leq d\varphi/dr\leq
B_1$. Similarly, for any stable vector $V\neq 0$ we have $-B_1\leq
d\varphi/dr\leq -B_1^{-1}$. As a result, the angles between
stable and unstable vectors are bounded away from zero. \label{lmB1}
\end{lemma}
{\em Proof}. The lemma follows from (\ref{dvarphidr}), the bound
$0<\kappa^- \leq\kappa_{\max}^-$ in Corollary~\ref{crkappa1}, and
a similar bound $-\kappa_{\max}^-\leq\kappa^+<0$ for strongly
convergent families. $\Box$\medskip
\noindent{\bf Convention} (on $B$'s), We denote by $B_i>0$
constants that only depend on the domain $Q$ and the bounds
on the function $p(x,y,\theta)$ and its derivatives. Such constants are
called {\em global constants}.\medskip
\noindent{\bf Remark}. All our claims about unstable vectors here
have their obvious counterparts for stable vectors, as in the
above lemma. For brevity, we will only state the claims for
unstable vectors. \medskip
\begin{lemma}[Uniform hyperbolicity - 2]
Let $V$ and $\tilde{V}$ be two unstable vectors at a point $X\in M$
and $T^n$ continuous at $X$. Then the angle between the unstable
vectors $DT^n(V)$ and $DT^n(\tilde{V})$ at the point $T^nX$ is less than
$C\lambda^n$, where $C>0$ and $\lambda\in (0,1)$ are global constants.
\end{lemma}
In other words, the cones made by unstable vectors shrink uniformly
and exponentially fast under $DT^n$ as $n\to\infty$.\medskip
{\em Proof}. Let $V$ and $\tilde{V}$ be tangent vectors to the
traces $\varphi=\varphi(r)$ and $\varphi=\tilde{\varphi}(r)$ of
two strongly unstable families $P_{st}$ and $\tilde{P}_{st}$.
According to (\ref{dvarphidr}), $|d\varphi/dr-d\tilde{\varphi}/dr|
\leq |\kappa^--\tilde{\kappa}^-|\cos\varphi$, hence it is
enough to prove that $|\kappa_n^--\tilde{\kappa}_n^-|\leq C\lambda^n$,
where $\kappa^-_n$ and $\tilde{\kappa}^-_n$ are taken at the point
$T^nX$ before the reflection. Note that $\Delta:=\kappa-\tilde{\kappa}$
satisfies $d\Delta/d\tau=-(\kappa+\tilde{\kappa}-h_{\theta})\Delta$
according to (\ref{kappatau}), and does not change
at reflections due to Lemma~\ref{lm+-}. Hence,
$|\Delta_{\tau}|\leq |\Delta_0|e^{-a\tau}$ where
$a=2\kappa_{\min}-\delta_0>0$. $\Box$\medskip
Now denote by $V_1=(dr_1,d\varphi_1)=DT(V)$ the image of
vector $V$ under $DT$. It is a tangent vector at $X_1=TX$. If $V$
is an unstable vector, then so is $V_1$. Let $V$ and $V_1$ be
tangent vectors to the traces left on $M$ by a strongly divergent
family $P_{st}$ at the points $X$ and $X_1$, respectively.
Denote by $L$ the length of the trajectory segment on the torus
between the points $X$ and $X_1$, and parametrize that segment by
the length parameter $\tau$, $0\leq\tau\leq L$. Denote by
$w^+,\kappa^+$, etc. the quantities introduced in Sect.~\ref{secH}
taken for the family $P_{st}$ immediately after the reflection at
the point $X$, and by $w^-_1,\kappa^-_1$, etc. the corresponding
quantities before the reflection at the point $X_1$.
\begin{lemma}
For any unstable vector $V$
\be
e^{-\delta_2}[1+\kappa^+L]
\leq\frac{|w_1^-|}{|w^+|}\leq
e^{\delta_2}[1+\kappa^+L]
% 1+(1-\delta_2)\kappa^+L
% \leq\frac{|w_1^-|}{|w^+|}\leq
% 1+(1+\delta_2)\kappa^+L
\label{w1-w+}
\ee
with some small $\delta_2>0$.
\label{lmw1-w+}
\end{lemma}
{\em Proof}. Combining (\ref{wtau}) and (\ref{kappatau2}) and
integrating with respect to $\tau$ from 0 to $L$ yields
(\ref{w1-w+}) with $\delta_2:=\delta_1L_{\max}$. $\Box$\medskip
Note that integrating from 0 to any $\tau0$
for every stable and unstable vector $V\neq 0$ due to
Lemma~\ref{lmB1}. Now (\ref{w1-w+}) can be rewritten as
\be
e^{-\delta_2}[1+\kappa^+L]
\leq\frac{|V_1|_p}{|V|_p}\leq
e^{\delta_2}[1+\kappa^+L]
\label{V1pVp}
\ee
because $|V_1|_p/|V|_p=|w_1^-|/|w^+|$, as it follows by
applying (\ref{drds}) to $V$ and $V_1$.
Note that in the pure billiard dynamics $\delta_2=0$, and we
recover a standard formula $|V_1|_p/|V|_p=1+\kappa^+L$, see
\cite{Si70}.
The inequality (\ref{V1pVp}) shows that the p-norm of unstable
vectors grows monotonically and exponentially in time (= the
number of collisions), i.e. for all $n\geq 1$
\be
|DT^n(V)|_p/|V|_p\geq\Lambda^n
\label{DTnVp}
\ee
where $\Lambda>1$ is a global constant, say
\be
\Lambda=1+\kappa_{\min}L_{\min}/2
\label{Lambda}
\ee
The p-metric
plays the role of the so called adapted metric of Axiom~A systems.
It also follows from (\ref{Deltakappa}) and (\ref{V1pVp}) that the
expansion of $V$ under $DT$ is mainly determined by $\cos\varphi$:
\be
\frac{B_2^{-1}}{\cos\varphi}\leq
\frac{|V_1|_p}{|V|_p}\leq\frac{B_2}{\cos\varphi}
\label{B2}
\ee
for some constant $B_2>0$.
We will primarily work with the Euclidean metric
$|V|=\sqrt{(dr)^2+(d\varphi)^2}$. It is clear that for stable and
unstable vectors $V\neq 0$, which satisfy Lemma~\ref{lmB1}, we
have
\be
1\leq\frac{|V|\cos\varphi}{|V|_p}\leq B_3
\label{B3}
\ee
for some constant $B_3>0$. Then (\ref{B2}) and (\ref{B3}) imply
\be
\frac{B_4^{-1}}{\cos\varphi_1}\leq
\frac{|DT(V)|}{|V|}\leq\frac{B_4}{\cos\varphi_1}
\label{B4}
\ee
for some constant $B_4>0$.
\begin{lemma}[Uniform hyperbolicity - 3]
For any unstable vector $V$ where $DT^n$ is defined
\be
|DT^n(V)|/|V|\geq B_5\Lambda^n
\label{DTnV}
\ee for global constants $\Lambda>1$ and $B_5>0$.
\label{lmuh3}
\end{lemma}
{\em Proof}. Indeed, due to (\ref{DTnVp}), (\ref{B2}) and (\ref{B3})
$$
|DT^n(V)|\geq |DT^n(V)|_p\geq \Lambda^{n-1}|DT(V)|_p\geq
\Lambda^{n-1}\frac{B_2^{-1}|V|_p}{\cos\varphi}\geq
\Lambda^{n-1}B_2^{-1}B_3^{-1}|V|
$$
\section{The properties of the billiard map $T:M\to M$}
\label{secBMT}
\setcounter{equation}{0}
Here we study stable and unstable curves, and singularity curves,
for the billiard map $T$ on the cross-section $M$. Certain
technical properties of those curves are necessary for the
construction and further study of SRB measures. In the theory of
dynamical systems, these properties are called {\em curvature
bounds, distortion bounds, absolute continuity, alignment} etc.
The proofs of these properties are, unfortunately, quite involved.
To make things worse, the proofs are not always available even in
the pure billiard case -- some of these facts are just known as
folklore, whose proofs have never been published. For the sake of
completeness, we provide here full proofs of all these
facts.\medskip
\noindent{\bf Definition}. A smooth curve $\gamma\subset M$
given by $\varphi=\varphi(r)$ is called an {\em unstable curve}
(or a {\em stable curve}) if it is
the trace of a strongly divergent (resp., strongly convergent)
family of trajectories.\medskip
Our ``invariance principle'' for strongly divergent families
implies that the class of unstable curves
is invariant under $T^n$, $n\geq 1$, and the class of stable
curves is invariant under $T^{-n}$, $n\geq 1$. We will refer to this
as the ``invariance principle'' for unstable curves.
\begin{lemma}[Curvature bounds]
There are constants $B_{\max}$ and $\bar{B}_{\max}$ such that for
any $C^2$ smooth unstable curve $\gamma$ its images $T^n\gamma$
satisfy $|d^2\varphi/dr^2|\leq B_{\max}$ eventually, for all
$n\geq n_{\gamma}$. Moreover, if $\gamma$ itself satisfies
$|d^2\varphi/dr^2|\leq B_{\max}$, then all its images $T^n\gamma$,
$n\geq 1$, satisfy $|d^2\varphi/dr^2|\leq \bar{B}_{\max}$.
\label{lmcur}
\end{lemma}
We note that a similar property for pure billiard dynamics
is known \cite{Y98,Ch99b}, but hardly a complete proof was
ever published. Our proof certainly covers the pure billiard case.
\medskip
{\em Proof}. Let $\varphi=\varphi(r)$ be an unstable curve, the
trace of a strongly divergent family $P_{st}$. Differentiating
(\ref{dvarphidr}) gives
$$
\frac{d^2\varphi}{dr^2}=\frac{dK(r)}{dr}+\frac{d\kappa^-}{dr}\cos\varphi-
\kappa^-\sin\varphi\frac{d\varphi}{dr}+\frac{dh^-}{dr}\sin\varphi+
h^-\cos\varphi\frac{d\varphi}{dr}
$$
Since $\partial Q$ is $C^3$ smooth, the term $dK/dr$ is
bounded. The term $\kappa^-$ is bounded by
Corollary~\ref{crkappa1}, and $d\varphi/dr$ is bounded by
Lemma~\ref{lmB1}. Now, using (\ref{p'h'}) and (\ref{t'}) gives
$dh^-/ds=h_R^-w^-+h_U^-w^-\tan\varphi$. Hence, due to (\ref{drds}),
$dh^-/dr=(dh^-/ds)/(dr/ds)=h^-_R\cos\varphi+h^-_U\sin\varphi$,
so $|dh^-/dr| \leq (4+\kappa_{\max}^-)\delta_0$.
