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Sinai billiards, decay of correlations, 
central limit theorem; invariance principle; law of iterated 
logarithms
---------------0510101530369
Content-Type: application/x-tex; name="abbrev.tex"
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\def\Prob{{\rm Prob}}
\def\Vol{{\rm Vol}}

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\def\fJ{\mathfrak{J}}
\def\fK{\mathfrak{K}}
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\def\fM{\mathfrak{M}}
\def\fP{\mathfrak{P}}
\def\fQ{\mathfrak{Q}}
\def\fR{\mathfrak{R}}
\def\fS{\mathfrak{S}}

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\def\proof{\noindent{\em Proof. }}

\newtheorem{theorem}{Theorem}[section]
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---------------0510101530369
Content-Type: application/x-tex; name="statall.tex"
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\documentclass[12pt,leqno]{article}
\usepackage{graphicx,psfrag,amssymb,amsmath,amsthm}
%\usepackage{showkeys}

%\textwidth6.25in \textheight8.5in \oddsidemargin.25in \topmargin0in

\input{abbrev.tex}

\numberwithin{equation}{section}

\begin{document}

\title{Advanced statistical properties of dispersing billiards}

\author{ N. Chernov$^1$}

\date{\today}

\maketitle

\footnotetext[1]{Department of Mathematics, University of Alabama
at Birmingham, Birmingham, AL 35294. Phone 205-934-2154; Fax
205-934-9025; Email:$\ $ chernov@math.uab.edu.}

\begin{abstract}
A new approach to statistical properties of hyperbolic dynamical
systems emerged recently; it was introduced by L.-S.~Young and
modified by D.~Dolgopyat. It is based on coupling method borrowed
from probability theory. We apply it here to one of the most
physically interesting models -- Sinai billiards. It allows us to
derive a series of new results, as well as make significant
improvements in the existing results. First we establish sharp
bounds on correlations (including multiple correlations). Then we
use our correlation bounds to obtain the central limit theorem
(CLT), the almost sure invariance principle (ASIP), the law of
iterated logarithms, and integral tests.
\end{abstract}

\noindent Keywords: Sinai billiards, decay of correlations,
central limit theorem, invariance principle, law of iterated
logarithms.

\section{Introduction}
\label{secI}

A billiard is a mechanical system in which a point particle moves
in a compact container $\cD$ and bounces off its boundary $\dcD$.
The dynamical properties of a billiard are determined by the shape
of $\dcD$, and they may vary greatly from completely regular
(integrable) to strongly chaotic. The first class of chaotic
billiards was introduced by Ya.~Sinai in 1970, see \cite{Si70},
who considered containers defined by
\beq \label{cD}
              \cD = \Tor^2 \setminus \cup_{i=1}^p \IB_i,
\eeq
where $\Tor^2$ denotes the unit 2-torus and $\IB_i \subset \Tor^2$
disjoint strictly convex domains (scatterers) with $C^3$ smooth
boundary whose curvature nowhere vanishes. Sinai proved
\cite{Si70} that the billiard flows and maps in such domains are
hyperbolic, ergodic, and K-mixing. He called these systems
\emph{dispersing billiards}, now they are known as \emph{Sinai
billiards}.

More advanced ergodic properties where established for Sinai
billiards as well: Bernoulliness was proved in \cite{GO}, Markov
partitions were constructed in \cite{BS80,BSC90}, estimates on
periodic points were obtained in \cite{St89,BSC90}. Statistical
properties were established fairly recently; these include
exponential decay of correlations \cite{Y98,Ch99}, Central Limit
Theorem (CLT) and Weak Invariance Principle (WIP), see
\cite{BS81,BSC91}. All these facts demonstrate that dispersing
billiards are strongly chaotic systems and can be placed in the
same category as Anosov and Axiom~A diffeomorphisms.

Traditional methods for proving statistical properties are based
on Markov partitions and symbolic dynamics
\cite{Bo75,Ru76,Ru78,Si68,Si72}. If a system admits a finite
Markov partition (this is the case for Anosov and Axiom~A maps),
then its symbolic system is a subshift of finite type. Then one
usually shows that the corresponding Perron-Frobenius operator
(acting on H\"older continuous densities) has a spectral gap, and
then derives all the above statistical properties (and more)
combining methods of functional analysis and probability
theory\footnote{This is a very efficient approach, but it relies
upon highly abstract elements of functional analysis (spectral
properties of pseudo-compact operators), which in a sense obscure
the dynamical content of the problem.}.

In the case of billiards, however, Markov partitions are never
finite, which rendered the symbolic representation inefficient and
the Perron-Frobenius operator unhandy. First attempts to
investigate statistical properties of dispersing billiards used
Markov approximations \cite{BS81,BSC91} and were technically
overcomplicated. They did produce the CLT and WIP but gave only
sub-optimal (sub-exponential) estimates on correlations.

Later Young developed \cite{Y98} a more efficient approach to
general hyperbolic maps with singularities, based on \emph{tower
construction} (which is a tractable version of countable Markov
partitions), and found a way to reemploy the Perron-Frobenius
operator. This produced an exponential bound on correlations for
dispersing billiards \cite{Y98,Ch99}. Still, the tower
construction is fairly complicated, and the use of the
Perron-Frobenius operator restricts the flexibility of the
arguments.

Then Young rederived \cite{Y99} exponential bound on correlations
for hyperbolic maps by a different method (borrowed from
probability theory) based on coupling of smooth measures, thus
bypassing symbolic formalism and the Perron-Frobenius operator
altogether. The underlying idea of this new approach is that the
images of different smooth measures on the phase space of billiard
map are getting closer together, and thus converge to a common
limit, the degree of `closeness' and the speed of convergence are
controlled by coupling. Her entire argument was intrinsically
dynamical and highly flexible, unlike earlier operator-based
proofs.

The elegance of Young's new method was recently demonstrated in
\cite{BL02} where it was applied to Anosov maps. Dolgopyat further
simplified \cite[Appendix A]{CD} the coupling method by replacing
smooth measures on phase space with one-dimensional measures on
unstable curves, which made the argument even more transparent and
almost elementary. Here we employ Dolgopyat's version of the
coupling method.

The paper is organized as follows. Section~2 contains the
necessary background on Sinai billiards. Section~3 describes the
coupling method and states its key tool -- Coupling Lemma (whose
proof is given in Appendix). In Section~4 we establish sharp
bounds on correlations for (dynamically) H\"older continuous
functions. These include bounds on multiple correlations.

In Section~5 we combine our estimates on correlations with a
general result by Philipp and Stout \cite{PSbook} to prove various
limit theorems: Central limit theorem (CLT), Weak Invariance
Principle (WIP), Almost Sure Invariance Principle (ASIP), Law of
Iterated Logarithms, and Integral Tests. The last three results
are actually new, in the billiard context.

It is interesting to note that all these probabilistic limit
theorems follow from sharp bounds on multiple correlations. The
earlier proofs of limit theorems (in particular, that of CLT, see
\cite{BS81,BSC91,Ch95,De89,Y98}) used bounds on correlations also,
but mainly relied upon some other (stronger) mixing properties of
the dynamics. One always wondered if the CLT could be derived
solely from correlation bounds. We demonstrate that this is indeed
possible: in fact, all our limit theorems formally follow from our
bounds on correlations, so that no other mixing properties of the
dynamics are necessary.

\medskip\noindent\textbf{Acknowledgements}. This work results from
the author's joint project with D.~Dolgopyat reported in
\cite{CD}, and many ideas belong to D.~Dolgopyat. Also,
R.~Markarian has read main proofs and made useful remarks. The
author was partially supported by NSF grant DMS-0354775.

\section{Preliminaries}
\label{SecP}

Here we recall basic facts about dispersing billiards. All of them
are well known, see \cite{Si70,BSC90,BSC91,Y98,Ch99,Ch05}.

Let $\cD \subset \Tor^2$ be a domain defined by (\ref{cD}) and
$\dcD = \cup_i \partial \IB_i$ its boundary. The billiard particle
moves inside $\cD$ with constant (unit) speed and bounces off
$\dcD$ according to the classical law ``the angle of incidence is
equal to the angle of reflection''.

The phase space of the billiard system is $\Omega = \cD \times
S^1$, and the billiards dynamics generates a flow $\Phi^t \colon
\Omega \to \Omega$. It is a Hamiltonian (contact) flow, and it
preserves Liouville (uniform) measure on $\Omega$.

At every reflection the velocity vector changes by the rule $v^+ =
v^- - 2\,\la v,n \ra\, n$, where $v^+$ and $v^-$ refer to the
postcollisional and precollisional velocities, respectively, and
$n$ denotes the inward unit normal vector to $\dcD$ at the
reflection point $q \in \dcD$. The family of postcollisional
velocity vectors with footpoints on $\dcD$ makes a 2D manifold
called the \emph{collision space}
$$
    \cM = \{x=(q,v)\in \Omega\colon\ q\in\dcD \,\ \la
      v,n\ra\geq 0\},
$$
where $\la \cdot \ra$ denotes the scalar product. The billiard
flow induces a billiard map (also called collision map) $\cF
\colon \cM \to \cM$.

Standard coordinates on $\cM$ are the arc length parameter $r$ on
the boundary $\dcD$ and the angle $\varphi \in [-\pi/2, \pi/2]$
between the vectors $v$ and $n$; the orientation of $r$ and
$\varphi$ is shown on Fig.~\ref{FigOrient}. Note that $\la v,n \ra
= \cos \varphi$. The space $\cM$ is the union of $p$ cylinders on
which $r$ is a cyclic (`horizontal') coordinate and $\varphi$ is a
linear (`vertical') coordinate. The map $\cF \colon \cM \to \cM$
preserves smooth measure $d\mu = c_\mu \cos \varphi\, \,dr \,
d\varphi$, where $c_{\mu}$ is the normalizing factor.

\begin{figure}[htb]
    \centering
    \psfrag{n}{\small $\varphi=0$}
    \psfrag{p}{\small $+\pi/2$}
    \psfrag{f}{\small $\varphi$}
    \psfrag{r}{\small $r$}
    \psfrag{m}{\small $-\pi/2$}
    \psfrag{z}{$0$}
    \psfrag{M}{$\cM$}
    \psfrag{u}{$\cC^u$}
    \psfrag{s}{$\cC^s$}
     \includegraphics[width=6in]{statall-1.eps}
    \caption{Orientation of $r$ and $\varphi$ and the
    collision space.}
    \label{FigOrient}
\end{figure}

For $x \in \cM$ denote by $\tau (x)$ the distance of the free path
between the collision points we at $x$ and $\cF (x)$. The flow
$\Phi^t$ can be represented as a suspension flow over the map $\cF
\colon \cM \to \cM$ under the ceiling function $\tau (x)$. Clearly
$\tau (x) \geq \tau_{\min} >0$, where $\tau_{\min}$ is the minimal
distance between the domains $\IB_i \subset \Tor^2$. If $\tau(x)$
is bounded above ($\tau(x) \leq \tau_{\max} < \infty$), then the
billiard is said to have \emph{finite horizon}.

The billiard map $\cF$ is hyperbolic. There is a family of
$D\cF$-invariant unstable cones $\cC^u_x \subset \cT_x \cM$ and a
family of $D\cF^{-1}$-invariant stable cones $\cC^s_x \subset
\cT_x \cM$. They can be defined so that $c_1 \leq d\varphi/dr \leq
c_2$ for all $(dr,d\varphi) \in \cC^u_x$ and $-c_2 \leq
d\varphi/dr \leq -c_1$ for all $(dr,d\varphi) \in \cC^s_x$, where
$0<c_1<c_2<\infty$ are constants.

A smooth curve $W \subset \cM$ is said to be \emph{unstable} (or
\emph{stable}) if at every point $x \in W$ the tangent space
$\cT_x W$ belongs to the unstable (stable) cone. Note that
unstable curves are increasing and stable curves are decreasing in
the $r\varphi$ coordinates, and their slopes are bounded away from
zero and infinity. Unstable curves are stretched by $\cF$, while
stable curves are stretched by $\cF^{-1}$.

For any point $x \in W$ on an unstable (or stable) curve $W
\subset \cM$ we denote by $\cJ_{W}\cF^n(x) = \|D_{x} \cF^{n}
(dx)\| / \|dx\|$, $dx \in \cT_x W$, the Jacobian of the
restriction of the map $\cF^n$ to $W$, at the point $x$. Then the
hyperbolicity means that there are constants $\Lambda =
\Lambda(\cD) >1$ and $C = C(\cD) >0$ such that for any unstable
curve $W^u$ and any stable curve $W^s$ and all $n \geq 1$ we have
$\cJ_{W^u} \cF^n (x) \geq C\Lambda^n$ and $\cJ_{W^s} \cF^{-n} (x)
\geq C\Lambda^n$. Note that $\Lambda$ and $C$ do not depend on
$W^u$ or $W^s$.

In general, the stretching is very nonuniform, as the stretching
factor grows to infinity near $\partial \cM = \{\cos\varphi =
0\}$. To control distortions of stable and unstable curves under
the action of $\cF$, it was proposed by Sinai \cite{BSC91} to
partition them by countably many lines that are parallel to
$\partial \cM$ and accumulate near $\partial \cM$. Let $k_0\geq 1$
be a large constant. For each $k \geq k_0$ we define two
\emph{homogeneity strips} $\IH_{\pm k} \subset \cM$ by
$$
    \IH_k=\{(r,\varphi)\colon
    \pi/2-k^{-2}<\varphi <\pi/2-(k+1)^{-2}\}
$$
and
$$
    \IH_{-k}=\{(r,\varphi)\colon
    -\pi/2+(k+1)^{-2}<\varphi < -\pi/2+k^{-2}\}.
$$
We also put $\IH_0=\{(r,\varphi) \colon -\pi/2+k_0^{-2} <\varphi <
\pi/2-k_0^{-2}\}$. Now $\cM$ is divided into homogeneity strips
$\IH_k$. Denote by $\IS_k = \{ (r,\varphi)\colon |\varphi| = \pi/2
- k^{-2}\}$ for $|k| \geq k_0$ the boundaries of the homogeneity
strips and put $\IS = \cup_{|k|\geq k_0} \IS_k$. A stable or
unstable curve $W\subset \cM$ is said to be \emph{weakly
homogeneous} if $W$ belongs to one strip $\IH_k$.

Now distortions can be characterized as follows. Let $W$ be an
unstable curve such that $W_n = \cF^{-n} (W)$ is a weakly
homogeneous unstable curve for all $0 \leq n \leq N-1$. Then we
have the following \emph{distortion bounds}, see \cite{Ch05}:
\beq  \label{distbound}
     C_1\leq
     e^{-C|W|^{1/3}} \leq
     \frac{\cJ_W \cF^{-n} (y)}{\cJ_W \cF^{-n} (z)}
    \leq e^{C|W|^{1/3}}\leq C_1,
\eeq
for every $y,z \in W$ and every $1 \leq n \leq N$; here $C,C_1>0$
are constants. Due to time reversibility, similar bounds hold for
stable curves.

