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\begin{document}
\title{Decay of correlations and dispersing billiards}
\author{N. Chernov
\\ Department of Mathematics\\
University of Alabama at Birmingham\\
Birmingham, AL 35294, USA\\
E-mail: chernov@vorteb.math.uab.edu
}
\date{\today}
\maketitle
\begin{abstract}
Dispersing billiards (or Sinai billiards) are
classical models of dynamical systems that
exhibit strong chaotic behavior but are
highly nonlinear and contain singularities.
It was a long standing conjecture that, due
to singularities, the rate of the decay of
correlations in dispersing billiards
(or the rate of mixing, or the
speed of relaxation to equilibrium)
is subexponential, i.e. slower than that
in Anosov and Axiom~A systems.
Recently, L.-S.~Young disproved this
conjecture -- she established an
exponential decay of correlations for a periodic
Lorentz gas with finite horizon. We prove the same
result for all the major classes of planar dispersing billiards,
including Lorentz gases without horizon and tables with
corner points.
We also design and prove a general theorem on the exponential
decay of correlations for smooth hyperbolic systems with
singularities, which is particularly convenient for physical
models like billiards.
\end{abstract}
{\em Keywords}: Decay of correlations, Sinai-Ruelle-Bowen measures,
dispersing billiards.
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\section{Introduction}
\label{secI}
\setcounter{equation}{0}
Strong statistical properties --
exponential decay of correlations (EDC) and
central limit theorem (CLT) --
for smooth uniformly hyperbolic
dynamical systems, namely Anosov and Axiom A diffeomorphisms,
have been proven by Ya.~Sinai, D.~Ruelle and R.~Bowen
in the seventies \cite{Si72,Ru,Bo75}. For piecewise
smooth or nonuniformly
hyperbolic systems, however, statistical properties are
often weaker -- the correlations decay slowly or the central
limit theorem fails, and, in any case, these properties
are very hard to prove.
We concentrate here on systems with uniform hyperbolicity, i.e.
such that one step expansion and contraction factors are
bounded away from unity, but we do not require smoothness
everywhere,
i.e. allow singularities. Well studied and physically
important systems of this kind are dispersing billiards, in
particular the periodic Lorentz gas.
L.~Bunimovich and Sinai \cite{BS81} obtained a CLT and
a subexponential (stretched exponential)
bound on correlations for a planar periodic
Lorentz gas with finite horizon
in 1981. These results were improved
and extended to other planar dispersing billiards in 1991,
see \cite{BSC91}, and to a multidimensional periodic
Lorentz gas with finite horizon in 1994, see \cite{Ch94}.
There was much of discussion in physical and mathematical
communities in the eighties about the actual rate of the
decay of correlations in dispersing billiards,
whether it is truly exponential or slower. Numerical
experiments produced inconclusive or contradictory estimates,
see a resent discussion and further references in \cite{GG}.
In early nineties, the `exponential' point of view
got the upper hand.
In 1992, the EDC was established for piecewise linear hyperbolic
2-D toral automorphisms by the author \cite{Ch92}. In 1994, Liverani
\cite{L} established the EDC for 2-D
piecewise smooth area-preserving
uniformly hyperbolic systems.
The singularities in the above papers consisted
of a finite number of smooth curves on which the dynamics
was discontinuous but
had one-sided derivatives (in particular, the derivatives
were uniformly bounded).
Those classes did not cover billiards, where derivatives are
always unbounded. Still the above results showed in principle
that singularities did not necessarily slow down the decay
of correlations.
A breakthrough occurred in 1996 when Young \cite{LSY}
proved the EDC for quite generic hyperbolic systems
with their Sinai-Ruelle-Bowen (SRB)
measures\footnote{For smooth maps that do not
preserve absolutely continuous measures,
SRB measures are the most physically
relevant invariant measures, see
discussions in \cite{LSY,Ch98}.} under one
assumption, the so called
exponential tail bound on return times to a selected
product-structure hyperbolic set.
She verified that assumption for two classes of
systems. One was that of 2-D
piecewise smooth uniformly hyperbolic maps,
and thereby Young extended Liverani's result \cite{L}
to non-area-preserving systems.
The author recently further extended this result
by Young to multidimensional maps \cite{Ch98}.
The second class in Young's paper \cite{LSY} consisted of
a planar periodic Lorentz gas with finite horizon.
Thus, Young
established, for the first time ever, an exponential bound
on correlations for a billiard model.
Our goal is to extend Young's result to other classes
of dispersing billiards. Before we do that, we
prove the EDC for uniformly hyperbolic systems with quite general
singularities. Our setup allows countably many singularity manifolds
and unbounded derivatives. We find sufficient conditions
under which correlations for SRB measures decay exponentially fast.
Since we actually prove, under our conditions, Young's
exponential tail bound mentioned above, our conditions are
more restrictive than Young's. On the other hand,
our conditions are a little easier to check than Young's:
in particular,
they involve only one iteration of the map rather
than all its positive iterations. After that, we show that
our conditions are mild enough to hold for the major classes
of planar dispersing billiards, including Lorentz gases without
horizon and tables with corner points.
Thus, we establish the EDC for all the main
types of planar dispersing billiards. This completely
settles the controversy over the decay of correlations
in billiards that has attracted so much attention in
the physical and mathematical communities.
The verification of our conditions for billiards
is not so hard a job, by the way, compared to
the sophisticated analysis of billiard dynamics done
in early papers \cite{BSC91,Ch94}. In a separate
paper, we plan to cover perturbed dispersing billiards
(subject to small external fields) and Lorentz gases
in any dimensions.
The paper is organized as follows. In Section~\ref{secSR}
we state our assumptions and the main theorem.
In Sections~\ref{secF}--\ref{secS} we prove the
main theorem. In Sections~\ref{secDB}--\ref{secCP}
we apply the main theorem to the planar dispersing
billiards. In Section~\ref{secFR} we make some remarks
and hints on the verification of the assumptions
of our main theorem.
\section{Statement of the main result}
\label{secSR}
\setcounter{equation}{0}
Let $M$ be an open subset in a $d$-dimensional
$C^\infty$ Riemannian manifold, such that $\bar{M}$ is compact
(the sets $M$ and $\bar{M}$ are not necessarily connected),
and let $\Gamma\subset \bar{M}$ be a closed subset.
We consider a map $T:\, M\setminus {\Gamma}\to M$,
which is a $C^2$ diffeomorphism of $M\setminus {\Gamma}$
onto its image.
The set $\Gamma$ will be referred to as the singularity set for $T$.
For $n\geq 1$ denote by
\be
{\Gamma}^{(n)}={\Gamma}\cup T^{-1}{\Gamma}\cup\cdots\cup T^{-n+1}{\Gamma}
\label{Gn}
\ee
the singularity set for $T^n$. Define
$$
M^+=\{x\in M: T^nx\notin {\Gamma},\, n\geq 0\},
\ \ \ \ \ \
M^-=\cap_{n> 0}T^n(M\setminus {\Gamma}^{(n)})
$$
and
$$
M^{0}=\cap_{n\geq 0}T^n(M^+)=M^+\cap M^-
$$
The sets $M^+$ and $M^-$ consist, respectively,
of points were all the future and past iterations
of $T$ are defined, and $M^{0}$ is the set of points
where all the iterations of $T$ are defined.
For any $\delta>0$ denote by ${\cal U}_{\delta}$
the $\delta$-neighborhood of the closed set
$\Gamma\cup\partial M$.
\medskip
\noindent
{\em Notation}.
We denote by $\rho$ the Riemannian metric in $M$ and by
$m$ the Lebesgue measure (volume) in $M$.
For any submanifold $W\subset M$ we denote by $\rho_W$
the metric on $W$ induced by the Riemannian metric in $M$,
by $m_W$ the Lebesgue measure on $W$ generated by $\rho_W$,
and by diam$W$ the diameter of $W$ in the $\rho_W$ metric.
\medskip
\noindent
{\bf Hyperbolicity}.
We assume that $T$ is fully and uniformly hyperbolic, i.e. there exist
two families of cones $C^u_x$ and $C^s_x$ in the tangent
spaces ${\cal T}_xM$, $x\in \bar{M}$, such
that $DT(C^u_x)\subset C^u_{Tx}$ and $DT(C^s_x)\supset
C^s_{Tx}$ whenever $DT$ exists, and
$$
|DT(v)|\geq \Lambda|v|\ \ \ \ \forall v\in C^u_x
$$
$$
|DT^{-1}(v)|\geq \Lambda|v|\ \ \ \ \forall v\in C^s_x
$$
with some constant $\Lambda>1$. These families of cones
are continuous on $\bar{M}$, their axes have the same dimensions
across the entire $\bar{M}$, and the angles between $C^u_x$
and $C^s_x$ are bound away from zero. Denote by $d_u$ and
$d_s$ the dimensions of the axes of $C^u_x$ and $C^s_x$,
respectively. The full hyperbolicity here means that
$d_u+d_s=\, $dim$M$.
For any $x\in M^+$ and $y\in M^-$ we set
$$
E^s_x=\cap_{n\geq 0}DT^{-n}(C^s_{T^nx}),\ \ \ \ \ \ \
E^u_y=\cap_{n\geq 0}DT^n(C^u_{T^{-n}y})
$$
respectively. It is standard, see, e.g., \cite{Pes92}, that
the subspaces $E^s_x$, $E^u_x$ are $DT$-invariant,
depend on $x$ continuously,
dim$E^{u,s}_x=d_{u,s}$, and $E^s_x\oplus E^u_x={\cal T}_xM$
for $x\in M^{0}$.
As a consequence, there can be no zero Lyapunov exponents
on $M^{0}$. The space $E^u_x$ is spanned by all vectors
with positive Lyapunov exponents, and $E^s_x$ by those
with negative Lyapunov exponents.
%We call $U$ a u-manifold if it is a smooth $d_u$-dimensional
%connected submanifold in $M$ of finite diameter (in the inner metric
%$\rho_U$) and at every $x\in U$ the tangent space
%${\cal T}_xU$ lies in $C^u_x$. Any u-manifold is expanded (locally)
%by $T$ in every direction at least by a factor $\Lambda$.
%Similarly, s-manifolds are defined.
We call a submanifold $W^u\subset M$ a local unstable manifold (LUM),
if $T^{-n}$ is defined and smooth on $W^u$ for all $n\geq 0$,
and $\forall x,y\in W^u$ we have $\rho(T^{-n}x,T^{-n}y)
\to 0$ as $n\to\infty$ exponentially fast. Similarly,
local stable manifolds (LSM), $W^s$, are defined. Obviously,
dim$W^{u,s}=d_{u,s}$.
We denote by $W^u(x)$, $W^s(x)$ local unstable and stable
manifolds containing $x$, respectively.
We primarily work with LUM's, and for
brevity we will denote them by just $W$, suppressing the
superscript $u$.
Denote by $J^u(x)=|{\rm det}(DT|E^u_x)|$ the jacobian
of the map $T$ restricted to $W(x)$ at $x$, i.e. the factor
of the volume expansion on the LUM $W(x)$ at the point $x$.
We assume the following standard properties
of unstable manifolds: \medskip
\noindent
{\bf Bounded curvature}.
The sectional curvature of any LUM $W$
is uniformly bounded by a constant $B\geq 0$.\medskip
\noindent
{\bf Distorsion bounds}. Let $x,y$ be in one connected
component of $W\setminus {\Gamma}^{(n-1)}$,
denote it by $V$. Then
\be
\log\prod_{i=0}^{n-1}\frac{J^u(T^ix)}{J^u(T^iy)}
\leq \varphi\left (\rho_{T^nV}(T^nx,T^ny)\right )
\label{distor1}
\ee
where $\varphi(\cdot )$ is some function, independent of $W$,
such that $\varphi(s)\to 0$ as $s\to 0$. \medskip
\noindent
{\bf Absolute continuity}.
Let $W_1,W_2$ be two sufficiently small LUM's,
such that any LSM $W^s$ intersects each of $W_1$
and $W_2$ in at most one point. Let $W_1'=\{x\in W_1:\,
W^s(x)\cap W_2\neq\emptyset\}$. Then we define a map
$h:W_1'\to W_2$
by sliding along stable manifolds. This map is often
called a holonomy map. We assume that it is absolutely continuous with
respect to the Lebesgue measures $m_{W_1}$ and $m_{W_2}$,
and its jacobian (at any density point of $W_1'$)
is bounded, i.e.
\be
1/C'\leq\frac{m_{W_2}(h(W_1'))}{m_{W_1}(W_1')}\leq C'
\label{ac}
\ee
with some $C'=C'(T)>0$. \medskip
\noindent
{\bf Non-branching of unstable manifolds}. LUM's are locally unique,
i.e. for any two LUM's $W^1(x),W^2(x)$ we have
$W^1(x)\cap B_{\varepsilon}(x)=W^2(x)\cap B_{\varepsilon}(x)$
for some $\varepsilon>0$.
Here $B_{\varepsilon}(x)$ is the $\varepsilon$-ball
centered at $x$.
Furthermore, let
$\{W^1_n\}$ and $\{W^2_n\}$ be two sequences of LUM's
that have a common limit point $x\in \bar{M}$, i.e.
$\rho(x,W^i_n)\to 0$ as $n\to\infty$ for $i=1,2$.
Assume also that
$\exists\varepsilon>0$ such that
$\rho(x,\partial W^i_n)>\varepsilon$
for all $n\geq 1$ and $i=1,2$. Then
$\rho_H(W^1_n\cap B_{\varepsilon}(x),
W^2_n\cap B_{\varepsilon}(x))\to 0$ as $n\to\infty$,
where
$$
\rho_H(A,B)=\max\{\sup_{x\in A}\rho(x,B),\sup_{y\in B}\rho(y,A)\}
$$
is the Hausdorff distance between sets.
\medskip
\noindent
{\bf u-SRB measures}. A unique probability measure $\nu_W$,
absolutely continuous with respect to the Lebesgue measure
$m_W$, is defined on any LUM $W$ by the
following equation:
\be
\frac{\rho_W(x)}{\rho_W(y)}=
\lim_{n\to\infty}
\prod_{i=1}^{n}\frac{J^u(T^{-i}y)}{J^u(T^{-i}x)}
\ \ \ \ \ \forall x,y\in W
\label{uSRB}
\ee
where $\rho_W(x)=d\nu_W/dm_W(x)$ is the density
of $\nu_W$ with respect to $m_W$.
The existence of the limit in (\ref{uSRB})
is guaranteed by (\ref{distor1}).
We call $\nu_W$ the u-SRB measure on $W$.
Observe that u-SRB measures are conditionally invariant
under $T$, i.e. for any submanifold $W_1\subset TW$,
the measure $T_{\ast}\nu_W|W_1$ (the image of $\nu_W$ under $T$
conditioned on $W_1$) coincides with $\nu_{W_1}$. \medskip
\noindent
{\bf SRB measure}. We assume that
the map $T$ preserves an ergodic Sinai-Bowen-Ruelle (SRB)
measure $\mu$. That is, there is an ergodic probability measure
$\mu$ on $M$ such that for $\mu$-a.e. $x\in M$ a LUM
$W(x)$ exists, and the conditional measure on $W(x)$ induced
by $\mu$ is absolutely continuous with respect to $m_{W(x)}$.
In fact, that conditional measure coincides with the u-SRB
measure $\nu_{W(x)}$. \medskip
\noindent
{\bf $\delta_0$-LUM's}. Let $\delta_0>0$.
We call $W$ a $\delta_0$-LUM if it is a LUM
and diam$\, W\leq\delta_0$. For an open subset
$V\subset W$ and $x\in V$ denote by $V(x)$ the connected
component of $V$ containing the point $x$. Let $n\geq 0$.
We call an open subset $V\subset W$
a $(\delta_0,n)$-subset if $V\cap\Gamma^{(n)}=\emptyset$
(i.e., the map $T^n$ is defined on $V$) and
diam$\, T^nV(x)\leq\delta_0$ for every
$x\in V$. Note that $T^nV$ is then a union of
$\delta_0$-LUM's. Define a function $r_{V,n}$ on $V$ by
\be
r_{V,n}(x)=\rho_{T^nV(x)}(T^nx,\partial T^nV(x))
\label{rVn}
\ee
Note that $r_{V,n}(x)$ is the radius of the largest
open ball in $T^nV(x)$ centered at $T^nx$. In particular,
$r_{W,0}(x)=\rho_W(x,\partial W)$.
\medskip
\noindent
{\bf Flatness and uniformity of LUM's}.
