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\begin{document}
\title{Dynamical Borel-Cantelli lemmas for Gibbs measures}
\author{N. Chernov$^\ast$ and D. Kleinbock$^\dagger$
%\\ Department of Mathematics\\
%University of Alabama at Birmingham\\
%Birmingham, AL 35294\\
%E-mail: chernov@vorteb.math.uab.edu\\
%Fax: 1-(205)-934-9025
}
\date{\today}
\maketitle
\begin{abstract}
Let $T:\, X\mapsto X$ be a deterministic dynamical system preserving a
probability measure $\mu$. A dynamical Borel-Cantelli lemma
asserts that for certain sequences of subsets $A_n\subset X$ and
$\mu$-almost every point $x\in X$ the inclusion $T^nx\in A_n$
holds for infinitely many $n$. We discuss here systems which are
either symbolic (topological) Markov chain or Anosov
diffeomorphisms preserving Gibbs measures. We find sufficient
conditions on sequences of cylinders and rectangles, respectively,
that ensure the dynamical Borel-Cantelli lemma.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
%\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{sublemma}[theorem]{Sublemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\footnotetext{$^\ast$ Partially supported by NSF grant
DMS-9732728.}
\footnotetext{$^\dagger$ Partially supported by NSF grant
DMS-9800607.}
\section{Introduction}
\label{secI}
\setcounter{equation}{0}
Let $T: X\mapsto X$ be a transformation preserving a probability
measure $\mu$. We use notation $\mu(f):=\int f\, d\mu$ for
integrable functions $f$ on $X$.
Let $A_n\subset X$ be a sequence of measurable sets. Put
$B_n=T^{-n}A_n$ and consider the set $$ \limsup_n
B_n=\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}B_n$$ of points which belong
to infinitely many $B_n$. A classical
Borel-Cantelli lemma in probability theory states:
\begin{lemma}[Borel-Cantelli]
{\rm (i)} If $\sum\mu(B_n)<\infty$, then $\mu(\limsup_n
B_n)=0$, i.e.~almost
every point $x\in X$ belongs to finitely many $B_n$.\\ {\rm (ii)}
If $\sum\mu(B_n)=\infty$ and $B_n$ are independent, then
$\mu(\limsup_n
B_n)=1$, i.e.~almost every point $x\in X$ belongs to infinitely
many $B_n$.
\end{lemma}
In terms of the transformation $T$, the lemma can be restated as
follows.
\begin{lemma}
{\rm (i)} If $\sum\mu(A_n)<\infty$, then for almost every point
$x\in X$ there are only finitely many $n$ such that $T^nx\in
A_n$.\\ {\rm (ii)} If $\sum\mu(A_n)=\infty$ and $T^{-n}A_n$ are
independent, then for almost every point $x\in X$ there are
infinitely many $n$ such that $T^nx\in A_n$. \label{lmBCT}
\end{lemma}
The second part of the lemma has a limited value for deterministic
dynamical systems, since one rarely works with purely independent
sets.
%Instead, for chaotic transformations, one may only expect that the
%sets $T^{-n}A_n$ and $T^{-m}A_m$ are asymptotically independent
%as $|m-n|\to\infty$. Under such conditions, the second part of the
%lemma can be proved.
This paper is devoted to extensions of the second part of the
lemma to certain dynamical systems -- Anosov diffeomorphisms and
topological Markov chains.
Below we always assume that $\sum_n\mu(A_n)=\infty$.
\medskip
\noindent {\bf Definition}. A sequence of subsets $A_n\subset X$
is called a {\em Borel-Cantelli} (BC) sequence if for $\mu$-a.e.~$x\in
X$ there are infinitely many $n$ such that $T^nx\in A_n$.
\medskip
Let
$$
\chi_n(x)=\chi_{T^{-n}A_n}(x)
$$
be the indicator of the set $B_n=T^{-n}A_n $. We set
$$
S_N(x)=\sum_{n=1}^N\chi_n(x)
$$
and
$$
E_N=\mu(S_N)=\sum_{n=1}^N\mu(A_n)\,.
$$
\noindent {\bf Definition}. A sequence of subsets $A_n\subset X$
is said to be a {\em strongly Borel-Cantelli} (sBC) sequence if
for $\mu$-a.e.~$x\in X$ we have $S_N(x)/E_N\to 1$ as $N\to\infty$.
\medskip
A stronger version of the classical Borel-Cantelli lemma is known,
see Theorem~6.6 in \cite{Durr}:
\begin{lemma} If $\sum\mu(B_n)=\infty$ and the events $B_n$ are
independent, then $S_N(x)/E_N\to 1$ almost surely as
$N\to\infty$. Moreover, the independence requirement can be
relaxed to the pairwise independence, i.e.~it is enough to require
$\mu(B_m\cap B_n)=\mu(B_m)\mu(B_n)$ for $m\neq n$.