Next we describe the singularities of $\cF$. We denote by $\cS_0 =
\partial \cM = \{\cos\varphi = 0\}$ the boundary of the collision
space (it consists of all `grazing' collisions). The map $\cF$
lacks smoothness on the set $\cS_1 = \cS_0 \cup \cF^{-1} (\cS_0)$
(we call it the \emph{singularity set} for $\cF$). In fact, $\cF$
is discontinuous on $\cS_1 \setminus \cS_0$. More generally, the
singularity sets for the maps $\cF^n$ and $\cF^{-n}$ are $\cS_n =
\cup_{i=0}^n \cF^{-i} (\cS_0)$ and $\cS_{-n} = \cup_{i=0}^n
\cF^{i} (\cS_0)$. For each $n \geq 1$, the set $\cS_{-n} \setminus
\cS_0$ is a finite or countable union of compact smooth unstable
curves (in fact, it is finite for billiards with finite horizon
and countable otherwise). Similarly, the set $\cS_{n} \setminus
\cS_0$ is a finite or countable union of compact smooth stable
curves.

In billiards without horizon, the smooth components of the
singularity sets $\cS_{1}$ and $\cS_{-1}$ accumulate near finitely
many points on $\cS_0$ and divide their neighborhoods into `cells'
described in \cite{BSC90,BSC91}.

Since, following Sinai, we partition stable and unstable curves by
the lines $\IS_k$, $|k| \geq k_0$, it is convenient to divide the
entire collision space $\cM$ into homogeneity strips $\IH_{k}$,
$|k| \geq k_0$, and $\IH_0$. So we get a new collision space $\MH$
constructed as a disjoint union of the closures of the
$\IH_{k}$'s; note that $\mu(\MH) = 1$ and $\partial \MH = \IS$.

The map $\cF\colon \cM \to \cM$ naturally acts on the new
collision space $\MH$. The map $\cF \colon \MH \to \MH$ lacks
smoothness on the `extended' singularity set $\cS_1 \cup \IS \cup
\cF^{-1} (\IS)$. Similarly, the map $\cF^n \colon \MH \to \MH$ is
not smooth on the `extended' singularity set
$$
   \SH_n \colon = \cS_n \cup
   \bigl(\cup_{m=0}^n \cF^{-m}
   (\IS)\bigr) .
$$
The inverse map $\cF^{-n} \colon \MH \to \MH$ is not smooth on the
`extended' singularity set
$$
   \SH_{-n} \colon = \cS_{-n} \cup
   \bigl(\cup_{m=0}^n \cF^{m}
   (\IS)\bigr) .
$$
For each $n \geq 1$, the set $\SH_{-n} \setminus \SH_0$ is a
countable union of compact smooth unstable curves. Similarly, the
set $\SH_{n} \setminus \SH_0$ is a countable union of compact
smooth stable curves. The set $\SH_0$ consists of parallel lines
that are neither stable nor unstable.

For each $n',n'' \geq 0$ the set $\cM \setminus (\SH_{-n'} \cup
\SH_{n''})$ is a countable union of open domains with piecewise
smooth boundaries (`curvilinear polygons'). Moreover, the interior
angles made by their boundary components do not exceed $\pi$
(i.e.\ those polygons are `convex', as far as the interior angles
are concerned); some interior angles may be equal to zero.

An unstable curve $W\subset \cM$ is said to be a homogeneous
unstable manifold (or, briefly, \emph{unstable H-manifold}) if
$\cF^{-n}(W)$ is a weakly homogeneous unstable curve for every
$n\geq 0$. Similarly, a \emph{stable H-manifold} is a curve
$W\subset \cM$ such that $\cF^{n}(W)$ is a weakly homogeneous
stable curve for every $n\geq 0$. For any unstable (stable)
H-manifold $W$ we have $|\cF^n W| \leq C \Lambda^{-|n|}$ for all
$n \leq 0$ (resp., $n \geq 0$); here $|W|$ denotes the length of
$W$.

Let $\xi^u_n$ denote the measurable partition of $\cM$ into the
connected components of the set $\cM \setminus \SH_{-n}$. Then
$\xi^u = \vee_{n\geq 1} \xi^u_n$ is the measurable partition of
$\cM$ into the (maximal) unstable H-manifolds (see a proof in
\cite{Ch05}). Similarly, if $\xi^s_n$ denotes the measurable
partition of $\cM$ into the connected components of the set $\cM
\setminus \SH_{n}$, then $\xi^s = \vee_{n\geq 1} \xi^s_n$ is the
measurable partition of $\cM$ into the (maximal) stable
H-manifolds. We denote by $W^u (x)$ and $W^s (x)$ the (maximal)
stable and unstable H-manifolds passing through the point $x \in
\cM$.

The conditional measures on unstable H-manifolds $W \subset \cM$
are absolutely continuous and their densities $\rho_{W^u}$ satisfy
\beq \label{uSRB}
   \frac{\rho_W(y)}{\rho_W(z)} = \lim_{n\to\infty}
   \frac{\cJ_{W}\cF^{-n}(y)}{\cJ_{W}\cF^{-n}(z)}
\eeq
for $y,z \in W$. For any unstable H-manifold $W \subset \cM$, the
unique probability density $\rho_W$ satisfying (\ref{uSRB}) is
called the \emph{u-SRB density}, and the corresponding probability
measure $\nu_W$ on $W$ is called the \emph{u-SRB} measure. The
distortion bounds (\ref{distbound}) imply the following bounds on
the u-SRB density, see \cite{Ch05}:
\beq \label{densbound1}
   \Bigl|\frac{d}{dx} \ln \rho_W (x)\Bigr|
   \leq \frac{C}{|W|^{2/3}},
\eeq
where $C = C(\cD) >0$ is a constant. It immediately follows that
\beq \label{densbound2}
     e^{-C|W(x,y)|^{1/3}} \leq
     \frac{\rho_W (x)}{\rho_W (y)}
    \leq e^{C|W(x,y)|^{1/3}},
\eeq
for all $x,y \in W$; here $W(x,y)$ denotes the segment of the
curve $W$ between the points $x$ and $y$.

For $\mu$-almost every point $x \in \cM$ there exist nonvanishing
stable and unstable H-manifolds $W^s (x)$ and $W^u (x)$ through
$x$. The point $x$ divides the H-manifold $W^s (x)$ (and $W^u
(x)$) into two segments, we denote by $r^s (x)$ (resp., $r^u (x)$)
the length of the shorter one. Then have the following estimate
(see, e.g., \cite{Ch05}):
\beq \label{Deps}
   \mu \{ x\colon \min\{r^u (x),r^s(x)\}
   < \varepsilon \}\leq C\varepsilon
\eeq
for some constant $C = C (\cD) >0$ and all $\varepsilon >0$.
Moreover, for any stable (or unstable) curve $W \subset \cM$ we
have
\beq \label{Deps1}
   m_W \{ x \in W\colon r^p (x)
   < \varepsilon \}\leq C\varepsilon
\eeq
where $p=u$ if $W$ is stable and $p=s$ if $W$ is unstable, and
$m_W$ denotes the Lebesgue measure on $W$. The following is known
as Sinai's Fundamental Theorem (it is actually a strengthened
version of \cite[Teorem 6.1]{Si70}):

\begin{theorem} \label{TmFTSinai}
Let $x \in \cM \setminus \cup_{n\geq 0} \SH_n$. Then for any $q>0$
and $A>0$ there exists an open neighborhood $\cU^u_x \subset \cM$
of $x$ such that for any unstable curve $W \subset \cU^u_x$
$$
   m_W\bigl( y\in W\colon \rHs(y)
   > A|W|\bigr)  \geq (1-q)\, m_W(W).
$$
Similarly, let $x \in \cM \setminus \cup_{n\geq 0} \SH_{-n}$. Then
for any $q>0$ and $A>0$ there exists an open neighborhood $\cU^s_x
\subset \cM$ of $x$ such that for any stable curve $W \subset
\cU^s_x$
$$
   m_W\bigl( y\in W\colon \rHu(y)
   > A|W|\bigr)  \geq (1-q)\, m_W(W).
$$
\end{theorem}


Next, billiard maps have the following absolute continuity
property. Let $W^1, W^2 \subset \cM$ be two unstable curves.
Denote $ W^i_{\ast} = \{x\in W^i \colon W^s(x) \cap W^{3-i} \neq
\emptyset\}$ for $i=1,2$. The map $\bh\colon W^1_{\ast} \to
W^2_{\ast}$ taking every point $x \in W^1_{\ast}$ to $\bar{x}=
W^s(x) \cap W^2$ (`sliding' it along the stable H-manifold) is
called the \emph{holonomy map}. It is absolutely continuous, and
its Jacobian (with respect to the Lebesgue measures on $W^1$ and
$W^2$) is given by
\beq \label{uAS}
   J \bh (x) = \lim_{n\to\infty}
   \frac{\cJ_{W^1}\cF^{n}(x)}{\cJ_{W^2}\cF^{n}\bigl(\bh(x)\bigr)}.
\eeq
It is uniformly bounded, $J \bh (x) \leq C$, where $C = C(\cD) >1$
is a constant. Moreover, if we put $\delta = \, {\rm dist}
(x,\bh(x))$ and denote by $\gamma$ the angle between the tangent
vectors to the curves $W^1$ and $W^2$ at the points $x$ and
$\bh(x)$, respectively, then $J \bh (x) \leq
A^{\gamma+\delta^{1/3}}$ for some constant $A = A(\cD) >1$, see
\cite{Ch05}.

The Jacobian $J\bh(x)$ is a continuous function on $W^1_{\ast}$,
but it is not smooth. Even for Anosov and Axiom~A systems, the
Jacobian of the holonomy map is only H\"older continuous. For
billiards, the H\"older continuity may fail, but a similar
property holds, it is sometimes called `dynamically defined
H\"older continuity' \cite[p.\ 597]{Y98}. To describe it, for any
$x, y \in \cM$ we denote by
\beq \label{s+}
    \bs_+(x,y) = \min \{ n\geq 0\colon y \notin \xi^s_n (x)\}
\eeq
the `separation time' (the first time when the images $\cF^n(x)$
and $\cF^n(y)$ lie in different connected components of the new
collision space $\MH$). Note that $\bs_+(x,y) = \infty$ iff $y \in
W^s (x)$. Observe that if $x$ and $y$ lie on one unstable curve $W
\subset \cM$, then
\beq \label{distbs+}
   |W(x,y)| \leq C \Lambda^{-\bs_+(x,y)}
\eeq
where $C = C(\cD) >0$ is a constant. Then the dynamical H\"older
continuity of the Jacobian of the holonomy map is expressed by
\beq \label{hDHC}
   |\ln J\bh(x) - \ln J\bh(y)|
   \leq C\btheta^{\bs_+(x,y)}.
\eeq
where $\btheta = \Lambda^{-1/6} <1$ and $C>0$ are constants, see
\cite{Ch05}.

Lastly we turn to the so called `growth lemma'. Let $W \subset
\MH$ be a weakly homogeneous unstable curve and $m_W$ the Lebesgue
measure on it. Its image $\cF (W) \subset \MH$ is a finite or
countable union of homogeneous unstable curves, which we call the
\emph{H-components} of $\cF (W)$. Inductively, we define the
H-components of $\cF^n (W)$ as the collection of the H-components
of $\cF (W_i)$, $i\geq 1$, where $W_i$ denote all the H-components
of $\cF^{n-1}(W)$. For every $x \in W$ the point $\cF^n(x)$
divides the H-component of $\cF^n(W)$ it belongs to into two
segments, and we denote by $r_n(x)$ the length of the shorter one.

Clearly, $r_n (x)$ is a function on $W$ that characterizes the
size of the H-components of $\cF^n (W)$. Note that $m_W \bigl(
r_0(x) < \varepsilon\bigr) = \min\{2\varepsilon, m_W(W)\}$, where
$m_W(W) = |W|$ is the length of $W$. The following statements are
proved in \cite{Ch99} and \cite[Lemma 3.10]{CD}:

\begin{lemma}[Growth Lemma] \label{TmGrow1}
There are constants $\hLambda \in (1, \Lambda)$, $\vartheta_1 \in
(0,1)$, and $c_1,c_2>0$, such that for any sufficiently short
unstable curve $W \subset \cM$, any $n\geq 0$ and $\varepsilon >0$
$$
   m_W \bigl(r_n(x) < \varepsilon\bigr) \leq
   c_1(\vartheta_1 \hLambda)^n\, m_W
   \bigl(r_0(x) < \varepsilon/\hLambda^n\bigr)
   + c_2 \varepsilon\, m_W(W).
$$
\end{lemma}

\begin{corollary} \label{TmGrow2}
There are constants $\varkappa >0$ and $c_3>0$, such that for all
$n \geq \varkappa \bigl| \ln |W|\bigr|$ and $\varepsilon>0$ we
have  $m_W\bigl(r_n(x) < \varepsilon\bigr) \leq c_3 \varepsilon\,
m_W(W)$,
\end{corollary}

\section{Coupling lemma} \label{SecCL}

We start with an observation that motivates the use of
one-dimensional measures on unstable curves, as proposed by
Dolgopyat.

Let us partition the collision space $\cM$ into small subdomains
(cells) $D_i \subset \cM$ and represent a smooth measure $\mu_0$
on $\cM$ as a weighted sum of its restrictions to those cells. Now
the image of a small domain $D \subset \cM$ under the map $\cF^n$
gets strongly expanded in the unstable direction, strongly
contracted in the stable direction, and possibly cut by
singularities into many pieces. Thus, $\cF^n(D)$ will soon look
like a union of one-dimensional curves, each of which resembles an
unstable manifold of the collision map $\cF$. Henceforth the
measure $\cF^n(\mu_0)$ will evolve as a weighted sum of smooth
measures on unstable curves.

Accordingly, we define a class of probability measures supported
on homogeneous unstable curves. A \emph{standard pair} $(W, \nu)$
is a homogeneous unstable curve $W \subset \cM$ with a probability
measure $\nu$ on it, whose density $\rho$ with respect to the
Lebesgue measure on $W$ is regular, see below.