It then remains to estimate the term $d\kappa^-/dr$. First, according
to (\ref{drds})
$$
\frac{d\kappa^-}{dr}=\frac{d\kappa^-}{ds}\Big /
\frac{dr}{ds}=[(\kappa')^-+(\dot{\kappa})^-t']\cdot\frac{\cos\varphi}{w^-}
$$
Substituting (\ref{kappat}) and (\ref{t'}) gives
$$
d\kappa^-/dr=(\kappa'/w)^-\cos\varphi-
[(\kappa^-)^2+(h^-)^2-h_R^-]\cdot (\sin\varphi-\alpha^-\cos\varphi)
$$
Here all the terms are bounded except, possibly, the term
$(\kappa'/w)^-$. So, it is enough to
prove that the quantity $ \Xi:=\kappa'/w$ is bounded by a
global constant before every reflection. Direct differentiation
and using (\ref{kappat}) yields
$$
d\kappa'/dt=d\dot{\kappa}/ds=-2p\kappa\kappa'
-p_{\theta}w\kappa^3-D_1w\kappa^2+ph_{\theta}\kappa'
$$
where $D_1$ is an expression involving first and second order
derivatives of the functions $p$ and $h$. All those derivatives
are bounded, since these functions are $C^2$ smooth on a compact
manifold $\cal M$, hence $|D_1|$ is bounded by a global constant.
Now, by using (\ref{wt}),
\begin{eqnarray}
d\Xi/dt&=&(d\kappa'/dt)/w-\kappa'\dot{w}/w^2\nonumber\\
&=&-3p\kappa\Xi+ph_{\theta}\Xi-p_{\theta}\kappa^3-D_1\kappa^2
\label{dXidt}
\end{eqnarray}
Now consider a reflection experienced by the family $P_{st}$ and
denote by $\Xi^-$ and $\Xi^+$ the values of $\Xi$ before and after
the reflection. Differentiating (\ref{kappa+-})-(\ref{Deltakappa})
gives
\be
\frac{d\kappa^+}{dr}=\frac{d\kappa^-}{dr}+\frac{2K(r)\sin\varphi}{\cos^2\varphi}
\cdot\frac{d\varphi}{dr}+\frac{2K'}{\cos\varphi}
+\frac{H_1}{\cos^2\varphi}-h_{\theta}^+\kappa^+\sin\varphi
\label{dkappa+dr}
\ee
Here we denote $K'=dK/dr$, which is bounded on $\partial Q$,
and $H_1$ is a small quantity, see below.\medskip
\noindent{\bf Convention} (on $D$'s and $H$'s), We will denote by
$D_i$ variable quantities whose absolute values are bounded above
by global constants, i.e. $|D_i|\leq B_i$ for some global constant
$B_i$. We will also denote by $H_i$ variable quantities whose
absolute values are bounded by some small constants depending on
$\delta_0$ in Assumption~B, i.e. $|H_i|\leq {\delta}^{\ast}_i$
where $\delta_i^{\ast}\to 0$ as $\delta_0\to 0$, i.e.
$\delta_i^{\ast}$ satisfy our convention on $\delta$'s.
\medskip
Note that
$d\kappa^+/dr=(d\kappa^+/ds)/(dr/ds)=[(\kappa')^++(\dot{\kappa})^+t']/(-w^+/\cos\varphi)$
and, similarly,
$d\kappa^-/dr=[(\kappa')^-+(\dot{\kappa})^-t']/(w^-/\cos\varphi)$,
where we used (\ref{drds}). Substituting these into
(\ref{dkappa+dr}) and using (\ref{kappat}), (\ref{dvarphidr}),
(\ref{kappa+-}), and (\ref{t'}) yields
\be
\Xi^+=-\Xi^-+\Delta\Xi
\label{Xi+-}
\ee
where
%\begin{eqnarray}
% \Delta\Xi=&-&\frac{2K(r)\sin\varphi}{\cos^3\varphi}\cdot\frac{d\varphi}{dr}
% -\frac{2K'}{\cos^2\varphi}-\frac{H_1}{\cos^3\varphi}+h_{\theta}^+\kappa^+\tan\varphi\nonumber\\
% &-&[(\kappa^+)^2+(h^+)^2-h_R^+](\tan\varphi+\alpha^+)\nonumber\\
% &+&[(\kappa^-)^2+(h^-)^2-h_R^-](\tan\varphi-\alpha^-)
% \label{Xi+Xi-}
%\end{eqnarray}
\be
\Delta\Xi=-\frac{6K^2(r)\sin\varphi}{\cos^3\varphi}
+\frac{D_2}{\cos^2\varphi}+\frac{H_2}{\cos^3\varphi}
\label{DeltaXi}
\ee
Here $D_2$ is an expression involving
$K',\kappa^-,\alpha^{\pm}$ and other bounded quantities.
Eqs. (\ref{dXidt}) and (\ref{Xi+-})-(\ref{DeltaXi}) completely
describe the evolution of the quantity $\Xi$ in time.
Since (\ref{dXidt}) is a linear differential equation,
we can decompose $\Xi=\Xi_1+\Xi_2$ so that
\be
d\Xi_1/dt=-3p\kappa\Xi_1+ph_{\theta}\Xi_1
\ \ \ \ {\rm and}\ \ \ \
d\Xi_2/dt=-3p\kappa\Xi_2+ph_{\theta}\Xi_2-p_{\theta}\kappa^3-D_1\kappa^2
\label{dXidt12}
\ee
and at every reflection
\be
\Xi^+_1=-\Xi^-_1
\ \ \ \ {\rm and}\ \ \ \
\Xi^+_2=-\Xi^-_2+\Delta\Xi
\label{Xi+-12}
\ee
Initially, at a time $t_0+0$ when the family $P_{st}$ just
leaves the curve $\gamma$ (its trace on $M$), we set
$\Xi_1(t_0+0)=\Xi(t_0+0)$ and $\Xi_2(t_0+0)=0$.
Now, since $|\Xi_1|$ does not change during reflections,
the first equation (\ref{dXidt12}) implies that
\be
|\Xi_1(t)|\leq |\Xi_1(t_0)|\cdot e^{-a(t-t_0)}
\ \ \ \ \ \ {\rm for}\ t>t_0
\label{Xi1}
\ee
where $a=3p_{\min}\kappa_{\min}-\delta_0>0$. Hence, the
component $\Xi_1$ converges to zero exponentially fast.
{\em Claim}. There is a global constant $B_6>0$ such that
$|\Xi_2(t-0)|\leq B_6$ for every moment of reflection $t>t_0$.
We prove the claim inductively. Suppose
$|\Xi_2(t_1-0)|\leq B_6$ before a reflection at some time $t_1>t_0$.
During the interval from $t_1$ to the next reflection,
$t_2$, we decompose $\Xi_2=\Xi_{21}+\Xi_{22}$ as in (\ref{dXidt12}),
so that
\be
d\Xi_{21}/dt=-3p\kappa\Xi_{21}+ph_{\theta}\Xi_{21}
\ \ \ \ {\rm and}\ \ \ \
d\Xi_{22}/dt=-3p\kappa\Xi_{22}+ph_{\theta}\Xi_{22}
-p_{\theta}\kappa^3-D_1\kappa^2
\label{dXidt2122}
\ee
and initially set $\Xi_{21}(t_1+0)=-\Xi_2(t_1-0)$ and $\Xi_{22}(t_1+0)=
\Delta\Xi$, where $\Delta\Xi$ is given by (\ref{DeltaXi}) and taken
at the reflection at $t_1$.
Similarly to (\ref{Xi1}), we now have
\be
|\Xi_{21}(t_2-0)|\leq |\Xi_2(t_1-0)|\cdot e^{-a(t_2-t_1)}
\label{Xi21}
\ee
The equation (\ref{DeltaXi}) shows that
$\Xi_{22}(t_1+0)=\Delta\Xi$ is
of order $O(1/\cos^3\varphi)=O(\kappa^3(t_1+0))$. It is then
convenient to ``link'' $\Xi_{22}$ with $\kappa^3$ and consider
the ratio $g(t):=\Xi_{22}(t)/\kappa^3(t)$. First, by
(\ref{Deltakappa}) and (\ref{DeltaXi})
$$
g(t_1+0)=\Xi_{22}(t_1+0)/\kappa^3(t_1+0)\leq B'
$$
with some global constant $B'$. Then, (\ref{dXidt2122}) and
(\ref{kappat}) imply
$$
dg/dt=-p_{\theta}+H_3g+D_3/\kappa
$$
Hence, $|g|$ stays bounded by a global constant between
the two reflections, i.e. $|g(t)|\leq B''$ for all $t_11$ is a large constant to be
specified later. For every $k\geq k_0$ we put
$$
I_k=\{(r,\varphi):\, \pi/2-k^{-2}<\varphi <\pi/2-(k+1)^{-2}\}
$$
and
$$
I_{-k}=\{(r,\varphi):\, -\pi/2+(k+1)^{-2}<\varphi < -\pi/2+k^{-2}\}
$$
and lastly
$$
I_0=\{(r,\varphi):\, -\pi/2+k_0^{-2}<\varphi < \pi/2-k_0^{-2}\}
$$
The domains $I_k$ are called homogeneity strips, they are
also used in the study of pure billiard systems
\cite{BSC91,Y98,Ch99b}.
We say that an unstable curve $\gamma\subset M$ is {\em homogeneous}
if it is entirely contained in one homogeneity strip $I_k$. Note that
if $\gamma$ is a homogeneous unstable curve, then for every point
$X=(r,\varphi)\in\gamma$ we have
\be
\cos\varphi\geq B_9^{-1}|\gamma|^{2/3}
\label{B9}
\ee
where $B_9>0$ is a global constant. Here and on $|\gamma|$
denotes the length of $\gamma$ in the Euclidean metric
$(dl)^2=(dr)^2+(d\varphi)^2$.
%We will also denote by dist$_{\gamma}(X,Y)$ the distance between
%points $X,Y\in\gamma$ along the curve $\gamma$.
Let $\gamma$ be an unstable curve, $X\in\gamma$ and $T^n$
continuous at $X$. Denote by $J_{\gamma,n}(X)$ the expansion
factor of the curve $\gamma$ under $T^n$ at the point $X$, i.e.
$J_{\gamma,n}(X):=|DT^nV|/|V|$
for any tangent vector $V$ to $\gamma$ at $X$.
\begin{lemma}[Distortion bounds]
Let $\gamma$ be an unstable curve on which $T^n$ is continuous.