\end{lemma}
In particular, if $B_n=T^{-n}A_n$ are pairwise independent, then
the sequence $\{A_n\}$ is an sBC sequence.
Consider the quantity
$$
R_{mn}=\mu(B_m\cap B_n)-\mu(B_m)\mu(B_n)=\mu(T^{-m}A_m\cap
T^{-n}A_n)-\mu(A_m)\mu(A_n)
$$
which characterizes the dependence of $B_m$ and $B_n$.
A sufficient condition for $\{A_n\}$ to be an sBC sequence, in
terms of $R_{mn}$, was first found by W.~Schmidt, see a proof by
Sprind\v{z}uk \cite{Sp}, in the context of Diophantine
approximations. It was recently adapted to dynamical systems by
D.~Kleinbock and G.~Margulis \cite{KM}:
\noindent{\bf (SP)} Assume that
$$
\exists C>0:\ \ \ \ \sum_{m,n=M}^NR_{mn}\leq C\cdot \sum_{n=M}^N\mu(A_n)
$$
for all $N\geq M\geq 1$.
\begin{theorem}[\cite{Sp}{\rm , Chapter I, Lemma 10, or} \cite{KM}{\rm , Lemma 2.6}]
If the sequence $\{A_n\}$ satisfies {\rm (SP)}, then it is an sBC
sequence; moreover, for a.e.~$x\in X$ one has
\be
S_N=E_N+O\Big(E_N^{1/2}\log^{3/2+\varepsilon}E_N\Big )\,.
\label{log}
\ee
\end{theorem}
W.~Philipp was first to derive the asymptotics (\ref{log}) in the context
of dynamical system, and he called it a quantitative Borel-Cantelli lemma
\cite{Ph}.\medskip
Note that there exist remarkable characterizations of
some ergodic properties of dynamical systems in terms of BC and sBC sequences.
We summarize these in the following
\begin{proposition} Let $T$ be a measure preserving transformation
of a probability space $(X,\mu)$. Then:\\
{\rm (i)} $T$ is ergodic $\iff$
every constant sequence $A_n\equiv A$, $\mu(A)>0$, is BC $\iff$ every
such sequence is sBC, i.e.~$S_N/E_N\to 1$ $\mu$-almost everywhere;\\
{\rm (ii)}
$T$ is weakly mixing $\iff$ every sequence $\{ A_n\}$ that only contains finitely
many distinct sets, none of them of measure zero, is BC $\iff$ for
every such sequence one has
$S_N/E_N\to 1$ in the $L^2$ metric, i.e.~$\mu(S_N/E_N-1)^2\to 0$;\\
{\rm (iii)} $T$
is lightly mixing\footnote{ $T$ is said to be
lightly mixing (see \cite{FT}) if for every two sets $A,B$ of positive
measure one has \linebreak
$\mu(T^{-n}A \cap B) > 0$ for large enough $n$; this condition lies
strictly between mixing and weak mixing.} $\iff$ every sequence that
only contains finitely
many distinct sets, possibly of measure zero, is BC.
\label{lmWM}
\end{proposition}
See Section~\ref{secPTMC} for the proof. Note that in part (ii),
the first equivalence was proved by
Y.~Guivarc'h and A.~Raugi (private communication); our proof is slightly
different. Part (iii) was pointed out
to us by
A.~del Junco. \medskip
Note also that there exist no measure-preserving system such that
every sequence $\{ A_n\}$ that only contains two distinct sets,
one of positive measure and the other of measure zero, is sBC.
This follows from a result of U. Krengel \cite{Kr}. On the other
hand, if $\mu$ has $K$ property, then any sequence that only
contains finitely many sets, none of them of measure zero, is sBC
(J.-P.~Conze, private communication).
%If the measure $\mu$ has $K$ property, then any sequence $\{
%A_n\}$ that only contains finitely many sets, none of them of
%measure zero, is an sBC sequence.
\medskip
%The part (ii) was proved by J.~Conze \cite{JC}.
It is important to mention that for any
(nontrivial) measure-preserving system $(X,\mu,T)$ there are
sequences of subsets
of $X$ (with divergent sum of measures) which are not BC. More precisely,
the following is true:
\begin{proposition} Let $(X,\mu)$ be a probability space. If $\mu$ is nontrivial
(that is, there are sets with measure strictly between $0$ and $1$), then
for any $\mu$-preserving transformation $T$ of $X$
there exists a sequence
$\{A_n\}$ of
measurable subsets of $X$ with $\sum_{n = 1}^\infty
\mu({A}_n)= \infty$ which is not BC. Furthermore, if $\mu$ is non-atomic, then
for any $\mu$-preserving transformation $T$ of $X$
there exists a sequence
$\{A_n\}$ of
measurable subsets of $X$ with $\sum_{n = 1}^\infty
\mu({A}_n)= \infty$ such that for
a.e.~$x\in X$ there are
at most finitely many $n$ for which $T^n x\in A_n$.