We will only work with $\delta_0$-LUM's for sufficiently
small values of $\delta_0$. For any such $\delta_0$-LUM $W$
the tangent spaces ${\cal T}_xW$ are almost parallel
at all points $x\in W$.
If $n\geq 1$ and $V\subset T^nW$ is another
$\delta_0$-LUM, then $T^n_{\ast}m_W|V$
(the $n$th iterate of $m_W$
conditioned on $V$) has an almost constant
density with respect to $m_{V}$, due to (\ref{distor1}).
The u-SBR measure $\nu_W$ is almost uniform
with respect to the Lebesgue measure $m_W$.
The smaller $\delta_0$, the more accurate these approximations
are, uniformly in $\delta_0$-LUM's $W$.
\medskip
We now turn to the key assumptions on the growth of unstable
manifolds that will ensure a fast decay of correlations.
\medskip
\noindent
{\bf Growth of unstable manifolds}. We assume that
there are constants $\alpha_0\in (0,1)$ and
$\beta_0,D_0,\kappa,\sigma,\zeta>0$ with the
following property. For any
sufficiently small $\delta_0,\delta>0$ and any
$\delta_0$-LUM $W$
there is an open $(\delta_0,0)$-subset
$V^0_{\delta}\subset W\cap{\cal U}_{\delta}$ and
an open $(\delta_0,1)$-subset
$V^1_{\delta}\subset W\setminus {\cal U}_{\delta}$
(one of these may be empty)
such that $m_W(W\setminus
(V^0_{\delta}\cup V^1_{\delta}))=0$ and $\forall\varepsilon>0$
\be
m_W(r_{V^1_{\delta},1}<\varepsilon)\leq \alpha_0\Lambda\cdot
m_W(r_{W,0}<\varepsilon/\Lambda)+\varepsilon\beta_0\delta_0^{-1}m_W(W)
\label{rgrowth11}
\ee
\be
m_W(r_{V^0_{\delta},0}<\varepsilon)\leq D_0\delta^{-\kappa}\,
m_W(r_{W,0}<\varepsilon)
\label{rgrowth10}
\ee
and
\be
m_W(V^0_{\delta})\leq D_0\, m_W(r_{W,0}<\zeta\delta^{\sigma})
\label{rw0}
\ee
We now state our main result, followed by the necessary
definitions.
\begin{theorem}
Let $T$ satisfy the above assumptions. If the system
$(T^n,\mu)$ is ergodic for all $n\geq 1$, then
the map $T$ has exponential decay of correlations (EDC)
and satisfies the central limit theorem (CLT) for H\"older
continuous functions on $M$.
\label{tmmain}
\end{theorem}
Let ${\cal H}_\eta$ be the class of H\"older continuous
functions on $M$ with exponent $\eta >0$:
$$
{\cal H}_\eta=\{f:\, M\to \IR\, |\,
\exists C>0:\, |f(x)-f(y)|\leq C\rho(x,y)^\eta ,\ \forall x,y\in M\}
$$
\noindent
{\bf Exponential decay of correlations}.
We say that
$(T,\mu)$ has exponential decay of correlations for H\"older
continuous functions if $\forall\eta>0$ $\exists\gamma
=\gamma(\eta)\in (0,1)$ such that $\forall f,g\in{\cal H}_\eta$
$\exists C=C(f,g)>0$ such that
$$
\left |
\int_M (f\circ T^n)g\, d\mu-
\int_M f\, d\mu\int_M g\, d\mu
\right |
\leq C\gamma^{|n|}\ \ \ \ \ \forall n\in\ZZ
$$
\noindent
{\bf Central limit theorem}.
We say that $(T,\mu)$ satisfies
central limit theorem (CLT) for H\"older
continuous functions if $\forall \eta>0,f\in{\cal H}_\eta$,
with $\int f\,d\mu=0$, $\exists \sigma_f\geq 0$ such that
$$
\frac{1}{\sqrt{n}}\sum_{i=0}^{n-1}f\circ T^i
\stackrel{\rm distr}{\longrightarrow}{\cal N}(0,\sigma_f^2)
$$
Furthermore, $\sigma_f=0$ iff $f=g\circ T-g$ for some $g\in L^2(\mu)$
\medskip
\section{Filtrations of unstable manifolds}
\label{secF}
\setcounter{equation}{0}
{\bf Existence of LUM's and LSM's}. For any $\varepsilon>0$, let
$$
M^{\pm}_{\Lambda,\varepsilon}=\{x\in M^{\pm}:\,
\rho(T^{\pm n}x,{\Gamma\cup\partial M})>\varepsilon \Lambda^{-n}\ \ \ \
\forall n\geq 0\}
$$
and
$$
M_{\Lambda}^{\pm}=\cup_{\varepsilon>0}M_{\Lambda,\varepsilon}^{\pm}
\ \ \ \ \ \ \ \
M_{\Lambda}^0=M_{\Lambda}^+\cap M_{\Lambda}^-
$$
The following fact is standard \cite{Pes92,LSY}: $\forall x\in
M^-_{\Lambda,\varepsilon}$ there is a LUM $W^u(x)$ such that
$\rho (x,\partial W^u(x))\geq\varepsilon$. Similarly,
$\forall x\in M^+_{\Lambda,\varepsilon}$ there is an
LSM $W^s(x)$ such that $\rho (x,\partial W^s(x))\geq\varepsilon$.
For $x\in M_{\Lambda,\varepsilon}^-$ we denote by $W^u_{\varepsilon}(x)$
the LUM which is a $\varepsilon$-ball centered at $x$ in the
$\rho_{W^u_{\varepsilon}(x)}$ metric, i.e.
$\rho_{W^u_{\varepsilon}(x)}(x,y)=\varepsilon$,
$\forall y\in\partial W^u_{\varepsilon}(x)$.
Similarly, $W^s_{\varepsilon}(x)$
is defined $\forall x\in M_{\Lambda}^+$. We will call
$W^s_{\varepsilon}(x)$ and $W^u_{\varepsilon}(x)$
stable and unstable {\em disks} of radius $\varepsilon$
through $x$, respectively.
\medskip
\noindent
{\bf Z-function}.
Let $W$ be a $\delta_0$-LUM, $n\geq 0$, and
$V\subset W$ an open $(\delta_0,n)$-subset of $W$.
We define the Z-function introduced in \cite{Ch98} by
\be
Z[W,V,n]=
\sup_{\varepsilon>0}\frac{m_W(x\in V:\, r_{V,n}(x)<\varepsilon)}
{\varepsilon\cdot m_W(W)}
\label{ZWVn}
\ee
The supremum here is not necessarily finite. It will be finite
if the boundary $\partial T^nV$ is `regular enough'.
In particular, if $\partial T^nV$ is piecewise smooth
(i.e., consists of a finite number of smooth compact
submanifolds of dimension $\leq d_u-1$), then
$Z[W,V,n]<\infty$, see e.g. \cite{Fed}.
In the case $m_W(W\setminus V)=0$,
the value of $Z[W,V,n]$ characterizes, in a certain way,
the `average size' of the components of $T^nV$ --
the larger they are the smaller $Z[W,V,n]$.
In particular, the value $Z[W,W,0]$
characterizes the size of $W$ in the following way.
\medskip
\noindent
{\em Examples}. Let $W$ be a ball of radius $r$, then
$Z[W,W,0]\sim r^{-1}$. Let $W$ be a cylinder
whose base is a ball of radius $r$ and height $h\gg r$,
then again $Z[W,W,0]\sim r^{-1}$. Let $W$ be a rectangular
box with dimensions $l_1\times l_2\times\cdots\times l_{d_u}$,
then $Z[W,W,0]\sim 1/\min\{l_1,\ldots, l_{d_u}\}$. \medskip
\noindent
{\em Notation}. Let $\delta_{\max}>0$ be so small that
$\alpha :=\alpha_0e^{6\varphi(\delta_{\max})}<1$. Denote also
$\beta :=\beta_0e^{6\varphi(\delta_{\max})}$ and
$D :=D_0e^{6\varphi(\delta_{\max})}$. We will
always assume that $\delta_0<\delta_{\max}$,
so that $\alpha <1$.
Next, put
$$
\bar{\beta}=2\beta/(1-\alpha)
$$
and
$$
a=-(\ln\alpha)^{-1}\ \ \ \ {\rm and}\ \ \ \
b=\max\{0,-\ln(\delta_0(1-\alpha)/\beta)/\ln\alpha\}
$$
We also put
\be
\delta_1=\delta_0/(2\bar{\beta})
\label{delta1}
\ee
\noindent
{\bf Convention of $\delta$'s}. We will define some
small parameters $\delta_i$, $i\geq 1$, so that each
$\delta_i$ will be a certain function of $\delta_{i-1}$.
Still, we can vary all of them together preserving
the specified relations between them, like the above (\ref{delta1}).
\medskip
\noindent
{\bf $\delta$-Filtration}. Let $\delta_0,\delta>0$ and
$W$ be a $\delta_0$-LUM.
Two sequences of open subsets $W=W_0^1\supset W_1^1
\supset W_2^1\supset\cdots$ and $W^0_n\subset W^1_n\setminus W^1_{n+1}$,
$n\geq 0$, are said to make a {\em $\delta$-filtration} of $W$,
denoted by $\{W^1_n,W^0_n\}$ if\footnote{In \cite{Ch98},
it was called a refined u-filtration.} $\forall n\geq 0$\\
(a) the sets $W_n^1$ and $W^0_n$
are $(\delta_0,n)$-subsets of $W$;\\
(b) $m_W(W^1_n\setminus (W^1_{n+1}\cup W^0_n))=0$.\\
(c) $T^nW^1_{n+1}\cap {\cal U}_{\delta\Lambda^{-n}}=\emptyset$
and $T^nW^0_n\subset {\cal U}_{\delta\Lambda^{-n}}$.
We put $W_{\infty}^1=\cap_{n\geq 0}W_n^1$.
Observe that $W_{\infty}^1\subset M_{\Lambda,\delta}^+$, and so
a stable disk $W^s_{\delta}(x)$ of radius $\delta$
exists at every point $x\in W_{\infty}^1$.
Put also $w_n^1=m_W(W_n^1)/m_W(W)$
and $w_n^0=m_W(W_n^0)/m_W(W)$.
Observe that $w_n^1=1-w_0^0-\cdots -w_{n-1}^0$ and
$w_n^1\searrow w_{\infty}^1:=
m_W(W^1_{\infty})/m_W(W)$ as $n\to\infty$.
\begin{theorem}
Let $W$ be a $\delta_0$-LUM and $\delta>0$.
Then there is a
$\delta$-filtration $(\{W_n^1\},\{W_n^0\})$ of $W$ such that \\
{\rm (i)} $\forall n\geq 1$ and $\forall\varepsilon>0$ we have
\be
m_W(r_{W^1_n,n}<\varepsilon)\leq (\alpha\Lambda)^n\cdot
m_W(r_{W,0}<\varepsilon/\Lambda^n)+\varepsilon\beta\delta_0^{-1}
(1+\alpha+\cdots +\alpha^{n-1})\, m_W(W)
\label{rgrowthn1}
\ee
Furthermore, $\forall n\geq 0$ and $\forall\varepsilon>0$
\be
m_W(r_{W^0_n,n}<\varepsilon)\leq D\delta^{-\kappa}\Lambda^{\kappa n}\,
m_W(r_{W^1_n,n}<\varepsilon)
\label{rgrowthn0}
\ee
and
\be
m_W(W^0_n)\leq D\, m_W(r_{W^1_n,n}<\zeta\delta^{\sigma}\Lambda^{-\sigma n})
\label{rwn}
\ee
{\rm (ii)} we have $\forall n\geq 1$
\be
Z[W,W^1_n,n]\leq \alpha^n Z[W,W,0]+\beta\delta_0^{-1} (1+\alpha+\cdots +\alpha^{n-1})
\label{rexpn}
\ee
{\rm (iii)} for any $n\geq 0$ we have $Z[W,W^0_n,n] \leq D
\delta^{-\kappa}\Lambda^{\kappa n}\cdot
Z[W,W^1_n,n]$;\\
{\rm (iv)} for any $n\geq 0$ we have
$w_n^0\leq D\zeta\delta^{\sigma}\Lambda^{-\sigma n}\cdot Z[W,W^1_n,n]$.
\label{tmexpr}
\end{theorem}
{\em Proof} of (\ref{rgrowthn1})-(\ref{rwn}) goes by induction on $n$.
The bound (\ref{rgrowthn1}) for $n=1$ and (\ref{rgrowthn0})-(\ref{rwn})
for $n=0$ follow from our assumptions (\ref{rgrowth11})-(\ref{rw0}),
respectively, after we set $W^1_1:=V^1_{\delta}$
and $W^0_0:=V^0_{\delta}$,
since $\alpha_0<\alpha$, $\beta_0<\beta$, and $D_00$
$$
m_{W_{n,j}'}(r_{V^1_{n,j},1}<\varepsilon)\leq \alpha_0\Lambda\cdot
m_{W_{n,j}'}(r_{W_{n,j}',0}<\varepsilon/\Lambda)
+\varepsilon\beta_0\delta_0^{-1}m_{W_{n,j}'}(W_{n,j}')
$$
$$
m_{W_{n,j}'}(r_{V^0_{n,j},0}<\varepsilon)\leq
D_0\delta^{-\kappa}\Lambda^{\kappa n}\,
m_{W_{n,j}'}(r_{W_{n,j}',0}<\varepsilon)
$$
and
$$
m_{W_{n,j}'}(V^0_{n,j})\leq D_0\,
m_{W_{n,j}'}(r_{W_{n,j}',0}<\zeta\delta^{\sigma}\Lambda^{-\sigma n})
$$
according to (\ref{rgrowth11})-(\ref{rw0}).
Using the distorsion bound (\ref{distor1})
and our definition of $\alpha,\beta,D$ yields
$$
m_{W_{n,j}}(r_{U^1_{n,j},n+1}<\varepsilon)\leq \alpha\Lambda\cdot
m_{W_{n,j}}(r_{W_{n,j},n}<\varepsilon/\Lambda)
+\varepsilon\beta\delta_0^{-1}m_{W_{n,j}}(W_{n,j})
$$
$$
m_{W_{n,j}}(r_{U^0_{n,j},n}<\varepsilon)\leq
D\delta^{-\kappa}\Lambda^{\kappa n}\,
m_{W_{n,j}}(r_{W_{n,j},n}<\varepsilon)
$$
and
$$
m_{W_{n,j}}(U^0_{n,j})\leq D\,
m_{W_{n,j}}(r_{W_{n,j},n}<\zeta\delta^{\sigma}\Lambda^{-\sigma n})
$$
where $U^1_{n,j}:=T^{-n}V^1_{n,j}$ and $U^0_{n,j}:=T^{-n}V^0_{n,j}$.
Summing up over $j$ gives
$$
m_W(r_{W^1_{n+1},n+1}<\varepsilon)\leq \alpha\Lambda\cdot
m_W(r_{W^1_n,n}<\varepsilon/\Lambda)
+\varepsilon\beta\delta_0^{-1}m_{W}(W^1_n)
$$
$$
m_W(r_{W^0_n,n}<\varepsilon)\leq
D\delta^{-\kappa}\Lambda^{\kappa n}\,
m_W(r_{W^1_n,n}<\varepsilon)
$$
and
$$
m_W(W^0_n)\leq D\,
m_W(r_{W^1_n,n}<\zeta\delta^{\sigma}\Lambda^{-\sigma n})
$$
where $W^1_{n+1}:=\cup_j U^1_{n,j}$ and $W^0_n:=\cup_j U^0_{n,j}$.
The bounds (\ref{rgrowthn0}) and (\ref{rwn}) for the
current value of $n$ are proved.
A direct use of (\ref{rgrowthn1}) with $\varepsilon$
replaced by $\varepsilon/\Lambda$, along with the obvious
bound $m_{W}(W^1_n)\leq m_W(W)$, gives (\ref{rgrowthn1})
with $n$ replaced by $n+1$. This
completes the inductive proof of (\ref{rgrowthn1}).