Assume that $\gamma_i:=T^i\gamma$ is a homogeneous unstable curve
for each $0\leq i\leq n$. Then for all $X,Y\in\gamma$
$$
|\ln J_{\gamma,n}(X)-\ln J_{\gamma,n}(Y)|\leq B_{10}
|\gamma_n|^b
$$
for some global constants $B_{10}>0$ and $b>0$ (in fact, $b=1/3$).
\label{lmdis}
\end{lemma}
We note that the corresponding property for pure billiard dynamics
is known \cite{Ch99b}, but only a proof of a somewhat weaker
statement was published \cite{BSC91}. Our proof covers
the pure billiard case, too.\medskip
{\em Proof}. Note that
$J_{\gamma,n}(X)=\prod_{i=0}^{n-1}J_{\gamma_i,1}(T^iX)$. Hence, it
is enough to prove the lemma for $n=1$, because $|\gamma_i|$ grows
exponentially in $i$ due to (\ref{DTnV}). So we put $n=1$.
Let $P_{st}$ be a strongly divergent family whose trace on $M$ is
the curve $\gamma$. We will use the notation adopted before
Lemma~\ref{lmw1-w+}. Consider $J_{\gamma,1}(X)$ as a function of
$X_1=(r_1,\varphi_1)=TX\in\gamma_1$, and parametrize $\gamma_1$ by
$r_1$. It is enough to prove that
\be
\left |\frac{d\ln J_{\gamma,1}}{dr_1}\right |
\leq \frac{B}{|\gamma_1|^{2/3}}
\label{dlnJg1}
\ee
for some global constant $B>0$. Then Lemma~\ref{lmdis} (with $n=1$)
would follow by integration over $\gamma_1$.
The bound (\ref{dlnJg1}), in turn, follows from
\be
\left |\frac{d\ln J_{\gamma,1}}{dr_1}\right |\leq
\frac{B\cos\varphi_1}{\cos\varphi}+\frac{B}{\cos\varphi_1}
\label{dJdr1}
\ee
with a global constant $B>0$, by
applying (\ref{B9}) to both $\gamma$ and $\gamma_1$, and because
$|\gamma|\geq B_4^{-1}|\gamma_1|\cos\varphi_1$, which follows
from (\ref{B4}).
We now prove (\ref{dJdr1}). We have
$|V|=|dr|\sqrt{1+({d\varphi}/{dr})^2}=
|ds|\,|w^+|(\cos\varphi)^{-1}\sqrt{1+({d\varphi}/{dr})^2}$,
and similarly for $|V_1|$, hence
$$
J_{\gamma,1}(X)=\frac{|V_1|}{|V|}=\frac{|w_1^-|}{|w^+|}\cdot
\frac{\cos\varphi}{\cos\varphi_1}\cdot
\frac{\sqrt{1+(d\varphi_1/dr_1)^2}}{\sqrt{1+(d\varphi/dr)^2}}
=J'\cdot J''\cdot J'''
$$
where $J',J'',J'''$ simply denote the first, second and third factors
in this expression. We bound them separately. First,
$$
\left |\frac{d\ln J'''}{dr_1}\right |\leq
\frac{|d\varphi_1/dr_1|\cdot |d^2\varphi_1/dr_1^2|}
{1+(d\varphi_1/dr_1)^2}+
\frac{|d\varphi/dr|\cdot |d^2\varphi/dr^2|}
{1+(d\varphi/dr)^2}
\cdot\left |\frac{dr}{dr_1}\right |
$$
Note that $|dr/dr_1|\leq B_4'\cos\varphi_1$
for some global constant $B_4'>0$ due to (\ref{B4}).
Hence, $|d\ln J'''/dr_1|$ is uniformly bounded due to
Lemmas~\ref{lmB1} and \ref{lmcur}.
Next,
$$
\left |\frac{d\ln J''}{dr_1}\right |\leq
\left |\frac{d\varphi_1/dr_1}{\cos{\varphi_1}}\right |+
\left |\frac{d\varphi/dr}{\cos{\varphi}}\right |
\cdot\left |\frac{dr}{dr_1}\right |\leq
\frac{B_1}{\cos{\varphi_1}}+
\frac{B_1B_4'\cos\varphi_1}{\cos{\varphi}}
$$
as required by (\ref{dJdr1}).
It remains to consider $\ln J'(X)=\int_{t_0}^{t_1}\kappa p\, dt$, cf.
(\ref{wt}), where $t_0$ and $t_1$ denote the moments of reflection
at $X$ and $X_1$, respectively. First,
$d\ln J'/dr_1=(d\ln J'/ds)/(dr_1/ds)$, and
\begin{eqnarray*}
d\ln J'/ds
&=&-\kappa^+p^+dt_0/ds+\kappa^-_1p_1^- dt_1/ds
+\int_{t_0}^{t_1} (\kappa p'+\kappa'p)\,dt\nonumber\\
&=&\kappa^+(w^+\tan\varphi+v^+)+\kappa^-_1(w_1^-\tan\varphi_1-v_1^-)
+\int_{t_0}^{t_1} (\kappa p'+\kappa'p)\,dt
% \label{dJ'ds}
\end{eqnarray*}
Note that $|w^+/w^-_1|\leq 2(\kappa^+L_{\min})^{-1}$
by (\ref{w1-w+}) and for all $w=w(t)$, $t_00$
\be
B_{11}^{-1}e^{-\delta_4 t}<|d\Phi^t(X)|0$ and a global constant $B_{11}$.
Also, let $P_{st}$ be a strongly convergent or divergent family
on a time interval $(t_1,t_2)$ that does not experience
singularities (grazing reflections) for $t_10$.
We now prove the boundedness of curvature. The above natural
parametrization $r=s$ does not satisfy our convention on
$\alpha$'s when $k$ is large. But for any point $X=(r,\varphi)\in\gamma$
we can reparametrize the outgoing family $P_{st}$,
$t>0$, with a new parameter $s$ so that $v^+=0$ and $w=1$ at $X$.
In this parametrization, as one can compute directly,
$$
(\kappa')^+=-\frac{dK(r)/dr+\sin\varphi_0dh^+/dr}{\cos^2\varphi_0}
+\frac{p\sin\varphi_0 [(\kappa^+)^2+(h^+)^2-h_R^+]^2}{\cos^3\varphi_0}
$$
Now we see that $(\kappa')^+=O(\cos^{-3}\varphi_0)=O((\kappa^+)^3)$,
so we are in the position of the proof of the claim in the proof
of Lemma~\ref{lmcur}. Just like then, we get a uniform bound
$\kappa'\leq B_6$ before the next reflection occurs. This proves
that the curvature of the curve $T\gamma$ is bounded by
a global constant. $\Box$\medskip
\begin{corollary}
Unstable curves are uniformly transversal to
the boundary $\partial M={\cal S}_0$
and to the components of the singularity set ${\cal S}_{-n}$, $n\geq 1$
(and to those of ${\cal D}_{-n}$, $n\geq 1$).
Stable curves are uniformly transversal to
the boundary $\partial M={\cal S}_0$
and to the components of the singularity set ${\cal S}_{n}$, $n\geq 1$
(and to those of ${\cal D}_n$, $n\geq 1$).
\end{corollary}
The following continuation property is standard
\cite{BSC91,Ch99b}:\medskip
\noindent{\bf Remark (Continuation property)}. Each endpoint, $X$,
of every smooth curve $\gamma\subset {\cal S}_{-n,0}$, $n\geq 1$,
lies either on ${\cal S}_0=
\partial M$ or on another smooth curve $\gamma'\subset
{\cal S}_{-n,0}$ that itself does not terminate at $X$. Hence,
each curve $\gamma\in {\cal S}_{-n,0}$ can be continued
monotonically up to ${\cal S}_0=\partial M$ by other curves in
${\cal S}_{-n,0}$.
\section{Growth of unstable curves}
\label{secGUC} \setcounter{equation}{0}
Here we discuss iterations of unstable curves under the action of
$T$. We prove a version of the so called ``growth lemma'', a key
element in the modern studies of ergodic and statistical
properties of hyperbolic dynamical systems.
Let $\gamma\subset M$ be an unstable curve of small length
$\varepsilon$ and $m\geq 1$. The map $T^m$ is defined on
$\gamma\setminus {\cal S}_{-m,0}$. By the ``invariance principle''
for unstable curves, the set $\gamma_m:= T^m(\gamma\setminus {\cal
S}_{-m,0})$ is a union of some unstable curves. Denote by
$K_m(\gamma)$ the number of those curves (connected components of
$\gamma_m$). By Lemma~\ref{lmuh3} (uniform hyperbolicity) the
total length of $\gamma_m$ is $\geq B_5\Lambda^m\varepsilon$.
However, the effect of growth of $\gamma_m$ with $m$ may be
effectively eliminated if $B_5\Lambda^m\ll K_m(\gamma)$. In that
case applying $T^m$ to $\gamma$ may produce nothing but a bunch of
curves that are even shorter that $\gamma$. If that happens
for all $m$, the very existence of SRB measures would be doubtful,
if not hopeless. Fortunately,
$K_m(\gamma)$ only grows linearly with $m$, provided $\varepsilon$
is small enough. We prove this below.
First, note that $K_m(\gamma)-1$ is the number of points of
intersection $\gamma\cap {\cal S}_{-m,-1}$. A point $X\in M$ where
$k\geq 2$ smooth curves of the set ${\cal S}_{-m,0}$ meet is
called a {\em multiple singularity point}, and $k$ is its {\em
multiplicity}. Denote by $K_m$ the maximal multiplicity of all
$X\in M$ for a given $m$.
\begin{lemma}
For each $m\geq 1$ there is an $\varepsilon_m>0$ such that for any
unstable curve $\gamma\subset M$ of length
$\varepsilon<\varepsilon_m$ we have $K_m(\gamma)\leq K_m$.
\label{lmmult0}
\end{lemma}
The lemma easily follows from the properties of unstable curves
and the singularity set ${\cal S}_{-m,0}$ proved in the previous
section.
\begin{lemma}[Multiplicity bound]
There is a global constant $C_0>0$ such that $K_m\leq C_0m$ for
all $m\geq 1$.
\label{lmmult}
\end{lemma}
We note that a linear bound on $K_m$ was first observed by
Bunimovich for pure billiard dynamics, see \cite{BSC91}. It is now
understood that it is the continuity of the flow $\Phi^t$ that
implies the linear bound on $K_m$. We give a proof of this fact
different from the original one in \cite{BSC91}.\medskip
{\em Proof}. If $K_m$ curves of ${\cal S}_{-m,0}$ meet at $X$,
then a neighborhood $U(X)$ of $X$ is divided by those curves into
some $L_m$ parts (sectors), and clearly $K_m\leq L_m$. We now will
show that $L_m\leq C_0m$ for some $C_0>0$.