\label{lmNBC}
\end{proposition}
See the end
of Section~\ref{secPTMC} for
the proof. Observe that a non-BC sequence can be easily constructed
when $T$ is
invertible: one can simply take $A_n=T^nA$,
where $0 < \mu(A) < 1$.
Therefore to prove the BC or sBC property for certain classes of
sequences it is necessary to impose certain restrictions on the
sets $A_n$, which, roughly speaking, guarantee that the sets $B_m$
and $B_n$ become nearly independent for large $|m-n|$.
\medskip
The first Borel-Cantelli lemma for deterministic
dynamical systems was proved in 1969 by W.~Philipp:
\begin{theorem}[\cite{Ph}]
Assume that $T(x)=\beta x$ (mod 1) with $\beta>1$, or
$T(x)=\{1/x\}$ (the Gauss transformation) and $\mu$ is the unique
$T$-invariant smooth measure on $[0,1]$. Then any sequence
$\{A_n\}$ of subintervals (with divergent sum of measures) is an sBC
sequence, and {\rm (\ref{log})} holds.
\label{tmP}
\end{theorem}
In particular, one can take any $x_0\in (0,1)$ and consider what
could be called
``a target shrinking
to $x_0$'' (terminology borrowed from \cite{HV}), i.e.~a sequence of
intervals $A_n = (x_0 - r_n, x_0 +
r_n)$ with $r_n\to 0$. Then almost all orbits $\{T^nx\}$ get into infinitely many such
intervals whenever $r_n$ decays slowly enough. This can be
thought of as a quantitative strengthening of density of almost all orbits
(cf.~the paper \cite{Bos} for a similar approach to the
rate of recurrence).
More generally, if $X$ is a metric space (e.g.~a Riemannian
manifold), one can try to prove that
any sequence
$\{A_n\}$ of balls in $X$ is BC or sBC; as in the example above, this
would imply that all points $x_0\in X$ can be ``well approximated'' by
orbit points $T^nx$ for almost all $x$.
D. Dolgopyat recently proved the following:
\begin{theorem}[\cite{Do}]
Let $T:X\mapsto X$ be an Anosov diffeomorphism with a smooth invariant
probability measure $\mu$. Then any sequence of round balls
(with divergent sum of measures) is sBC.
\label{tmD}
\end{theorem}
Another example of a dynamical Borel-Cantelli lemma is given in the
paper \cite{KM}, where the following theorem was essentially proved:
\begin{theorem}[\cite{KM}]
Let $G$ be a connected semisimple center-free Lie
group without compact factors, $\Gamma$ an irreducible
lattice in $G$, $\mu$ the normalized Haar measure on $\ggm$, $g$ a
partially hyperbolic element of $G$, and let $T$ be the left shift
$T(x) = gx$, $x\in \ggm$. Let $\{A_n\}$ be a sequence of subsets of
$\ggm$ with divergent sum of measures and ``uniformly regular
boundaries'', namely, such that for some $\delta>
0$ and $0 < c < 1$ one has
\be
\mu(\delta{\rm -neighborhood \ of\ }A_n) \le c\mu(A_n) {\rm \ \ \ for\
all\ }n\,.
\label{ur}
\ee
Then there exist positive $C_1, C_2$ such that for $\mu$-a.e.~$x\in
\ggm$ one has
$$
C_1 \le \liminf_{N\to\infty}S_N(x)/E_N \le
\limsup_{N\to\infty}S_N(x)/E_N \le C_2\,;
$$
in particular, $\{A_n\}$ is a BC sequence.
\label{tmKM}
\end{theorem}
It is shown in \cite{KM} that the above condition (\ref{ur}) is
satisfied if $\ggm$ is not compact and the sets $\{A_n\}$ are
{\em complements} of balls centered in a fixed point $x_0\in\ggm$.
This way one gets a description of {\em growth} of almost all orbits
$T^nx$ as follows: if a sequence $R_n$ increases slowly enough, then
for almost all $x$ one has dist$(x_0,T^nx) \ge R_n$ for infinitely
many $n$. This has important applications to geometry and
number theory.
When this paper was under preparation, we learned that J.-P.~Conze
and A.~Raugi \cite{CR} proved a dynamical Borel-Cantelli lemma for
certain Markov processes and one-sided topological Markov chains
with Gibbs measures.
\section{Statement of results}
\label{secSR}
\setcounter{equation}{0}
Our paper deals with Anosov diffeomorphisms and the corresponding
symbolic systems -- topological Markov chains.
Let $T:X\mapsto X$ be a transitive Anosov diffeomorphism. Let ${\cal
R}=\{R_1,\ldots,R_M\}$ be a finite Markov partition of $X$, and
$\bf A$ the corresponding transition matrix of zeroes and ones.
For definitions and basic facts on Markov partitions, see
\cite{Bo75,ChS}.