Next, the parts (ii)-(iv) follow directly from
(\ref{rgrowthn1})-(\ref{rwn}), respectively, upon
dividing by $m_W(W)$ and using (\ref{rVn}). $\Box$. \medskip
\noindent
{\bf Remark}. The proofs of the above theorem would
go through even for slightly smaller values of $\alpha,\beta,D$:
$\alpha=\alpha_0e^{2\varphi(\delta_0)}$,
$\beta=\beta_0e^{2\varphi(\delta_0)}$, and
$D=D_0e^{2\varphi(\delta_0)}$. Our choice of $\alpha,\beta,D$
allows us to extend the above theorem to absolutely continuous measures
on $W$ whose density with respect to the Lebesgue measure $m_W$
is close enough to a constant. Precisely, if $\tilde{m}_W$ is
a measure on $W$ with density $\tilde{\rho}(x)=d\tilde{m}_W/dm_W(x)$,
then we can replace $m_W$ with $\tilde{m}_W$ in (\ref{ZWVn})
and in the above theorem provided
$\tilde{\rho}(x)/\tilde{\rho}(y)\leq e^{2\varphi(\delta_0)}$,
$\forall x,y\in W$. In particular, this holds for
the u-SRB measure $\tilde{m}_W=\nu_W$.
\begin{corollary}
Let $\bar{Z}_W=\max\{Z[W,W,0],\bar{\beta}/\delta_0\}$.
Then \\
{\rm (i)} $Z[W,W^1_n,n]\leq \bar{Z}_W$ and
$Z[W,W^0_n,n]\leq D\delta^{-\kappa}\Lambda^{\kappa n}\,\bar{Z}_W$
for all $n\geq 0$;\\
{\rm (ii)} $Z[W,W^1_n,n]\leq\bar{\beta}/\delta_0=(2\delta_1)^{-1}$
for all $n\geq a\ln Z[W,W,0]+b$;\\
{\rm (iii)}
$w_n^0\leq D\zeta\delta^{\sigma}\Lambda^{-\sigma n}\bar{Z}_W$
for all $n\geq 0$;\\
{\rm (iv)} $w_n^1\geq 1-D\zeta\delta^{\sigma}\bar{Z}_W/(1-\Lambda^{-\sigma})$
for all $n\geq 1$; \\
{\rm (v)} $m_W(W_{\infty}^1)\geq m_W(W)
\cdot \left [1-D\zeta\delta^{\sigma}\bar{Z}_W/(1-\Lambda^{-\sigma})\right ]$
\label{crwzn}
\end{corollary}
\noindent
{\bf Modified Z-function}.
The values $Z[W,W_n^1,n]$ and $Z[W,W_n^0,n]$ do not characterize
the average size of the components of $T^nW^1_n$ or $T^nW^0_n$,
respectively, since
$W^1_n$ and $W^0_n$ are not subsets of full measure
in $W$. To characterize the average sizes of the
components of any $(\delta_0,n)$-subset $V\subset W$
we will also use the quantity
\be
Z[V,n]:=
\sup_{\varepsilon>0}\frac{m_W(x\in V:\, r_{V,n}(x)<\varepsilon)}
{\varepsilon\cdot m_W(V)}
=Z[W,V,n]\times\frac{m_W(W)}{m_W(V)}
\label{ZVn}
\ee
This value depends on $V$ and $n$, but not on $W$.
It characterizes the average size of the components of $T^nV$.
Accordingly, the values of
$$
Z[W_n^1,n]=Z[W,W_n^1,n]/w_n^1
\ \ \ \ \ {\rm and}\ \ \ \ \
Z[W_n^0,n]=Z[W,W_n^0,n]/w_n^0
$$
characterize the average size of the components of
$T^nW^1_n$ or $T^nW^0_n$, respectively.
\medskip
\noindent
{\bf Special case}.
In our further arguments, the set $W^1_{\infty}$ will be
often very dense in $W$, so that $w^1_{\infty}>0.9$. We call
this a special case, and corollary~\ref{crwzn} then implies
that for all
$n\geq a\ln Z[W,W,0]+b$ we have $Z[W^1_n,n]\leq 0.6/\delta_1$.
We will say then that the components of $T^nW^1_n$ are
large enough, on the average.\medskip
\noindent
{\bf Remark}. The values of $Z[W,W_n,n]$,
$Z[W,W^1_n,n]$, $Z[W,W^0_n,n]$,
$w^1_n$, and $w^0_n$ above will certainly
not change if we replace the Lebesgue measure $m_W$
by any measure proportional to it. It is also
straightforward that all the above results
extend to countable disjoint unions of
$\delta_0$-LUM's with finite measures that are
linear combinations of the Lebesgue measures on individual components.
Precisely, let $W=\cup_k W^{(k)}$ be a countable union of pairwise
disjoint $\delta_0$-LUM's
and let $\hat{m}_W=\sum_k u_k m_{W^{(k)}}$,
with some $u_k>0$, be a finite measure on $W$. Then $Z[W,V,n]$ is
still defined by (\ref{ZWVn}), with $m_W$ replaced by $\hat{m}_W$,
for any set $V=\cup_k V^{(k)}$,
where $V^{(k)}$ are some open $(\delta_0,n)$-subsets of $U^{(k)}$.
The definition of $\delta$-filtration and the proof of Theorem~\ref{tmexpr}
go through with only minor obvious changes.
\medskip
\noindent
{\bf Final Remark}. Let $W'$ be a $\delta_0$-LUM,
$k\geq 1$, and $V'\subset W'$ an open
$(\delta_0,k)$-subset. Then $W=T^kV'$ is a finite
or countable union of $\delta_0$-LUM's. The measure
$\tilde{m}_W:=T^k_{\ast}m_{W'}|W$ on $W$ is almost uniform
(proportional to the Lebesgue measure $m_W$) on each component of $W$.
Actually, its density differs from a constant by less than
$e^{2\varphi(\delta_0)}$, according to (\ref{distor1}).
Due to the remark after Theorem~\ref{tmexpr},
all the above results will then apply to $(W,\tilde{m}_W)$,
instead of $(W,m_W)$.
\medskip
The following proposition generalizes the above special case.
Its proof goes like the proof of Proposition 4.4 in \cite{Ch98},
with obvious modifications.
\begin{proposition}
Let $(\{W_n^1\},\{W_n^0\})$ be a $\delta$-filtration of
a $\delta_0$-LUM $W$ satisfying Theorem~\ref{tmexpr}, such that
$w^1_{\infty}=p>0$. Then for all $n\geq a_1(\ln Z[W,W,0]-\ln p)
+b_1$ we have $m_W(W^1_{\infty})/m_W(W^1_n)\geq 0.9$
and $Z[W^1_n,n]\leq 0.6/\delta_1$, i.e. the components of
$T^nW^1_n$ will be large enough, on the average.
Here $a_1=a+(\sigma\ln \Lambda)^{-1}$ and $b_1$ is another constant
determined by $\alpha,\beta,\delta_0,\Lambda,\zeta,D$.
\label{prwp}
\end{proposition}
\noindent
{\bf Final Remark (Part 2)}. The above proposition also applies
to any pair $(W,\tilde{m}_W)$ described in Final Remark before the
proposition. Likewise, some further results stated and proved
for $\delta_0$-LUM's $W$ with Lebesgue measures $m_W$,
will also apply to measures $\tilde{m}_W=T^k_{\ast}m_{T^{-k}W}$
on $W$ for any $k\geq 1$.
\section{Rectangles}
\label{secR}
\setcounter{equation}{0}
Here we mostly repeat, in a brief manner, the constructions of
\cite{Ch98}, Section 5. \medskip
\noindent
{\bf Rectangles and subrectangles}. A subset $R\subset M^0$
is called a {\em rectangle} if
$\exists\varepsilon>0$ such that for any $x,y\in R$
there is an LSM $W^s(x)$ and an LUM $W^u(y)$,
both of diameter $\leq\varepsilon$, that
meet in exactly one point, which also belongs
in $R$. We denote that point by $[x,y]=W^s(x)\cap W^u(y)$.
A subrectangle $R'\subset R$ is called a u-subrectangle if
$W^u(x)\cap R=W^u(x)\cap R'$
for all $x\in R'$. Similarly, s-subrectangles are defined.
We say that a rectangle $R'$ u-crosses another rectangle
$R$ if $R'\cap R$ is a u-subrectangle in $R$ and an
s-subrectangle in $R'$. \medskip
\noindent
{\bf s-Shadowing and s-distance}.
Let $x\in M$ and $r\in (0,\delta_0)$. We denote by $S_r(x)$
any s-manifold that is a ball of radius $r$ centered at $x$ in
its own metric, $\rho_{S_r(x)}$. By that we mean $\rho_{S_r(x)}
(x,y)=r$, $\forall y\in\partial S_r(x)$.
We call such $S_r(x)$ an {\em s-disk}.
In order to define s-disks also around points close to $\partial M$
we extend the cone families $C^u$ and $C^s$ continuously
beyond the boundaries of $M$ into
the $\delta_0$-neighborhood of $M$.
Then s-disks $S_r(x)$ exist $\forall x\in M, \forall r\in (0,\delta_0)$.
Let $W$ be a $\delta_0$-LUM, and $x\in M$.
Clearly, any s-disk $S_{\delta_0}(x)$ can meet $W$ in at most
one point. We call
$$
H_x(W)=\{y\in W:\, y=S_{\delta_0}(x)\cap W\ \
{\rm for}\ \ {\rm some}\ \ S_{\delta_0}(x)\}
$$
the {\em s-shadow} of $x$ on $W$.
We say that a point $x\in M$ is overshadowed by a LUM
$W$ if $\forall S_{\delta_0}(x)$ we have $S_{\delta_0}(x)
\cap W\neq\emptyset$. We call
$$
\rho^s(x,W)=\sup_{S_{\delta_0}(x)}\rho_{S_{\delta_0}(x)}
(x,S_{\delta_0}(x)\cap W)
$$
the {\em s-distance} from $x$ to $W$.
Let $W,W'$ be two $\delta_0$-LUM's. We call
$$
H_W(W')=\cup_{x\in W}H_x(W')
$$
the s-shadow of $W$ on $W'$. We say that $W'$ overshadows
$W$ if it overshadows every point $x\in W$. In this case
we define
$$
\rho^s(W,W')=\sup_{x\in W}\rho^s(x,W')
$$
the s-distance from $W$ to $W'$.
We assume that $\delta_0$, and hence
$\delta_1=\delta_0/(2\bar{\beta})$, are small
enough, so that
$$
A_{\delta_1}\stackrel{\rm def}{=}\{x\in M:\, {\rm the}\
{\rm unstable}\ {\rm disk}\ \
W^u_{\delta_1}(x)\ \ {\rm exists}\}\neq\emptyset
$$
Let $z\in A_{\delta_1}$. Consider $W(z):=W^u_{\delta_1/3}(z)$,
the `central part' of the existing unstable disk $W^u_{\delta_1}(z)$.
It is a $\delta_0$-LUM, and a perfect ball in its own
metric. It is easy to compute that for a perfect ball $W$ of
radius $\delta$ in $\IR^{d_u}$ one has $Z[W,W,0]=d_u/\delta$.
Since the manifolds $W(z)$, $z\in A_{\delta_1}$, actually have
some (bounded) sectional curvature, $Z[W(z),W(z),0]$ might
be larger than $3d_u/\delta_1$, but if $\delta_1$ is small enough,
we will have \cite{Ch98}
\be
Z[W(z),W(z),0]\leq 4d_u/\delta_1
\label{Z1}
\ee
for all $z\in A_{\delta_1}$.
Now let $\delta_2$ be defined by
\be
\frac{\delta_2^{\sigma}}{\delta_1} =
\frac{1-\Lambda^{-\sigma}}{40\, D\zeta d_u}
\label{d12}
\ee
For any $z\in A_{\delta_1}$ fix one
$\delta_2$-filtration $(\{W_n^1(z)\},\{W_n^0(z)\})$
of $W(z)$ satisfying Theorem~\ref{tmexpr}.
Recall that $\forall x\in W^1_{\infty}(z)$ a stable disk
$W^s_{\delta_2}(x)$ exists, cf. Sect.~\ref{secF}.
The following lemmas are consequences of (\ref{Z1}), (\ref{d12})
and the parts (ii), (v) of Corollary~\ref{crwzn},
see proofs in \cite{Ch98}.
\begin{lemma}
$m_{W(z)}(W^1_{\infty}(z))\geq 0.9\cdot m_{W(z)}(W(z))$.
\label{lm0.9}
\end{lemma}
\begin{lemma}
$\forall n\geq n_0':=a\ln(16d_u)+\max\{1,a\ln[\beta\delta_0^{-1}/(1-\alpha)]\}$
we have \\
{\rm (i)} $Z[W(z),W^1_n(z),n]<(2\delta_1)^{-1}$ and
$Z[W_n^1(z),n]<0.6/\delta_1$;\\
{\rm (ii)} $
m_{W(z)}(x\in W_n^1(z):\, r_{W_n^1(z),n}(x)>\delta_1)
> 0.4\cdot m_{W(z)}(W_n^1(z))
> 0.4\cdot m_{W(z)}(W_{\infty}^1(z))$.\\
In other words, (ii) means that at least 40\% of the points in
$T^nW^1_n(z)$ (with respect to the measure induced by $m_{W(z)}$)
lie a distance $\geq\delta_1$ away from the boundaries of $T^nW^1_n(z)$.
\label{lm40}
\end{lemma}
\noindent
{\bf Remark}. Let $z\in A_{\delta_1}$. For a moment, let
$W(z)=W_{\varepsilon}^u(z)$ be the stable disk of any
radius $\varepsilon\in (\delta_1/3,\delta_1)$.
That disk $W(z)$ is larger than $W^u_{\delta_1/3}(z)$,
and so (\ref{Z1}) still holds. Therefore, the statements
(i) and (ii) of the above lemma hold as well. Furthermore,
if, again for a moment, we decrease $\delta_2$ thus making
the ratio $\delta_2^{\sigma}/\delta_1$ even smaller than the one
specified by (\ref{d12}),
then Lemma~\ref{lm0.9} will still hold, and then so will (i)
and (ii) of Lemma~\ref{lm40}.
\medskip
Let $\delta_3\ll\delta_2$, to be specified later. The
following proposition is proved in \cite{Ch98}, Proposition 5.3.
\begin{proposition}
Let $W$ be a $\delta_0$-LUM, and $W'$ another
$\delta_0$-LUM that overshadows $W$ and $\rho^s(W,W')\leq\delta_3$.
Let $(\{W_n^1\},\{W_n^0\})$ be a
$\delta_2$-filtration of $W$. Then $\forall n\geq 1$ and any connected
component $V$ of $W^1_n$ there is a connected domain $V'\subset W'
\setminus \Gamma^{(n)}$ such that the $\delta_0$-LUM $T^nV'$
overshadows the $\delta_0$-LUM $T^nV$,
and $\rho^s(T^nV,T^nV')\leq\delta_3\Lambda^{-n}$.
\label{pruu}
\end{proposition}
\noindent
{\bf Canonical rectangles}.
For any $z\in A_{\delta_1}$ we define a `canonical'
rectangle $R(z)$ as follows:
$y\in R(z)$ iff $y= W^s_{\delta_2}(x)\cap W$ for some
$x\in W^1_{\infty}(z)$ and for some LUM $W$ that
overshadows $W(z)=W^u_{\delta_1/3}(z)$, and such that
$\rho^s(W(z),W)\leq\delta_3$.
Observe that if $\delta_3/\delta_20$ is
determined by the minimum angle between the stable and
unstable cone families, then every $W$ that overshadows
$W(z)$ and is $\delta_3$-close to it in the
above sense will meet all stable disks $W^s_{\delta_2}(x)$,
$x\in W^1_{\infty}(z)$. In that case $R(z)$ will be a rectangle, indeed.
We fix $\delta_3/\delta_2$ now as follows:
\be
\delta_3/\delta_2= \min\{c',1-\Lambda^{-1}, 1/3\}
\label{d21}
\ee
For any connected subdomain $V\subset W(z)$ the set
$R_V(z):=\{y\in R(z):\, W^s(y)\cap V\neq\emptyset\}$ is
an s-subrectangle in $R(z)$ ``based on $V$''.
For $n\geq 1$, the partition of $W^1_n(z)$ into connected
components, $\{V\}$, induces a partition of $R(z)$ into
s-subrectangles $\{R_V(z)\}$ that are based on those components.
If $R_V(z)$ is one of those s-subrectangles,
then Proposition~\ref{pruu} implies
that $T^nR_V(z)$ is a rectangle.
\begin{lemma}
For any $\delta_3>0$ there is a $\delta_4>0$
such that $\forall z,z'\in A_{\delta_1}$
such that $\rho(z,z')<\delta_4$, the LUM $W^u_{\delta_1/2}(z')$
overshadows the LUM $W(z)=W^u_{\delta_1/3}(z)$, and $\rho^s
(W(z),W^u_{\delta_1/2}(z'))\leq\delta_3/2$. Likewise,
the LUM $W^u_{\delta_1}(z)$ overshadows the LUM $W^u_{\delta_1/2}(z')$,
and $\rho^s(W^u_{\delta_1/2}(z'),W^u_{\delta_1}(z))\leq\delta_3/2$.