On each of the $L_m$ parts of $U(X)$ the map $T^m$ is continuous
and can be extended by continuity to the point $X$. Thus, $T^mX$
can be defined in $L_m$ different ways. To see exactly how that
happens, first note that the real time trajectory $\Phi^tX$ is
well and uniquely defined for all $t>0$. This trajectory may be
tangent to $\partial Q$ at one or more points. We call such points
tangent (grazing) reflections. Now, the $L_m>1$ different versions
of $T^m$ at $X$ are possible precisely when the trajectory
$\Phi^tX$ has tangential reflections: each of those reflections
can be counted as either a ``hit'' (making an iteration of $T$) or
a ``miss'' (skipping it in the construction of $T$).
Note that the real time elapsed until the $m$th iteration of $T$
(in any of its versions) is less than $m\tau_{\max}$. Hence, there
can be no more that $C_1m$ reflections (both tangential and
regular ones) involved in the construction of $T^m$ at $X$, where
$C_1=\tau_{\max}/\tau_{\min}$ is a global constant. Let
$\tilde{m}\leq C_1m$ be the number of tangential reflections among
the first $C_1m$ reflections on the trajectory $\Phi^tX$. It seems
that, with a choice of hit or miss at every tangential reflection,
we would have up to $2^{\tilde{m}}$ versions of $T^m$ at $X$. That
would be too many for us. Fortunately, relatively few sequences of
hits and misses materialize, as we show next.
Note that there can be no more than $C_1$ tangential reflections
in a row. Consider a string of $p$ consecutive tangential
reflections on the trajectory $\Phi^tX$, $t>0$, with $1\leq p\leq
C_1$. Let $Y'=\Phi^{t'}X\in M$ be the last regular reflection
point on the trajectory $\Phi^tX$ before the above string. If
there are previous tangential reflections on $\Phi^tX$, $00$, so they are
increasing curves in the $r,\varphi$ coordinates (by
Lemma~\ref{lmSn}). Now, there are at most $2^p$ possible hit/miss
sequences on the string of $p$ tangential reflections that we have
right after the point $Y'$. Accordingly, $U(Y')$ is divided into
$\leq 2^p$ parts (sectors) along some curves in ${\cal S}_{-p,-1}$, which
are decreasing curves (by Lemma~\ref{lmSn}). So, we have two
partitions of $U(Y')$: one into $L'$ sectors by increasing curves,
and the other into $\leq 2^p$ sectors by decreasing curves. These
two partitions combined divide $U(Y')$ into no more than $L'+2^p$
parts, as it is clear from Fig.~2. So, each string of $p$
consecutive tangential reflections adds $\leq 2^p$ (i.e., $\leq
2^{C_1}$) parts to the partition of $U(X)$ by ${\cal S}_{-m,-1}$.
Hence, $L_m\leq 2^{C_1}C_1m$. $\Box$\medskip
\setlength{\unitlength}{0.012in}
%\begin{figure}
\begin{picture}(300,140)(0,0)
\put(150,70){\circle*{2}}
\put(150,70){\line(2,1){50}}
\put(150,70){\line(1,1){36}}
\put(150,70){\line(1,2){20}}
\put(150,70){\line(-1,-1){36}}
\put(150,70){\line(-1,-2){20}}
\put(150,70){\line(-2,-3){26}}
\put(150,70){\line(-3,-2){44}}
\put(150,70){\line(-2,-1){50}}
\put(150,70){\line(3,-1){54}}
\put(150,70){\line(2,-1){48}}
\put(150,70){\line(-1,1){36}}
\put(150,70){\line(-1,2){20}}
\put(150,70){\line(-2,1){50}}
\put(149,52){$Y'$}
\put(99,27){$L'$}
\put(102,105){$2^p$}
\put(30,0){Figure 2. The partition of the neighborhood $U(Y')$.}
\end{picture}
%\end{figure}
\vspace*{1cm}
Lemmas~\ref{lmmult0}-\ref{lmmult} effectively guarantee the growth
of sufficiently short unstable curves under $T^m$. Precisely, if
$m$ is large enough, and the unstable curve $\gamma$ is short
enough, then the expansion factor $B_5\Lambda^m$ of $\gamma$ under
$T^m$ is larger than the ``cutting factor'' $K_m(\gamma)\leq
C_0m$.
%As far as ${\cal D}_0$ is concerned, the situation seems to be
%even worse: an unstable curve $\gamma$ near $\partial M$ can be
%divided by ${\cal D}_0$ into arbitrary many or even infinitely
%many homogeneous curves. However, the proximity of $\gamma$ to
%$\partial Q$ implies very strong expansion of $T^{-1}\gamma$ under
%$T$ by (\ref{??4.8}). In fact, the preimage of $\gamma$ is so
%strongly expanded under the previous iteration of $T$ that the
%subsequent subdivision of $\gamma$ by ${\cal D}_0$ cannot
%eliminate the overall expansion effect.
We can now proceed exactly as in \cite{Ch99b}. A scheme developed
there for the pure billiard dynamics perfectly works for us here,
it can be repeated almost word by word. We refer the reader to
\cite{Ch99b} and only describe certain major steps in the scheme
necessary for our further analysis.
We start by cutting $M$ along the boundaries of the homogeneity
strips $I_k$ thus making $M=\cup_kI_k$ a disconnected countable
union of strips $I_k$. This makes the map $T$ discontinuous on the
set $\Gamma={\cal S}_{-1}\cup {\cal D}_{-1}$. Note that after
cutting $M$ into these strips, any connected unstable curve
$\gamma\subset M$ will be automatically homogeneous.
Then we fix a higher iteration $T_1=T^m$ of the map $T$, with $m$
picked so that $C_0m < B_5\Lambda^m-1$. The map $T_1$ uniformly
expands unstable vectors: $|DT_1(V)|\geq\Lambda_1|V|$ with
$\Lambda_1:=B_5\Lambda^m>1$ for all unstable vectors $V$ by Lemma
\ref{lmuh3}. The map $T_1$ has singularity set
$\Gamma_1=\Gamma\cup T^{-1}\Gamma\cup\cdots\cup
T^{-m+1}\Gamma={\cal S}_{-m,-1}\cup{\cal D}_{-m,-1}$. Note also
that $\Lambda_1>K_m+1$ by Lemma~\ref{lmmult}, so that $T_1$
expands sufficiently short unstable curves faster than the
singularity set ${\cal S}_{-m,-1}$ breaks them into pieces.
%In order to state our version of the ``growth lemma'' we need some
%notation adopted in \cite{Ch99b}.
For any smooth curve $\gamma\subset M$ we denote by $\rho_\gamma$
the metric on $\gamma$ induced by the Euclidean metric on $M$ and
by $m_\gamma$ the Lebesgue measure on $\gamma$ generated by
$\rho_\gamma$. Note that $m_{\gamma}(\gamma)=|\gamma|$ is the
length of the curve $\gamma$.
An important remark is now in order. Let $\gamma$ be a homogeneous
unstable curve, $n\geq 1$, and $\xi\subset T_1^n\gamma$ any
connected (and hence homogeneous and unstable) curve. Consider the
measure $m^{(n)}_{\xi}:=T_{1,\ast}^n m_{\gamma}|_{\xi}$, i.e. the
image of the Lebesgue measure $m_{\gamma}$ under $T_1^n=T^{mn}$
conditioned on $\xi$. It is a probability measure on $\xi$
absolutely continuous with respect to the Lebesgue measure
$m_{\xi}$, and its density $f^{(n)}_{\xi}=dm^{(n)}_{\xi}/dm_{\xi}$
satisfies
\be
\frac{f^{(n)}_\xi(X)}{f^{(n)}_\xi(Y)}=
\frac{J_{\gamma,mn}(T^{-mn}Y)}{J_{\gamma,mn}(T^{-mn}X)}
\ \ \ \ \ {\rm for\ all}\ X,Y\in \xi
\label{nSRB}
\ee
Lemma~\ref{lmdis} (distortion bounds) implies that
\be
|\ln f^{(n)}_\xi(X)-\ln f^{(n)}_\xi(Y)|\leq B_{10} |\xi|^{1/3}
\label{lnff}
\ee
\noindent{\bf Key Remark}. By making $|\xi|$ smaller, we can make
the the density $f^{(n)}_{\xi}$ almost constant on $\xi$,
uniformly in $\xi$, $\gamma$ and $n$. In what follows we only work
with unstable curves of small length, less than some $\rho_0>0$.
We will assume that $\rho_0$ is small enough, hence all the
measures $m^{(n)}_{\xi}$ on curves $\xi\subset T_1^n\gamma$ will
be almost uniform.\medskip
For $n\geq 1$ denote by $\Gamma_1^{(n)}=\Gamma_1\cup
T_1^{-1}\Gamma_1\cup\cdots\cup T_1^{n-1}\Gamma_1$ the singularity
set for $T_1^n$. For any $\delta>0$ let ${\cal U}_{\delta}$ denote
the $\delta$-neighborhood of the set $\Gamma_1\cup\partial M$.
Let $\rho_0>0$, $n\geq 0$, and $\gamma\subset M$ an unstable
curve (which is automatically homogeneous). Let $\xi\subset\gamma$
be a disjoint union of open subintervals of $\gamma$, and for
every $X\in\xi$ denote by $\xi(X)$ the subinterval of $\xi$
containing the point $X$. We call $\xi$ a $(\rho_0,n)$-subset
(of $\gamma$) if for every $X\in\xi$ the set $T_1^n\xi(X)$ is a
single homogeneous unstable curve of length $\leq\rho_0$ (in
particular, $\xi$ does not intersect the set $\Gamma_1^{(n)}$).
Define a function $r_{\xi,n}$ on $\xi$ by
\be
r_{\xi,n}(X)=\rho_{T_1^n\xi(x)}(T_1^nX,\partial T_1^n\xi(X))
\label{rVn}
\ee
which is simply the distance from $T_1^nX$ to the nearest
endpoint of the curve $T_1^n\xi(X)$ (measured along this curve).