The regularity of the density $\rho(x)$ should be comparable to
the regularity of the map $\cF$, the latter is expressed by two
key estimates -- distortion bounds and absolute continuity. While
distortions are fairly smooth, the Jacobian of the holonomy map is
only `dynamically H\"older continuous' (\ref{hDHC}). Accordingly,
we say that a density $\rho(x)$ on a homogeneous unstable curve $W
\subset \cM$ is regular if
\beq \label{DHC}
       |\ln \rho(x) - \ln \rho(y)|
       \leq C_{\rm r}\, \btheta^{\bs_+ (x,y)}.
\eeq
Here $C_{\rm r}>0$ is a sufficiently large constant. Observe that
$\rho$ is uniformly bounded:
\beq \label{rhomaxmin}
    \max_{x\in W} \rho (x) / \min_{x \in W} \rho(x)
    \leq \, {\rm const}\, = e^{C_{\rm r}}.
\eeq
The condition (\ref{DHC}) will not be altered if we multiply the
density $\rho(x)$ by a constant. Therefore, given a standard pair
$(W,\nu)$, any subcurve $W' \subset W$ with the conditional
measure induced by $\nu$ on it will make a standard pair. It is
easy to see that any unstable H-manifold $W^u$ with the u-SRB
measure $\nu_{W^u}$ on it makes a standard pair.

The class of standard pairs is invariant under $\cF$ in the
following sense:

\begin{proposition} \label{PrStPair}
Let $(W, \nu)$ be a standard pair. For each $n \geq 0$, denote by
$W_{i,n}$ the H-components of $\cF^n(W)$. Then $\cF^n(\nu) =
\sum_i c_{i,n} \nu_{i,n}$ where $\sum_i c_{i,n} = 1$ and each
$(W_{i,n}, \nu_{i,n})$ is a standard pair.
\end{proposition}

\proof By induction, it is enough to prove this for $n=1$. Let
$W_{i,1}$ be an H-component of $\cF(W)$ and $x,y \in W_{i,1}$.
Denote $x_1 = \cF^{-1} (x)$ and $y_1 = \cF^{-1} (y)$. Observe that
$\bs_+(x,y) = \bs_+(x_1,y_1) -1$. Now using (\ref{distbound}) and
(\ref{distbs+}) gives
\begin{align*}
       |\ln \rho_{i,n}(x) - \ln \rho_{i,n}(y)| &\leq
       |\ln \rho_{i,n}(x_1) - \ln \rho_{i,n}(y_1)|\\
       &\quad +|\ln\cJ_W \cF^{-1}(x) - \ln\cJ_W \cF^{-1}(y)|\\
      & \leq C_{\rm r} \btheta^{\bs_+ (x_1,y_1)} + C|W_{i,1}|^{1/3}\\
      & \leq C_{\rm r} \btheta\, \btheta^{\bs_+ (x,y)}
      + C'\btheta^{\bs_+(x,y)}
\end{align*}
for some constant $C' = C'(\cD) >0$. Thus it is enough to assume
that $C_{\rm r}$ is so large that $C_{\rm r} \btheta + C' \leq
C_{\rm r}$. \qed \medskip

We see that $\cF^n$ transforms the measure $\nu$ from a standard
pair into a weighted sum of measures on finitely or countably many
standard pairs. Motivated by this observation, we introduce even
more general families of standard pairs:

A \emph{standard family} is an arbitrary (countable or
uncountable) family $\cG = \{ (W_\alpha, \nu_{\alpha}) \}$,
$\alpha \in \fA$, of standard pairs with a probability factor
measure $\lambda_\cG$ on the index set $\fA$. Such a family
induces a probability measure $\mu_\cG$ on the union $\cup_\alpha
W_\alpha$ (and thus on $\cM$) defined by
$$
    \mu_\cG (B) =
    \int \nu_\alpha (B \cap W_\alpha) \,
    d\lambda_\cG (\alpha)
    \qquad \forall B \subset \cM.
$$
Proposition~\ref{PrStPair} now simply says that $\cF^n$ transforms
a standard pair into a countable standard family (whose factor
measure is defined by the sequence of the coefficients
$\{c_{i,n}\}$). Similarly, any standard family $\cG$ is mapped by
$\cF^n$ into another standard family $\cG_n = \cF^n (\cG)$. It is
easy to see that $\mu_{\cG_n} = \cF^n (\mu_\cG)$.

It will be important to control the size of curves $W_\alpha$ in a
standard family $\cG = \{ (W_\alpha, \nu_{\alpha}) \}$. For every
$\alpha \in \fA$ and $x \in W_\alpha$, the point $x$ divides the
curve $W_\alpha$ into two parts, and we denote by $r_\cG (x)$ the
length of the shorter one (in the Euclidean metric). This defines
a function $r_\cG$ on\footnote{In all our subsequent proofs, the
curves $W_\alpha$ of every standard family will be disjoint;
however in this general definition we need not assume this: the
function $r_\cG$ is simply defined on every curve $W_\alpha$
separately.} $\cup_\alpha W_\alpha$. Now we denote
$$
   \cZ_\cG =
   \sup_{\varepsilon>0}
   \frac{\mu_{\cG} (r_\cG < \varepsilon)}{\varepsilon}
   =\sup_{\varepsilon>0}
   \frac{\int \nu_\alpha \bigl(x\in W_\alpha
   \colon r_\cG (x) < \varepsilon\bigr) \,
    d\lambda_\cG (\alpha)}{\varepsilon}.
$$
If $\cG$ consists of a single standard pair $(W,\nu)$ and $\nu$ is
is the normalized Lebesgue measure on $W$, then $\cZ_\cG = 2/|W|$.
If $\nu$ is an arbitrary regular density, then $\cZ_\cG \sim
1/|W|$, in the sense that  $C_1 < \cZ_\cG |W| < C_2$, where $C_1=
C_1(\cD)>0$ and $C_2= C_2(\cD)>0$ are constants.

Now for an arbitrary standard family $\cG = \{ (W_\alpha,
\nu_{\alpha}) \}$ we have
\beq \label{ExcZint}
   \cZ_\cG \sim \int
   \frac{d\lambda_\cG (\alpha)}{|W_\alpha|}
\eeq
(in this formula, either both quantities are finite or both
are infinite). We will only consider standard families $\cG$ with
$\cZ_\cG < \infty$. Next we investigate how the quantity
$\cZ_{\cG_n}$, where $\cG_n = \cF^n (\cG)$, changes with $n$.

Let $\cG$ be a standard family consisting of a single standard
pair $(W,\nu)$. Let $\cG_n = \cF^n (\cG)$. The growth
lemma~\ref{TmGrow1} and (\ref{rhomaxmin}) imply that for all $n
\geq 0$ and $\varepsilon >0$
\begin{align} \label{Ggrow1}
   \nu \bigl(r_{\cG_n} (\cF^n x) < \varepsilon\bigr) &\leq
   {\rm const}\bigl[ (\vartheta_1 \hLambda)^n\, \nu
   \bigl(r_\cG (x) < \varepsilon/\hLambda^n\bigr)
   + \varepsilon \bigr] \nonumber\\
    &\leq {\rm const} \bigl[ \vartheta_1^n
   \varepsilon/|W| + \varepsilon \bigr]
\end{align}
where the constants depend on the table $\cD$ only.

\begin{proposition} \label{PrGgrow1}
Let $\cG = \{ (W_\alpha, \nu_{\alpha}) \}$, $\alpha \in \fA$, be a
standard family and $\cG_n = \cF^n (\cG)$. Then for all $n \geq 0$
and $\varepsilon >0$ we have $\cZ_{\cG_n} \leq c_1 \vartheta_1^n
\cZ_\cG + c_2$ for some constants $c_i = c_i (\cD) >0$, $i=1,2$.
\end{proposition}

\proof It is enough to integrate (\ref{Ggrow1}) with respect to
the factor measure $\lambda_\cG$ and use (\ref{ExcZint}). \qed
\medskip

We see that if $\cZ_\cG$ is very large, the sequence $\cZ_{\cG_n}$
will decrease exponentially fast until it goes under a certain
threshold, say $c_1+c_2$.

\begin{corollary} \label{CrGgrow2}
For all $n \geq \varkappa \ln \cZ_\cG$ we have $\cZ_{\cG_n} \leq
c_3$ for some constants $\varkappa, c_3>0$.
\end{corollary}

The reader should notice similarities between
\ref{PrGgrow1}--\ref{CrGgrow2} and \ref{TmGrow1}--\ref{TmGrow2}.

Motivated by these facts, we introduce the notion of a proper
family. Let $C_{\rm p}>0$ be a sufficiently large constant (the
subscript $\rm p$ in $C_{\rm p}$ stands for `proper'). A standard
family $\cG$ is said to be \emph{proper} if $\cZ_\cG \leq C_{\rm
p}$. A family consisting of a single standard pair $(W,\nu)$ is
proper iff $|W| \geq c_0 = \, {\rm const} >0$, i.e.\ iff the curve
$W$ is not too short ($c_0$ can be made arbitrarily small by
choosing $C_{\rm p}$ sufficiently large). We call such $(W,\nu)$
\emph{proper standard pairs}.

The partition $\xi^u$ of $\cM$ into maximal unstable H-manifolds
with u-SRB measures on them and the factor measure induced by
$\mu$ makes a (special) standard family $\cE$. For this family
$\mu_\cE = \mu$, of course. Note also that $\cE$ is mapped by
$\cF$ into itself. It follows from (\ref{Deps1}) that the family
$\cE$ is proper.

More generally, let $\{W\}$ be a smooth foliation of $\cM$ into
unstable curves that stretch from $\varphi = -\pi/2$ to $\varphi =
\pi/2$. Dividing them by the homogeneity lines $\IS_k$, $|k| \geq
k_0$, gives a smooth foliation of $\cM$ into homogeneous unstable
curves $\{W_\alpha\}$. The measure $\mu$ induces smooth
conditional measures $\{\nu_\alpha\}$ on $\{W_\alpha\}$ and a
factor measure on the index set. It is easy to check that the so
defined standard family $\cG$ is proper and $\mu_\cG = \mu$.

Next we turn to the coupling construction. Let $(W_1, \nu_1)$ and
$(W_2, \nu_2)$ be two standard pairs. For a large $n$, their
images $\cF^n W_1$ and $\cF^n W_2$ consist of many H-components
scattered all over $\cM$. Some H-components $W' \subset \cF^n W_1$
of the first image may lie close to some H-components $W'' \subset
\cF^n W_2$ of the other image. Then certain points $x' \in W'$ can
be joined by stable manifolds with points $x'' \in W''$, so that
their further iterations will get closer together exponentially
fast. In this way we can `link' the measures they carry and
eventually show that the asymptotic behavior of the two measures
$\cF^n (\nu_1)$ and $\cF^n (\nu_2)$ becomes identical.

This argument may run into an obvious problem, though: the
H-components $W'$ and $W''$ may carry different amount (mass) of
the corresponding measures. To resolve this problem we may, so to
say, couple a heavy piece with several light ones. This can be
done by splitting a heavy piece into several `thinner' curves,
each coupled to a different partner.

To implement this idea, it is convenient to split each original
curve $W_1$ and $W_2$ into uncountably many `fibers'. To this end,
given a standard pair $(W, \nu)$, we consider $\hW \colon =W
\times [0,1]$ and equip $\hW$ with a probability measure $\hnu$
defined by
\begin{equation}
\label{RecMes}
    d\hnu(x,t)=d\nu(x) \, dt = \rho(x)\, dx\, dt
\end{equation}
where $\rho(x)$ is the density of $\nu$ and $0\leq t\leq 1$. We
call $\hW$ a {\it rectangle} with {\it base} $W$. The map $\cF^n$
can be naturally defined on $\hW$ by $\cF^n(x,t)=(\cF^n x,t)$ and
any function $f$ initially defined on $W$ can be also extended to
$\hW$ by $f(x,t)=f(x)$.

Next, recall that $\cF^n (W_1)$ and $\cF^n (W_2)$ are, generally,
countable standard families. Thus, in the above construction we
may start with two standard families, rather than two standard
pairs, and couple the images of measures initially defined on the
two families. In that case we need to split \emph{every} unstable
curve $W$ in \emph{each} original family into uncountable many
`fibers' by constructing a rectangle over $W$.

Given a standard family $\cG = (W_\alpha, \nu_\alpha)$, $\alpha
\in \fA$, with a factor measure $\lambda_\cG$, we denote by $\hcG
= (\hW_\alpha, \hnu_\alpha)$ the family of the corresponding
rectangles, with $\hW_\alpha$ being the rectangle with base
$W_\alpha$, equip $\hcG$ with the same factor measure
$\lambda_\cG$, and denote by $\hmu_\cG$ the induced measure on the
union $\cup_\alpha \hW_\alpha$.

The following lemma is the key instrument of the coupling method:

\begin{lemma}[Coupling Lemma] \label{LmCoupl}
Let $\cG = (W_\alpha, \nu_\alpha)$, $\alpha \in \fA$, and $\cE =
(W_\beta, \nu_\beta)$, $\beta \in \fB$, be two proper standard
families. Then there exist a bijection (a coupling map) $\Theta
\colon \cup_\alpha \hW_\alpha \to \cup_\beta \hW_\beta$ that
preserves measure, i.e.\ $\Theta (\hmu_{\cG}) = \hmu_{\cE}$, and a
(coupling time) function $\Upsilon: \cup_\alpha \hW_\alpha\to \IN$
such that

{\rm A.} Let $(x,t) \in \hW_\alpha$, $\alpha \in \fA$, and $\Theta
(x,t) = (y,s) \in \hW_\beta$, $\beta \in \fB$. Denote $m= \Upsilon
(x,t) \in \IN$. Then the points $\cF^m (x)$ and $\cF^m (y)$ lie on
the same stable H-manifold $W^s \subset \cM$.

{\rm B.} There is a uniform exponential tail bound on the function
$\Upsilon$: we have $\hmu_{\cG_1}\bigl( \Upsilon>n\bigr) \leq
C_\Upsilon \vartheta_\Upsilon^n$ for some constants $C_\Upsilon =
C_\Upsilon (\cD)>0$ and $\vartheta_\Upsilon = \vartheta_\Upsilon
(\cD) <1$.
\end{lemma}

We postpone the proof of Coupling Lemma and turn to statistical
properties for dispersing billiards. First we introduce a class of
appropriate functions on $\cM$ (observables). They will be
characterized by dynamically defined H\"older continuity.

Similar to the (future) separation time $\bs_+(x,y)$ defined by
(\ref{s+}), we introduce the \emph{past separation time}:
\beq \label{s-}
   \bs_-(x,y) = \min\{n\geq 0\colon y\notin \xi^u_{n} (x)\}
\eeq
(the first time in the past when the preimages $\cF^{-n} (x)$ and
$\cF^{-n} (y)$ lie in different connected components of the `new'
collision space $\MH$). If $x$ and $y$ lie on one stable curve $W
\subset \cM$, then
\beq \label{distbs-}
   \dist(x,y) \leq C \Lambda^{-\bs_-(x,y)}
\eeq
for some constant $C = C(\cD) >0$. We say that a function $f
\colon \cM \to \IR$ is \emph{dynamically H\"older continuous} if
there are $\vartheta_f \in (0,1)$ and $K_f >0$ such that for any
$x$ and $y$ lying on one unstable curve
\beq \label{DHC+}
    |f(x) - f(y)| \leq K_f \vartheta_f^{\bs_+ (x,y)}
\eeq
and for any $x$ and $y$ lying on one stable curve
\beq \label{DHC-}
    |f(x) - f(y)| \leq K_f \vartheta_f^{\bs_- (x,y)}
\eeq
We denote the space of such functions by $\cH$. The class of
dynamically H\"older continuous functions contains the class of
(ordinary) H\"older continuous functions; the latter are
characterized by
\beq \label{HolderCont}
   |f(x)-f(y)| \leq
   C_f[\dist(x,y)]^{\alpha_f}\qquad \forall x,y \in \cM,
\eeq
here $\alpha_f \in (0,1]$ is the H\"older exponent and the minimal
$C_f >0$ satisfying (\ref{HolderCont}) is the H\"older norm of
$f$. Indeed, any H\"older continuous function $f \colon \cM \to
\IR$ is dynamically H\"older continuous with $\vartheta_f =
\Lambda^{-\alpha_f}$.