The matrix $\bf A$ is transitive, i.e.~${\bf A}^K$ is completely
positive for some $K\geq 1$. Let $\Sigma=\Sigma_{\bf A}$ be the
{\em topological Markov chain} for $\bf A$, i.e.~a set of doubly
infinite sequences $\uom=\{\omega_i\}_{i=-\infty}^{\infty}\in
\{1,\ldots,M\}^{\ZZ}$ defined by
$$
\Sigma=\{\uom\in\{1,\ldots,M\}^{\ZZ}:\
{\bf A}_{\omega_i\omega_{i+1}}=1\ \ \ \forall i\in\ZZ\}\,.
$$
The set $\Sigma$ equipped with the product topology is a
compact space, and there is a left shift homeomorphism
$\sigma:\Sigma\mapsto\Sigma$ defined by
$(\sigma\uom)_i=\omega_{i+1}$. Let $\pi:\Sigma\mapsto X$ be
the projection defined by
$$
\pi(\uom)=\cap_{i=-\infty}^{\infty}T^{-i}R_{\omega_i}\,.
$$
Then $\pi$ is a continuous surjection and
$\pi\circ\sigma=T\circ\pi$. Fix an $a\in (0,1)$ and let $d_a$ be a
metric on $\Sigma$ defined by $d_a(\uom,\uom')=a^n$ where
$n=\min\{n:\, \omega_i=\omega_i',\ \forall |i|0$.
There is a sequence of cylinders $\{C_n\}$ with $\sum \mu(C_n) =
\infty$ which is defined on $\{\varepsilon|\Lambda_n|\}$-centered (or,
alternatively, $\{\varepsilon|\Lambda_n|\}$-aligned) intervals
$\Lambda_n\subset\ZZ$
and is not a BC sequence. Moreover,
for a.e.~$\uom\in\Sigma$ there are only finitely many
$n$ such that $\sigma^n\uom\in C_n$.
\label{tm3}
\end{theorem}
%Recall that we always assume $\sum\mu(C_n)=\infty$.
Theorems~\ref{tm2} and \ref{tm3} show that it is not enough,
even for the BC property, that the cylinders are `relatively well'
centered or aligned.
Consider a one-sided topological Markov chain
$\sigma:\Sigma^+\mapsto\Sigma^+$ defined on the space of one-sided
sequences
$$
\Sigma^+=\{\uom\in\{1,\ldots,M\}^{\ZZ_+}:\
{\bf A}_{\omega_i\omega_{i+1}}=1\ \ \ \forall i\in\ZZ_+\}\,;
$$
here $\ZZ_+=\{0,1,2,\ldots\}$. Note that the shift $\sigma$
preserves $\Sigma^+$ but is not invertible, every sequence $\uom$
may have up to $M$ preimages. One-sided topological Markov chains
give symbolic representation for piecewise smooth expanding
interval maps satisfying the Markov condition.
Theorems~\ref{tm1}--\ref{tm3} apply to one-sided topologically
mixing Markov chains without change. Note, however, that all the
cylinders must be defined on intervals $\Lambda\subset\ZZ_+$. In
particular, our theorems hold for cylinders defined on intervals
that are $D$-aligned, $\{l_n\}$-aligned and
$\{\varepsilon|\Lambda_n|\}$-aligned, respectively\footnote{Note that in
this case the result of Theorem~\ref{tm1} can be derived from a
recent manuscript by Conze and Raugi \cite{CR}.}. Consider the metric
$d_a^+$ on $\Sigma^+$ given by $d_a^+(\uom,\uom')=a^n$ where
$n=\min\{n:\, \omega_i=\omega_i',\ \forall i0$.
%We consider proper rectangles,
%i.e.~such that $R=\overline{{\rm int}\, R}$, or equivalently,
%$W^u_z(R)=\overline{{\rm int}\, W^u_z(R)}$ and
%$W^s_z(R)=\overline{{\rm int}\, W^s_z(R)}$ for all $z\in R$ (here
%the closure and intersection are taken with respect to the internal
%metrics on those submanifolds).
Our rectangles are not necessarily connected.
Our main assumption must be some sort of `roundness' of
rectangles, the necessity of which we explained above.
For any $\varepsilon>0$ put
$$
W^u_z(R,\varepsilon)=\{x\in W^u_z(R):\,
{\rm dist}(x,\partial W^u_z(R))<\varepsilon\}
$$
and
\be
R^u_z(\varepsilon)=[W^u_z(R,\varepsilon),W^s_z]\,.
\label{Ru}
\ee
This is a sort of $\varepsilon$-neighborhood of the
stable boundary $\partial^sR$. Similarly, the $\varepsilon$-neighborhood
of the unstable boundary $\partial^uR$ is defined, call it
$R^s_z(\varepsilon)$.
Now fix another constant $\varepsilon_0 \in (0,\varepsilon_1)$
and some constants $C_0>0$, $\gamma>0$.