In particular, note that $r_{\gamma,0}(X)=\rho_\gamma(X,\partial
\gamma)$. We will use shorthand
$m_{\gamma}(r_{\xi,n}<\varepsilon)$ for $m_{\gamma}(X\in\xi:\,
r_{\xi,n}(X)<\varepsilon)$
\medskip
\begin{proposition}[``Growth lemma'']
There is a global constant $\alpha_0\in (0,1)$ and positive global
constants $\beta_0,\beta_1,\beta_2,\kappa,\sigma,\zeta$ with the
following property. For any sufficiently small $\rho_0,\delta>0$
and any homogeneous unstable curve $\gamma\subset M$ of length
$\leq\rho_0$, there is an open $(\rho_0,0)$-subset
$\xi^0_{\delta}\subset \gamma\cap{\cal U}_{\delta}$ and an open
$(\rho_0,1)$-subset $\xi^1_{\delta}\subset \gamma\setminus {\cal
U}_{\delta}$ (one of these subsets may be empty) such that
$m_{\gamma}(\gamma\setminus (\xi^0_{\delta}\cup
\xi^1_{\delta}))=0$ and for all $\varepsilon>0$ we have
\be
m_\gamma(r_{\xi^1_{\delta},1}<\varepsilon)\leq \alpha_0\Lambda_1\cdot
m_\gamma(r_{\gamma,0}<\varepsilon/\Lambda_1)
+\varepsilon\beta_0\rho_0^{-1}m_\gamma(\gamma)
\label{rgrowth11}
\ee
\be
m_\gamma(r_{\xi^0_{\delta},0}<\varepsilon)\leq \beta_1\delta^{-\kappa}\,
m_\gamma(r_{\gamma,0}<\varepsilon)
\label{rgrowth10}
\ee
and
\be
m_\gamma(\xi^0_{\delta})=m_{\gamma}(\gamma\cap{\cal U}_{\delta})\leq
\beta_2\, m_\gamma(r_{\gamma,0}<\zeta\delta^{\sigma})
\label{rw0}
\ee
\label{prgrowth}
\end{proposition}
A general meaning of the above inequalities is the following:
(\ref{rgrowth11}) ensures that the curves in the set
$T_1\xi_{\delta}^1$ are, on the average, long enough; (\ref{rw0})
asserts that the total measure of the set $\xi_{\delta}^0$ is
small enough; and (\ref{rgrowth10}) guarantees that the connected
components of $\xi_{\delta}^0$ are not too tiny (hence, they will
grow under $T^n_1$ fast enough).
The proof of this proposition repeats word by word the proof of an
identical proposition for the pure billiard case. That proof was
given in \cite{Ch99b} (see the proof of the estimates (2.6)--(2.8)
in Section~7 there). It was based on certain facts about billiards
which were all listed in \cite{Ch99b}. Here we have proved
the corresponding
facts for our model in Sections~\ref{secH} and \ref{secBMT}. We
even tried to use similar notation for the convenience of the
reader. Thus here we can refer to \cite{Ch99b} for the
proof of the above proposition.
\begin{corollary}
For any sufficiently small $\rho_0>0$ and any homogeneous
unstable curve $\gamma\subset M$ of length $\leq\rho_0$ there
is an open $(\rho_0,1)$-subset $\xi^1\subset \gamma$ such that
$m_{\gamma}(\gamma\setminus \xi^1)=0$ and for all
$\varepsilon>0$ we have
\be
m_\gamma(r_{\xi^1,1}<\varepsilon)\leq \alpha_0\Lambda_1\cdot
m_\gamma(r_{\gamma,0}<\varepsilon/\Lambda_1)
+\varepsilon\beta_0\rho_0^{-1}m_\gamma(\gamma)
\label{exp1nu}
\ee Also, for any $n\geq 2$ there is an open $(\rho_0,n)$-subset
$\xi^n\subset \gamma$ such that $m_{\gamma}(\gamma\setminus
\xi^n)=0$ and for all $\varepsilon>0$ we have
\begin{eqnarray}
m_\gamma(r_{\xi^n,n}<\varepsilon) &\leq& (\alpha_1\Lambda_1)^n\cdot
m_\gamma(r_{\gamma,0}<\varepsilon/\Lambda_1^n)
+\varepsilon\beta_3\rho_0^{-1}(1+\alpha_1+\cdots +\alpha_1^{n-1})
m_\gamma(\gamma)\nonumber\\
&\leq& \alpha_1^n\varepsilon
+\varepsilon\beta_3\rho_0^{-1}(1-\alpha_1)^{-1}
m_\gamma(\gamma) \label{expnnu}
\end{eqnarray}
Lastly, for all sufficiently small $\delta>0$ we have
\begin{eqnarray}
m_\gamma(\gamma\cap T_1^{-n}{\cal U}_{\delta}) &\leq&
\beta_4\, m_\gamma(r_{\xi^n,n}<\zeta\delta^{\sigma})\nonumber\\
&\leq& \beta_4\alpha_1^n\zeta\delta^{\sigma}
+\beta_4\zeta\delta^{\sigma}\beta_3\rho_0^{-1}(1-\alpha_1)^{-1}
m_\gamma(\gamma)
\label{delnnu}
\end{eqnarray}
Here $\alpha_1\in(\alpha_0,1)$ and $\beta_3>\beta_0$,
$\beta_4>\beta_2$ are some global constants.
\end{corollary}
{\em Proof}. The bound (\ref{exp1nu}) follows from
(\ref{rgrowth11}) by taking the limit $\delta\to 0$. The bound
(\ref{expnnu}) follows from (\ref{exp1nu}) by induction on $n$,
this induction argument was explained in detail on pp. 432--433 in
\cite{Ch99a}. The first inequality in (\ref{delnnu}) is obtained
by applying (\ref{rw0}) to every connected curve in $T_1^n\xi^n$,
where $\xi^n$ is the set involved in (\ref{expnnu}). The second
inequality in (\ref{delnnu}) then follows directly from the bound
(\ref{expnnu}).
We note that the necessity to slightly increase the constants
$\alpha_0,\beta_0,\beta_2$ (to $\alpha_1,\beta_3,\beta_4$
respectively) results from the slight nonuniformity of the measure
$m^{(n)}_{\xi}$ with respect to the Lebesgue measure $m_{\xi}$ on
every connected component $\xi$ of the set $T_1^n\xi^n$. In view
of our Key Remark, we can make $\rho_0>0$ small enough, so that
the increase of $\alpha_0$ will be small, hence $\alpha_1$ will be
still less than one, because the requirement $\alpha_1<1$ is
crucial. $\Box$\medskip
Now we fix a $\rho_0>0$ satisfying Proposition~\ref{prgrowth}. We
also fix a small $q\in (0,1)$ and let
$\rho_1=\rho_0q(1-\alpha_1)/4\beta_3$. For any homogeneous
unstable curve $\gamma\subset M$ of length $\leq\rho_0$ and $n\geq
1$ let $\xi^n\subset\gamma$ be the set involved in (\ref{expnnu}).
Denote
$$
\xi^n(\rho_1)=\{X\in\xi^n:\, |T_1^n\xi^n(X)|\geq \rho_1\}
$$
In other words, $T_1^n\xi^n(\rho_1)$ will be the union of long
enough (longer than $\rho_1$) components of $T_1^n\xi^n$. A direct
calculation based on (\ref{expnnu}) yields:
\begin{corollary}
For all $n\geq n(\gamma):=\ln
m_{\gamma}(\gamma)/\ln\alpha_1+\ln(q/\rho_1)/\ln\alpha_1$ we have
$$
m_{\gamma}(\xi^n(\rho_1))\geq (1-q)\, m_{\gamma}(\gamma)
$$
\label{cr55}
\end{corollary}
This means that in the set $T_1^n\gamma$, sufficiently long
components (longer than $\rho_1$) will be prevalent after
$n(\gamma)$ iterations of $T_1$. Note that
$\rho_0,\rho_1,q$ are global constants (independent of the force
$\bf F$).
We complete this section with the construction of stable and
unstable manifolds.
An unstable curve $\gamma\subset M$ is called an {unstable fiber}
(or unstable manifold) if for all $n\geq 1$ the map $T^{-n}$ is
defined on $\gamma$ and $T^{-n}\gamma$ is also an unstable curve.
Likewise, $\gamma$ is a stable fiber if $T^n\gamma$ is a stable
curve for all $n\geq 0$.
Note that for an unstable fiber $\gamma$ we have
diam$(T^{-n}\gamma)\to 0$ as $n\to\infty$. Similarly, for a stable
fiber $\gamma$ we have diam$(T^{n}\gamma)\to 0$ as $n\to\infty$.
The above notion corresponds to a standard definition of stable
and unstable manifolds for hyperbolic dynamical systems. It is not
very helpful in the case of billiards, because of the lack of
proper distortion bounds. Such bounds are only available on
homogeneous stable and unstable curves, as we have seen in
Section~\ref{secBMT}. Hence, we adopt the following: \medskip
\noindent{\bf Definition}. An unstable curve $\gamma\subset M$ is
called an unstable homogeneous fiber, or {\em h-fiber}, if for all
$n\geq 0$ the curve $T^{-n}\gamma$ is a homogeneous unstable
curve. Similarly, $\gamma\subset M$ is a stable h-fiber if for all
$n\geq 0$ the curve $T^n\gamma$ is a homogeneous stable curve.
\medskip
Clearly, stable and unstable h-fibers are automatically ordinary
stable and unstable fibers. But generally, h-fibers are shorter
than ordinary fibers. In other words, an ordinary fiber can be a
union (finite or countable) of h-fibers.
We now prove that h-fibers exist and are abundant in $M$. The
hyperbolicity of the flow $\Phi^t$ or the map $T$ does not
automatically provide the existence of h-fibers, though, because
both the flow and the map have singularities.
For $\varepsilon>0$, denote by ${\cal U}^-_{\varepsilon}$ the
$\varepsilon$-neighborhood of ${\cal S}_0\cup{\cal
S}_{-1}\cup{\cal D}_0$, and by ${\cal U}^+_{\varepsilon}$ the
$\varepsilon$-neighborhood of ${\cal S}_0\cup{\cal S}_{1}\cup{\cal
D}_0$. Let
$$
M_{\varepsilon}^{\pm}
=\{X\in M:\, T^{\pm n}X\notin{\cal U}^{\pm}_{\varepsilon\Lambda^{-n}}
\ \ \ {\rm for}\ \ {\rm all}\ \ n\geq 1\}
$$
(here and on $\Lambda$ is the global constant defined by (\ref{Lambda})).
The following is standard \cite{Pes92,Y98,Ch99a}:\medskip
\noindent{\bf Fact}. For every point $X\in M^-_{\varepsilon}$, an
unstable h-fiber $\gamma^u(X)$ exists and stretches by at least
$c_0\varepsilon$ in both directions from $X$ (where $c_0>0$ is a
global constant). Similarly, for every point $X\in
M^+_{\varepsilon}$, a stable h-fiber $\gamma^s(X)$ exists and
stretches by at least $c_0\varepsilon$ in both directions from
$X$.