Furthermore, suppose that $f$ is piecewise H\"older continuous,
i.e.\ there are $n_1,n_2 \geq 0$ such that $f$ is H\"older
continuous on every connected component of the set $\cM \setminus
(\SH_{n_1} \cup \SH_{-n_2})$, i.e.\ (\ref{HolderCont}) holds, with
the same $C_f$ and $\alpha_f$, whenever $x$ and $y$ belong in the
same component. Then again $f$ is dynamically H\"older continuous.

It is easy to verify by direct inspection that in billiards with
finite horizon the return time function $\tau(x)$ is H\"older
continuous on the connected components of the set $\cM \setminus
\cS_{1}$, hence it is dynamically H\"older continuous.

\medskip\noindent{\sc Remark}. Occasionally we will deal
with functions satisfying only one of the conditions (\ref{DHC+})
and (\ref{DHC-}). We denote the space of functions satisfying
(\ref{DHC+}) by $\cH^+$, and those satisfying (\ref{DHC-}) by
$\cH^-$.


\section{Equidistribution and decay of correlations} \label{SecEDC}

First we use Coupling Lemma to show that, given a proper standard
family $\cG$, the images $\cF^n (\mu_\cG)$ of its measure
$\mu_\cG$ weakly converge, as $n \to \infty$, to the
$\cF$-invariant measure $\mu$; furthermore, in a certain sense the
speed of convergence is exponential. We call this property
\emph{equidistribution}.

\begin{theorem}[Equidistribution] \label{TmEqDist}
Let $\cG$ be a proper standard family. For any dynamically
H\"older continuous function $f \in \cH$ and $n \geq 0$
\begin{equation}
\label{BEqEq}
         \biggl| \int_\cM f\circ \cF^n\, d\mu_\cG
         - \int_\cM f\, d\mu\,\biggr|
         \leq B_f \theta_f^n
\end{equation}
where $B_f = 2C_\Upsilon \bigl(K_f + \|f\|_\infty\bigr)$ and
\beq \label{thetaf}
   \theta_f = \bigl[ \max\{\vartheta_\Upsilon,\vartheta_f\}\bigr]^{1/2}<1.
\eeq
\end{theorem}

\proof Recall that there is a proper $\cF$-invariant family $\cE$
such that $\mu = \mu_\cE$. The coupling lemma~\ref{LmCoupl} gives
us a coupling map $\Theta$ between the families $\cG$ and $\cE$
and the corresponding coupling time function $\Upsilon$. Then
\begin{align} \label{threeDelta}
         \Delta\colon &=
         \int_\cM f\circ \cF^n\, d\mu_\cG
         - \int_\cM f\circ \cF^n\, d\mu_\cE \nonumber\\
         &= \int_{\hcG} f(\cF^n(x,t))\, d\hmu_\cG
         - \int_{\hcE} f(\cF^n(y,s))\, d\hmu_\cE \nonumber\\
         &= \int_{\hcG} \bigl[f(\cF^n(x,t))-
         f(\cF^n(\Theta(x,t)))\bigr]\, d\hmu_\cG.
\end{align}
Note that if $\Theta(x,t) = (y,s)$ and $m \colon = \Upsilon(x,t)
\leq n$, then $\bs_-(\cF^nx, \cF^ny) > n-m$, hence
\beq \label{ffdiff}
     \bigl|f(\cF^n(x,t)-
         f(\cF^n(\Theta(x,t)))\bigr| \leq K_f \vartheta_f^{n-m}.
\eeq
Now the last integral in (\ref{threeDelta}) can be decomposed as
\beq \label{int...decomp}
   \int_{\hcG} [\dots]=\int_{\Upsilon \leq n/2} [\dots]
   +\int_{\Upsilon>n/2} [\dots]=I+\RmII.
\eeq
The clause A of Lemma~\ref{LmCoupl} and (\ref{ffdiff}) imply that
$|I|\leq K_f \vartheta_f^{n/2}$ and the clause B implies that
$|\RmII|\leq 2\|f\|_\infty C_\Upsilon \vartheta_\Upsilon^{n/2}$.
\qed \medskip

Theorem~\ref{TmEqDist} can be extended to `multiple' observables,
i.e.\ observations made at multiple moments of time. Let $f_0,f_1,
\ldots, f_k \in \cH$ be dynamically H\"older continuous functions
with the same $\vartheta_f = \vartheta_{f_i}$, the same $K_f =
K_{f_i}$, and the same $\|f\|_\infty = \|f_i\|_\infty$, $0\leq
i\leq k$. (For example, we can take $f_0 = f_1 = \cdots = f_k =
f$.) Consider the product $\tilde{f} = f_0\cdot (f_1\circ
\cF)\cdot (f_2\circ\cF^2) \cdots (f_k \circ \cF^k)$.

\begin{theorem} \label{TmEqDist1}
Let $\cG$ be a proper standard family. Then for $n\geq 0$
\begin{equation}
\label{BEqEq1}
         \biggl| \int_\cM \tilde{f}\circ \cF^n\, d\mu_\cG
         - \int_\cM \tilde{f}\, d\mu\,\biggr|
         \leq B_{\tilde{f}}\, \theta_f^n
\end{equation}
where $\theta_f <1$ is as in (\ref{thetaf}) and $B_{\tilde{f}} =
2C_\Upsilon \|f\|_{\infty}^k(1-\vartheta_f)^{-1} \bigl(K_f +
\|f\|_\infty\bigr)$.
\end{theorem}

\proof The argument is almost identical to the proof of
Theorem~\ref{TmEqDist}, except (\ref{ffdiff}) must be replaced by
\begin{align*}
     \bigl|\tilde{f}(\cF^n(x,t))-
     \tilde{f}(\cF^n(\Theta(x,t)))\bigr|
     &\leq K_f \|f\|_\infty^k
     \bigl(\vartheta_f^{n-m}
     +\cdots+\vartheta_f^{n-m+k}\bigr)\\
     &\leq K_f \|f\|_\infty^k (1-\vartheta_f)^{-1}
     \vartheta_f^{n-m}.\qquad\qed
\end{align*}

\noindent{\sc Remark}. We used consecutive time moments
$0,1,\ldots,k$ for simplicity. The statement (and the proof) will
not change if we consider any increasing sequence of time moments
$0<t_1<t_2<\cdots <t_k$.

\medskip\noindent{\sc Remark}. In Theorems~\ref{TmEqDist} and
\ref{TmEqDist1} we assumed that the initial standard family $\cG$
was proper. If it is not, then we will have to wait until its
image $\cF^m \cG$ becomes proper for some $m \geq 1$ and then
apply these theorems with $n$ replaced by $n-m$. If $\cZ_\cG <
\infty$, then the waiting time is $m= \varkappa \ln \cZ_\cG$
iterations of $\cF$, according to Corollary~\ref{CrGgrow2}. If
$\cG$ consists of a single standard pair $(W,\nu)$, the waiting
time is $m= \varkappa \bigl| \ln |W|\bigr|$.

\medskip\noindent{\sc Remark}.
In Theorems~\ref{TmEqDist} and \ref{TmEqDist1} it is enough to
assume that $f \in \cH^-$ (respectively, $f_0, \ldots, f_k \in
\cH^-$), cf.\ Remark in the end of the previous section.

Next we derive an exponential bound on correlations for
dynamically H\"older continuous observables. Similar bounds were
obtained earlier \cite{Y98,Ch99}, but ours is sharper; its
advantage will be apparent in the next section. For brevity, we
use notation $\la f \ra = \int_{\cM} f\, d\mu$.

\begin{theorem}[Exponential decay of correlations]\label{TmEDC}
For every pair of dynamically H\"older continuous functions $f,g
\in \cH$ and $n \geq 0$
\beq  \label{expbound}
        \bigl| \la f\cdot (g\circ\cF^n)\ra
        -\la f\ra \la g\ra \bigr|
        \leq B_{f,g}\, \theta_{f,g}^{n}
\eeq
where
\beq \label{thetafg}
       \theta_{f,g} =
       \bigl[ \max\bigl\{\vartheta_\Upsilon,\vartheta_f,
       \vartheta_g,e^{-1/\varkappa} \bigr\}\bigr]^{1/4}<1,
\eeq
where $\varkappa>0$ is the constant of Theorem~\ref{TmGrow2},
\beq  \label{Bfg}
    B_{f,g} = C_0 \bigl(K_f\|g\|_\infty+
    K_g\|f\|_\infty+\|f\|_{\infty}\|g\|_\infty\bigr),
\eeq
and $C_0 = C_0(\cD) >0$ is a constant.
\end{theorem}

\proof We again use the proper standard family $\cE = \{ W_\alpha,
\nu_\alpha \}$, $\alpha \in \fA$, such that that $\mu_\cE = \mu$.
We can write
$$
   \la f\cdot (g\circ\cF^n)\ra =
   \int \bigl(f\circ\cF^{-n/4}\bigr)\,
   \bigl(g\circ\cF^{3n/4}\bigr)\,d\mu_\cE.
$$
Let $\bar{f}$ denote the conditional expectation of
$f\circ\cF^{-n/4}$ on the unstable H-manifolds $W_\alpha$ with
respect to the u-SRB measure $\nu_\alpha$, i.e.
$$
   \bar{f} (x) = \int_{W_\alpha}
   f\circ\cF^{-n/4} \, d\nu_\alpha
   \qquad \forall x \in W_\alpha
   \quad \forall \alpha \in \fA.
$$
Clearly the function $f\circ\cF^{-n/4}$ is almost constant on each
H-manifold $W_\alpha$. Precisely, due to the dynamical H\"older
continuity of $f$
\beq \label{barf-f}
   \sup_{W_\alpha} f\circ\cF^{-n/4} -
   \inf_{W_\alpha} f\circ\cF^{-n/4}
   \leq K_f \vartheta_f^{n/4},
\eeq
hence $\sup_{x \in \cM} |\bar{f}(x) - f\circ\cF^{-n/4}(x) | \leq
K_f \vartheta_f^{n/4}$. Note also that $\la f \ra = \la \bar{f}
\ra$, thus
\begin{align*}
        \Delta \colon &= \la f\cdot (g\circ\cF^n)\ra
        -\la f\ra \la g\ra \\
        &= \la \bar{f} \cdot (g\circ\cF^{3n/4})\ra
        -\la \bar{f} \ra \la g\ra + \delta_1,
\end{align*}
where $|\delta_1| \leq K_f \|g\|_\infty \vartheta_f^{n/4}$. Since
the function $\bar{f}$ is constant on every H-manifold $W_\alpha$,
we denote its value by $\bar{f}_\alpha$ and use
Theorem~\ref{TmEqDist} and the second remark after
Theorem~\ref{TmEqDist1} to obtain
\beq \label{intWbeta}
   \int_{W_\alpha} \bar{f} \cdot (g\circ\cF^{3n/4})\, d\nu_\alpha =
   \bar{f}_\alpha \la g \ra + \delta_\alpha
\eeq
where
\begin{align*}
  |\delta_\alpha| &\leq 2 \min \Bigl\{
  \|f\|_\infty \|g\|_\infty ,\\
  &\qquad \|f\|_\infty C_\Upsilon \bigl(K_g+\|g\|_\infty\bigr)
  \theta_g^{3n/4 - \varkappa| \ln|W_\alpha||}\Bigr\}
\end{align*}
and $\theta_g = \bigl[ \max\{ \vartheta_\Upsilon, \vartheta_g\}
\bigr]^{1/2}$. Integrating (\ref{intWbeta}) with respect to the
factor measure $\lambda_\cE$ of the family $\cE$ gives
$$
   \la \bar{f} \cdot (g\circ\cF^{3n/4})\ra
   =\la \bar{f} \ra \la g\ra + \int_\fA \delta_\alpha \,
   d\lambda_\cE(\alpha).
$$
It remains to estimate the last term here. Since $\cE$ is a proper
family, we have $\mu\bigl(\cup W_\alpha \colon |W_\alpha| <
e^{-\frac{n}{4\varkappa}}\bigr) \leq C_{\rm
p}e^{-\frac{n}{4\varkappa}}$. For H-manifolds $W_\alpha$
satisfying $|W_\alpha| \geq e^{-\frac{n}{4\varkappa}}$ we have
$3n/4 - \varkappa \bigl| \ln|W_\alpha| \bigr| \geq n/2$. Therefore
\begin{align*}
  \biggl| \int \delta_\alpha \,
   d\lambda_\cE(\alpha)\biggr|
   &\leq 2C_{\rm p} \|f\|_\infty \|g\|_\infty e^{-\frac{n}{4\varkappa}}\\
   &\quad +2\|f\|_\infty C_\Upsilon \bigl(K_g+\|g\|_\infty\bigr)
   \theta_g^{n/2}.
\end{align*}
Theorem~\ref{TmEDC} is now proved. \qed \medskip

In this theorem, it is enough to assume that $f \in \cH^+$ and $g
\in \cH^-$, cf.\ our earlier remarks. This observation leads to an
important corollary. Suppose $f$ is constant on every unstable
H-manifold $W^u$ and $g$ is constant on every stable H-manifold
$W^s$. In that case we may assume that $K_f=K_g=0$ and
$\theta_f=\theta_g=0$, and therefore
\beq \label{corrWWconst}
        \bigl| \la f\cdot (g\circ\cF^n)\ra
        -\la f\ra \la g\ra \bigr|
        \leq C_0 \|f\|_{\infty} \|g\|_{\infty}
        \theta_0^{n}
\eeq
for all $n \geq 0$, where $\theta_0 =\bigl[
\max\bigl\{\vartheta_\Upsilon, e^{-1/\varkappa} \bigr\}
\bigr]^{1/4}<1$. This follows from (\ref{expbound})--(\ref{Bfg}).

\begin{corollary} \label{CrmuAB}
Let $A,B \subset \cM$ be two measurable sets such that $A = \cup
W^u$ is the union of \emph{some} unstable H-manifolds and $B =
\cup W^s$ be the union of \emph{some} stable H-manifolds (here we
mean \emph{maximal} H-manifolds, cf.\ Section~\ref{SecP}). Then
\beq \label{muAB}
    \bigl|\mu\bigl(A \cap \cF^{-n}(B)\bigr) - \mu(A)\mu(B)\bigr|
    \leq C_0 \theta_0^n.
\eeq
\end{corollary}