\label{lmzz'}
\end{lemma}
{\em Proof}. It is enough to prove the first statement, the
second one is completely similar. We actually need to prove
that $\forall x\in W(z)$ we have
$\rho^s(x,W^u_{\delta_1/2}(z'))\leq\delta_3/2$.
Assume that this is not the case, i.e. $\forall\delta_4>0$
$\exists z,z'\in A_{\delta_1}$ such that $\rho(z,z')<
\delta_4$ and $\exists x\in W(z)$ such that
$\rho^s(x,W^u_{\delta_1/2}(z'))>\delta_3/2$.
We take a sequence $\delta_4=1/n$, $n\geq 1$,
and the corresponding points $z_n,z_n'$. Due
to the compactness of $\bar{M}$, there is a
subsequence $n_k$ such that $\exists z_{\infty}:=\lim_k z_{n_k}=
\lim_k z_{n_k}'\in\bar{M}$.
This clearly contradicts our assumption on non-branching of
unstable manifolds.
$\Box$\medskip
Let $n_0''=\min\{n\geq 1:\, \Lambda^n>2\}$. The following
proposition is proved in \cite{Ch98}, Proposition~5.3.
\begin{proposition}
Let $z\in A_{\delta_1}$ and $n\geq n_0''$. Let $V$ be a connected component
of $W^1_n(z)$ and $x\in V$ such that $r_{V,n}(x)>\delta_1$
and $\rho(T^nx,z')<\delta_4$ for some $z'\in A_{\delta_1}$.
Then the rectangle $T^nR_V(z)$ u-crosses the rectangle $R(z')$,
i.e. $T^nR_V(z)\cap R(z')$ is {\rm (i)} a u-subrectangle in
$R(z')$ and {\rm (ii)} an s-subrectangle in $T^nR_V(z)$.
\label{prret}
\end{proposition}
\section{Rectangular structure, return times, and tail bound}
\label{secS}
\setcounter{equation}{0}
The constructions in this section mostly repeat those in
\cite{Ch98}, Sections 6,7. \medskip
Consider the SRB measure $\mu$. Clearly, if $\delta_0$
is small enough, then $\exists z_1\in A_{\delta_1}$
such that $\mu(R(z_1))>0$.
We fix such a $\delta_0$ and one such
$z_1\in A_{\delta_1}$. We then denote, for brevity,
$R=R(z_1)$, $W=W(z_1)$, $W^1_{\infty}=W^1_{\infty}(z_1)$, etc.
Let ${\cal Z}=\{z_1,z_2,\ldots,z_p\}$ be a finite $\delta_4$-dense
subset of $A_{\delta_1}$ containing the above point $z_1$.
We call ${\cal R}=\cup_i R(z_i)$ the rectangular
structure. It is a finite union of rectangles that are
likely to overlap and may not cover $M$ or even the support
of $\mu$.
We will partition the set $W_{\infty}^1$ into a countable
collection of subsets $W_{\infty,k}^1$, $k\geq 0$,
such that for every $k\geq 1$ there is an integer $r_k\geq 1$ such that
for the s-subrectangle $R_k\subset R$ based
on\footnote{By the s-subrectangle $R_k\subset R$
based on $W^1_{\infty,k}$ we mean the set
$R_k=\{x\in R:\, W^s(x)\cap W^1_{\infty}\in W^1_{\infty,k}\}$.}
$W^1_{\infty,k}$ the set $T^{r_k}(R_k)$ will be
a u-subrectangle in some $R(z_i)$, $z_i\in {\cal Z}$.
This fact is considered as a {\em proper return}
(of $R_k$ into ${\cal R}$, under $r_k$ iterations of $T$).
We define the return time function $r(x)$ on $W^1_{\infty}$ by
$r(x)=r_k$ for $x\in W^1_{\infty,k}$, $k\geq 1$, and
$r(x)=\infty$ for $x\in W^1_{\infty,0}$.
We call the sets $W^1_{\infty,k}$ for $k\geq 1$ {\em gaskets},
cf. \cite{Ch98}, and $W^1_{\infty,0}$ the {\em leftover set}.
The following theorem immediately follows from
Young \cite{LSY}, see also \cite{Ch98}, Section 6.
\begin{theorem}
Assume that $(T^n,\mu)$ is ergodic for all
$n\geq 1$, and $\mu(R)>0$. If
\be
m_W\{r(x)>n\}\leq C\theta^n\ \ \ \ \forall n\geq 1
\label{tail}
\ee
for some $C>0$, $\theta\in (0,1)$,
then the system $(T,\mu)$ satisfies EDC and CLT.
\label{tmY2}
\end{theorem}
The construction of the partition $W^1_{\infty}=\cup_k W^1_{\infty,k}$
consists of several steps. \medskip
\noindent
{\bf Initial growth.}
First, we take $n_1=\max\{n_0',n_0''\}$.
According to Lemma~\ref{lm40}, we have \\
(a) $Z[W,W_{n_1}^1,n_1]<(2\delta_1)^{-1}$ and $Z[W_{n_1}^1,n_1]<0.6/\delta_1$,
i.e. the components of $T^{n_1}W^1_{n_1}$ are large enough,
on the average, and \\
(b) $m_{W}\{x\in W^1_{n_1}:\,
r_{W^1_n,n}(x)\geq \delta_1\} \geq 0.4\, m_W(W^1_{n_1})$,
i.e. at least 40\% of the points in $T^{n_1}W^1_{n_1}$ (with respect to
the measure induced by $m_W$) lie a distance $\geq\delta_1$
away from $\partial T^{n_1}W^1_{n_1}$.\\
(Recall that (b) actually follows from (a).)
Let $W^g:=T^{n_1}W^1_{n_1}$, and $\tilde{m}_{W^g}=T^{n_1}_{\ast}
m_W|W^g$ the induced measure on $W^g$.
For every connected component $V\subset W^g$ such that
$\exists x_V\in V:\, \rho_V(x_V,\partial V)\geq\delta_1$ we arbitrarily
fix one such point $x_V$. Then $x_V\in A_{\delta_1}$, and
$\exists z_V\in {\cal Z}$ such that $\rho(x_V,z_V)<\delta_4$.
We fix one such $z_V$, too.
Then we label the set $T^{-n_1}(V\cap R(z_V))$ as one of our
gaskets $W^1_{\infty,k}$, and we define $r_k=n_1$ on it. According
to Proposition~\ref{prret}, $T^{r_k}(R_k)$ is a u-subrectangle
in $R(z_V)$. As in \cite{Ch98},
we will call the set $V\cap R(z_V)$ a gasket, too.
The next lemma follows from Lemmas~\ref{lm0.9}
and \ref{lm40}, along with the absolute continuity (\ref{ac}),
see \cite{Ch98}.
\begin{lemma}
There is a $q=q(T)>0$ such that, independently of the choice
of the points $x_V$ and $z_V$ in the components
$V\subset W^g$, the just defined gaskets
$W^1_{\infty,k}$ satisfy
$$
m_W\left (\cup W^1_{\infty,k}\right )\geq q\, m_W(W^1_{n_1})
$$
\label{lmq}
\end{lemma}
In other words, a certain fraction ($\geq q$) of $W^g$ returns
at the $n_1$-th iteration. This is the earliest return.
The definition of further returns requires more careful
considerations to avoid possible overlaps of gaskets,
as it is explained in \cite{Ch98}. \medskip
\noindent
{\bf Capture}.
Every connected component $V$ of $W^g$ where a point
$x_V$ is picked is now subdivided
into two connected sets: $V^c:=W^u_{\delta_1/2}(x_V)$
and $V^f:=V\setminus V^c$. The gasket
$V\cap R(z_V)$ defined above lies wholly
in $V^c$, see \cite{Ch98}. We say that $V^c$ is `captured' at the
$n_1$-th iteration, and the set
$V^f$, is `free to move'. Let
$W^f=\cup_{V\subset W^g}V^f$.
The set $W^{f}$
contains no points of the previously defined gaskets.
For $n\geq 0$, let $W^{f,1}_n:=W^f\cap T^{n_1}W^1_{n_1+n}$ and
$W^{f,0}_n:=$int$\, (W^{f,1}_n\setminus W^{f,1}_{n+1})$ for $n\geq 0$.
It is easy to see that the sets $\{W^{f,1}_n,W^{f,0}_n\}$ make a
$(\delta_2\Lambda^{-n_1})$-filtration
of the manifold $W^f$ and satisfies
Theorem~\ref{tmexpr}.
It was shown in \cite{Ch98}, Sect.~6, that
$Z[W^{f,1}_{n_2},n_2]<0.6/\delta_1$ for $n_2:
=[-\ln 9.6/\ln\alpha]+1$.
In other words, it takes a fixed number of iterations,
$n_2$, to recover the lost average size of the freely
moving manifold, $T^nW^{f,1}_n$, $n\geq 0$,
after the removal of the
captured parts from $W^g$. As soon as this is done,
i.e. at the iteration $n=n_2$,
at least 40\% of the image $T^{n}W^{f,1}_n$,
will lie a distance $\geq\delta_1$ from its boundary,
just as in the claim (b) above.
Next, we inductively repeat the above procedure of
picking points $x_V,z_V$ in the large components $V$
of the freely moving manifold, defining new gaskets
$V\cap R(z_V)$, capturing disks containing the
newly defined gaskets, etc., see \cite{Ch98}.
According to Lemma~\ref{lmq},
the points of the freely moving manifold are being
captured at an exponential rate: at least a fraction
$q>0$ of them is captured every $n_2$ iterations of $T$.
Let $t_0(x)$, $x\in W^1_{\infty}$, be the number
of iterations it takes to capture the image of the point $x$.
Lemma~\ref{lmq} implies that
\be
m_W(t_0(x)>n)/m_W(W^1_{\infty})\leq C_0\theta_0^n
\label{t0}
\ee
with $\theta_0=q^{1/n_2}<1$ and some $C_0>0$. In
particular, $t_0(x)<\infty$ for a.e. $x\in W^1_{\infty}$. \medskip
\noindent
{\bf Release}.
Next, we take care of the captured parts of the manifolds $T^nW^1_n$,
$n\geq 1$.
Let $B^c\subset T^{n_c}W^1_{n_c}$ be a connected part captured
at the $n_c$-th iteration of $T$, $n_c\geq n_1$. Then
$B^c$ is a perfect ball of radius $\delta_1/2$ in
some connected component of $T^{n_c}W^1_{n_c}$. It carries the measure
$\tilde{m}_{B^c}=T^{n_c}_{\ast}m_W|B^c$.
The center $x_c$ of the disk $B^c$ belongs in $A_{\delta_1}$,
and there is a point $z_c\in {\cal Z}$ such that $\rho(x_c,z_c)
<\delta_4$ and such that the set $B^c_R:=B^c\cap R(z_c)$ makes a new
gasket at the moment of capture.
Let $B^c_{\infty}:=B^c\cap T^{n_c}W^1_{\infty}$.
Denote $B^c_n=B^c\cap T^{n_c}W^1_{n_c+n}$ for $n\geq 0$.
Observe that
\be
Z[B^c,B^c,0]\leq 4d_u/\delta_1
\ \ \ \ \ {\rm and}\ \ \ \ \
Z[B^c_n,n]<0.6/\delta_1\ \
\forall n\geq n_0'
\label{ZBc}
\ee
according to the remark after Lemma~\ref{lm40}.
In other words, it takes $n_0'$ iterations of $T$ to make
the components of $T^nB^c_n$ large enough, on the average.
In order to define a new gasket in any large component $V$
of $T^nB^c_n$ and avoid possible overlaps with the image
$T^nB^c_R$ of the old gasket $B^c_R$,
we will make sure that $V$ contains no points
of $T^nB_R^c$. We define
a `point release time', $f(x)$, for points
$x\in B^c_{\infty}\setminus B^c_R$. A point $x$
will be released if $T^{f(x)}(x)$ is sufficiently
far from $T^{f(x)}B_R^c$.
The definition of the release time is different
for points of different type: \medskip
{\em Type I points} are such that there is an LSM
$W^s(x)$ meeting the manifold $W^u_{\delta_1}(z_c)$
in one point, call it $h(x)$. Then $h(x)\notin
W^1_{\infty}(z_c)$, otherwise $x$ would have belonged in $B^c_R$.
Hence, either $h(x)\in W^u_{\delta_1}(z_c)\setminus
W^u_{\delta_1/3}(z_c)$ or $h(x)\in W^0_m(z_c)$ for some
$m=m(x)\geq 0$.
In the former case, we set $m(x)=0$ and $\varepsilon(x)=
\rho(h(x),W^u_{\delta_1/3}(z_c))$. In the latter case
we set $\varepsilon(x)=\rho(T^mh(x),\partial T^mW^0_m(z_c))$.
We now define the release time to be $f(x)=m(x)+\log_{\Lambda}
(\delta_0/\varepsilon(x))$, one formula for both cases. \medskip
{\em Type II points} have no local stable manifolds that
extend to $W^u_{\delta_1}(z_c)$. Let $x\in B^c_{\infty}$
be such a point. According to the second
statement in Lemma~\ref{lmzz'}, $\rho^s(x,W^u_{\delta_1}(z_c))
\leq\delta_3/2$. Hence, no local stable manifold $W^s(x)$
contains a stable disk of radius $\delta_3/2$
around $x$. Therefore, $x\notin M^+_{\Lambda,\delta_3/2}$,
see Section~\ref{secF}. Let then $m=m(x)=\min\{
m'>0:\, \rho(T^{m'}x,\Gamma\cup\partial M)\leq\delta_3 \Lambda^{-m'}/2\}$.
We claim that, on the component of $T^{m}B^c_{m}$ containing
$T^{m}x$, there are no points of $T^{m}B_R^c$ in the
$(\delta_2 \Lambda^{-m}/2)$-neighborhood of $T^{m}x$. Indeed, if some point
$y\in T^{m}B_R^c$ were there, its LSM $W^s(y)$ would
contain a point $y'\in T^{m}W^1_{\infty}(z_c)$, which is at distance
$\leq \delta_3 \Lambda^{-m}$ from $y$. Then $\rho(y',\Gamma\cup\partial M)
\leq \delta_2 \Lambda^{-m}$, since $\delta_3/\delta_2\leq 1/3$.
This, however, contradicts the definition
of $W^1_{\infty}(z_c)$, cf. Section~\ref{secF}. We now define
the release time to be $f(x)=2m(x)+\log_{\Lambda}(2\delta_0
/\delta_2)$. \medskip
For any point $x\in B^c_{\infty}\setminus B^c_R$
of either type and any $n\geq f(x)$ the point
$T^nx$ should be at least the distance
$\delta_0$ from $T^nB_R^c$ (measured along $T^nB^c_n$), so that the
component of $T^nB^c_n$ containing $T^nx$ does not intersect
$T^nB_R^c$ at all.
Therefore, we are free to define new gaskets
and capture new disks on any component $V\subset T^nB^c_n$ that
contains at least one released point, i.e. such that $\exists x\in T^{-n}V:
\, f(x)\leq n$.
We can only define a gasket, however, if $\exists x\in V:\,
\rho_V(x,\partial V)\geq\delta_1$, i.e. if $V$ is large enough.
Hence the next step. \medskip
\noindent
{\bf Growth}.
To control the size of the components of $T^nB^c_n$, we
collect, for every $n\geq 0$, the components $V\subset T^nB^c_n$
released at the $n$-th iterations. We say that $V$ is released
at the $n$-th iteration if at least one point of $V$ is
released at this iteration, and none of the points of
the component of $T^{i}B_i^c$ that contains $T^{-(n-i)}V$
is released at the $i$-th iteration for any $i=0,\ldots,n-1$.
In that case we define another function,
the `component release time', $s(x)=n$, on $B^c_{\infty}
\cap T^{-n}V$.
Observe that $s(x)$ is defined for each $x\in B^c_{\infty}
\setminus B_R^c$ and $s(x)\leq f(x)$.
For any $s\geq 0$ let
\be
\tilde{W}=\tilde{W}(s)=\cup\{V\subset T^{s}B^c_s:\, s(x)=s
\ \ \forall x\in B^c_{\infty}\cap T^{-s}V\}
\label{Wstilde}
\ee
be the union of the components of $T^{s}B^c_{s}$
released exactly at the $s$-th iteration.