\medskip
In the notation of the previous section, we have
$r_{\gamma^u(X),0}(X)\geq c_0\varepsilon$ for every $X\in
M^-_{\varepsilon}$, and $r_{\gamma^s(X),0}(X)\geq c_0\varepsilon$
for every $X\in M^+_{\varepsilon}$.
\begin{proposition}
For $\nu_0$-almost every point $X\in M$ there are stable and
unstable h-fibers $\gamma^u(X)$ and $\gamma^s(X)$ through $X$.
Moreover,
$$
\nu_0(X:\, r_{\gamma^u(X),0}(X)\leq \varepsilon)\leq C\varepsilon
\ \ {\rm and}\ \
\nu_0(X:\, r_{\gamma^s(X),0}(X)\leq \varepsilon)\leq C\varepsilon
$$
for some global constant $C>0$. In particular, the union of
h-fibers shorter than $\varepsilon$ has $\nu_0$-measure less than
${\rm const}\cdot\varepsilon$. \label{prf}
\end{proposition}
{\em Proof}. Since the set ${\cal S}_{0}\cup{\cal S}_{\pm 1}$ is a
finite union of smooth compact curves, the $\nu_0$ measure of its
$\varepsilon$-neighborhood is less than const$\cdot\varepsilon$. A
similar fact for the set ${\cal D}_0$ can be verified by direct
inspection. Then $\nu_0({\cal U}^-_{\varepsilon})\leq
B'\varepsilon$ for some global constant $B'$. Due to
(\ref{DTB11}), for all $n\geq 1$ we have $\nu_0(T^n{\cal
U}^-_{\varepsilon\Lambda^{-n}})\leq B'B_{11}
\varepsilon(e^{-\delta_4}\Lambda)^{-n}$. Therefore,
$\nu_0(M^-_{\varepsilon})\geq 1-B\varepsilon$ for some global
constant $B$. A similar bound holds for $M^+_{\varepsilon}$. Now
the proposition follows from the above fact. $\Box$\medskip
We record a few standard facts about h-fibers, which follow from
the properties proved in Sections~\ref{secH}-\ref{secBMT}, in the
same way as in the pure billiard case \cite{BSC91}:\\ (1) if a
sequence of h-fibers $\gamma_n^u$, $n\geq 1$, converges to a curve
$\gamma$ in the $C^0$ metric, then $\gamma$ is an h-fiber.\\ (2)
For every point $x\in M_{\varepsilon}^-$ the h-fiber $\gamma^u(X)$
is unique, i.e. h-fibers do not cross each other or branch out.
The same holds for every $X\in M_{\varepsilon}^+$ and
$\gamma^s(X)$.
\section{A Sinai-Ruelle-Bowen measure for the map $T$}
%for the map $T$ and the flow $\Phi^t$}
\label{secSRBM} \setcounter{equation}{0}
For any unstable h-fiber $\gamma\subset M$, a unique probability
measure $\nu_{\gamma}$, absolutely continuous with respect to the
Lebesgue measure $m_\gamma$ with density
$f_\gamma=d\nu_\gamma/dm_\gamma$, is defined by the following
condition:
\be
\frac{f_\gamma(X)}{f_\gamma(Y)}=
\lim_{n\to\infty}
\frac{J_{T^{-n}\gamma,n}(T^{-n}Y)}{J_{T^{-n}\gamma,n}(T^{-n}X)}
\ \ \ \ \ {\rm for\ all}\ X,Y\in \gamma
\label{uSRB}
\ee
(compare this to (\ref{nSRB})). The existence of the the limit
(\ref{uSRB}) is guaranteed by Lemma~\ref{lmdis} (distorsion
bounds). We call $\nu_\gamma$ the u-SRB measure on $\gamma$.
Observe that u-SRB measures are conditionally invariant under $T$,
i.e. for any subsegment $\gamma_1\subset T\gamma$, the measure
$T_{\ast}\nu_\gamma|\gamma_1$ (the image of $\nu_\gamma$ under $T$
conditioned on $\gamma_1$) coincides with $\nu_{\gamma_1}$.
Note that the density $f_{\gamma}$ is a pointwise limit of the
densities $f^{(n)}_{\gamma}$ introduced in the previous section,
as $n\to\infty$. The bound (\ref{lnff}) implies a similar bound
for $f_{\gamma}$. So, according to our Key Remark, all the u-SRB
densities are almost constant on unstable h-fibers of length
$\leq\rho_0$.
\medskip\noindent{\bf Definition}. A $T$-invariant ergodic
probability measure $\nu$ on $M$ is called a Sinai-Ruelle-Bowen
(SRB) measure if its conditional distributions on unstable
h-fibers are absolutely continuous. In that case the conditional
measure $\nu|_{\gamma}$ is the u-SRB measures $\nu_\gamma$ on
every unstable h-fiber. \medskip
The significance of SRB measures lies in the following facts. For
any SRB measure $\nu$ there is a set $B\subset M$ of positive
Lebesgue measure (called sometimes the basin of attraction) such
that for every $X\in B$ and any continuous function $f:\, M\to\IR$
$$
\frac{f(X)+f(TX)+\cdots +f(T^{n-1}X)}{n}\to\int_Mf(X)\, d\nu
$$
as $n\to\infty$. Thus, the measure $\nu$ describes the
distribution of trajectories of points $X\in B$, which are
physically observable (detectable) since $\nu_0(B)>0$. Hence, SRB
measures are physically observable.
The first goal of this section is to prove the existence and
finiteness of SRB measures. We first prove a similar claim for the
map $T_1=T^m$ introduced in the previous section.
In \cite{Pes92}, Pesin found sufficient conditions for the
existence of SRB measures for a wide class of hyperbolic maps with
singularities (he called them generalized hyperbolic attractors),
which included the class we study here. We restate Pesin's
existence theorem in our notation. Denote by $m$ the Lebesgue
measure on $M$.
\medskip
\begin{theorem}[see \cite{Pes92}]
The map $T_1$ admits at least one and at most countably many SRB
measures, provided the following two conditions hold. First, there
are constants $C_1>0,q_1>0$ such that for all $\varepsilon>0,n\geq
1$
\be
m(T_1^{-n}{\cal U}_{\varepsilon})\leq C_1\varepsilon^{q_1}
\label{P0}
\ee
Second, there is an unstable h-fiber $\gamma\subset M$ and
constants $C_2>0,q_2>0$ such that for all $\varepsilon>0,n\geq 1$
\be
m_{\gamma}(\gamma\cap T_1^{-n}{\cal U}_{\varepsilon})
\leq C_2\varepsilon^{q_2}
\label{P1}
\ee
Each SRB measure is K-mixing and Bernoulli, up to a finite
cycle.
\end{theorem}
Recall that ${\cal U}_{\varepsilon}$ stands for the
$\varepsilon$-neighborhood of the set $\Gamma_1\cup\partial M$.
Later Sataev \cite{Sat} showed that the number of SRB measures
is finite under two additional conditions: there
are constants $C_3>0,q_3>0$ such that for every homogeneous
unstable curve $\gamma\subset M$ there are $n_{\gamma}\geq 1$ and
$C_{\gamma}>0$ such that for all $\varepsilon>0$
\be
m_\gamma(\gamma\cap T_1^{-n}{\cal U}_{\varepsilon})
\leq C_\gamma\varepsilon^{q_3}\, m_\gamma(\gamma)\ \ \ \ {\rm for\ all}\ \ n>0
\label{S1}
\ee
and
\be
m_\gamma(\gamma\cap T_1^{-n}{\cal U}_{\varepsilon})
\leq C_3\varepsilon^{q_3}\, m_\gamma(\gamma)\ \ \ \ {\rm for\ all}\ \
n>n_\gamma
\label{S2}
\ee
We now verify Pesin's and Sataev's conditions.
\begin{proposition}
The map $T_1$ satisfies (\ref{P0})-(\ref{S2}). Hence, $T_1$ admits
at least one and at most finitely many SRB measures. Every SRB
measure is K-mixing and Bernoulli, up to a finite cycle.
\label{prT1PS}
\end{proposition}
{\em Proof}. We foliate $M$ by smooth unstable curves whose
collection we denote by $\Gamma^\ast=\{\gamma\}$. We require that
the length of each $\gamma\in\Gamma^{\ast}$ be $\rho_0$ (except
for the corners of $M$ and narrow strips $I_k$, where the curves
are necessarily shorter). Let $m^\ast_\gamma$ be the conditional
measure on each $\gamma\in\Gamma^{\ast}$ induced by the Lebesgue
measure $m$ on $M$, and $m^{\ast}$ the factor measure on $\Gamma^\ast$.
If the foliation is smooth enough and $\rho_0$ small enough, then
every $m_\gamma^\ast$ will have almost uniform density with
respect to the Lebesgue measure $m_\gamma$. In fact, the curves
$\gamma$ can be chosen as parallel line segments, then the
measures $m_\gamma^\ast$ will be exactly uniform. Now the
condition (\ref{P0}) easily follows from the bound (\ref{delnnu})
by integration over $\Gamma^{\ast}$ with respect to the factor measure
$m^{\ast}$, which is a straightforward calculation. The conditions
(\ref{P1}) and (\ref{S1}) are direct consequences of
(\ref{delnnu}). Lastly, the inequality (\ref{S2}) follows from
(\ref{delnnu}) whenever $\alpha_1^n\ln m_{\gamma}(\gamma)/\ln\alpha_1$. $\Box$\medskip
\begin{proposition}
The map $T$ admits at least one and at most finitely many SRB
measures. Every SRB measure is K-mixing and Bernoulli, up to a
finite cycle. \label{prTPS}
\end{proposition}
{\em Proof}. Let $\nu$ be an SRB measure for the map $T_1=T^m$.
Then the measure $(\nu+T_{\ast}\nu+\cdots +T^{m-1}_{\ast}\nu)/m$
will be an SRB measure for the map $T$, hence the existence part.
Now, let $\nu$ be an SRB measure for $T$. If it is ergodic for
$T_1$, then it is an SRB measure for $T_1$. Otherwise $\nu$ has at
most $m$ ergodic components (with respect to $T_1$), each of which
is an SRB measure for $T_1$. This proves Proposition~\ref{prTPS}.