Theorem~\ref{TmEqDist} can be extended to `multiple' correlations,
i.e.\ correlations between observations made at multiple moments
of time. This sharply improve earlier results \cite{CDe,D}. Let
$f_0, f_1, \ldots, f_r \in \cH$ and $g_0, g_1, \ldots, g_k \in
\cH$ be two sets of dynamically H\"older continuous functions. We
suppose $f$'s have identical parameters $\vartheta_f =
\vartheta_{f_i}$, $K_f = K_{f_i}$, and $\|f\|_\infty =
\|f_i\|_\infty$ for all $0\leq i\leq r$. Similarly, let $g$'s have
identical parameters $\vartheta_g = \vartheta_{g_i}$, $K_g =
K_{g_i}$, and $\|g\|_\infty = \|g_i\|_\infty$ for all $0\leq i\leq
k$. Consider two products $\tilde{f} = f_0\cdot (f_1\circ
\cF^{-1})\cdot (f_2\circ\cF^{-2}) \cdots (f_r \circ \cF^{-r})$ and
$\tilde{g} = g_0\cdot (g_1\circ \cF)\cdot (g_2\circ\cF^2) \cdots
(g_k \circ \cF^k)$.

\begin{theorem}[Exponential decay of multiple correlations]\label{TmEDCM}
For all $n > 0$
\beq  \label{expbound2}
        \bigl| \la \tilde{f}\cdot
        (\tilde{g} \circ\cF^n)\ra
        -\la \tilde{f}\ra \la \tilde{g}\ra \bigr|
        \leq B_{\tilde{f},\tilde{g}}\, \theta_{f,g}^{|n|}
\eeq
where $\theta_{f,g}$ is as in (\ref{thetafg}) and
\beq  \label{Bfg2}
    B_{\tilde{f},\tilde{g}} = C_0
    \|f\|_\infty^r \|g\|_\infty^k \biggl[
    \frac{K_f\|g\|_\infty}{1-\vartheta_f}+
    \frac{K_g\|f\|_\infty}{1-\vartheta_g}+
    \|f\|_{\infty}\|g\|_\infty\biggr]
\eeq
and $C_0 = C_0(\cD) >0$ is a constant.
\end{theorem}

\proof The argument is almost identical to the proof of
Theorem~\ref{TmEDC}, with a few modifications. First, we note
obvious bounds: $\|\tilde{f}\|_\infty \leq \|f\|_\infty^{r+1}$ and
$\|\tilde{g}\|_\infty \leq \|g\|_\infty^{k+1}$. Second,
(\ref{barf-f}) is replaced with
\begin{align*}
   \sup_{W_\alpha} \tilde{f}\circ\cF^{-n/4} -
   \inf_{W_\alpha} \tilde{f}\circ\cF^{-n/4}
   &\leq K_f \|f\|_\infty^r \bigl(\vartheta_f^{n/4}
   +\cdots+ \vartheta_f^{n/4+r}\bigr)\\
   &\leq K_f \|f\|_\infty^r (1-\vartheta_f)^{-1}
   \vartheta_f^{n/4}.
\end{align*}
Lastly, Theorem~\ref{TmEqDist1} must be used instead of
\ref{TmEqDist}. \qed

\medskip \noindent{\sc Remark}. We used consecutive time moments
$-r,\ldots,-1,0$ and $n,n+1,\ldots,n+k$ for simplicity. The
statement (and the proof) will be the same for any two increasing
sequences of time moments $t_{-r}<\cdots<t_{-1}<0$ and
$n<t_1<\cdots<t_k$ separated by a `time gap' of length $n$.

\section{Limit theorems}

First we recall relevant definitions. Given a function $f \colon
\cM \to \IR$, we denote its partial sums by $S_n =
\sum_{i=0}^{n-1} f\circ F^i$. Assume that $\la f \ra = 0$. We say
that $f$ satisfies the Central Limit Theorem (CLT) if
\beq \label{CLT}
   \lim_{n\to\infty} \mu
   \left \{\frac{S_n}{\sqrt{n}}\leq
   z\right \}
   =\frac{1}{\sqrt{2\pi}\sigma}
   \int_{-\infty}^z e^{-\frac{s^2}{2\sigma^2}}\, ds
\eeq
for all $-\infty <z< \infty$. Here $\sigma=\sigma_f\geq 0$ is
related to correlations by
\beq \label{CLTsigma}
   \sigma_f^2= \sum_{n=-\infty}^{\infty}\la f \cdot
   (f \circ \cF^n) \ra
   =\la f^2 \ra+2\sum_{n=1}^{\infty}\la f \cdot
   (f \circ \cF^n) \ra.
\eeq
In addition, the degenerate case $\sigma_f^2=0$ occurs if and only
if the function $f$ is cohomologous to zero, i.e.\ $f=g-g\circ F$
for some $g\in L^2(M)$.

Next, for $N\geq 1$ and $x\in M$ consider continuous function $W_N
(s;x)$ of $s \in [0,1]$ defined by
$$
    W_N\Bigl(\frac nN ;x\Bigr)=
    \frac{S_n(x)}{\sigma_f\sqrt{N}}
$$
at rational points $s=n/N$ and by linear interpolation in between.
The invariant measure $\mu$ and the family $\{W_N(s;x), x \in M\}$
induce a probability measure on the space of continuous functions
on $[0,1]$.

We say that $f$ satisfies \emph{Weak Invariance Principle} (WIP)
if the above measure weakly converges, as $N\to\infty$, to the
Wiener measure. We say that $f$ satisfies \emph{Almost Sure
Invariance Principle} (ASIP) if there is a standard Wiener process
(a Brownian motion) $B(s;x)$ on $M$ with respect to the measure
$\mu$ so that for some $\lambda>0$
$$
       |W_N(s;x)-B(s;x)|= \cO (N^{-\lambda})
$$
for $\mu$-almost all $x\in M$. The invariance principle thus
asserts that $S_n$, after a proper rescaling of space and time,
converges to the Wiener process.

In this section we prove that every dynamically H\"older
continuous observable $f \in \cH$ satisfies the CLT, WIP, and
ASIP, along with a few other limit laws. We use a general theorem
established by Philipp and Stout \cite{PSbook}, which we present
here in a form adapted to our needs.

Let $\xi_0$ be a finite or countable partition of $\cM$. Denote
$\xi_n = \cF^n (\xi_0)$ and $\xi_m^n = \xi_m \vee \cdots \vee
\xi_n$ for all $m<n$ (we allow $m= -\infty$ and $n =\infty$). Each
partition $\xi_m^n$ has its $\sigma$-algebra $\fF_m^n$ consisting
of measurable $\xi_m^n$-sets. We assume that
$\xi_{-\infty}^{\infty}$ is the maximal partition, i.e.\
$\fF_{-\infty}^{\infty}$ is the full $\sigma$-algebra of all
measurable sets in $\cM$.

Let $f \colon \cM \to \IR$ be a measurable function. For $m \geq
1$, denote by $\bar{f}_m = E \bigl(f| \fF_{-m}^m\bigr)$ the
conditional expectation of $f$ on the $\sigma$-algebra
$\fF_{-m}^m$, i.e.\
$$
   \bar{f}_m(x) = \frac{1}{\mu(B)} \int_B f\, d\mu,
   \qquad \forall x \in B\quad \forall B \in \xi_{-m}^m.
$$
The following theorem is proved in \cite{PSbook}, see Theorem 7.1
and Remark on p.~81 in that book.

\begin{theorem}[\cite{PSbook}] \label{TmPS}
Suppose that there exist constants $0< \delta <2$ and $C>0$ such
that
\beq \label{PS1}
     \la |f(x)|^{2+\delta} \ra < \infty
\eeq
and for all $m \geq 1$
\beq \label{PS2}
     \la |f(x) - \bar{f}_m|^{2+\delta} \ra
     \leq C m^{-(2+7/\delta)}.
\eeq
Moreover, suppose that $\sigma_f^2 >0$ and
\beq \label{PS3}
    {\rm Var}\, S_n = n\sigma_f^2 +
    \cO\bigl(n^{1-\delta/30}\bigr)
\eeq
as $n \to \infty$. Finally, suppose that for all $n \geq 1$
\beq \label{PS4}
  |\mu(A \cap B) - \mu(A)\mu(B)| \leq C n^{-168(1+2/\delta)}
\eeq
for all $A \in \fF_{-\infty}^0$ and $B \in \fF_n^{\infty}$. Then
the sequence $\cX_n = f \circ \cF^n$ satisfies the ASIP and the
WIP.
\end{theorem}

The ASIP actually implies the WIP, as well as many other limit
theorems. We only mention some and refer the reader to
\cite[Section 1]{PSbook} for more.

\begin{corollary}[Integral Tests]
Let $\phi(t)$ be a positive nondecreasing real-valued function.
Then
$$
  \mu\Bigl(\frac{S_n-n\la f \ra}{\sqrt{n}}
  > \phi(n)\ \ {\rm infinitely}\ {\rm often}\Bigr) =
  0\ {\rm or}\ 1
$$
according as the integral $\int_1^{\infty} \frac{\phi(t)}{t}\,
e^{-\phi^2(t)/2}\,dt$ converges or diverges. Furthermore, put $M_n
= \max_{1 \leq i \leq n} |S_i - i\la f \ra|$. Then
$$
  \mu\bigl(M_n/\sqrt{n} < \phi^{-1}(n)\ \
  {\rm infinitely}\ {\rm often}\bigr) =
  0\ {\rm or}\ 1
$$
according as the integral $\int_1^{\infty} \frac{\phi^2(t)}{t}\,
e^{-8\pi^{-2}\phi^2(t)}\,dt$ converges or diverges.
\end{corollary}

\begin{corollary}[Law of Iterated Logarithm]
For $\mu$-almost every point $x \in \cM$
$$
   \limsup_{n\to\infty}
   \frac{S_n-n\la f \ra}{\sqrt{2n \sigma_f^2
   \log \log n}} = 1.
$$
\end{corollary}

It remains to prove the following:

\begin{lemma}
The collision map $\cF \colon \cM \to \cM$ for dispersing
billiards and dynamically H\"older continuous observables $f \in
\cH$ such that $\sigma_f^2 >0$ satisfy all the condition of
Theorem~\ref{TmPS}.
\end{lemma}

\proof Observe that (\ref{PS1}) holds for any $\delta \in (0,2)$
because $\|f\|_\infty < \infty$. Next, the exponential decay of
correlations easily implies an even stronger property than
(\ref{PS3}): ${\rm Var}\, S_n = n\sigma_f^2 +\cO(1)$, see
\cite[Section 3]{Ch95}. To prove (\ref{PS2}) and (\ref{PS4}) we
need to construct the partition $\xi_0$.

Let $\xi_0$ be the partition of $\cM$ into the connected
components of the set $\cM \setminus \SH$. That is, the elements
of $\xi_0$ are domains on which the map $\cF \colon \MH \to \MH$
is smooth. Recall that every element of $\xi_0$ is a curvilinear
polygon, bounded by stable curves, and the interior angles of
these polygons do not exceed $\pi$.

Then $\xi_1 = \cF (\xi_0)$ is the partition of $\cM$ into the
connected components of the set $\cM \setminus \SH_{-1}$, so its
elements are similar curvilinear polygons bounded by unstable
curves. It is easy to see that for any $A \in \xi_0$ and $B \in
\xi_1$ the intersection $A \cap B$ is a curvilinear polygon, i.e.\
the elements of $\xi_0^1 = \xi_0 \vee \xi_1$ are the connected
components of the set $\cM \setminus (\SH \cup \SH_{-1})$.

By induction, one can verify directly that for any $p,q>0$ the
elements of the partition $\xi_{-p}^q$ are the connected
components of the set $\cM \setminus (\SH_{p+1} \cup \SH_{-q})$.
Therefore, the elements of the partition $\xi_{-m}^m$ are the
connected components of the set $\cM \setminus (\SH_{m+1} \cup
\SH_{-m})$. Also, they are curvilinear polygons bounded by stable
and unstable curves, and their interior angles do not exceed
$\pi$.

Let $A \in \xi_{-m}^m$. Obviously, for any two points $x,y \in A$
there is a point $z\in A$ such that $x$ and $z$ belong to one
unstable curve and $y$ and $z$ belong to one stable curve.
Therefore, $\bs_+ (x,z) > m$ and $\bs_-(y,z) > m$, thus $|f(x) -
f(y)| \leq 2K_f \vartheta_f^m$. It follows that $\la |f(x) -
\bar{f}_m|^{2+\delta} \ra \leq \, {\rm const}\,
\vartheta_f^{(2+\delta)m}$. This implies (\ref{PS2}).

Lastly we prove the main hypothesis (\ref{PS4}). Observe that
$\xi_1^{\infty}$ coincides with the partition $\xi^u$ of $\cM$
into (maximal) unstable H-manifolds introduced in
Section~\ref{SecP}. Similarly, $\xi_{-\infty}^0$ coincides with
the partition $\xi^s$ of $\cM$ into (maximal) stable H-manifolds.
Hence any set $A \in \fF_{-\infty}^0$ is a union of some stable
H-manifolds. Similarly, for any set $B \in \fF_n^{\infty}$ its
preimage $\cF^{-n+1} (B) \in \fF_1^{\infty}$ will be a union of
some unstable H-manifolds. Now Corollary~\ref{CrmuAB} easily
implies that $|\mu(A \cap B) - \mu(A)\mu(B)| \leq  C_0
\theta_0^{n-1}$. This proves (\ref{PS4}). \qed

%\appendix
\section*{Appendix}
\renewcommand{\theequation}{A.\arabic{equation}}
\renewcommand{\thetheorem}{A.\arabic{theorem}}
\setcounter{equation}{0} \setcounter{theorem}{0}


Here we prove the coupling lemma~\ref{LmCoupl}. Our argument is
quite lengthy but fairly transparent and completely dynamical.

Any closed region $\fQ \subset \cM$ bounded by two unstable
H-manifolds and two stable H-manifolds will be called a
\emph{solid rectangle}. Its boundary consists of four smooth
curves, which we naturally call the \emph{u-sides} and
\emph{s-sides}. If an unstable H-manifold $W^u \subset \cM$
crosses both s-sides of $\fQ$, we say that it \emph{fully crosses}
the solid rectangle $\fQ$. Similar notions apply to stable
H-manifolds.