The manifold $\tilde{W}$ carries the measure $\tilde{m}_{\tilde{W}}=
T^{s}_{\ast}\tilde{m}_{B^c}|\tilde{W}$. Consider open sets
$\tilde{W}^1_n:=\tilde{W}\cap T^{s}B^c_{s+n}$ and
$\tilde{W}^0_n:={\rm int}\,(\tilde{W}^1_n\setminus\tilde{W}^1_{n+1})$,
$n\geq 0$. It is easy to see that they make a refined
$(\delta_2 \Lambda^{-n_c-s})$-filtration of $\tilde{W}$
satisfying Theorem~\ref{tmexpr}. Denote then
\be
p(s)=\tilde{m}_{\tilde{W}}(\tilde{W}^1_{\infty})/
\tilde{m}_{\tilde{W}}(\tilde{W})=
\tilde{m}_{\tilde{W}}(\tilde{W}\cap T^sB^c_{\infty})/
\tilde{m}_{\tilde{W}}(\tilde{W})
\label{ps}
\ee
If $p(s)=0$, we can simply disregard such a $\tilde{W}=\tilde{W}(s)$.
If $p(s)>0$, then Proposition~\ref{prwp} applies to
$(\tilde{W},\tilde{m}_{\tilde{W}})$, according to
Final Remark (Part 2). Hence, $\exists n\geq 1$
such that $Z[\tilde{W}^1_n,n]\leq 0.6/\delta_1$, i.e. the
components of $T^n\tilde{W}^1_n$ are large enough, on the average.
Let $g$ be the minimum of such $n$'s. We call $g$ the `growth time'
and define another function, $g(x)=g$ on
$B^c_{\infty}\cap T^{-s}\tilde{W}$
(note that $g(x)$ is a constant function on $B^c_{\infty}\cap
T^{-s}\tilde{W}$, and it only depends on $s$, so we will
also write it as $g(s)$).
Consider now the manifold $\hat{W}=T^{g}\tilde{W}^1_{g}$
and the measure $\tilde{m}_{\hat{W}}=T_{\ast}^{g}
\tilde{m}_{\tilde{W}}|\hat{W}$ on it.
Denote $\hat{W}^1_{\infty}=T^g(\tilde{W}^1_{\infty})
=T^g(\tilde{W}\cap T^sB^c_{\infty})$.
According to Proposition~\ref{prwp}, \\
(c) $\tilde{m}_{\hat{W}}(\hat{W}^1_{\infty})
>0.9\,\tilde{m}_{\hat{W}}(\hat{W})$, and \\
(d) $Z[\hat{W},\hat{W},0]\leq 0.6/\delta_1$, so that
at least 40\% of the points in $\hat{W}$ (with respect
to the measure $\tilde{m}_{\hat{W}}$) lie a distance
$\geq\delta_1$ away from $\partial\hat{W}$. \\
Next, we define new gaskets
and capture new disks on the large components of $\hat{W}$,
as we did to $W^g$ early in this section, and repeat
the procedure `initial growth' applying it to $\hat{W}$.
Let $t(x)$ be the `capture time'
for $x\in \hat{W}^1_{\infty}$, i.e. the minimum of
$t\geq 0$ such that $T^tx$ belongs in a captured disk.
The next lemma follows from the properties (c)
and (d) of the manifold $\hat{W}$ just like Lemma~\ref{lmq}
and (\ref{t0}) followed from the similar properties of
the manifold $W^g$:
\begin{lemma}
We have $\tilde{m}_{\hat{W}}(t(x)>n)/\tilde{m}_{\hat{W}}
(\hat{W}^1_{\infty})\leq C_0\theta_0^n$ with the same
constants as in (\ref{t0}).
\label{lmt}
\end{lemma}
We emphasize that our construction of gaskets
is inductive. For a.e. point $x\in W^1_{\infty}$,
the cycle `growth$\to$capture$\to$release$\to$growth$\ldots$'
repeats until the point returns to $\cal R$ at some moment of capture.
If it never returns, we put it into the leftover set
$W^1_{\infty,0}$ and define $r(x)=\infty$.
This concludes our definition of the partition
$W^1_{\infty}=\cup_k W^1_{\infty,k}$
and the return time $r(x)$.
\medskip
\noindent
{\bf Exponential tail bound}.
We now turn to the proof of the exponential tail bound (\ref{tail}).
First, we show that the points of any captured disk $B^c$
are released at an exponential rate.
\begin{lemma}
There are $C_1>0$ and $\theta_1\in (0,1)$ such that
for every captured disk $B^c$ we have $\tilde{m}_{B^c}
(f(x)>n)/\tilde{m}_{B^c}(B^c)0$.
In view of the absolute continuity (\ref{ac}), it is enough
to estimate the measure $m_{W^u_{\delta_1}(z_c)}
\{h(x):\, f(x)>n\}$. Fix an $r\in (0,(1+\kappa)^{-1})$.
The measure of the set $\{h(x):\, m(x)>rn\}$ is
exponentially small in $n$ due to the part (iii)
of Corollary~\ref{crwzn} and (\ref{Z1}).
Next, for every $0\leq m\leq rn$, we have
$$
m_{W^u_{\delta_1}(z_c)}\{h(x):\,
m(x)=m\ \&\ \varepsilon(x)<\delta_0 \Lambda^{-(1-r)n}\}
\leq 4d_uD\delta_0\delta_1^{-1}\delta_2^{-\kappa}\Lambda^{-(1-r-r\kappa)n}
$$
based on the definition of $Z[W,W^0_m,m]$, the part (i)
of Corollary~\ref{crwzn} and (\ref{Z1}).
Due to our choice of $r$, the right hand side is
exponentially small in $n$, uniformly in $m$.
Thus, the points of type I obey our claim.
For any point $x$ of type II with $m(x)=m$,
observe that $T^m x\in {\cal U}_{\delta_3 \Lambda^{-m}/2}$
and $T^kx\notin{\cal U}_{\delta_3 \Lambda^{-k}/2}$ for
all $k=0,\ldots, m-1$. Denote $U=B^c$ and consider
a $(\delta_3/2)$-filtration $\{U^1_n,U^0_n\}$ of $U$
satisfying Theorem~\ref{tmexpr}.
Then $x\in U^0_m$, and $\tilde{m}_{B^c}(U^0_m)$
is exponentially small
in $m$ by the part (iii) of Corollary~\ref{crwzn}
and (\ref{ZBc}). $\Box$.\medskip
The next lemma is proven in \cite{Ch98}, Lemma~7.2.
It shows that the released components in the images
of any captured disk $B^c$ grow at an exponential rate:
\begin{lemma}
There are $C_2>0$ and $\theta_2\in (0,1)$
such that for every captured disk $B^c$ we have
$\tilde{m}_{B^c}(s(x)+g(x)>n)
0$
the return time (the length of the free path till the
next collision), see details in \cite{BSC91}. The coordinates
on $M'$ are denoted by $(r,\varphi)$, where $r\in\partial Q$
is the arc length parameter and $\varphi\in [-\pi/2,\pi/2]$
is the angle of reflection. The map $T$ preserves the smooth
measure $d\mu=c_{\mu}\,\cos\varphi\,dr\, d\varphi$, where
$c_{\mu}$ is the normalizing constant. It is known that
$\mu$ is an SRB measure, the system $(T,\mu)$ is ergodic,
mixing, K-mixing and Bernoulli \cite{Si70,GO}.
\begin{theorem}
The billiard ball map $(T,\mu)$ enjoys exponential decay of correlations.
\label{tmb}
\end{theorem}
\noindent
{\bf Discontinuity curves}. Let $S_0=\partial Q\times
\{\varphi=\pm\pi/2\}$ be the natural boundary of $M'$.
Put $S_n=T^nS_0$ for all $n\in\ZZ$, and $S_{m,n}=
\cup_{i=m}^n S_i$ for $-\infty\leq m\leq n\leq\infty$.
Each $S_n$ is a finite union of $C^2$-curves whose slope,
in the $r,\varphi$ coordinates,
is positive for $n\geq 1$ and negative for $n\leq -1$.
The sets $S_{-n,0}$ and $S_{0,n}$ consist of discontinuity
curves for $T^n$ and $T^{-n}$, respectively. The following
{\em continuation property} is important (see also Sect.~\ref{secFR}):
each endpoint, $x_0$,
of every smooth curve $\gamma\subset S_{-m,0}$, $m\geq 1$,
lies either on $\partial M'$ or on another smooth curve
$\gamma'\subset S_{-m,0}$ that itself does not terminate at $x_0$.
Hence, each curve $\gamma\in S_{-m,0}$ can be continued
up to $\partial M'$
by other curves in $S_{-m,0}$.
\medskip
\noindent
{\bf Invariant cones and Alignment}. Identifying the tangent space
at each point with the $(r,\varphi)$-plane, the derivative
$DT$ maps the cone $\{r\varphi\geq 0\}$ strictly into
itself. We call the $DT$-image
of $\{r\varphi\geq 0\}$ the unstable cone $C^u$.
Similarly, $DT^{-1}$ maps $\{r\varphi\leq 0\}$
strictly into itself, and $C^s$ is defined
accordingly. These two families of cones are
$DT$-invariant in the sense of Sect.~\ref{secSR}.
The tangent vectors to the curves in $S_m$ belong in
unstable cones for $m\geq 1$ and in stable cones
for $m\leq -1$, this property is often referred
to as Alignment.
\medskip
\noindent
{\bf Transversality}.
The angles between stable and unstable cones are
bounded away from zero. This follows from the
fact that the edges of the cones $C^u$ and $C^s$
are uniformly bounded away from the $r$- and
$\varphi$-axes.
In other words, for any tangent vector $v=(dr,d\varphi)$
in either stable or unstable cone we have
\be
00$ mean some positive constants.
More precisely, let $x=(r,\varphi)\in M'$ and $v=(dr,d\varphi)$
be a tangent vector at $x$. We put
\be
{\cal B}(x,v)=\frac{1}{\cos\varphi}
\left (\frac{d\varphi}{dr}+{\cal K}(r)\right )
\label{calB}
\ee
where ${\cal K}(r)>0$ is the curvature of the boundary
$\partial Q$ at the point of reflection, $r$. Denote by
${\cal K}_{\min}>0$ and ${\cal K}_{\max}>0$ the minimum
and maximum of the curvature of $\partial Q$. Also, for the
class of billiards discussed here, $\tau(x)\geq\tau_{\min}>0$.
The value of ${\cal B}(x,v)$ represents the curvature of the orthogonal
cross-section of the bundle of the outgoing velocity vectors
specified by the points $(r+\varepsilon\, dr,\varphi+
\varepsilon\, d\varphi)$, $\varepsilon\approx 0$, see \cite{BSC90,BSC91}.
Put $x_1=(r_1,\varphi_1)=Tx$ and $v_1=(dr_1,d\varphi_1)=DT(v)$.
It follows from the mirror equation of geometric optics,
see \cite{BSC90,BSC91}, that
\be
{\cal B}(x_1,v_1)=\frac{2{\cal K}(r_1)}{\cos\varphi_1}
+\frac{1}{\tau(x)+{\cal B}^{-1}(x,v)}
\label{mirror}
\ee
Hence,
\be
\frac{d\varphi_1}{dr_1}={\cal K}(r_1)+
\cos\varphi_1\left ( \tau(x)+\cos\varphi
\left (\frac{d\varphi}{dr}+{\cal K}(r)\right )^{-1}\right )^{-1}
% \frac{\cos\varphi_1\, (d\varphi/dr+{\cal K}(r))}
% {\cos\varphi+\tau(x)(d\varphi/dr+{\cal K}(r))}
\label{10}
\ee
This proves (\ref{B1}) with $B_1=\max\{{\cal K}_{\min}^{-1},
{\cal K}_{\max}+\tau_{\min}^{-1}\}$. Note that for billiard
tables with corner points $\tau_{\min}=0$, and so the upper
bound in (\ref{B1}) fails, see also Sect.~\ref{secCP}.
\medskip
\noindent
{\bf Hyperbolicity}.
The expansion and contraction of tangent vectors
can be conveniently described in a pseudometric
that is loosely called p-metric \cite{BSC91,LSY}.
If $v=(dr,d\varphi)$ is
a vector in either stable or unstable cone, then
its p-norm is defined by
\be
|v|_p=\cos\varphi\, |dr|
\label{pmetric}
\ee
In this norm, $DT$ is uniformly hyperbolic:
\be
|DT(v)|_p\geq \Lambda|v|_p\ \ \forall v\in C^u,
\ \ \ \ {\rm and}\ \ \ \
|DT^{-1}(v)|_p\geq \Lambda|v|_p\ \ \forall v\in C^s
\label{DTv}
\ee
with some constant $\Lambda>1$.
More precisely, the expansion factor of unstable
vectors $v=(dr,d\varphi)\in C^u_x$ is given by
\cite{BSC91}
\be
\frac{|DT(v)|_p}{|v|_p}=1+\tau(x){\cal B}(x,v)
=1+\frac{\tau(x)}{\cos\varphi}
\left (\frac{d\varphi}{dr}+{\cal K}(r)\right )
\label{DTvu}
\ee
so we can set $\Lambda=1+\tau_{\min}(B_1^{-1}+{\cal K}_{\min})$.
Clearly, the expansion factor is mainly determined by $\cos\varphi$:
\be
\frac{B_2^{-1}\tau_{\min}}{\cos\varphi}\leq
\frac{|DT(v)|_p}{|v|_p} \leq
\frac{B_2\tau(x)}{\cos\varphi}
\ \ \ \ \forall v\in C^u
\label{B2}
\ee
In particular, the derivative $DT$ in the p-metric
is unbounded near $S_0$, where $\cos\varphi\approx 0$.
\medskip
\noindent
{\bf ``Homegeneity strips'' and the definition of $M$}. The unboundedness
of $DT$ near $S_0$ makes the distorsion control in the sense of
(\ref{distor1}) particularly difficult for billiards. One has
to partition the neighborhood of $S_0$ into countably many narrow
strips parallel to $S_0$ in each of which the control is possible.
We fix a large $k_0\geq 1$ and for each $k\geq k_0$ define ``homogeneity
strips''
$$
I_k=\{(r,\varphi):\, \pi/2-k^{-2}<\varphi <\pi/2-(k+1)^{-2}\}
$$
and
$$
I_{-k}=\{(r,\varphi):\, -\pi/2+(k+1)^{-2}<\varphi < -\pi/2+k^{-2}\}
$$
We put
$$
I_0=\{(r,\varphi):\, -\pi/2+k_0^{-2}<\varphi < \pi/2-k_0^{-2}\}
$$
The exact choice of $k_0$ will be made later.
Now we define an open subset $M\subset M'$,
on which $T$ will satisfy all our assumptions.
We put $M=\cup I_k$. Moreover, it is convenient to
consider $I_k$ as
regions in the $(r,\varphi)$ plane with disjoint closures
(as if we cut $M'$ along the boundaries of $I_k$ and
moved the strips $I_k$ apart from each other). The map
$T$ restricted on $M$ has the singularity set $\Gamma:=
S_{-1}\cup T^{-1}(\cup_k\partial I_k)$. Since the boundaries
of $I_k$ are parallel to $S_0$, their preimages under $T$
have tangent vectors in stable cones, so that the above
Alignment holds for the curves in $\Gamma$, just as for $S_{-1}$.
It also holds for all the curves in $\Gamma^{(n)}$, $n\geq 1$,
which are defined by (\ref{Gn}). We will denote also by $T$
the restriction of the original billiard map $T$ on $M$.
\medskip
\noindent
{\bf Stable and unstable fibers}. It is known that
stable and unstable manifolds, or fibers, for the map $T$ on $M$
(in the sense of Section~\ref{secSR}) exist at a.e.
point $x\in M$. In \cite{BSC91}, they were called
`homogeneous fibers'. The boundedness of the curvature
of both stable and unstable fibers, as well as the
absolute continuity (\ref{ac}) are standard facts,
see \cite{BSC91,LSY}.
The distorsion bound (\ref{distor1}) requires some extra
work. In \cite{BSC91}, Proposition~A1.1(d), it was established
that the left hand side of (\ref{distor1}) is uniformly bounded above.
This is a little less than we now require in (\ref{distor1}).
However, a careful analysis of the argument in \cite{BSC91} reveals
that in fact more was proved there:
\be
\log\prod_{i=0}^{n-1}\frac{J^u(T^ix)}{J^u(T^iy)}
\leq {\rm const}\cdot \left [{\rm dist}(T^nx,T^ny)\right ]^a
\label{distor2}
\ee
for some $a>0$, in the notation of (\ref{distor1}).