\medskip
\noindent{\bf Definition}. We say that a rectangle $R$ is
{\em u-quasiround} if for some $z\in R$ \\
(i) the set $W^u_z(R)$ has (external) diameter $\leq\varepsilon_1$
and internal diameter $\geq\varepsilon_0$ (note that this set
will be perfectly round if $\varepsilon_0=\varepsilon_1$);
\\
(ii) For all $\varepsilon>0$
\be
\mu( R^u_z(\varepsilon))\leq C_0|\ln\varepsilon|^{-1-\gamma}\mu(R)
\label{lne}
\ee
Similarly, {\em s-quasiround} rectangles are defined.
\medskip
Note that the definition of u- and s-quasiroundness
depends on the pre-fixed constants $\varepsilon_1,\varepsilon_0,
C_0,\gamma$.
The choice of $z$ in this definition is not important,
since the same properties will also holds for all $z\in R$,
with possibly slightly different values of
$\varepsilon_1,\varepsilon_0$ and $C_0$. The exact values of
$\varepsilon_1,\varepsilon_0,C_0,\gamma$ may affect some constants
in our estimates, but otherwise will be irrelevant.
Note that if the set $\partial W^u_z(R)$ is
smooth or piecewise smooth and the measure
on $W^u_z$ induced by $\mu$ is smooth, then
$\mu( R^u_z(\varepsilon))\leq {\rm const}\cdot\varepsilon\mu(R)$.
It is quite common in hyperbolic dynamics to assume that the
measure of $\varepsilon$-neighborhoods of boundaries
or singularities is bounded by const$\cdot\varepsilon^a$
for some $a>0$. Our bound (\ref{lne}) is milder than that.
Next, we need to consider arbitrary small rectangles that satisfy some
sort of roundness condition.
\medskip
\noindent{\bf Definition}. We call a rectangle $R$ {\em
eventually quasiround} (EQR) if there are two integers $k^-\leq
k^+$ such that $T^{k^+}(R)$ is u-quasiround and $T^{k^-}(R)$ is
s-quasiround. \medskip
The integers $k^{\pm}$ may not be uniquely defined
for a rectangle $R$, but each of them is defined by $R$
up to a small additive
depending on the ratio $\varepsilon_1/\varepsilon_0$,
so the choice of $k^{\pm}$ for a given $R$ will not be important.
EQR rectangles in the Anosov setting play a role similar to that
of cylinders for TMC's, and the numbers $k^-$, $k^+$ correspond to
the endpoints of cylinders. Note, however, that EQR rectangles are
not generated by any Markov partitions. On the other hand, we
impose the regularity condition (\ref{lne}) on the boundary of EQR
rectangles, while no such condition was assumed for cylinders.
Note that if dim$\, X=2$, then stable and unstable manifolds are
one-dimensional, and, with appropriate choice of $\varepsilon_0$,
$\varepsilon_1$, every connected rectangle is EQR. Indeed, the
property (i) obviously holds, while the property (ii) follows
from our Lemma~\ref{lmF2} in Section~\ref{secPAD}.\medskip
\noindent{\bf Definition}. We say that two EQR rectangles
$R_1,R_2$ with the corresponding integers $k_1^-,k_1^+$ and
$k_2^-,k_2^+$ characterizing their quasiroundness are $D$-{\em
nested} for $D\geq 0$ if either $[k_1^-,k_1^+]\subset
[k_2^--D,k_2^++D]$ or $[k_2^-,k_2^+]\subset [k_1^--D,k_1^++D]$.
\medskip
\begin{theorem}
Let $T:X\mapsto X$ be an Anosov diffeomorphism with a Gibbs measure
$\mu$ defined by a H\"older continuous potential $\varphi$ on $X$,
and $D\geq 0$ a constant. Let $\{R_n\}$ be a sequence of EQR
rectangles. Assume that for all $m,n\geq
1$ the rectangles $R_m,R_n$ are $D$-nested. Then $\{R_n\}$
satisfies {\rm (SP)} and hence, if in addition $\sum \mu(R_n)
= \infty$,
it is an sBC sequence and {\rm (\ref{log})} holds. \label{tm4}
\end{theorem}
\noindent{\bf Examples}.\\ 3. If a sequence of EQR rectangles
$R_n$ satisfies the condition
\be
|k^-_n+k^+_n|\leq D=\, {\rm const}
\label{kn+-}
\ee
then it is an sBC sequence and verifies (\ref{log}).\\ 4. In
particular, if $T$ is a linear 2-D toral automorphism and $\mu$
the Lebesgue measure, then any sequence of connected rectangles
with uniformly bounded ratio of stable and unstable sides (which
is sometimes called `aspect ratio') satisfies the condition
(\ref{kn+-}) and hence the conclusion of Theorem~\ref{tm4} holds. \\ 5. Let
$T:X\mapsto X$ be
the baker's transformation of the unit square $X=[0,1]\times
[0,1]$ and $\mu$ the Lebesgue measure. Note that $T$ is
discontinuous but still admits a finite Markov partition. Then any
sequence of balls with diverging measures is a BC sequence.