Given a solid rectangle $\fQ \subset \cM$, denote by $\fR = \fR
(\fQ)$ the set of points $x \in \fQ$ such that \emph{both}
H-manifolds $W^u (x)$ and $W^s (x)$ fully cross $\fQ$. The set
$\fR$ is a closed nowhere dense Cantor-like subset of $\fQ$ that
has a natural direct-product structure. More generally, a closed
subset $\fR \subset \cM$ is called a \emph{Cantor rectangle} (or
just \emph{rectangle}) if for any $x,y\in \fR$ the intersection
$W^u (x) \cap W^s (y)$ consists of one point that belongs to
$\fR$.

Let $\fR$ be a rectangle and $z \in \fR$. The set $W^u (z) \cap
\fR$ is closed and lies on an increasing curve, thus it has two
extreme points, call them $x_1$ and $x_2$. Similarly, let $y_1$
and $y_2$ denote the two extreme points of the set $W^s (z) \cap
\fR$. Then the two stable H-manifolds $W^s (x_i)$, $i=1,2$, and
two unstable H-manifolds $W^u (y_i)$, $i=1,2$ enclose a solid
rectangle $\fQ$ containing $\fR$. We denote it by $\fQ (\fR)$ and
call it the \emph{hull} of the rectangle $\fR$.

Let $\fR$ be a rectangle. A subset $\fR_1 \subset \fR$ is called a
\emph{u-subrectangle} if $W^u (x) \cap \fR = W^u (x) \cap \fR_1$
for any point $x \in \fR_1$. Similarly, $\fR_2 \subset \fR$ is
called an \emph{s-subrectangle} if $W^s (x) \cap \fR = W^s (x)
\cap \fR_1$ for any point $x \in \fR_2$.

The image $\cF^n (\fR)$ of a rectangle $\fR$ for any $n \in \IZ$
is a finite or countable union of rectangles $\{\fR_i\}$. For $n
>0$, their preimages $\cF^{-n} (\fR_i)$ are s-subrectangles in
$\fR$. For $n <0$, they are u-subrectangles in $\fR$.

Due to absolute continuity, $\mu(\fR) >0$ if and only if for any
(and hence, for every) point $z \in \fR$ we have\footnote{We
denote by $|W|$ the length of a curve $W$, thus for any subset $B
\subset W$ the expression $|B|$ means the one-dimensional Lebesgue
measure of $B$.} $|W^u (z) \cap \fR| >0$ and $|W^s (z) \cap \fR|
>0$. We will only deal with rectangles of positive measure. For
any $x \in \fR$ we call
$$
   \rho^u(\fR) = \inf_{x\in\fR}
   \frac{|W^u(x) \cap \fR|}{|W^u(x) \cap \fQ(\fR)|}
$$
the (minimal) \emph{u-density} of $\fR$. Similarly the (minimal)
\emph{s-density} $\rho^s (\fR)$ is defined and we call $\rho (\fR)
= \min \{ \rho^u(\fR), \rho^s (\fR) \}$ the (minimal)
\emph{density} of the rectangle $\fR$. Due to the compactness of
$\fR$, $\mu(\fR)>0$ if and only if $\rho(\fR)>0$.

\begin{proposition} \label{PrRects}
For any point $x \in \cM$ that has non-vanishing H-manifolds $W^u
(x)$ and $W^s (x)$, there is a closed rectangle $\fR \ni x$ of
positive measure. Moreover, for any $\delta>0$, we can find a
rectangle $\fR \ni x$ with density $\rho (\fR) > 1-\delta$ and
such that the point $x$ divides the curves $W^u(x) \cap \fQ(\fR)$
and $W^s(x) \cap \fQ(\fR)$ in the ratio between $0.5-\delta$ and
$0.5+\delta$, i.e.\ $x$ is almost a geometric center of $\fQ
(\fR)$.
\end{proposition}

\proof Since $W^u (x)$ and $W^s (x)$ exist, we have $x \in \cM
\setminus \bigl( \cup_{n=-\infty}^\infty \SH_n \bigr)$. Now the
claim easily follows from Sinai's Fundamental
Theorem~\ref{TmFTSinai}. \qed \medskip

Next we construct a special rectangle $\fR_\ast$ whose stable
manifolds will be used to `connect' (or `couple') points of $\cF^n
(\cup_\alpha W_\alpha)$ with those of $\cF^n (\cup_\beta
W_\beta)$. The rectangle $\fR_\ast$, like a magnet, will `attract'
the H-components of the images of our proper standard families.

Let $W^u \subset \cM$ be an unstable H-manifold. Recall that $r_n
(x)$ denotes the distance from the point $\cF^n (x)$ to the
nearest endpoint of the H-component of $\cF^n (W^u)$ that contains
the point $\cF^n (x)$. Also, recall that $r^s (x)$ denotes the
distance, measured along the (maximal) stable H-manifold $W^s
(x)$, from $x$ to the nearest endpoint of $W^s (x)$. It is a
rather standard fact of hyperbolic dynamics that
\beq \label{rHslower}
   r^s (x) \geq \min_{n\geq 0} \tC^{-1}
   \Lambda^{n}\, r_n (x),
\eeq
where $\Lambda >1$ is the minimal expansion factor and $\tC =
\tC(\cD) >0$ is a constant.

Now let $\tW \subset W^u$ be subcurve (to be chosen later). Given
$\kappa>0$, we now put
\beq \label{Wkappa}
    \tW_{\kappa} \colon = \tW \setminus
    \cup_{n\geq 0} \{x\in \tW\colon
    r_n(x) < \tC\Lambda^{-n}\kappa\}.
\eeq
Note that the subset $\tW_\kappa \subset \tW$ is closed. Due to
(\ref{rHslower}), we have $r^s (x) \geq \kappa$ for every $x\in
\tW_{\kappa}$. We denote by $\fS_\kappa(\tW) = \{ W^s(x)\colon x
\in \tW_\kappa \}$ the set of the corresponding stable H-manifolds
(all of them extend the distance $\geq \kappa$ from $\tW$ on both
sides).

Since $\tW_\kappa$ is a closed set, it has two extreme points,
$x_1$ and $x_2$, on the curve $\tW$, which correspond to two
extreme H-manifolds $W^s (x_1)$ and $W^s (x_2)$ in the family
$\fS_\kappa (\tW)$. We say that an unstable curve $W$ \emph{fully
crosses} our family $\fS_\kappa (\tW)$, if it crosses \emph{all}
the H-manifolds $W^s \in \fS_\kappa (\tW)$.

Denote by $\cG^u_\kappa(\tW)$ the family of all unstable
H-manifolds $W^u \subset \cM$ that fully cross the family
$\fS_\kappa (\tW)$. Lastly let
$$
   \fR_\kappa(\tW) = \cup_{W^s \in \fS_\kappa(\tW)}
   \cup_{W^u \in \cG^u_\kappa(\tW)}
   W^s \cap W^u
$$
denote the rectangle made by our two families of stable and
unstable H-manifolds.

\begin{proposition} \label{PrRectsB}
For any $\delta>0$ there are a subcurve $\tW \subset W^u$ and a
$\kappa>0$ such that the rectangle $\fR_\ast = \fR_\kappa(\tW)$
has density $\rho (\fR_\ast) > 1-\delta$.
\end{proposition}

\proof First let $\tW = W^s$. It follows from (\ref{Deps1}) that
$\bigl|\tW \setminus \cup_{\kappa>0} \tW_\kappa \bigr|=0$. Choose
$\kappa>0$ such that $|\tW_\kappa|>0$ and pick a Lebesgue density
point $z\in \tW_\kappa$ on the curve $\tW$. Now reduce $\tW$ so
that it becomes a small neighborhood of $z$. The rest of the proof
is the same as in Proposition~\ref{PrRects}. \qed
\medskip

From now on we choose a small $\delta>0$ and fix the corresponding
$\tW \subset W^u$ and $\kappa >0$ and the special rectangle
$\fR_\ast = \fR_\kappa(\tW)$ constructed in the above proposition.
We denote $\fS = \fS_\kappa (\tW)$, for brevity, and slightly
abusing notation we also denote by $\fS$ the union of all the
H-manifolds $W^s \in \fS$. Note that $\tW_\kappa = \tW \cap \fS$.
For any unstable curve $W$ that fully crosses $\fS$ we set
$W_\kappa \colon = W \cap \fS$.

Next, for any standard pair $(W, \nu)$ and any $n \geq 0$ denote
by $W_{n,i}$ all the H-components of $\cF^n (W)$ that fully cross
$\fS$ and put
\beq \label{Wstar}
    W_{n,\ast} = \cup_i \cF^{-n} (W_{n,i} \cap \fS).
\eeq
The following is proved in \cite[Theorem 3.13]{BSC91} :

\begin{proposition} \label{PrWstar}
There are constants $n_1 \geq 1$ and $d_1>0$ such that for any
proper standard pair and any $n\geq n_1$ we have $\nu (W_{n,\ast})
\geq d_1$.
\end{proposition}

This easily extends to proper standard families, with obvious
notation:

\begin{corollary} \label{CrMagnet}
There are constants $n_0 \geq 1$ and $d_0>0$ such that for any
proper standard family $\cG = \{ (W_\alpha ,\nu_\alpha) \}$ and
any $n\geq n_0$ we have $\mu_\cG \bigl(\cup_\alpha
W_{\alpha,n,\ast}\bigr) \geq d_0$.
\end{corollary}

\noindent{\sc Remark}. In Proposition~\ref{PrWstar} we assumed
that the standard pair $(W, \nu)$ was proper. If it is not, then
we have to wait until its image $\cF^m (W)$ becomes a proper
standard family for some $m \geq 1$ and then apply
Corollary~\ref{CrMagnet}, with $n_0$ replaced by $n_0 +m$. Recall
that the waiting time here is $m= \varkappa \bigl| \ln |W|\bigr|$.

Let $\cP = (W, \nu)$ be a standard pair such that $W$ fully
crosses the family $\fS$ (the magnet) constructed in the previous
section. Then $W_\kappa = W \cap \fS$ is a closed nowhere dense
set on the curve $W$, and its complement $W \setminus W_{\kappa}$
consists of infinitely many intervals; we call them \emph{gaps in}
$W_{\kappa}$. These gaps naturally correspond to the connected
components of the set $\tW \setminus \tW_{\kappa}$ (gaps in
$\tW_{\kappa}$) which are created by the points $x \in \tW$
satisfying $r_n (x) < \tC\Lambda^{-n}\kappa$, in accordance with
(\ref{Wkappa}).

Let $\tV \subset \tW \setminus \tW_{\kappa}$ be an interval. We
call
$$
    n = \min \{ i \geq 1\colon r_i (x) <
      \tC\Lambda^{-i}\kappa\ \ \text{for some}\ x \in \tV \}
$$
the \emph{rank} of $\tV$. Clearly every gap in $\tW_k$ has a rank.
If rank$\, \tV =n$, then $\cF^{n-1} (\tV)$ is a curve of length
$\geq C\Lambda^{-n}$ for some constant $C = C(\cD) >0$. Indeed,
consider the H-component of $\cF^{n-1} (\tW)$ containing
$\cF^{n-1} (\tV)$; it intersects the $(C \Lambda^{-n}
)$-neighborhood of the singularity set $\SH$ for some constant
$C>0$; then it has to cross this neighborhood completely (for if
it terminates somewhere inside that neighborhood, then it must
have been torn by the singularities some time earlier).

Now every gap $V \subset W \setminus W_{\kappa}$ corresponds to a
gap $\tV \subset \tW \setminus \tW_{\kappa}$ that has some rank $n
\geq 1$; in this case we also say that $V$ itself has rank $n$.
The endpoints of $V$ are linked to those of $\tV$ by stable
H-manifolds $\WHs \in \fS$, hence their images $\cF^{n-1} (V)$ and
$\cF^{n-1} (\tV)$ must be exponentially close to each other (the
distance between them is $\ll \Lambda^{-n}$). Therefore $\bigl|
\cF^{n-1} (V) \bigr| \geq \frac 12 C\Lambda^{-n}$. It follows that
the set $\cF^{n(2+ \varkappa \ln \Lambda)} (V)$ will be a proper
standard family, in accordance with Remark after
Corollary~\ref{CrMagnet}.

Accordingly, we define \emph{recovery time} function $\br_\cP (x)$
on $W \setminus W_\kappa$ by setting $\br_\cP (x) = [n(2+
\varkappa \ln \Lambda)]$, where $n$ is the rank of the gap $V
\subset W \setminus W_{\kappa}$ containing the point $x$ (note
that the function $\br_\cP (x)$ is now constant on every gap).

Corollary~\ref{TmGrow2} implies that for all $n \geq 1$
\beq \label{lWtheta}
   \nu \bigl( x\in W \setminus W_\kappa\colon
   \br_\cP (x) > n\bigr) \leq \,{\rm const}\,
   \Lambda^{-n} \, \nu (W \setminus W_\kappa).
\eeq
Next, let $\bs_\cP (x)$ be another function on $W \setminus
W_\kappa$ (with values in $\IN$) such that
\beq \label{bs1}
      \bs_\cP (x) \equiv \,{\rm const}
      \qquad \text{on every gap}\ V \subset W
          \setminus W_{\kappa}
\eeq
and
\beq \label{bs2}
    \bs_\cP (x) \geq \br_\cP (x) + n_0.
\eeq
Since both functions $\br_\cP$ and $\bs_\cP$ are constant on each
gap $V$, we will occasionally denote their values by $\br_\cP (V)$
and $\bs_\cP (V)$, respectively.

Now applying Corollary~\ref{CrMagnet} to every gap $V \subset W
\setminus W_\kappa$ (more precisely, to its image $\cF^{\br_\cP
(V)} (V)$, cf.\ Remark after Corollary~\ref{CrMagnet}.) gives
\beq \label{Vmagnet}
    \nu \bigl(V_{\bs_\cP (V),\ast}\bigr) \geq d_0\, \nu (V),
\eeq
in the notation of (\ref{Wstar}). In other words at time $n =
\bs_\cP (V)$ the $d_0$-fraction of the image $\cF^n (V)$ will be
`on the magnet'. At this time the image $\cF^n (V)$ may be
`stopped' and part of it (which intersects the magnet $\fS$) may
be coupled with the corresponding image of another proper standard
family (this will be done in the next section). We call $\bs_\cP$
the \emph{stopping time} function.