$\Box$\medskip
The following proposition gives Theorem~\ref{tmmain2}
modulo Theorem~\ref{tmmain1}, whose proof is yet
to be completed.
\begin{proposition}
Each SRB measure $\nu$ of the map $T_{\bf F}$ enjoys the
exponential decay of correlations (\ref{EDC}) and satisfies the
central limit theorem (\ref{CLT}). The correlation bound
(\ref{EDC}) is uniform in $\bf F$. \label{prEDCCLT}
\end{proposition}
{\em Proof}. This follows from a general theorem proved in
\cite{Ch99b}. That theorem is stated for generic hyperbolic maps
satisfying certain assumptions. All the assumptions have been
already verified in Sections~\ref{secH}-\ref{secGUC}. The
uniformity of the correlation bound follows from the fact that all
the constants in the crucial estimates in
Sections~\ref{secH}-\ref{secGUC} (most notably, in the ``growth
lemma'' \ref{prgrowth}) are global, i.e. independent of $\bf F$.
Thus we obtain Proposition~\ref{prEDCCLT}. $\Box$\medskip
The uniqueness of an SRB measure for $T$ requires a more elaborate
argument. We recall that the space $M$ in the coordinates
$(r,\varphi)$ does not depend on the force $\bf F$ in
(\ref{emotion}). So we consider all the maps $T=T_{\bf F}$ as
defined on the same space $M$. For ${\bf F}=0$, we get the
billiard map $T_0$. Recall that the space $M$ is cut into
countably many strips, $I_k$, hence all stable and unstable
curves will be automatically homogeneous.
The classes of stable and unstable curves depend on $\bf F$, but
only slightly, as it follows from our definitions in Sections~\ref{secH}
and \ref{secBMT}. For simplicity, we intersect these classes over all
relevant $\bf F$'s. Hence, from now on, stable and unstable curves
mean such curves for all relevant maps $T=T_{\bf F}$. On the
contrary, stable and unstable h-fibers depend on $\bf F$ strongly
(not only their directions, but even more their sizes), so we will
denote them by $\gamma^{s,u}_{\bf F}(X)$, respectively, for $X\in M$.
%Note that $\gamma^{s,u}_0(X)$ are then h-fibers for the billiard map $T_0$.
For any $\rho>0$, consider the class ${\cal C}_u(\rho)$ of unstable
curves $\gamma\subset M$ of length $\geq\rho$. Denote by
$\overline{{\cal C}_u(\rho)}$ its closure in the Hausdorff metric.
Recall that the Hausdorff metric defines the distance between
two compact subsets $A,B\subset M$ by
$$
{\rm dist}(A,B)=\max\{\max_{X\in A}\,{\rm dist}(X,B),
\max_{Y\in B}\,{\rm dist}(Y,A)\}
$$
(it is just the $C^0$ metric if restricted to continuous
curves in $M$). Recall that our unstable curves are at least $C^2$,
their tangent vectors satisfy the uniform bound in Lemma~\ref{lmB1}
and their curvature is uniformly bounded by Lemma~\ref{lmcur}.
Therefore, all the curves in the class $\overline{{\cal C}_u(\rho)}$
are of length $\geq\rho$, at least $C^1$
(but not necessarily $C^2$), and their tangent vectors
satisfy the same bound in Lemma~\ref{lmB1}. We will
call curves $\gamma\subset\cup_{\rho>0}\overline{{\cal C}_u(\rho)}$
{\em generalized unstable curves}. Similarly, generalized
stable curves are defined, and we denote their class respectively by
$\cup_{\rho>0}\overline{{\cal C}_s(\rho)}$.
We call a rhombus $R\subset M$ a domain bounded by two unstable
curves and two stable curves (called the sides of $R$). We say that
a generalized unstable curve $\gamma$ {\em straddles} $R$ if $\gamma\subset R$
and the endpoints of $\gamma$ lie on the (opposite) stable sides
of $R$. We say that $\gamma$ {\em properly crosses} $R$ if
$\gamma$ intersects the middle half of each stable side
of $R$ and the points of intersection divide $\gamma$ into three
parts of which the smallest one is $\gamma\cap R$.
Similar notion are defined for generalized stable curves.
For a rhombus $R$, let $R_{\bf F}^{\ast}$ be the set of points
$X\in R$ such that both $\gamma^u_{\bf F}(X)$ and
$\gamma^s_{\bf F}(X)$ properly cross $R$.
\begin{lemma}
There is a rhombus $R\subset M$ such that
$\nu_0(R_0^{\ast})>0$.
\end{lemma}
This easily follows from Proposition~\ref{prf}. $\Box$\medskip
Note that we do not claim that $\nu_0(R_{\bf F}^{\ast})>0$ for all
$\bf F$, or even for any ${\bf F}\neq 0$. This will follow
from our further results, see Corollary~\ref{crRu}, etc.
We fix a rhombus $R$ that satisfies the above lemma. Since
it does not depend on the force $\bf F$, it is a ``global''
object, just like our constants $B_i$ in the previous sections.
%We can also assume that the sides of the rhombus $R$ are nearly equal.
For any generalized unstable curve $\gamma$ and $n\geq 1$
let $\gamma_{\bf F}(n)$ denote the union of intervals
$\xi\subset\gamma$ such that $T_{\bf F}^n\xi$ is one
generalized unstable curve that straddles $R$. Also,
let $\gamma_{\bf F}^p(n)$ denote the union of intervals
$\xi\subset\gamma$ such that $T_{\bf F}^n\xi$ is one
generalized unstable curve that properly crosses $R$.
\begin{lemma}
There are global constants $\tilde{n}=\tilde{n}(\rho_1,R)\geq 1$ and
$\tilde{\beta}_1=\tilde{\beta}_1(\rho_1,R)>0$ such that for every
generalized unstable curve $\gamma\subset M$ of length
$\geq\rho_1$ and all $n\geq\tilde{n}$
$$
m_{\gamma}(\gamma_0^p(n))\geq\tilde{\beta}_1\,
m_{\gamma}(\gamma)
$$
\label{lmbetap}
\end{lemma}
In other words, for all $n\geq\tilde{n}$
the image $T_0^n\gamma$ will contain a certain positive
fraction (characterized by $\tilde{\beta}_1$)
of curves that properly cross the rhombus $R$.
This lemma is proved in \cite{BSC91} (see Theorem 3.13 there)
under the additional assumption that $\gamma$ is an h-fiber.
However, the past images $T_0^{-k}\gamma$, $k>0$, are not involved
in that theorem or its proof, and, clearly,
there is no difference between
unstable h-fibers and generalized unstable curves as far as their
forward iterations are concerned. Thus, Theorem~3.13 in
\cite{BSC91} extends to generalized unstable curves.
\begin{lemma}
For any force $\bf F$ satisfying Assumptions A and B and every
generalized unstable curve $\gamma\subset M$ of length
$\geq\rho_1$ and all $n\in [\tilde{n},\tilde{n}+m]$
$$
m_{\gamma}(\gamma_{\bf F}(n))\geq\tilde{\beta}_2\,
m_{\gamma}(\gamma)
% \label{betas}
$$
where $\beta_2>0$ is a global constant,
and $m$ is again the fixed power of $T$, i.e. such that $T_1=T^m$.
\label{lmbetas}
\end{lemma}
In other words, for every $n=\tilde{n},\ldots,\tilde{n}+m$ the
image $T_{\bf F}^n\gamma$ will contain a certain positive fraction
of curves that straddle the rhombus $R$.\medskip
{\em Proof}. Let $\gamma$ be a generalized unstable curve and
$n\in [\tilde{n},\tilde{n}+m]$. Consider the curves $\xi\in
T_0^n\gamma$ that properly cross $R$ (such curves exist by
Lemma~\ref{lmbetap}). Let $\bf F$ be small enough, so that the map
$T_{\bf F}$ is a small enough perturbation of $T_0$, in particular
the singularity sets of these maps are close enough to each other.
Let also $\gamma'\subset M$ be a generalized unstable curve
sufficiently close to $\gamma$ in the Hausdorff metric. We claim
that if ${\bf F}\approx 0$ and $\gamma'\approx\gamma$, then to
every curve $\xi\in T_0^n\gamma$ that properly crosses $R$ there
corresponds a curve $\xi'\in T_{\bf F}^n\gamma'$ that is close to
$\xi$ and has almost the same length. We emphasize that we first
fix $\gamma$ and $n$, and then assume that ${\bf F}\approx 0$ and
$\gamma'\approx\gamma$, for the given $\gamma$ and $n$. Note that
$\xi'$ will be one curve (not broken by singularities), because of
the continuation property from the end of Section~\ref{secBMT}.
One can easily see that, by that property,
if any long enough generalized unstable
curve $\gamma$ intersects the singularity set for $T_{0}$, then
any generalized unstable curve $\gamma'$ close enough to $\gamma$
(in the Hausdorff metric) intersects the singularity set for
$T_{\bf F}$ with ${\bf F}\approx 0$,
and vice versa. This justifies our claim. Now, since
$\xi'$ is close to $\xi$, and $\xi$ properly crosses $R$, then
$\xi'$ crosses both stable sides of $R$, and so the curve
$\xi'\cap R$ straddles $R$.
Thus, given $\gamma$ and $n$, there is an open neighborhood ${\cal
V}(\gamma,n)$ of the curve $\gamma$ in the class $\overline{{\cal
C}_u(\rho_1)}$ equipped with the Hausdorff metric and a
$\delta_0(\gamma,n)>0$ such that any curve $\gamma'\subset {\cal
V}(\gamma,n)$ satisfies
\be
m_{\gamma'}(\gamma_{\bf F}'(n))\geq\tilde{\beta}_2\,
m_{\gamma'}(\gamma')
\label{betas}
\ee
for some global constant $\tilde{\beta}_2>0$
and all $\bf F$'s that satisfy Assumptions~A and B with
$\delta_0<\delta_0(\gamma,n)$. The finite intersection
$$
{\cal V}(\gamma):=\cap_{n=\tilde{n}}^{\tilde{n}+m}
{\cal V}(\gamma,n)
$$
is also an open neighborhood of the curve $\gamma$
in the class $\overline{{\cal C}_u(\rho_1)}$. Any curve
$\gamma'\subset {\cal V}(\gamma)$ satisfies the inequality
(\ref{betas}) for all $n\in [\tilde{n},\tilde{n}+m]$ and
with all $\bf F$'s that satisfy Assumptions~A and B with
$$
\delta_0<\delta_0(\gamma):=
\min_{\tilde{n}\leq n\leq\tilde{n}+m}\delta_0(\gamma,n)
$$
Since the class $\overline{{\cal C}_u(\rho_1)}$ is obviously
compact in the Hausdorff metric, there is a finite cover of
$\overline{{\cal C}_u(\rho_1)}$ by some ${\cal V}(\gamma_j)$,
$1\leq j\leq J$. This proves the lemma for all forces satisfying
Assumptions~A and B with
$$
\delta_0<\delta_{\ast}:=\min_{1\leq j\leq J}\delta_0(\gamma_j)
$$
$\Box$\medskip
\noindent{\bf Remark}. Note that in the proof of
Lemma~\ref{lmbetas} we have put a new restriction
$\delta_0<\delta_{\ast}$ on $\delta_0$ that enters Assumption~B.