Indeed, in each ball $B\subset X$ one can find a `dyadic' square
$R\subset B$ such that $\mu(R)\geq 0.1\mu(B)$. Dyadic squares
correspond to 0-centered cylinders in the symbolic space, so one
can apply Theorem~\ref{tm1} and obtain the sBC property for the
dyadic squares, which implies (at least) the BC property for the
original balls.\medskip
Next, we generalize Example 4 to nonlinear Anosov diffeomorphisms.
Let $T:\, X\mapsto X$, dim$\, X=2$, be an Anosov diffeomorphism of a surface.
Recall that in this case every connected rectangle $R\subset X$ is EQR.
For a connected rectangle $R$ we denote
$$
d^u(R)=\sup_{z\in R}|W^u_z(R)|
\ \ \ \ \ \ {\rm and}\ \ \ \ \ \
d^s(R)=\sup_{z\in R}|W^s_z(R)|\,,
$$
where $|W^u|$, $|W^s|$ stand for the Lebesgue measures
(lengths) of the corresponding curves $W^u, W^s$. Let $B\geq 1$.
We say that a rectangle $R$ has a $B$-{\em bounded aspect ratio}
if
$$
B^{-1}\leq d^u(R)/d^s(R)\leq B\,.
$$
Note that rectangles with $B$-bounded aspect ratio are, in the
geometric sense, close to squares (i.e., `round'). This geometric
version of roundness is somewhat more preferable and easier to
check than the dynamical roundness assumed by (\ref{kn+-}).
\begin{theorem}
Let $T:\, X\mapsto X$, dim$\, X=2$, be an Anosov diffeomorphism with
a Gibbs measure $\mu$ defined by a H\"older continuous potential
$\varphi$ on $X$, and $B\geq 1$ a constant. Let $\{R_n\}$ be a
sequence of connected rectangles with (uniformly) $B$-bounded aspect ratio.
Then $\{R_n\}$ satisfies {\rm (SP)} and hence, if in addition $\sum \mu(R_n)
= \infty$,
it is an sBC sequence and {\rm (\ref{log})} holds. \label{tm5}
\end{theorem}
The extensions of Theorems~\ref{tm2} and \ref{tm3} to EQR
rectangles can also be obtained but are hardly worth pursuing,
because the examples of cylinders constructed in \ref{tm2}
and \ref{tm3} can be simply projected on $X$ and produce the
corresponding examples of rectangles.
\section{Proofs for topological Markov chains}
\label{secPTMC}
\setcounter{equation}{0}
The following facts about Gibbs measures are standard:
\noindent
{\bf Fact 1} \ For any cylinder $C$ defined on an interval $\Lambda$
$$
c_1\theta_1^{|\Lambda|}\leq\mu(C)\leq c_2\theta_2^{|\Lambda|}\,,
$$
where $c_1,c_2>0$ and $\theta_1,\theta_2\in (0,1)$ only depend
on the Gibbs measure $\mu$.
\noindent
{\bf Fact 2} \ Let $C_1\subset C$ be cylinders
defined on intervals $\Lambda_1,\Lambda$ (note that
in this case $\Lambda_1\supset\Lambda$), then
$$
c_1\theta_1^{|\Lambda_1|-|\Lambda|}
\leq\mu(C_1)/\mu(C)\leq c_2\theta_2^{|\Lambda_1|-|\Lambda|}\,.
$$
\noindent
{\bf Fact 3} \ Let $C_1,C_2$ be cylinders defined on disjoint
intervals $[n_1^-,n_1^+]$ and $[n_2^-,n_2^+]$ in $\ZZ$.
Assume, without loss of generality that $n_1^+0$ and $\theta_3\in (0,1)$ only depend on the Gibbs
measure $\mu$.\medskip
Facts 1 and 2 can be proved with the help of a normalized
potential for the Gibbs measure $\mu$, see \cite{ChS}. Fact 3 is
proved by R.~Bowen in \cite{Bo75}.
\begin{lemma}
If $C_1$, $C_2$ are cylinders defined on intervals $[n_1^-,n_1^+]$
and $[n_2^-,n_2^+]$ in $\ZZ$ such that $n_1^+\leq n_2^+$, then
$$
|\mu(C_1\cap C_2)-\mu(C_1)\mu(C_2)|\leq
c_4\theta_4^{n_2^+-n_1^+}\mu(C_1)\,,
$$
where $c_4>0$ and $\theta_4\in (0,1)$ only depend on the Gibbs
measure $\mu$. Similarly, if $n_2^-\leq n_1^-$, then
$$
|\mu(C_1\cap C_2)-\mu(C_1)\mu(C_2)|\leq
c_4\theta_4^{n_1^--n_2^-}\mu(C_1)\,.