Observe that the stopping time $\bs_\cP$ is not yet completely
specified by (\ref{bs1}) and (\ref{bs2}); that is the purpose of
the next proposition.

\begin{proposition} \label{PrStop}
We can define the stopping time function $\bs_\cP (x)$ on $W
\setminus W_\kappa$ so that for all $n \in \IN$
\beq \label{stop1}
  \frac{\nu \bigl(x\in W \setminus W_\kappa\colon
   \bs_\cP (x)=n\bigr)}{\nu (W \setminus W_\kappa)} =q_n ,
\eeq
where $\{ q_n \}$ is a sequence satisfying
\beq \label{stop2}
   \sum q_n =1\quad{\rm and}\quad
   q_n < \,{\rm const}\, \theta^{n}
\eeq
for some $\theta \in (\Lambda^{-1},1)$. Furthermore, the sequence
$\{q_n\}$ is independent of $\cP$, i.e.\ it is the same for all
standard pairs $\cP = (W, \nu)$ such that $W$ fully crosses the
family $\fS$.
\end{proposition}

\proof Due to (\ref{lWtheta}), it is easy to define $\bs_\cP$ so
that for all $n>0$
$$
  \frac{\nu \bigl(x\in W \setminus W_\kappa\colon
   \bs_\cP (x)=n\bigr)}{\nu (W \setminus W_\kappa)}
   \leq \, {\rm const}\, \theta^{n}.
$$
We still have a considerable flexibility in defining the function
$\bs_\cP$, and we want to adjust it so that it will satisfy
(\ref{stop1}) with a sequence $\{q_n\}$ being independent of
$\cP$. To this end we split every gap $V$ into an uncountable
family of `thinner' curves with the help of rectangles described
in Section~\ref{SecCL}. Precisely, we replace each gap $V$ with a
rectangle $V \times [0,1]$. Then we can divide the latter into
subrectangles $V \times I_j$, where $I_j \subset [0,1]$ are some
subintervals, and define $\bs_\cP$ so that it takes a different
value on each subrectangle $I_j$. The sizes of the subintervals
$I_j \subset [0,1]$ must be selected to ensure (\ref{stop1}), as
well as (\ref{stop2}).

Since our $\bs_\cP$ is now constant on every subrectangle $V
\times I_j$, the latter can be then collapsed to a curve $V_j$,
which geometrically coincides with the gap $V$, but carries a
measure different from $\nu$; precisely, it carries the measure
$\nu$ multiplied by $|I_j|$. In the end we will have a countable
family of curves $\{V_j\}$ and the function $\bs_\cP$ will be
constant on each of them, as desired. \qed \medskip

We now turn to the construction of the coupling map $\Theta \colon
\cup_\alpha \hW_\alpha \to \cup_\beta \hW_\beta$ for
Lemma~\ref{LmCoupl}. This will be done recursively, in an
algorithm-like manner. The first two steps of our construction
will be described in detail, and then it will be clear how it
proceeds.

Recall that we are given two proper standard families $\cG =
(W_\alpha, \nu_\alpha)$, $\alpha \in \fA$, and $\cE = (W_\beta,
\nu_\beta)$, $\beta \in \fB$, with the corresponding measures
$\mu_\cG$ and $\mu_\cE$. We denote by $\hcG = (\hW_\alpha,
\hnu_\alpha)$ and $\hcE = (\hW_\beta, \hnu_\beta)$ the respective
families of rectangles, and we have two probability measures
$\hmu_\cG$ and $\hmu_\cE$ on the unions $\cup_\alpha \hW_\alpha$
and $\cup_\beta \hW_\beta$, respectively.

We define the first stopping time function $\bs_0$ on the unions
$\cup_\alpha \hW_\alpha$ and $\cup_\beta \hW_\beta$ to be constant
$\bs_0 (x,t) \equiv n_0$. At time $\bs_0=n_0$ some of the
H-components of the images $\cF^{\bs_0} \bigl( \cup_\alpha
W_\alpha\ \bigr)$ and $\cF^{\bs_0} \bigl( \cup_\beta W_\beta\
\bigr)$ will fully cross the magnet $\fS$; and the total measure
of the respective intersections with $\fS$ will be $\geq d_0$ due
to Corollary~\ref{CrMagnet}.

For every H-component $W_{\alpha,\bs_0,i}$ of $\cF^{\bs_0}
(W_\alpha)$ that fully crosses $\fS$ we consider the corresponding
rectangle $\hW_{\alpha,\bs_0,i} = W_{\alpha,\bs_0,i} \times
[0,1]$. We now split off a subrectangle $W_{\alpha,\bs_0,i} \times
[0,\btau_{\alpha,i}]$ with some $0< \btau_{\alpha,i} \leq 0.5$ so
that
\beq \label{d0/2}
       \hmu_\cG \bigl( \cup_\alpha \tW_{\alpha,1}\bigr)
       = d \colon = d_0/2,
\eeq
where
\begin{align*}
   \tW_{\alpha,1} = \bigl\{(x,t) \in \hW_\alpha
   &\colon \cF^{\bs_0} (x) \in W_{\alpha,\bs_0,i} \cap \fS\\
   & \&\ t\in [0,\btau_{\alpha,i}]\ \ \text{for some}\ i\bigr\}
\end{align*}
This can be done easily due to Corollary~\ref{CrMagnet}: if
$\mu_\cG \bigl(\cup_\alpha W_{\alpha,\bs_0,\ast}\bigr) = d_0$ in
Corollary~\ref{CrMagnet}, we simply set $\btau_{\alpha,i} = 0.5$
for all $\alpha$ and $i$; if the inequality in
Corollary~\ref{CrMagnet} is strict, we have ``too much of a good
thing'', then we lower some of $\btau_{\alpha,i}$'s to make
(\ref{d0/2}) exact.

We will put tildas over $W$'s that denote subsets of $\cup_\alpha
\hW_\alpha$ on which we are currently defining the coupling map
$\Theta$. At the first step of our construction, $\Theta$ must
take points $(x,t) \in \cup_\alpha \tW_{\alpha,1}$ to points
$(y,s) \in \cup_\beta \hW_\beta$ such that $\cF^{\bs_0} (x)$ and
$\cF^{\bs_0} (y)$ lie on the same stable H-manifold $W^s \in \fS$.
It also must preserve measure (i.e., take $\hmu_\cG$ to
$\hmu_\cE$). To correctly define $\Theta$ on the set $\cup_\alpha
\tW_{\alpha,1}$ we will first describe its image, which we will
denote by $\cup_\beta \tW_{\beta,1}$ (here the index $1$ refers,
of course, to the \emph{first} step of our construction).

One may be tempted to define $\tW_{\beta,1}$'s in the same way as
we defined $\tW_{\alpha,1}$'s above. In that case the sets
$\cup_\beta \tW_{\beta,1}$ and $\cup_\alpha \tW_{\alpha,1}$ would
have the same overall measure ($=d$), and their
$\cF^{\bs_0}$-images would lie on the same stable H-manifolds $W^s
\in \fS$, but this may not suffice, as such a map may not preserve
measure. Indeed, for some $W^s \in \fS$ the intersections $W^s
\cap \cF^{\bs_0} (\cup_\alpha \tW_{\alpha,1})$ and $W^s \cap
\cF^{\bs_0} (\cup_\beta \tW_{\beta,1})$ may carry different
`amount' of measures $\hmu_\cG$ and $\hmu_\cE$, respectively.
There are two possible reasons for this `mismatch': (i) the
densities of our measures may vary along our H-components and (ii)
the Jacobian of the holonomy map may also vary and differ from one

To deal with the possible 'mismatch', we first assume, without
loss of generality, that the diameter of the `special rectangle'
$\fR_\ast$ is very small; so that the corresponding oscillations
of the densities are at least very small (say, the ratio of the
densities at different points on the same H-component is between
0.99 and 1.01), and the Jacobian of the holonomy map takes values
in a narrow interval around one, say, in $[0.99,1.01]$.

We now define the set $\cup_\beta \tW_{\beta,1}$ as follows. For
every H-component $W_{\beta,\bs_0,j} \subset \cF^{\bs_0}
(W_\beta)$ that fully crosses the magnet $\fS$ we will construct a
function $\btau_{\beta,j} (y)$ on $W_{\beta,\bs_0,j} \cap \fS$,
with values in the interval $[0,0.6]$, and then put
\begin{align*}
   \tW_{\beta,1} = \bigl\{(y,t) \in \hW_\beta
   &\colon \cF^{\bs_0} (y) \in W_{\beta,\bs_0,j} \cap \fS\\
   & \&\ t\in [0,\btau_{\beta,j} (\cF^{\bs_0}y) ]\ \
   \text{for some}\ j\bigr\}.
\end{align*}
The functions $\btau_{\beta,j}$ can be constructed so that for
every stable H-manifold $W^s \in \fS$ the intersections  $W^s \cap
\cF^{\bs_0} (\cup_\alpha \tW_{\alpha,1})$ and $W^s \cap
\cF^{\bs_0} (\cup_\beta \tW_{\beta,1})$ carry the same `amount' of
measures $\hmu_\cG$ and $\hmu_\cE$, respectively. This is the
reason why we need to give some room to the functions
$\btau_{\beta,j}$ so that their values can be adjusted
accordingly, this is why we allow them to go up to $0.6$ (as
compared to $\btau_{\alpha,i}$ that took values $\leq 0.5$).

Also, since the densities of measures for standard pairs are
dynamically H\"older continuous (\ref{DHC}), and so is the
Jacobian of the holonomy map, it follows that the functions
$\btau_{\beta,j} (y)$ will be dynamically H\"older continuous as
well, i.e.\ they will satisfy
\beq \label{tauHc}
   |\ln \btau_{\beta,j}(y)-\ln \btau_{\beta,j}(z)| \leq
   C_0 \btheta^{s_+(y,z)}
\eeq
for some constant $C_0 >0$.

We now naturally define the coupling map $\Theta \colon
\cup_\alpha \tW_{\alpha,1} \to \cup_\beta \tW_{\beta,1}$ that
preserves measure and couples points whose $\cF^{\bs_0}$-images
lie on the same stable manifold of the $\fS$ family. We note that
\beq \label{d0over2}
   \hmu_{\cG} \bigl(
   \cup_\alpha \tW_{\alpha,1} \bigr)
   =\hmu_{\cE} \bigl(
   \cup_\beta \tW_{\beta,1} \bigr) = d,
\eeq
where the constant $d = d_0 /2$ was introduced in (\ref{d0/2}).
Lastly we set the coupling time function $\Upsilon (x,t)=\bs_0$ on
$\cup_\alpha \tW_{\alpha,1}$. This concludes the first step of our
construction of the coupling map $\Theta$.

Before we move on to the second step, we need to `inventory' the
remaining (uncoupled) parts of the families $\cG$ and $\cE$ and
represent their images at time $\bs_0$ by unions of some
rectangles. To this end we first define a constant function
$\btau_{\alpha,i} (x)$ on every H-component $W_{\alpha,\bs_0,i}$
of $\cF^{\bs_0} (W_\alpha)$ that fully crosses $\fS$ so that
$\btau_{\alpha,i} (x) \equiv \btau_{\alpha,i}$, where
$\btau_{\alpha,i}$ is the constant chosen earlier (before equation
(\ref{d0/2})).

On the contrary, the function $\btau_{\beta,j} (y)$ defined
earlier on the Cantor-like subset $W_{\beta,\bs_0,j} \cap \fS
\subset W_{\beta,\bs_0,j}$ is \emph{not} constant. We now extend
it to the entire curve $W_{\beta,\bs_0,j}$ by linear interpolation
(we make it linear on every gap $V \subset W_{\beta,\bs_0,j}
\setminus \fS$ and overall continuous). The graph of
$\btau_{\beta,j}$ divides the rectangle $W_{\beta,\bs_0,j} \times
[0,1]$ into two parts (`subrectangles', each has one irregular
side, see Fig.~\ref{FigHolder}). It is easy to verify that the
function $\btau_{\beta,j} (y)$, after its extension to
$W_{\beta,\bs_0,j}$, is still dynamically H\"older continuous in
the sense of (\ref{tauHc}).

\begin{figure}[htb]
    \centering
    \psfrag{t}{$t$}
    \psfrag{j}{$\btau_{\beta,j}(y)$}
    \psfrag{0}{$0$}
    \psfrag{1}{$1$}
    \includegraphics{statall-3.eps}
    \caption{The partition of a rectangle over an H-component
    $W_{\beta,\bs_0,j}$: the irregular line in the middle is
    the graph of the function  $\btau_{\beta,j}(y)$; it separates
    the `upper subrectangle' (of the second type) from
    lower trapezoids (of the third type).}
    \label{FigHolder}
\end{figure}

Now the `uncoupled' sets $\cup_\alpha \cF^{\bs_0} \bigl(
\hW_\alpha \setminus \tW_{\alpha,1} \bigr)$ and $\cup_\beta
\cF^{\bs_0} \bigl( \hW_\beta \setminus \tW_{\beta,1} \bigr)$
consist of connected components of three types.

\emph{First}, there are rectangles corresponding to the
H-components of $\cF^{\bs_0} (W_\alpha)$ and $\cF^{\bs_0}
(W_\beta)$ that do not fully cross the magnet $\fS$, we did not
modify them in any way.

\emph{Second}, the `upper subrectangles'
\beq \label{upperG}
 \{(x,t)\colon x\in W_{\alpha,\bs_0,i}\ \&\
 t\in [\btau_{\alpha,i}(x),1]\}
\eeq
and similar regions
\beq \label{upperE}
 \{(y,t)\colon y\in W_{\beta,\bs_0,j}\ \&\
 t\in [\btau_{\beta,j}(x),1]\}
\eeq
(the latter are not genuine subrectangles, they have one `jagged'
side as shown on Fig.~\ref{FigHolder}). All of the regions
(\ref{upperG})--(\ref{upperE}) have sufficiently long bases
(longer than the size of the special rectangle $\fR_\ast$ in its
unstable direction).

\emph{Third}, the `lower subrectangles'
$$
 \{(x,t)\colon x\in V\ \&\
 t\in [0,\btau_{\alpha,i}(x)]\}
$$
constructed over gaps $V \subset W_{\alpha,\bs_0,i} \setminus \fS$
and similar regions
$$
 \{(y,t)\colon y\in V'\ \&\
 t\in [0,\btau_{\beta,j}(y)]\}
$$
constructed over gaps $V' \subset W_{\beta,\bs_0,j} \setminus \fS$
(the latter are trapezoids as shown on Fig.~\ref{FigHolder}).

Next we `rectify' the irregular sides of the rectangles of the
second and third type by a simple algorithm: it consists of
stretching of each vertical (i.e.\ parallel to the $t$ axis) fiber
inside every rectangle accordingly. More precisely, given a
`rectangle' $\hW_1 = \{(x,t) \colon x\in W\ \&\ t\in [0,\btau
(x)]\}$, where $W$ is an unstable curve and $\btau(x) \colon W \to
[0,1]$ is a continuous function, equipped with a probability
measure $d\hnu (x,t) = \rho (x)\, dx\, dt$, we transform the
interval $[0,\btau(x)]$ onto the unit interval $[0,1]$ linearly at
every point $x\in W$, and thus obtain a full-height rectangle $\hW
= W \times [0,1]$ with measure
\beq \label{rho2}
   d\hnu_1 (x,t)= \rho_1(x)\, dx\, dt, \qquad
   \rho_1 (x) = \btau(x)\, \rho(x).
\eeq
Recall that the `ceiling function' $\btau(x)$  is dynamically
H\"older continuous for the rectangles of the second and third
type, and so is every regular density $\rho (x)$ according to
(\ref{DHC}). Hence the new density $\rho_1 (x)$ defined by
(\ref{rho2}) will be also dynamically H\"older continuous,
precisely
\beq \label{rhoHcnew}
       |\ln \rho_1 (x) - \ln \rho_1 (y)|
       \leq (C_0 + C_{\rm r})\, \btheta^{\bs_+ (x,y)}.
\eeq
Of course the constant $C = C_0 + C_{\rm r}$ is larger than
$C_{\rm r}$ in (\ref{DHC}), so the density $\rho_1 (x)$ is not
necessarily regular (yet). But its images will smooth out and
become regular in $m_0 \geq 1$ iterations of $\cF$, where $m_0
\geq 1$ is a constant (this follows from distortion bounds).
Moreover, by making $C_{\rm r}$ larger, if necessary, we can even
ensure that $m_0 =1$, i.e.\ the densities regularize right away.