Since the argument in \cite{BSC91} is lengthy,
we will not repeat it here, besides all the necessary details
are there, in the proof of Proposition~A1.1(d) in \cite{BSC91}.
Next we prove the non-branching of unstable fibers.
Let a sequence of LUM's $\{W_n\}$ have a limit point $x$,
and $\rho(x,\partial W_n)>\varepsilon$ for some $\varepsilon>0$.
Then the curves $W_n\cap B_{\varepsilon}(x)$ converge
in the Hausdorff metric to a LUM of length $2\varepsilon$
through $x$, see \cite{BSC91}. The uniqueness of the LUM
$W(x)$ through $x$ implies the non-branching of
unstable fibers.
It is also standard that u-SRB measures on unstable fibers for
dispersing billiards exist and the invariant measure $\mu$
is SRB measure.
It then remains to verify our main assumption,
on the growth of unstable fibers. This requires some
extra work and switching to a higher iterate of $T$.
\section{Smooth billiards with finite horizon}
\label{secFH}
\setcounter{equation}{0}
Here we make an additional assumption on $Q$, that
it has ``finite horizon'',
i.e. the free path between successive collisions
is uniformly bounded: $\tau(x)\leq\tau_{\max}<\infty$.
For this subclass of dispersing billiards
Theorem~\ref{tmb} was actually proved by Young \cite{LSY}. We
will prove it here by the techniques developed in
the previous sections.
\medskip
\noindent
{\bf Expansion rates in Euclidean metric}. Despite the
convenience of the p-metric (\ref{pmetric}), we will
work in the Euclidean metric $|v|=(dr^2+d\varphi^2)^{1/2}$
for the reasons explained below.
First, due to (\ref{B1}) for any stable or unstable vector $v$
\be
1\leq \frac{|v| \cos\varphi}{|v|_p}\leq B_3<\infty
\label{B3}
\ee
with $B_3=(1+B_1^2)^{1/2}$. Hence, (\ref{B2}) implies
\be
\frac{B_4^{-1}}{\cos\varphi_1}\leq
\frac{|DT(v)|}{|v|} \leq
\frac{B_4}{\cos\varphi_1}
\ \ \ \ \forall v\in C^u
\label{B4}
\ee
with $B_4=B_2B_3\max\{\tau_{\max},\tau_{\min}^{-1}\}$,
here again $(r_1,\varphi_1)=Tx$.
The reason why we prefer the Euclidean metric $|\cdot |$ to
the p-metric is related to the specific mechanism of growth
of unstable fibers under $T$ in the presence of countably many
singularity lines in $\Gamma$. Let an unstable fibers $W$ be cut into very
many, in the worst case countable many, pieces by the set $\Gamma$.
Then small pieces of $W\setminus\Gamma$ are mapped into the
vicinity of $S_0$, where $\cos\varphi$ is small.
In the p-metric, these pieces will experience
strong growth at the next iteration of $T$,
due to (\ref{B2}). This will allow them to recover in size,
as we will show later. Note, however, a time delay:
the recovery occurs one iteration {\em after} (!) the cutting.
For this technical reason, the map $T$ on $M$ in the p-metric
has no chance to satisfy the assumption (\ref{rgrowth11}).
In the Euclidean metric, the growth occurs {\em simultaneously}
(!) with cutting, as it follows from (\ref{B4}). This makes the
verification of (\ref{rgrowth11}) possible.
The expansion factors for unstable vectors under $T$ in the
Euclidean metric are not bounded from 1, however.
This is one reason why we need to consider a higher power of $T$:
\begin{lemma}
There is an $m_1\geq 1$ such that for any $m>m_1$
and any point $x\in M$
$$
|DT^{m}(v)|\geq \Lambda^{m-m_1}|v|\ \ \forall v\in C^u,
\ \ \ \ {\rm and}\ \ \ \
|DT^{-m}(v)|\geq \Lambda^{m-m_1}|v|\ \ \forall v\in C^s
$$
where these derivatives exist.
\label{lmmm}
\end{lemma}
{\em Proof}. Combining (\ref{B3}), (\ref{B2}) and (\ref{DTv}) yields
$$
|DT^{m}(v)|\geq |DT^{m}(v)|_p
\geq \Lambda^{m-1}|DT(v)|_p
\geq \frac{\Lambda^{m-1}\tau_{\min}|v|_p}{B_2\cos\varphi}
\geq \frac{\Lambda^{m-1}\tau_{\min}|v|}{B_2B_3}
$$
Hence it is enough to take any $m_1$ such that $\Lambda^{m_1-1}
>B_2B_3/\tau_{\min}$. The stable vectors are treated similarly.
$\Box$\medskip
Denote by $|W|_{\max}$ the maximal length of LUM's in $M$.
\medskip
\noindent
{\bf Accumulation of singularity lines}. There are two
sources of accumulation of the components of the set
$\Gamma$ that can cut LUM's into arbitrary many pieces.
First, the set $\cup T^{-1}(\cup_k\partial I_k)$ consists
of countably many curves stretching approximately parallel to some
curves in $S_{-1}$ and approaching them. So, each set
$T^{-1}I_k$, $k\neq 0$, is a narrow strip with curvilinear
boundaries. The expansion of unstable fibers in these
strips can be estimated by (\ref{B4}). More precisely,
let $W\subset T^{-1}I_k$ be a LUM, for some $k\neq 0$.
Then the expansion factor, $J^u(x)$, on $W$ satisfies
\be
00$.
We fix an $m_2$ such that $Am+B+1 < \Lambda^{m-m_1}$
for any $m\geq m_2$.
\medskip
\noindent
{\bf A higher iteration of the map $T$}.
It is enough to establish exponential decay of correlations
for the system $(T^m,\mu)$ with any particular $m\geq 1$,
see Proposition~\ref{prTm} in Section~\ref{secFR}.
We now fix $m:=\min\{m_1,m_2\}+1$ and let $T_1:=T^m$.
Note that $S_{-m,0}$ is the set of singularity
curves for the map $T_1$ on $M'$. The map $T_1$
restricted on $M$ has singularity set
$\Gamma_1:=\Gamma^{(m)}$,
where $\Gamma^{(m)}$ is defined in terms of $\Gamma$
by (\ref{Gn}).
The map $T_1$ has the same stable and unstable cones and
the same LUM's and LSM's as does $T$. Thus, the Alignment
and Transversality hold for $T_1$ as well. Lemma~\ref{lmmm} implies that
$$
|DT_1(v)|\geq \Lambda_1|v|\ \ \forall v\in C^u,
\ \ \ \ {\rm and}\ \ \ \
|DT_1^{-1}(v)|\geq \Lambda_1|v|\ \ \forall v\in C^s
$$
with $\Lambda_1:=\Lambda^{m-m_1}>1$, so the map $T_1$ is uniformly
hyperbolic in the Euclidean metric. Our choice of $m$ also ensures that
\be
\Lambda_1 > K_m+1
\label{LKm}
\ee
It remains to verify our main assumption,
the one on the growth of unstable fibers,
but before we introduce a handy indexing
system.
\medskip
\noindent
{\bf Indexing system}.
Let $\delta_0>0$ and
$W$ be a $\delta_0$-LUM. If $\delta_0$ is small enough,
then $W$ crosses at most $K_m$ curves of the set $S_{-m}$,
so the set $W\setminus S_{-m}$ consists of at most $K_m+1$
connected curves, call them $W_1,\ldots,W_p$ with $p\leq K_m+1$.
On each of $W_j$ the map $T_1$ (as a map on $M'$) is smooth,
but any $W_j$'s may be cut into arbitrary many or countably
many pieces by other curves in $\Gamma_1$, which are the
preimages of the boundaries of $I_k$. Let $\Delta\subset W$
be a connected component of the set $W\setminus\Gamma_1$. It can
be uniquely identified with the $(m+1)$-tuple
$(k_1,\ldots,k_m;j)$ such that $\Delta\subset W_j$
and $T^i\Delta\subset I_{k_i}$ for $1\leq i\leq m$.
We will then write $\Delta=\Delta(k_1,\ldots,k_m;j)$.
Of course, some strings $(k_1,\ldots,k_m;j)$ may not
correspond to any piece of $W$, for such strings
$\Delta(k_1,\ldots,k_m;j)=\emptyset$.
Denote by $J^u_1(x)=J^u(x)\cdots J^u(T^{m-1}x)$ the
expansion factor of the unstable subspace $E^u_x$
under $DT_1$. Let $|\Delta|=m_{\Delta}(\Delta)$
be the Euclidean length
of a LUM $\Delta$. We record two important facts:\\
(a) For every point $x\in\Delta(k_1,\ldots,k_m;j)$ we have
$$
J^u_1(x)\geq L_{k_1,\ldots,k_m}:
=\max\left \{\Lambda_1,\, B_6\prod_{k_i\neq 0}k_i^2\right \}
$$
where $B_6=\left (\max\{B_4,B_5\}\right )^{-m}$.
This follows from (\ref{B4}) and (\ref{B5}).\\
(b) For each $\Delta(k_1,\ldots,k_m;j)$ we have
$$
|\Delta(k_1,\ldots,k_m;j)|\leq
M_{k_1,\ldots,k_m}:=
\min\left\{ |W|,\, B_7\prod_{k_i\neq 0}k_i^{-2}\right\}
$$
where $B_7=B_6^{-1}|W|_{\max}$.
This follows from the previous fact.
Next, put
$$
\theta_0:=2\sum_{k=k_0}^{\infty}k^{-2}\leq 4/k_0
$$
\noindent
{\bf Growth of unstable fibers}. Let $W$ be a $\delta_0$-LUM
and $\delta>0$ be small. Due to the Transversality, the angles
between $W$ and the curves of $\Gamma_1$ that cross $W$
are uniformly bounded away from zero. For each connected
component $\Delta\subset W\setminus\Gamma_1$ put
$\Delta^0=\Delta\cap{\cal U}_{\delta}$ and
$\Delta^1={\rm int}(\Delta\setminus{\cal U}_{\delta})$, where
${\cal U}_{\delta}$ is the $\delta$-neighborhood of
$\Gamma_1\cup\partial M$. Due to the Transversality and
Continuation properties, the set $\Delta^0$ consists
of two subintervals adjacent to the endpoints of $\Delta$
(they may overlap and cover $\Delta$, of course).
The set $\Delta^1$ is either empty or a subinterval of $\Delta$.
We put $W^1=\cup_{\Delta\subset W\setminus\Gamma_1}\Delta^1$.
For each $\Delta^1$ the set $T_1(\Delta_1\cap\{r_{W^1,1}<\varepsilon\})$
is the union of two subintervals of $T_1\Delta^1$ of length
$\varepsilon$ adjacent to the endpoint of $T_1\Delta^1$.
Using the above indexing system gives
\begin{eqnarray}
m_W(r_{W^1,1}<\varepsilon)
&\leq&\sum_{k_1,\ldots,k_m,j}
2\varepsilon L_{k_1,\ldots,k_m}^{-1}\nonumber\\
&\leq& 2\varepsilon p\left [\Lambda_1^{-1}+B_6
(\theta_0+\theta_0^2+\cdots +\theta_0^m)\right ]\nonumber\\
&\leq& 2\varepsilon (K_m+1)\left (\Lambda_1^{-1}+B_6m\theta_0\right )
\label{mW11}
\end{eqnarray}
We now assume that $k_0$ is large enough so that
$$
\alpha_0:=(K_m+1)(\Lambda_1^{-1}+B_6m\theta_0)<1
$$
and thus get
$$
m_W(r_{W^1,1}<\varepsilon) \leq \min\{ |W|,2\alpha_0\varepsilon\}
$$
The first term on the right hand side of (\ref{rgrowth11}) is
equal to
$$
\alpha_0\Lambda_1\min\{|W|,2\varepsilon/\Lambda_1\}
=\min\{\alpha_0\Lambda_1|W|,2\alpha_0\varepsilon\}
$$
Since $\alpha_0\Lambda_1>1$, we get
\be
m_W(r_{W^1,1}<\varepsilon) \leq
\alpha_0\Lambda_1\cdot m_W\left (r_{W,0}<\varepsilon/\Lambda_1\right )
\label{alp0}
\ee
Next, to obtain an open $(\delta_0,1)$-subset $V^1_{\delta}$ of $W^1$,
one needs to further subdivide the intervals $\Delta^1\subset W$
such that $|T_1\Delta^1|>\delta_0$. Each such LUM
$T_1\Delta^1$ we divide into $s_{\Delta}$ equal subintervals
of length $\leq\delta_0$,
with $s_{\Delta}\leq |T_1\Delta^1|/\delta_0$.
If $|T_1\Delta^1|<\delta_0$, then we set $s_{\Delta}=0$
and leave $\Delta^1$ unchanged.
Then union of the preimages under $T_1$ of the above
intervals will make $V^1_{\delta}$.
Now we must estimate the measure of the
$\varepsilon$-neighborhood of the additional endpoints
of the subintervals of $T_1\Delta^1$. This gives
\begin{eqnarray*}
m_W(r_{V^1_{\delta},1}<\varepsilon) -
m_W(r_{W^1,1}<\varepsilon)
&\leq&\sum_{\Delta\subset W\setminus\Gamma_1}
2s_{\Delta}\varepsilon| B_9\Delta^1|/|T_1\Delta^1|\\
&\leq&\sum_{\Delta\subset W\setminus\Gamma_1}
2B_9\varepsilon|\Delta^1|/\delta_0\\
&\leq& 2B_9\varepsilon\delta_0^{-1}|W|
\end{eqnarray*}
Here $B_9=\exp({\rm const}\cdot |W|_{\max}^a)$ is an upper bound
on distorsions on LUM's, see (\ref{distor2}).
Combining the above bound with (\ref{alp0})
completes the proof of (\ref{rgrowth11}) with $\beta_0=2B_9$.
We now prove (\ref{rgrowth10}). It is enough to consider
$\varepsilon<|W|/2$, so that the right hand side of
(\ref{rgrowth10}) equals $2D_0\delta^{-\kappa}\varepsilon$.
We can put $V^0_{\delta}=W\setminus\overline{V^1_{\delta}}$. Then
the left hand side of (\ref{rgrowth10}) does not exceed
$2J_{\delta}\varepsilon$, where $J_{\delta}$ is the
number of nonempty connected components of the set
$\overline{V^0_{\delta}}$, which is at most
the number of connected components of $W\setminus\Gamma_1$
of length $>2\delta$. Hence, clearly $J_{\delta}\leq |W|/\delta
\leq\delta_0/\delta$. This proves (\ref{rgrowth10})
with $\kappa=1$.
Lastly, we prove the inequality (\ref{rw0}).
Again, let $\Delta$ be a connected component of
$W\setminus\Gamma_1$ and $\Delta^0$, $\Delta^1$
be defined as above, with the set $\Delta^0$ consisting
of two subintervals adjacent to the endpoints of $\Delta$.
Since the angles between $W$ and curves in $\Gamma_1\cup\partial M$
are bounded away from zero, each of these subintervals has
length between $\delta$ and $B_8\delta$,
where $B_8$ depends on the minimum angle between LUM's and
curves in $\Gamma_1\cup\partial M$.
Now, the right hand side of (\ref{rw0})
equals $D_0\min\{|W|,2\zeta\delta^{\sigma}\}$.
So, it is enough to show that
$m_W(V^0_{\delta})\leq B\delta^{\sigma}$
for some $B,\sigma>0$. We have
\begin{eqnarray*}
m_W(V^0_{\delta})
&\leq&\sum_{\Delta\subset W\setminus\Gamma_1}\min\{2B_8\delta,|\Delta|\}\\
&\leq&\sum_{k_1,\ldots,k_m,j}\min\{2B_8\delta,M_{k_1,\ldots,k_m}\}\\
&\leq& {\rm const}\cdot\delta+
{\rm const}\cdot{\sum_{k_1,\ldots,k_m}}^{\kern-0.7em{\ast}}
\min\left\{ \delta, \prod_{k_i\neq 0}k_i^{-2}\right\}
\end{eqnarray*}
where $\sum^{\ast}$ is taken over $m$-tuples that contain
at least one nonzero index $k_i\neq 0$. The following lemma,
which is proved in Appendix,
completes the proof of (\ref{rw0}) with $\sigma=(2m)^{-1}$.
\begin{lemma}
Let $\delta>0$ and $m\geq 1$. Then
$$
\sum_{k_1,\ldots,k_m\geq 2}
\min\left\{ \delta,(k_1\cdots k_m)^{-2}\right\}
\leq B(m)\cdot\delta^{1/2m}
$$
\label{lmk2}
\end{lemma}
\section{Smooth billiards without horizon}
\label{secWH}
\setcounter{equation}{0}
Here we relax the assumption on `finite horizon',
i.e. allow arbitrarily long free runs between consecutive
collisions. In particular, this is always the case
when $\partial Q$ is just one closed curve on $\IT^2$.