$$
\label{lmCC}
\end{lemma}
{\em Proof}. The first claim follows from Facts 1 and 3 if $n_1^+<
n_2^-$ and from Facts 1 and 2 if $n_1^+\geq n_2^-$. The second claim
is symmetric to the first. $\Box$\medskip
{\em Proof of Theorem~\ref{tm1}}. We estimate the quantity
$R_{mn}=\mu(C_m\cap\sigma^{m-n}C_n)-\mu(C_n)\mu(C_m)$. Without
loss of generality, assume that
the interval $\Lambda_m$ is ``nested'' in $\Lambda_n$,
i.e.~$\Lambda_m$ lies in the $D$-neighborhood
of $\Lambda_n$. Note that we do not assume
that $m \ge n$ or $m \le n$. By Lemma~\ref{lmCC}, we have $|R_{mn}|\leq
c_4\theta_4^{|m-n|-D}\mu(C_m)$. Summing up over all $n$ satisfying
our nesting condition (that $\Lambda_m$ is ``nested'' in $\Lambda_n$)
gives a quantity bounded by const$\cdot\mu(C_m)$. Now summing up over
$m=M,\ldots,N$ proves (SP). $\Box$\medskip
In the following proofs of Theorems~\ref{tm2} and \ref{tm3} we use
a special construction. Let $T$ be a measure preserving transformation
(invertible or not) of a probability space $(X,\mu)$, and let $\{\tilde{A}_k\}$ be a sequence of
measurable subsets of $X$ and $\{l_k\}$ a sequence of natural
numbers. Put $s_0=0$
and $s_k=l_1+\cdots +l_k$ for $k\geq 1$. Consider a new sequence
of sets $\{{A}_n\}$ defined as follows:
$$
T^{1-l_1}\tilde{A}_1,T^{2-l_1}\tilde{A}_1,\ldots,
T^{-1}\tilde{A}_1,\tilde{A}_1,
T^{1-l_2}\tilde{A}_2,\ldots,T^{-1}\tilde{A}_2,\tilde{A}_2,
T^{1-l_3}\tilde{A}_3,\ldots,T^{-1}\tilde{A}_3,\tilde{A}_3,\ldots
$$
Note that the $n$th set in this sequence is
\be
A_n=T^{n-s_k}\tilde{A}_k\,,
\label{CnCk}
\ee
where $k$ is defined by $s_{k-1}0$, which violates (SP). $\Box$\medskip
We write $a_n\approx b_n$ for two sequences of numbers $\{a_n\}$ and
$\{b_n\}$ if there are constants $00$ (called {\em expansivity
constant}) such that
$$
\forall k\in\ZZ\ \ d(T^kx,T^ky)<\delta\ \ \ \
\Leftrightarrow\ \ \ \ x=y\,.
$$
In fact, due to the hyperbolicity of $T$, for some $C>0$ and $0 <
\theta < 1$ one has
\be
\forall |k|\leq n\ \ d(T^kx,T^ky)<\delta\ \ \ \
\Rightarrow\ \ \ \ d(x,y)0$,
we say that $\bf x$ is an $\alpha$-{\em pseudo-orbit} if
$$
d(T^kx,x_{k+1})<\alpha\ \ \ \ {\rm whenever}\ \ \ \ k,k+1\in\Lambda\,.
$$
We say that the orbit of $x\in X$ $\beta$-{\em shadows} $\bf x$ if
$$
d(T^kx,x_k)<\beta\ \ \ \ \ \ \forall k\in\Lambda\,.
$$
\medskip
\noindent
{\bf Shadowing lemma}. For any $\beta>0$ there is an $\alpha>0$
such that every $\alpha$-pseudoorbit is $\beta$-shadowed by a true
orbit of some $x\in X$.
Note that if $\Lambda=\ZZ$ and $\beta<\delta/2$,
then the true orbit shadowing $\bf x$ is unique
by the expansivity. We fix a
$\beta<\delta/2$ and this fixes the corresponding
$\alpha>0$.
Note that if the pseudoorbit is periodic, then it is shadowed
by a true periodic orbit with the same period.
%\noindent{\bf Interpolation}.
Given $\alpha>0$, there is an integer $K>0$ such that for every
$x,y\in X$ and $n\geq K$ there is a $z\in X$ such that
$$
d(z,x)<\alpha\ \ \ \ \ {\rm and}\ \ \ \ \ d(T^nz,y)<\alpha\,,
$$
which follows from the topological transitivity of $T$.
(Note that our choice of $\alpha$ made above also fixes $K$.)
Using this remark, we can interpolate (concatenate)
several $\alpha$-pseudoorbits defined on intervals
of $\ZZ$ separated by gaps of lengths $\geq K$
in the following way.