Thus the remaining (uncoupled) sets
\beq \label{remain1}
  \cup_\alpha \cF^{\bs_0} \bigl( \hW_\alpha \setminus
  \tW_{\alpha,1} \bigr)
  \quad \text{and} \quad
  \cup_\beta \cF^{\bs_0} \bigl(
  \hW_\beta \setminus \tW_{\beta,1} \bigr)
\eeq
are unions of rectangles of the full (unit) height. We denote the
families of those rectangles by $\hcG_1 = \{ \hW_{\alpha,1} \}$
and $\hcE_1 = \{ \hW_{\beta,1} \}$, respectively. (Note that the
$\cF^{\bs_0}$-image of an `old rectangle' $\hW_\alpha$ may contain
countably many `new' rectangles $\hW_{\alpha,1}$, thus we have to
reindex our families, so the new indices $\alpha$ and $\beta$ may
not correspond to the old ones used for the original families
$\cG$ and $\cE$, but this will cause no harm.)

So we get two new families $\hcG_1$ and $\hcE_1$, each carries a
measure induced by the $\cF^{\bs_0}$-image of the original measure
($\hmu_\cG$ or $\hmu_\cE$). Conditioning the induced measures on
the new families $\hcG_1$ and $\hcE_1$ gives two probability
measures on them, we call them $\hmu_{\hcG_1}$ and
$\hmu_{\hcE_1}$, respectively. The densities of the new measures
$\hmu_{\hcG_1}$ and $\hmu_{\hcE_1}$ may not be regular, but their
images become regular in just $\leq m_0$ iterations on $\cF$ (or
even in just one step, see above), so we disregard this slight
inconvenience.

The main complication is that the new families $\hcG_1$ and
$\hcE_1$ may not be proper, because they contain myriad of
arbitrarily small rectangles of the third type created in the gaps
of the magnet $\fS$. Of course, if we condition the measures
$\hmu_{\hcG_1}$ and $\hmu_{\hcE_1}$ onto the union of rectangles
of the first and second type, then the so reduced families (albeit
not necessarily proper either) will obviously recover and become
proper standard families in just a few iterations of $\cF$, we
leave the verification of this simple fact to the reader; so we
may assume that the rectangles of the first and second type make a
proper standard family already.

However, on the rectangles of the third type, the recovery time
may vary greatly. We define the stopping time function $\bs_1
(x,t)$ on the rectangles of the third type as described in
Proposition~\ref{PrStop}. In particular, the function $\bs_1$
takes values in $\IN$, is constant on every rectangle, and it
corresponds to the time when the image of the rectangle become a
proper standard family plus an extra $n_0$ iterations of $\cF$.

We need also to define the stopping time function $\bs_1$ on the
rectangles of the first and the second types, so that it takes
values in $\IN$, is constant on every rectangle, and its overall
distribution on \emph{all} rectangles matches the one described in
Proposition~\ref{PrStop}, i.e.\
\beq \label{ddqn}
   \hmu_{\hcG_1} \bigl( \cup_\alpha \hW_{\alpha,1} \colon
   \bs_1 = n \bigr) = q_n \qquad \forall n\in \IN
\eeq
and
\beq \label{ddqnE}
   \hmu_{\hcE_1} \bigl( \cup_\beta \hW_{\beta,1} \colon
   \bs_1 = n \bigr) = q_n \qquad \forall n\in \IN.
\eeq
with the same sequence $\{q_n\}$ as in
(\ref{stop1})--(\ref{stop2}).

In order to define such a function $\bs_1$ and ensure (\ref{ddqn})
and (\ref{ddqnE}) we may need to split some rectangles
$\hW_{\alpha,1}$ and $\hW_{\beta,1}$ of the first and second type
into `thinner' subrectangles, as we did in the proof of
Proposition~\ref{PrStop}, and define $\bs_1$ separately on every
subrectangle. Since the family of rectangles of the first and
second type is proper already, this task is much simpler than the
proof of Proposition~\ref{PrStop}, so we leave details to the
reader.

Now we are in a position very similar to the one we were in at
beginning of this section. For every rectangle $\hW_{\alpha,1}$,
the set $\cF^{\bs_1} (\hW_{\alpha,1})$ with the induced measure
will be a proper family, it will contain H-components fully
crossing the magnet $\fS$, and their intersections with $\fS$ will
have a relative measure $\geq d_0$ due to
Corollary~\ref{CrMagnet}:
\beq \label{d0d0}
   \frac{\hmu_{\hcG_1} (\hW_{\alpha,1,\bs_1,\ast})}
   {\hmu_{\hcG_1} (\hW_{\alpha,1})}
    \geq d_0
    \quad \text{and} \quad
   \frac{\hmu_{\hcE_1} (\hW_{\beta,1,\bs_1,\ast})}
   {\hmu_{\hcE_1} (\hW_{\beta,1})}
    \geq d_0
\eeq
for every $\alpha$ and $\beta$, in the notation of (\ref{Wstar})
and Proposition~\ref{PrWstar}. We note that $\bs_1$ is constant on
every rectangle $\hW_{\alpha,1}$ and $\hW_{\beta,1}$, so these
notation make sense.

Of course, the value of $\bs_1$ in (\ref{d0d0}) depends on
$\alpha$ (or $\beta$). In what follows, we will group rectangles
$\hW_{\alpha,1}$ and $\hW_{\beta,1}$ on which the function $\bs_1$
takes the same value. In particular, we have
\beq \label{mdYd0}
     \hmu_{\hcG_1} \bigl(\cup_\alpha
     \hW_{\alpha,1,\bs_1,\ast} \colon
     \bs_1 = n \bigr)
     \geq d_0\, \hmu_{\hcG_1} \bigl(
     \cup_\alpha \hW_{\alpha,1} \colon
      \bs_1 = n \bigr)
     = d_0q_n
\eeq
and, similarly,
\beq \label{mdYd0E}
     \hmu_{\hcE_1} \bigl(\cup_\beta
     \hW_{\beta,1,\bs_1,\ast} \colon
     \bs_1 = n \bigr)
     \geq d_0\, \hmu_{\hcE_1} \bigl(
     \cup_\beta \hW_{\beta,1} \colon
      \bs_1 = n \bigr)
     = d_0q_n
\eeq
due to (\ref{ddqn})--(\ref{d0d0}).

Next, for every $n \geq 1$ we again consider all the rectangles
$\{ \hW_{\alpha,1} \}$ and $\{ \hW_{\beta,1} \}$ on which the
function $\bs_1$ takes value $n$. Their images at time $n$ will
contain H-components that fully cross the magnet $\fS$, and the
relative measure of their intersections with $\fS$ will be $\geq
d_0$ due to (\ref{mdYd0})--(\ref{mdYd0E}). At that time we apply
our coupling procedure described in the first step and then link
(`couple') their subsets of relative measure $d= d_0/2$, according
to (\ref{d0over2}), so that
\beq \label{dqnGE}
     \hmu_{\hcG_1} \bigl( \hcG_2^{(n)} \bigr) =
     \hmu_{\hcE_1} \bigl( \hcE_2^{(n)} \bigr) = dq_n,
\eeq
where
$$
    \hcG_2^{(n)} \colon =
     \bigl\{ (x,t) \in \cup_\alpha
     \hW_{\alpha,1,\bs_1,\ast} \colon
     \bs_1 = n \ \&\  \cF^n (x,t)\
     \text{is coupled} \bigr\}
$$
and
$$
    \hcE_2^{(n)} \colon =
     \bigl\{ (y,t) \in \cup_\beta
     \hW_{\beta,1,\bs_1,\ast} \colon
     \bs_1 = n \ \&\  \cF^n (y,t)\
     \text{is coupled} \bigr\}.
$$
Doing this for all $n \geq 1$ constitutes the second step of our
construction. We denote by $\hcG_2 = \cup_n \cF^{-\bs_0} \bigl(
\hcG_2^{(n)} \bigr)$ and $\hcE_2 = \cup_n \cF^{-\bs_0} \bigl(
\hcE_2^{(n)} \bigr)$ the preimages of all the subsets `coupled'
during the second step of the construction. Note that $\hcG_2
\subset \hcG$ and $\hcE_2 \subset \hcE$. The coupling map $\Theta$
is thus extended to $\Theta \colon \hcG_2 \to \hcE_2$. We also
define the coupling time function $\Upsilon$ on the set $\hcG_2$
by
$$
   \Upsilon(x,t) = \bs_0(x,t) + \bs_1 \bigl(\cF^{\bs_0} (x,t)\bigr).
$$
It should be clear now how the construction of the coupling map
proceeds as the above steps are repeated recursively.

Finally, we prove the clause B of Coupling Lemma~\ref{LmCoupl}
(this will also imply that the coupling map $\Theta$ is defined
almost everywhere on the standard family $\hcG$). First we
summarize the results of the previous constructions. For each $k
\geq 1$, at the $k$th step we define the stopping time function
$\bs_{k-1}$ on the sets $\cup_\alpha \hW_{\alpha,k-1}$ and
$\cup_\beta \hW_{\beta,k-1}$ of yet uncoupled points. Then we
`couple' some points of their images $\cup_\alpha \cF^{\bs_{k-1}}
\bigl( \hW_{\alpha,k-1} \bigr)$ and $\cup_\beta \cF^{\bs_{k-1}}
\bigl( \hW_{\beta,k-1} \bigr)$. Then we denote by $\hcG_k$ and
$\hcE_k$ the preimages of just `coupled' subsets, on the original
families $\cG$ and $\cE$. Lastly we extend the coupling time
function $\Upsilon$ to the set $\hcE_k$ by
$$
   \Upsilon(z) = \bs_0(z) + \bs_1 (\cF^{\bs_0} z)
   +\cdots+\bs_{k-1}(\cF^{\bs_0+\cdots+\bs_{k-2}}z),
$$
where $z = (x,t) \in \hcG_k$. Observe that the point
$\cF^{\Upsilon (z)} (z)$ and its partner $\cF^{\Upsilon (z)}
\bigl(\Theta (z) \bigr)$ lie on the same stable H-manifold, which
proves the claim A of Lemma~\ref{LmCoupl} (assuming that $\Theta$
is indeed defined almost everywhere).

Next we rewrite the `measure' relations
(\ref{ddqn})--(\ref{ddqnE}) and (\ref{dqnGE}) for the $k$th step.
For brevity, we identify the set $\hcG_k$ and $\hcE_k$ with their
images, i.e.\ we consider all our stopping time functions as
defined on the original families $\hcG$ and $\hcE$. Then
(\ref{ddqn})--(\ref{ddqnE}) generalize to the following
`conditional probability' formula
\beq \label{msss1}
   \hmu_\cY (\bs_{k}=n/\bs_{k-1}=n_{k-1},
   \ldots,\bs_1=n_1,\bs_0=n_0)=q_n
\eeq
where $\cY = \cG$ or $\cE$; and (\ref{dqnGE}) generalize to
another `conditional probability' formula
\beq \label{msss2}
   \hmu_\cY \bigl(\hcY_k/\bs_{k-1}=n_{k-1},\ldots,\bs_1=n_1,
   \bs_0=n_0\bigr) = d,
\eeq
where again $\cY = \cG$ or $\cE$.

The following argument is standard in the studies of random walks
(but we do not assume here that the reader is familiar with it).
Let
\beq \label{barpn}
    \bar{p}_n \colon = \hmu_\cG \bigl( (x,t)\in
    \cup_\alpha \hW_\alpha \colon
    \Upsilon (x,t) = n \bigr)
\eeq
denote the fraction of points coupled exactly at time $n$ (i.e.,
at the $n$th iteration of $\cF$, rather than at the $n$th step of
our construction). For example, $\bar{p}_i=0$ for $i<n_0$ and
$\bar{p}_{n_0}=d$. Then, due to (\ref{msss2}), $p_n \colon =
\bar{p}_n/d$ will be the fraction of points \emph{stopped} at time
$n$, i.e. \
$$
   p_n = \hmu_\cG \bigl( (x,t) \in \cup_\alpha \hW_\alpha
   \colon \bs_0+\bs_1+\cdots+\bs_k=n\ \text{for some}\ k\bigr).
$$
Due to (\ref{msss1}) and (\ref{msss2}) we have the following
`convolution law':
\beq \label{convolution}
  p_{n+n_0} = (1-d)\Bigl( q_{n}+ (1-d)
  \sum_{i=1}^{n-1} q_{n-i} \, p_{n_0+i}\Bigr)
  \qquad \forall n\geq 1.
\eeq
Its verification is rather straightforward, and we leave it to the
reader as an enjoyable exercise.

Now consider two complex analytic functions
$$
  P(z)=\sum_{n=1}^\infty p_{n_0+n}z^n
  \quad\text{and}\quad
  Q(z)=\sum_{n=1}^\infty q_{n}z^n.
$$
Since $|p_n| \leq 1$ and $|q_n| \leq 1$, these functions are
defined at least on the open unit disk $\{ z \colon |z| <1\}$ in
the complex plane $\IC$. Moreover, (\ref{stop2}) easily implies
that $|Q(z)| \leq 1$ for all $|z|\leq 1$ and that $Q(z)$ is
analytic in a slightly larger complex disk $\{z\colon |z| <
\theta^{-1} \}$, here $\theta^{-1}>1$.

Equation (\ref{convolution}) implies $P(z)=(1-d)\,
Q(z)+(1-d)^2\,P(z)\,Q(z)$, hence
$$
   P(z)=\frac{(1-d)\,Q(z)}{1-(1-d)^2\,Q(z)}.
$$
We see that $P(z)$ is analytic in some complex disk of radius
greater than one, i.e.\ in $\{z\colon |z| < 1+\delta\}$ for some
$\delta>0$, hence $|p_n| \leq \,{\rm const} (1+\delta')^{-n}$ for
any $\delta' < \delta$. A similar exponential bound then follows
for $\bar{p}_n = d\, p_n$ introduced in (\ref{barpn}). Coupling
Lemma~\ref{LmCoupl} is proved. \qed


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\end{document}
\end
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