In this case, the singularity set $S_{m}\subset M'$
for each $m\neq 0$
is a countable (not finite!) union of smooth curves.
These curves accumulate in the vicinities of a finite
number of points $\omega_1,\ldots,\omega_s\in S_0$,
whose trajectories only contain grazing (tangent)
reflections at the boundary $\partial Q$, so that
their velocity vectors never change. The finite set
$\Omega=\{\omega_1,\ldots,\omega_s\}$ is $T$-invariant.
Moreover, for any open set $U\supset\Omega$ there is
another open set $V\supset\Omega$ such that $TV\subset U$.
\medskip
\noindent
{\bf Cell structure of $M'$}.
The structure of the singularity curves $S_{-1}$
near the points $\omega_1,\ldots,\omega_s$ is described
in \cite{BSC90,BSC91} in great detail. We will need the
following facts here:\\
(a) The curves $S_{-1}$ partition $M'$ into a
countable number of connected regions, which
we call {\em cells}. The neighborhood of each point
$\omega_j\in\Omega$ contains
infinitely many small cells whose sizes
decrease as they approach $\omega_j$. Small cells
near each $\omega_j$ can be naturally labelled
${\cal C}_{j,t}$ with $t=1,2,\ldots$, see \cite{BSC90,BSC91}.
(b) For each cell ${\cal C}_{j,t}$ and every $x\in {\cal C}_{j,t}$
we have ${\rm const}_1\cdot t\leq \tau(x)\leq {\rm const}_2\cdot t$.
Hence, the expansion factor at $x$ satisfies
\be
\frac{B_{10}^{-1}t}{\cos\varphi_1}\leq
\frac{|DT(v)|}{|v|}\leq
\frac{B_{10}t}{\cos\varphi_1}
\ \ \ \ \forall v\in C^u
\label{DTvi}
\ee
where $(r_1,\varphi_1)=Tx$, as in (\ref{B4}). This follows
from (\ref{B2}) and (\ref{B3}).\\
(c) For each small cell ${\cal C}_{j,t}$ and every
$x\in {\cal C}_{j,t}$ we have
$\cos\varphi_1\leq B_{11}t^{-1/2}$, where again $(r_1,\varphi_1)=Tx$.
(A similar bound holds for $\cos\varphi$, but we will not need it.)
\medskip
\noindent
{\em Convention}. In all that follows, we only consider sufficiently
small cells, with numbers $t\geq t_0$, where $t_0$ is large and
will be fixed later. Put ${\cal C}=\cup_{j=1}^s\cup_{t\geq t_0}
{\cal C}_{j,t}$. This set is small, and its complement $M'\setminus{\cal C}$
makes `most of ' $M'$. We need not label any cells in
$M'\setminus{\cal C}$.
\medskip
Now, we will repeat the arguments of Section~\ref{secFH}, working
out the necessary modifications. The bound (\ref{B3}) still holds.
The bound (\ref{B4}) holds for all $x\notin{\cal C}$,
i.e. in the `main part' of $M'$, where $\tau(x)$ is bounded.
For $x\in{\cal C}$ we have the bound (\ref{DTvi}).
Lemma~\ref{lmmm} still holds, because its proof only uses
the lower bound on $\tau(x)$.
The analysis of the accumulation of singularity lines has to
be supplemented now, since the curves of $S_{-1}$ additionally
accumulate near each point $\omega_j\in\Omega$. The bound (\ref{B5})
holds for all $x\in (M'\setminus{\cal C})\cap T^{-1}I_k$, $k\neq 0$,
whereas for each $x\in {\cal C}_{j,t}\cap T^{-1}I_k$, $k\neq 0$, we have
\be
00$ such that any $\delta_0$-LUM $W\subset M'\setminus V$
crosses at most $K_m$ curves of the set $S_{-m}$. In this
case we call $W_1,\ldots, W_p$, $p\leq K_m+1$, the
connected components of $W\setminus S_{-m}$. If
$W\subset V$, then $T^iW\subset U$ for all $0\leq i\leq m$,
so $T^iW$ can only cross the boundaries of some small cells.
In this case we put $W_1=W$. In either case, each
connected component $\Delta'$ of the set $W\setminus
S_{-m}$ can be uniquely identified with the
$(m+1)$-tuple $(l_1,\ldots,l_m;j)$ such that
$\Delta'\subset W_j$ and $l_i$, $1\leq i\leq m$,
is defined by $T^{i-1}\Delta'\subset {\cal C}_{j_i,l_i}$
with some $1\leq j_i\leq s$ if $T^{i-1}\Delta'\subset{\cal C}$,
and $l_i=2$ otherwise.
Now, each connected component $\Delta$ of the set
$W\setminus\Gamma_1$, where $\Gamma_1=\Gamma^{(m)}$
is defined as in sect.~\ref{secFH}, can
be uniquely identified with the $(2m+1)$-tuple
$(l_1,k_1,\ldots,l_m,k_m;j)$ where
$(k_1,\ldots,k_m)$ are defined as in Sect.~\ref{secFH}.
We will then write $\Delta=\Delta(l_1,k_1,\ldots,l_m,k_m;j)$.
We record three important facts:\\
(d) For every point $x\in\Delta(l_1,k_1,\ldots,l_m,k_m;j)$ we have
$$
J^u_1(x)\geq L_{l_1,k_1,\ldots,l_m,k_m}:
=\max\left \{\Lambda_1,\, B_{13}\prod_{l_i,k_i\neq 0}k_i^2l_i\right \}
$$
This follows from (\ref{B17a}) and (\ref{B17b}).\\
(e) For each $\Delta(l_1,k_1,\ldots,l_m,k_m;j)$ we have
$$
|\Delta(k_1,\ldots,k_m;j)|\leq
M_{l_1,k_1,\ldots,l_m,k_m}:=
\min\left\{ |W|,\, B_{14}\prod_{l_i,k_i\neq 0}k_i^{-2}l_i^{-1}\right\}
$$
(f) For each $k_i\neq 0$ we have $l_i\leq \chi k_i^4$
and for each $k_i=0$ we have $l_i\leq\chi k_0^4$,
with $\chi=2B_{11}^2$. This follows from the fact (c) above.
We now assume that $t_0>\chi k_0^4$. Then, in our indexing
system, for each $k_i=0$ we have $l_i=2$. Next, put
\begin{eqnarray*}
\theta_1:&=& 2\sum_{k=k_0}^{\infty}
\sum_{l=[\chi k_0^4]}^{[\chi k^4]}k^{-2}l^{-1}\\
&\leq& {\rm const}\cdot\sum_{k=k_0}^{\infty}k^{-2}\ln k\\
&\leq& {\rm const}\cdot k_0^{-1}\ln k_0
\end{eqnarray*}
\noindent
{\bf Growth of unstable fibers}.
We now proceed with the proofs of (\ref{rgrowth11})-(\ref{rw0})
using the same notation as in Section~\ref{secFH}.
As in (\ref{mW11}), we have
\begin{eqnarray*}
m_W(r_{W^1,1}<\varepsilon)
&\leq&\sum_{l_1,k_1,\ldots,l_m,k_m,j}
2\varepsilon L_{l_1,k_1,\ldots,l_1,k_m}^{-1}\\
&\leq& 2\varepsilon p\left [\Lambda_1^{-1}+B_{13}
(\theta_1+\theta_1^2+\cdots +\theta_1^m)\right ]
\end{eqnarray*}
We now assume that $k_0$ is large enough so that
$$
\alpha_0:=(K_m+1)(\Lambda_1^{-1}+B_{13}m\theta_1)<1
$$
This fixes our choice of $k_0$, and hence $t_0$. After that
we complete the proof of (\ref{rgrowth11}) as
in Sect.~\ref{secFH}, word by word.
The proof of (\ref{rgrowth10}) does not change.
To prove (\ref{rw0}) as in Sect.~\ref{secFH}, we note that
\begin{eqnarray*}
m_W(V^0_{\delta})
&\leq&\sum_{l_1,k_1,\ldots,l_m,k_m,j}
\min\{2B_8\delta,M_{l_1,k_1,\ldots,l_m,k_m}\}\\
&\leq& {\rm const}\cdot\delta+
{\rm const}\cdot{\sum_{l_1,k_1,\ldots,l_m,k_m}}^{\kern-1.5em{\ast}}
\min\left\{ \delta, \prod_{l_i,k_i\neq 0}l_i^{-1}k_i^{-2}\right\}
\end{eqnarray*}
where $\sum^{\ast}$ is taken over $2m$-tuples
that contain at least one nonzero index $k_i\neq 0$.
The following lemma, which is proved in Appendix,
completes the proof of (\ref{rw0}) with $\sigma=(6m+1)^{-1}$.
\begin{lemma}
Let $\delta>0$ and $m\geq 1$. Then
$$
\sum_{k_1,\ldots,k_m\geq 2}
\sum_{l_1=2}^{[\chi k_1^4]}\cdots\sum_{l_m=2}^{[\chi k_m^4]}
\min\left\{ \delta,(l_1\cdots l_m)^{-1}(k_1\cdots k_m)^{-2}\right\}
\leq B(m)\cdot\delta^{1/(6m+1)}
$$
\label{lmk2l}
\end{lemma}
\section{Dispersing billiard tables with corner points}
\label{secCP}
\setcounter{equation}{0}
In this section we consider dispersing billiard
tables $Q\subset \IR^2$. They necessarily have corner points,
i.e. intersections of smooth curves of $\partial Q$. We
assume, as usual \cite{BSC90,BSC91},
that all such intersections are transversal, i.e.
the angle made by the sides of $Q$ at each corner point
is positive. By the way, this is widely believed to be
a necessary assumption for exponential decay of correlation,
because otherwise the decay seems to be polynomial \cite{M}.
\medskip
\noindent
{\bf New singularity lines}.
Let $\hat{r}_1,\ldots,\hat{r}_t$ be the $r$-cooridanes of the corner
points of $\partial Q$. Put $V_0=\{(r,\varphi)\in M':\,
r=\hat{r}_1,\ldots,\hat{r}_t\}$. It is convenient
to cut $M'$ along the segments $\{r=\hat{r}_i\}$,
$1\leq i\leq t$, that make $V_0$ and then think of $M'$ as
a union of disjoint rectangles (each bounded by two $S_0$ segments
and two $V_0$ segments) and cylinders (each bounded by two $S_0$
closed curves), see \cite{BSC90,BSC91}. Then $S_0\cup V_0$
will be the natural boundary of $M'$.
We use the notations $V_m=T^{-m}V_0$ and $V_{m,n}$
in the same way as $S_m$ and $S_{m,n}$, Sect.~\ref{secDB}. Then
the singularity set for $T^m$, $m\geq 1$, is $S_{-m,0}\cup V_{-m,0}$.
This set has the continuation property, see Sect.~\ref{secDB}.
The Alignment holds as well, i.e. all the tangent vectors to $V_m$
are in unstable cones for $m>0$ and in the stable cones
for $m<0$.
Denote by $K_m$ the multiplicity of $S_{-m,0}\cup V_{-m,0}$,
i.e. the maximal number of smooth curves of this set that
intersect or terminate at any one point of $M'$. Unlike
the previous sections, it is not known for the present
class of billiards how fast $K_m$ grows with $m$. We have
to assume that it does not grow too fast. Specifically,
there is a large enough $m$ such that
\be
K_m < \Lambda_0^{m-m_3}-1
\label{Km3}
\ee
where the constants $\Lambda_0>1$ and $m_3$ are defined below.
Similar bounds are
commonly assumed in the literature \cite{BSC90,BSC91,L,LSY}.
The bound (\ref{Km3}) is widely believed to hold for generic
billiard tables \cite{BSC90}, even though this is not known.
There will be no more assumptions on the region $Q$
in this section.
A detailed study of billiard tables with corner points
was done in \cite{BSC90,BSC91}. We will recall the
necessary facts.
\medskip
\noindent
{\bf Corner series}.
The new phenomenon here is the existence of series
of two or more consecutive reflections near
a corner point. During those series, the free paths
are short, i.e. $\tau(x)\approx 0$, and so the expansion of unstable
vectors, even in the p-metric, is weak, due to (\ref{DTvu}).
Let us fix a sufficiently small $\varepsilon>0$, and call a series
of consecutive reflections a {\em corner series} if
they all occur in the $\varepsilon$-neighborhood of one
corner point. Three facts make the analysis easier: \\
(a) The number of reflections in any corner series
is uniformly bounded above (by some $m_0\geq 1$). So,
there is a constant $\tau_{\min}'>0$ such that for each
$x\in M$ there is an $i\in\{0,\ldots,m_0-1\}$ such that
$\tau(T^ix)\geq\tau_{\min}'$. \\
(b) Each corner series contains at most one grazing
reflection, and that
reflection is necessarily the first or
the last one in the series. So, there is a
constant $c_0>0$ such that in each corner series
$T^ix=(r_i,\varphi_i)$, $0\leq i\leq g$,
we have $\cos\varphi_i>c_0$ for all $i$'s,
except possibly one, and that exceptional one
is either $i=0$ or $i=g$. \\
(c) The curvature of LUM's, LSM's and all smooth
curves in $S_m\cup V_m$, $m\in\ZZ$, is uniformly bounded above.
We call a corner series $T^ix$, $0\leq i\leq g$, with
no grazing reflections (i.e., such that $\cos\varphi_i>c_0$
for all $0\leq i\leq g$)
a {\em regular} one. Corner series with the first
grazing reflection ($\cos\varphi_0\tau_{\min}'>0$,
with
$$
\Lambda:=1+\tau_{\min}'(B_1+{\cal K}_{\min})>1
$$
In particular,
the expansion and contraction is uniform for the map $T^{m_0}$:
\be
|DT^{m_0}(v)|_p\geq \Lambda_0^{m_0}|v|_p\ \ \forall v\in C^u,
\ \ \ \ {\rm and}\ \ \ \
|DT^{-m_0}(v)|_p\geq \Lambda_0^{m_0}|v|_p\ \ \forall v\in C^s
\label{DTvm0}
\ee
with
$$
\Lambda_0:=\Lambda^{1/m_0}>1
$$
Next, the homogeneity strips $I_k$ and the region $M$
are defines exactly as in Sect.~\ref{secDB}. The properties
of stable and unstable fibers and SRB measure described
in the end of Sect.~\ref{secDB} are valid without change.
\medskip
\noindent
{\bf Expansion rates in Euclidean metric}.
Despite certain deterioration of hyperbolicity
in terms of the p-metric, it does not get any
worse in terms of the Euclidean metric, as the
following lemma shows, cf. (\ref{B4}).
\begin{lemma}
Let $x=(r,\varphi)\in M$ and $Tx=(r_1,\varphi_1)\in M$.