This restriction is probably much more severe than any of the
restrictions on $\delta_0$ we needed before. Therefore, the
uniqueness of the SRB measure probably holds for much smaller
forces $\bf F$ than the hyperbolicity of $T_{\bf F}$ and the
existence and finiteness of SRB measures do. We therefore expect
that in physical models where $\bf F$ changes from ${\bf F}=0$
continuously (such as by increasing the strength of an electrical
field \cite{CELS}), one first observes a unique non-smooth SRB
measure, then a finite collection of SRB measures, and then
non-SRB stationary states. Such experiments were done, for
example, in \cite{DM}. This discussion is related to the
physically important issue of the range of applicability of the
linear response theory -- see van Kampen's objections \cite{vK}
and some counterarguments in \cite{CELS}.
\medskip
Lemma~\ref{lmbetas} and Corollary~\ref{cr55} easily imply the
following two corollaries.
\begin{corollary}
There are global constants $\tilde{n}_1\geq 1$ and
$\tilde{\beta}_3>0$ such that for any force
$\bf F$ satisfying Assumptions A and B with
$\delta_0<\delta_{\ast}$ and every
generalized unstable curve $\gamma\subset M$ of length
$\geq\rho_1$ and all $n\geq\tilde{n}_1$
$$
m_{\gamma}(\gamma_{\bf F}(n))\geq\tilde{\beta}_3\,
m_{\gamma}(\gamma)
% \label{betas}
$$
\label{crbetas1}
\end{corollary}
The main difference from Lemma~\ref{lmbetas} is that now
{\em all} $n\geq\tilde{n}_1$ are covered, rather than
$n\in [\tilde{n},\tilde{n}+m]$.
\begin{corollary}
There are global constants $\tilde{n}_2\geq 1$ and
$\tilde{\beta}_4>0$ such that for any force
$\bf F$ satisfying Assumptions A and B with
$\delta_0<\delta_{\ast}$ and every
generalized unstable curve $\gamma\subset M$ of length
$|\gamma|=\varepsilon>0$ and all
\be
n\geq \tilde{n}(\varepsilon):=\ln\varepsilon/\ln\alpha_1+\tilde{n}_2
\label{neps}
\ee
we have
\be
m_{\gamma}(\gamma_{\bf F}(n))\geq\tilde{\beta}_4\,
m_{\gamma}(\gamma)
\label{beta2}
\ee
\label{crbetas2}
\end{corollary}
Let $R^u_{\bf F}$ be the set of points $X\in R$ such that
the unstable h-fiber $\gamma^u(X)\cap R$ straddles $R$.
\begin{corollary}
There is a global constant $\tilde{\beta}_R>0$ such that for any
$T_{\bf F}$ with $\delta_0<\delta_{\ast}$ and any SRB measure
$\nu$ of $T_{\bf F}$ we have $\nu(R^u_{\bf F})>\tilde{\beta}_R$.
Furthermore, let $\nu$ be not mixing, so that by
Proposition~\ref{prTPS}, $T_{\bf F}$ permutes a finite number of
subsets $X_1,\ldots,X_k\subset M$ on each of which $T_{\bf F}^k$
is mixing. In this case we have $\nu(R^u_{\bf F}\cap X_i)>0$ for
every $i=1,\ldots,k$. \label{crRu}
\end{corollary}
\begin{proposition}
For any force $\bf F$ satisfying Assumptions A and B with
$\delta_0<\delta_{\ast}$ the SRB measure of the map $T_{\bf F}$
is unique and mixing. \label{pruniq}
\end{proposition}
We first adopt a definition.
\medskip\noindent{\bf Definition}.
Let ${\gamma}^s_{\bf F}$ be a stable h-fiber. For $\varepsilon>0$,
let $\Gamma_{\varepsilon}({\gamma}^s_{\bf F})$ denote the union of
all stable h-fibers in $M$ that are $\varepsilon$-close to
${\gamma}^s_{\bf F}$ in the Hausdorff metric. We call
$\gamma^s_{\bf F}$ a {\em density h-fiber} if for every
$\varepsilon>0$ the set $\Gamma_{\varepsilon}(\gamma^s_{\bf F})$
has positive Lebesgue measure in $M$. Note that in this case for
any generalized unstable curve $\xi\subset M$ that crosses
$\gamma^s_{\bf F}$, the set
$\xi\cap\Gamma_{\varepsilon}(\gamma^s_{\bf F})$ has positive
$m_{\xi}$ measure, by the absolute continuity Lemma~\ref{lmabsc}.
Similarly, we introduce unstable density h-fibers.
\begin{lemma}
For each map $T_{\bf F}$ there are density h-fibers.
In fact, their union has full Lebesgue measure.
If $\gamma^s_{\bf F}$ is a density h-fiber, then
all the connected components of $T^{-n}\gamma^s_{\bf F}$
are density h-fibers, too, for every $n\geq 1$.
\end{lemma}
{\em Proof}. The first two claims follows from
Proposition~\ref{prf}. To prove the last one, we se
$n=1$ and note that $T_{\bf F} ^{-1}$ is
piecewise smooth and its singularities are unstable
curves with the continuation property. Then we use
indunction on $n$. $\Box$\medskip
{\em Proof of Proposition~\ref{pruniq}}.
Let $\gamma^s_{\bf F}$ be a density h-fiber.
By the above Lemma and
Corollary~\ref{crbetas2} (actually applied to stable curves),
there exist density h-fibers in $T^{-n}\gamma^s_{\bf F}$ for some
$n\geq 1$ that straddle the rhombus $R$. This, along with
Corollary~\ref{crRu}, proves Proposition~\ref{pruniq}. $\Box$\medskip
\begin{proposition}
For any force $\bf F$ satisfying Assumptions A and B with
$\delta_0<\delta_{\ast}$ the SRB measure $\nu$ of the map
$T_{\bf F}$ is positive on open sets. Moreover, for every small
round disk $D\subset M$ we have
$$
\nu(D)\geq c_1[\nu_0(D)]^{1+\delta_5}
% \label{betas}
$$
for some global constant $c_1>0$ and small constant
$\delta_5>0$ depending on $\delta_0$ (i.e., $\delta_5\to 0$ as
$\delta_0\to 0$). \label{prsigma1}
\end{proposition}
{\em Proof}. Since the disk $D$ is connected, it belongs in one
homogeneity strip $I_k$, and so the quantity $\cos\varphi$ does
not vary too much over $D$, i.e. the measure $\nu_0$ is almost
proportional to the Lebesgue measure $m$ on $D$. We can find a
rhombus $R_D\subset D$ whose opposite sides are parallel straight
lines and which is big enough so that, say, $\nu_0(R_D)\geq
\nu_0(D)/10$. Now, we foliate the rhombus $R_D$ by parallel stable
segments $\gamma$ that straddle $R_D$ and are parallel to the
stable sides of $R_D$. Hence, all $\gamma$'s in our foliation have
the same length, $\varepsilon$. Note that $\varepsilon\geq
c_2\sqrt{\nu_0(D)}$ with a global constant $c_2>0$. For any
$\gamma$ in our foliation of $R_D$ and $n\geq 1$ let $\gamma_{\bf
F}(-n)$ denote the union of intervals $\xi\subset\gamma$ such that
$T_{\bf F}^{-n}\xi$ is one stable curve that straddles the rhombus
$R$ (fixed earlier). Corollary~\ref{crbetas2}
(actually, its dual statement for stable curves)
implies that $m_{\gamma}(\gamma_{\bf
F}(-n))>\tilde{\beta}_4\,m_{\gamma}(\gamma)$ for all
$n\geq\tilde{n}:=\tilde{n}(\varepsilon)$. Consider the set
$$
R_D(-n):=\cup_{\gamma}\gamma_{\bf F}(-n)
$$
where the union is taken over all $\gamma$ in our foliation of
$R_D$. Our previous estimates imply that
$m(R_D(-n))\geq\tilde{\beta}_4\, m(R_D)$ and hence
$\nu_0(R_D(-n))\geq\tilde{\beta}_5\,\nu_0(R_D)$ for all
$n\geq\tilde{n}$, and with the global constant
$\tilde{\beta}_5=\tilde{\beta}_5/2$.
Now the volume compression bounds (\ref{DTB11}) imply
$$
\nu_0(T_{\bf F}^{-n}R_D(-n))\geq
B_{11}^{-1}e^{-\delta_4n}\tilde{\beta}_5\,\nu_0(R_D)
$$
for all $n\geq\tilde{n}$.
We put $n=\tilde{n}=\tilde{n}(\varepsilon)$
given by (\ref{neps}) and obtain
$$
\nu_0(T_{\bf F}^{-\tilde{n}}R_D(-\tilde{n}))
\geq c'\varepsilon^{\delta_6}\nu_0(R_D)
\geq c''\,[\nu_0(R_D)]^{1+\delta_6/2}
$$
with some positive global constants $c',c''$ and a small
constant $\delta_6=\delta_4/\ln\alpha_1$.
We put $\delta_5=\delta_6/2$.
Next observe that the set $T_{\bf F}^{-\tilde{n}}R_D(-\tilde{n})$
is a union of stable curves that straddle our fixed rhombus $R$.
Lemma~\ref{lmabsc} (absolute continuity) then implies that for any
unstable curve $\xi$ that straddles $R$ we have
$$
m_{\xi}(\xi\cap T_{\bf F}^{-\tilde{n}}R_D(-\tilde{n}))
\geq c'''\,[\nu_0(R_D)]^{1+\delta_5}m_{\xi}(\xi)
$$
with a global constant $c'''>0$. This bound combined with
Corollary~\ref{crRu} yields
$$
\nu(T_{\bf F}^{-\tilde{n}}R_D(-\tilde{n}))
\geq \tilde{c}\, [\nu_0(R_D)]^{1+\delta_5}
$$
for some global constant $\tilde{c}>0$ and
the SRB measure $\nu$ of the map $T_{\bf F}$.
The $T_{\bf F}$-invariance of $\nu$ completes the proof
of Proposition~\ref{prsigma1}.
$\Box$\medskip
Theorems~\ref{tmmain1} and \ref{tmmain2} are now proved.
\medskip
{\bf Acknowledgment}. The author is grateful to H. van den Bedem
and L.-S.~Young for their interest to this work and useful discussions.
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\end{document}
\end