Then
\be
\frac{B_{16}^{-1}}{\cos\varphi_1}
\leq\frac{|DT(v)|}{|v|}\leq
\frac{B_{16}}{\cos\varphi_1}
\ \ \ \ \ \forall v\in C^u
\label{B77}
\ee
\end{lemma}
{\em Proof}. We will prove the lower bound, the proof of the
upper bound is completely similar. (We will only need the
lower bound, anyway.) Denote $v=(dr,d\varphi)$ and $DT(v)=
(dr_1,d\varphi_1)$. First of all,
$$
\frac{|DT(v)|}{|v|}=
\frac{(dr_1^2+d\varphi_1^2)^{1/2}}{(dr^2+d\varphi^2)^{1/2}}=
\frac{(1+(d\varphi_1/dr_1)^2)^{1/2}}{(1+(d\varphi/dr)^2)^{1/2}}
\cdot\frac{\cos\varphi}{\cos\varphi_1}
\cdot\frac{|DT(v)|_p}{|v|_p}
$$
Next, we use the lower bound in (\ref{B1}), which always holds,
recall that $B_3=(1+B_1^2)^{1/2}$, and then substitute (\ref{DTvu}):
\begin{eqnarray*}
\frac{|DT(v)|}{|v|}
&\geq&\frac{d\varphi_1/dr_1}{B_3\cdot d\varphi/dr}
\cdot\frac{\cos\varphi}{\cos\varphi_1}
\cdot\frac{|DT(v)|_p}{|v|_p}\\
&\geq&\frac{d\varphi_1/dr_1}{B_3\cdot d\varphi/dr}
\cdot\frac{\cos\varphi}{\cos\varphi_1}
\cdot\left (1+\frac{\tau(x)}{\cos\varphi}\frac{d\varphi}{dr}\right )\\
&=&\frac{d\varphi_1/dr_1}{B_3\, \cos\varphi_1}
\cdot\left (\left (\frac{d\varphi}{dr}\right )^{-1}\cdot\cos\varphi
+\tau(x)\right )
\end{eqnarray*}
Now, consider three cases:\\
{\em Case} 1: $\tau(x)>\tau_{\min}'$. Clearly, (\ref{B77}) holds
with $B_{16}=B_1B_3/\tau_{\min}'$.\\
Observe that if $\tau(x)<\tau_{\min}'$, then
the points $x$ and $Tx$ belong in one corner series.\\
{\em Case} 2: the corner series containing $x$ and $Tx$ is regular
or right-singular. Then $d\varphi/dr\leq B_1$ and $\cos\varphi\geq c_0$,
so (\ref{B77}) holds with $B_{16}=B_1^2B_3/c_0$. \\
{\em Case} 3: the points $x$ and $Tx$ belong in a left-singular
corner series. Then we have two subcases: \\
(3a) $x$ is the first point in that corner series. Then
$d\varphi/dr\leq B_1$ and $d\varphi_1/dr_1\geq B_{15}^{-1}
(\tau(x)+\cos\varphi )^{-1}$ by (\ref{B15}). Hence,
(\ref{B77}) holds with $B_{16}=B_1B_3B_{15}$. \\
(3b) $x$ is not the first point of the corner series,
which then starts at some other point, call it
$T^{-j}x=(\tilde{r},\tilde{\varphi})$, $1\leq j\leq m_0$.
Denote $t=\sum_{i=1}^j\tau(T^{-i}x)$. Now,
$$
d\varphi_1/dr_1\geq B_{15}^{-1}(t+\tau(x)+\cos\tilde{\varphi})^{-1}
\ \ \ \ {\rm and}\ \ \ \
d\varphi/dr\leq B_{15}(t+\cos\tilde{\varphi})^{-1}
$$
due to (\ref{B15}), and also $\cos\varphi\geq c_0$. Now (\ref{B77})
follows with $B_{16}=B_3B_{15}^2/c_0$.
We now set $B_{16}=\max\{B_1B_3/\tau_{\min}',
B_1^2B_3/c_0,B_1B_3B_{15},B_3B_{15}^2/c_0\}$.
The lemma is proved. $\Box$\medskip
Next, we prove an analogue of Lemma~\ref{lmmm}.
\begin{lemma}
There is an $m_3\geq 1$ such that for any $m>m_3$ and
any point $x\in M\setminus S_{-m,0}\cup V_{-m,0}$
$$
|DT^m(v)|\geq\Lambda_0^{m-m_3}|v|\ \ \ \ \forall v\in C^u
$$
A similar bound holds for stable vectors.
\label{lmmm2}
\end{lemma}
{\em Proof}. Let $j_1=1+\min\{i\geq 0: \tau(T^ix)\geq\tau_{\min}'\}$.
and $j_2=1+\min\{i\geq j_1: \tau(T^ix)\geq\tau_{\min}'\}$. Note that
$j_1\leq m_0+1$ and $j_2\leq 2m_0+2$. Note also that the points
$T^{j_1}x$ and $T^{j_1-1}x$ cannot belong in one corner series,
so the vector $DT^{j_1}v$ satisfies the upper bounds in (\ref{B1})
and (\ref{B3}). Due to (\ref{DTv0}) and (\ref{DTvm0}), we have
\be
|DT^m(v)|\geq |DT^m(v)|_p\geq \Lambda_0^{m-j_2-m_0}|DT^{j_2}(v)|_p
\label{line1}
\ee
Next, we need the following standard estimate for dispersing billiards:
\begin{sublemma}
Let $x=(r,\varphi)\in M$ and $v=(dr,d\varphi)\in C^u_x$. Then
for any $n\geq 1$
$$
\frac{|DT^n(v)|_p}{|v|_p}\geq
1+\frac{\tau(x)+\cdots+\tau(T^{n-1}x)}{\cos\varphi}
\left (\frac{d\varphi}{dr}+{\cal K}(r)\right )
$$
\end{sublemma}
{\em Proof}. For all $0\leq i\leq n$, denote $T^ix=x_i=(r_i,\varphi_i)$,
$\tau_i=\tau(x_i)$ and ${\cal B}_i={\cal B}(x_i,DT^i(v))$, cf. (\ref{calB}).
It follows from (\ref{mirror}) that
${\cal B}_i^{-1}\leq\tau_{i-1}+{\cal B}_{i-1}^{-1}$, and so
$$
{\cal B}_{i}^{-1}\leq\tau_0+\cdots +\tau_{i-1}+{\cal B}_{0}^{-1}
$$
Now, due to (\ref{DTvu}),
$$
\frac{|DT^{i+1}(v)|_p}{|DT^i(v)|_p}=
1+\tau_i{\cal B}_i\geq
\frac{\tau_0+\cdots+\tau_i+{\cal B}_0^{-1}}
{\tau_0+\cdots+\tau_{i-1}+{\cal B}_0^{-1}}
$$
Multiplying this estimate for all $i=0,\ldots,n-1$ gives
$$
\frac{|DT^{n}(v)|_p}{|v|_p}\geq
\frac{\tau_0+\cdots+\tau_{n-1}+{\cal B}_0^{-1}}{{\cal B}_0^{-1}}
$$
which proves the sublemma. $\Box$ \medskip
We now complete the proof of Lemma~\ref{lmmm2}. Let
$T^{j_1}x=(r_{j_1},\varphi_{j_1})$. We subsequently use
the sublemma, the lower bound in (\ref{B1}), the
upper bound in (\ref{B3}) for the vector $DT^{j_1}(v)$,
and the lower bound in (\ref{B77}):
\begin{eqnarray*}
|DT^{j_2}(v)|_p
&\geq&
\frac{\tau(T^{j_1}x)+\cdots +\tau(T^{j_2-1}x)}{B_1\cos\varphi_{j_1}}
|DT^{j_1}(v)|_p\\
&\geq&
\frac{\tau_{\min}'}{B_1B_3}
|DT^{j_1}(v)|\\
&\geq&
\frac{\tau_{\min}'}{B_1B_3B_{16}^{j_1}}|v|
\end{eqnarray*}
Recall that $j_1,j_2\leq m_0+1$.
So, it is enough to take any $m_3$ such that
$\Lambda_0^{m_3-2m_0-1}>B_1B_3B_{16}^{m_0+1}/\tau_{\min}'$.
Lemma~\ref{lmmm2} is proved. $\Box$ \medskip
\noindent
{\bf Accumulation of singularity lines and the map $T_1$}.
As in Section~\ref{secFH}, the boundaries of the regions
$T^{-1}I_k$, $k\geq 0$, accumulate near some curves of
$S_{-1}$. For each LUM $W\subset T^{-1}I_k$, $k\geq 0$,
we again have the estimate (\ref{B5}) (with a different
value of $B_5$, though) since it follows
from (\ref{B77}).
We now fix a sufficiently large $m>m_3$ for which (\ref{Km3})
holds. Let $T_1:=T^m$. Then Lemma~\ref{lmmm2} implies that
$$
|DT_1(v)|\geq\Lambda_1|v|\ \ \ \ \forall v\in C^u
\ \ \ \ {\rm and} \ \ \ \
|DT_1^{-1}(v)|\geq\Lambda_1|v|\ \ \ \ \forall v\in C^s
$$
with $\Lambda_1:=\Lambda_0^{m-m_3}>1$. also, (\ref{Km3}) implies
$$
\Lambda_1>K_m+1
$$
Note also that the set $S_{-m,0}\cup V_{-m,0}$ is a finite union
of smooth compact curves.
We are now in exactly the same position as in Section~\ref{secFH}.
So, the indexing system used in that section and the proofs
of (\ref{rgrowth11})-(\ref{rgrowth10}) go through without change.
The proof of (\ref{rw0}) requires a correction, though,
because now some curves in $\partial M$ (specifically, the segments of $V_0$)
are not uniformly transversal to unstable fibers. As a result,
for some LUM's $W$ the set ${\cal U}_{\delta}$ may cover on $W$
an interval longer than const$\cdot\delta$, unlike what we had
in Sect.~\ref{secFH}.
To overcome this problem, we invoke a useful estimate, proved in
\cite{BSC90}, Lemma 2.7: for any LUM $W$ and any point $x=(r,\varphi)\in W$
the tangent vector $(dr,d\varphi)$ to $W$ satisfies
$$
\frac{d\varphi}{dr}\leq\frac{B_{17}}{|r-r_0|^{1/2}}
$$
where $(r_0,\varphi_0)$ is the endpoint of $W$
closets to $x$. Hence, $|\varphi-\varphi_0|\leq 2B_{17}|r-r_0|^{1/2}$,
so that the $\delta$-neighborhood of $V_0$ can only cover an interval
on $W$ of length $\leq B_{18}\delta^{1/2}$. We now finish the
proof of (\ref{rw0}) in a manner similar to that of Sect.~\ref{secFH}:
\begin{eqnarray*}
m_W(V^0_{\delta})
&\leq&\sum_{\Delta\subset W\setminus\Gamma_1}\min\{2B_{18}\delta^{1/2},|\Delta|\}\\
&\leq&\sum_{k_1,\ldots,k_m,j}\min\{2B_{18}\delta^{1/2},M_{k_1,\ldots,k_m}\}\\
&\leq& {\rm const}\cdot\delta^{1/2}+
{\rm const}\cdot{\sum_{k_1,\ldots,k_m}}^{\kern-0.7em{\ast}}
\min\left\{ \delta^{1/2}, \prod_{k_i\neq 0}k_i^{-2}\right\}
\end{eqnarray*}
where $\sum^{\ast}$ is taken over $m$-tuples that contain
at least one nonzero index $k_i\neq 0$. The bound (\ref{rw0})
now follows from Lemma~\ref{lmk2}, but with $\sigma=(4m)^{-1}$
rather than $\sigma=(2m)^{-1}$.
\section{Final remarks and discussion}
\label{secFR}
\setcounter{equation}{0}
Theorem~\ref{tmmain} obviously holds for functions that
are only H\"older continuous on the connected components
of the set $M\setminus\Gamma^{(m)}$ for some $m\geq 1$.
Moreover, it can be naturally extended to a wider class
of the so called piecewise H\"older continuous functions,
as defined in \cite{BSC91,Ch94}.
In applications, it is often enough to prove Theorem~\ref{tmmain}
for any power, $T^m$, of the map $T$:
\begin{proposition}
Let $m\geq 2$. Assume that the map $T^l$ is H\"older continuous
(with some exponent $\eta_l>0$) on every connected component of
$M\setminus\Gamma^{(l)}$ for each $l=1,\ldots, m$. If $T^m$
enjoys exponential decay of correlations, then so does $T$.
\label{prTm}
\end{proposition}
{\em Proof}. Let $n\geq 1$, and $n=km+l$ with some $0\leq l\leq m-1$.
Let $f,g\in {\cal H}_{\eta}$. Then
\begin{eqnarray}
\int_M (f\circ T^n)g\, d\mu
&=&
\int_M \left (f\circ T^n-f\circ T^{km}\right )g\, d\mu +
\int_M (f\circ T^{km})g\, d\mu \nonumber\\
&=&
\int_M (h_l\circ T^{km})g\, d\mu +
\int_M (f\circ T^{km})g\, d\mu
\label{fgh}
\end{eqnarray}
where $h_l=f\circ T^l-f$. The function $h_l$ is H\"older
continuous (with exponent $\eta_l\eta>0$) on each connected
component of $M\setminus\Gamma^{(l)}$. Since $l$ takes a
finite number of values, both integrals
in (\ref{fgh}) are exponentially small in $k$. $\Box$
\medskip
Lastly, we discuss the assumptions of our main theorem~\ref{tmmain}.
First, the assumption on the existence of an ergodic SRB measure
$\mu$ does not seem to be necessary. Indeed, it can be often proved
under various general assumptions similar to ours,
see \cite{Pes92,Sat92,LSY},
and the proof is normally easier than that of statistical
properties of $\mu$. We intentionally left out this problem
in the paper, in order to focus on the EDC and CLT.
Note, however, that the other assumptions in Section~\ref{secSR}
do not logically imply the existence of SRB measures,
as the following example shows.
{\em Example}. Let $R=\{(x,y):\, 01\}$ be an
open strip in $\IR^2$, and let
$M'=\{(s,t):\, 0\leq s\leq 1,\, 0\leq t\leq 1\}$
with the identification of $s=0$ and $s=1$ be a closed cylinder.
Let $T_1:R\to R$ be given by $(x,y)\to (x/3+1/3,2y-1)$ and $T_2:
R\to M'$ be defined by $s=y$ (mod 1) and $t=e^{-y}+
x(e^{-y-1}-e^{-y})$. Then $M=T_2(R)$ is an open subset
of $M'$, and the map $T=T_2\circ T_1\circ T_2^{-1}$
takes $M$ to $M$. It satisfies all the assumptions
of Section~\ref{secSR} (other than the existence of an SRB measure),
with $\Gamma=\emptyset$, but has no SRB measure.
We now comment on our main assumptions
(\ref{rgrowth11})--(\ref{rw0}). They are proved
in \cite{Ch98} in the case where $\Gamma\cup\partial M$
was a finite union of smooth compact hypersurfaces, and
$T$ had one-sided derivatives on $\Gamma\cup\partial M$.
Assume now that $\Gamma\cup\partial M$ consists of a {\em countable} number
of smooth compact hypersurfaces. Three additional assumptions,
all valid for billiard systems, may significantly simplify the
proof of (\ref{rgrowth11})--(\ref{rw0}):
\medskip
\noindent
{\em Bounded curvature}. If the sectional curvature of
the smooth components of $\Gamma\cup\partial M$ is uniformly
bounded, then one can approximate them by hyperplanes
(since they are almost flat on the small scale of our
$\delta_0$-LUM's).
\medskip
\noindent
{\em Continuation}. Assume that each boundary point, $x_0$,
of every smooth component $\gamma\subset\Gamma$
lies either on $\partial M$ or on another smooth component
$\gamma'\subset\Gamma$ that itself does not terminate at $x_0$.
Also, assume that for each point $x\in M$ there is a neighborhood
$V(x)$ that intersect only a finite number of smooth components
of $\Gamma$ (i.e., infinitely many components of $\Gamma$ can
only accumulate near $\partial M$).
\medskip
\noindent
{\em Transversality}. The tangent planes to $\Gamma\cup\partial M$
and unstable cones are uniformly transversal, i.e. the angles between
them (properly defined in \cite{Ch98}) are bounded away from zero.
\smallskip
The above properties imply the following.
Let $W$ be a LUM, and $x\in W\cap {\cal U}_{\delta}$. Then $x$ lies
in a $(B\delta)$-neighborhood of the set $(W\cap\Gamma)\cup\partial W$.
Here the constant $B>0$ is determined by the minimum angle between
the tangent planes to $\Gamma\cup\partial M$ and unstable cones.
This property allows to work with the $(B\delta)$-neighborhood of
the intersection $W\cap\Gamma$
when proving (\ref{rgrowth11})--(\ref{rw0}). This is exactly what
we did in Sections~\ref{secFH}--\ref{secWH}, as well as in \cite{Ch98}.
Lastly, in the case dim$E^u=d_u=1$, the assumption (\ref{rgrowth10})
always holds, and our proof in Section~\ref{secFH} applies.
\section*{Appendix}
\label{secA}
\setcounter{section}{1}
\setcounter{equation}{0}
\setcounter{theorem}{0}
\renewcommand{\thetheorem}{\Alph{section}.\arabic{theorem}}
\renewcommand{\theequation}{\Alph{section}.\arabic{equation}}
Here we provide the proofs of the technical estimates
in Lemmas~\ref{lmk2} and \ref{lmk2l}. We denote by Vol$_m$ the
$m$-dimensional volume in $\IR^m$.
\begin{sublemma}
Let $A>1$ and $m=1,2,\ldots$. Consider the region
$R_m(A)\subset \IR^m$ defined by
$$
R_m(A)=\{x_1,\ldots,x_m\geq 1,\,
x_1\cdots x_m1$ and $m\geq 1$, $k\geq 2$. Consider the region
$R_{2m}(A,B,k)\subset \IR^{2m}$ defined by
$$
R_{2m}(A,B,k)=\{x_1,y_1,\ldots,x_m,y_m\geq 1,\,
x_1\cdots x_m