\medskip
\noindent{\bf Specification}. Let $\alpha$-pseudoorbits
${\bf x}_j$ be defined on disjoint intervals of $\ZZ$ separated by gaps
of length $\geq K$. Then the ${\bf x}_j$ are all $\beta$-shadowed
by one true orbit of some $x\in X$.
One can also find a periodic orbit that $\beta$-shadows all
${\bf x}_j$, with period $P:=i_{\max}-i_{\min}+K$, where $i_{\max}$
and $i_{\min}$ are the maximum and the minimum points of the
union of the intervals of $\ZZ$ on which the pseudoorbits
${\bf x}_j$ are defined.
Due to the expansivity, the number of periodic orbits of period $P$
in the above construction is less than some $L$ independent of
the lengths of the intervals of $\ZZ$ where the pseudoorbits are
defined. The value of $L$ only depends on the number of these
intervals and the lengths of gaps between them. In our further
arguments, we will interpolate no more than four pseudoorbits
at a time, and the gaps between them will never exceed $2K$,
so we just fix the corresponding constant $L$.
Now, let $g:X\mapsto \IR$ be a H\"older continuous function. The
bound (\ref{exp1}) implies the following.
\medskip
\noindent{\bf Approximation of sums along orbits}.
There is a constant $B=B(g)$ such that
$$
\forall k\in [p,q]\ \ \ d(f^kx,f^ky)<\delta\ \ \ \ \
\Rightarrow\ \ \ \ \ \left |\sum_{k=p}^qg(T^kx)
-\sum_{k=p}^qg(T^ky)\right |\leq B\,.
$$
Furthermore, let the specification property be used to shadow
two finite orbits $\{T^kx'\}$, $k\in\Lambda'$, and
$\{T^kx''\}$, $k\in\Lambda''$, with
$$
K\leq\, {\rm dist}(\Lambda',\Lambda'')\leq 2K\,,
$$
by a periodic orbit of $z$ of period
$$
P=|\Lambda'|+|\Lambda''|+\,{\rm dist}(\Lambda',\Lambda'')+K\,,
$$
then
\be
\left |\sum_{k\in\Lambda'}g(T^kx')+\sum_{k\in\Lambda''}g(T^kx'')
-\sum_{k=1}^{P}g(T^kz)\right |\leq B':=2B+3K||g||_{\infty}\,.
\label{3B}
\ee
Note that $B'$ is a constant, just like $B$, independent of the
lengths of the intervals $\Lambda',\Lambda''$.
For $n\geq 1$, let
$$
{\rm Fix}(T^n,X)=\{x\in X:\, T^nx=x\}
$$
be the set of periodic points of period $n$ in $X$.
\medskip
\noindent{\bf Periodic orbit approximation of Gibbs measures}.
Let $\mu$ be a Gibbs measure corresponding to a H\"older
continuous potential $\varphi:X\mapsto \IR$. For each $n\geq 1$, let
$\mu_n$ be an atomic probability measure concentrated on
Fix$(T^n,X)$ that assigns weight
\be
\mu_n(x)=Z_n^{-1}\exp[\varphi(x)+\varphi(Tx)+\cdots+\varphi(T^{n-1}x)]
\label{mun}
\ee
to each point $x\in\,$Fix$(T^n,X)$ (here $Z_n$ is a normalizing factor).
Then $\mu_n$ weakly converges to $\mu$ as $n\to\infty$.
\medskip
\noindent{\bf Variational principle}.
Let $\varphi:X\mapsto\IR$ be a continuous function and $P_\varphi$ its
topological pressure. Then
\be
\sup_{\nu}[h_{\nu}(T)+\nu(\varphi)]=P_\varphi\,,
\label{varpr}
\ee
where the supremum is taken over all $T$-invariant probability
measures $\nu$ on $X$, and $h_{\nu}(T)$ is the Kolmogorov-Sinai
entropy of $\nu$. Any measure $\nu$ that turns (\ref{varpr}) into
an equality is called an equilibrium state for $\varphi$.
Equilibrium states exist for every continuous function $\varphi$.
If $\varphi$ is
H\"older continuous on $X$, the equilibrium state is unique and
coincides with the Gibbs measure for the potential $\varphi$.
We now prove a few technical lemmas. Let $\mu$ be a Gibbs measure
on $X$ corresponding to a H\"older continuous potential $\varphi$.
We generalize our notation of Section~\ref{secPTMC}
by writing for any two variable quantities $A$ and $B$
$$
A\approx B\ \ \ \ \ \Leftrightarrow \ \ \ \ \ 00$.
Then $P:=\lim_{n\to\infty}a_n/n$ exists.
Furthermore, $|a_n-Pn|\leq 2R$ for all $n$.
\end{sublemma}
{\em Proof}.
Fix an $m\geq 1$. For $n\geq 1$, write $n=km+l$
with $0\leq l\leq m-1$. Then it follows by induction
on $k$ that $|a_n-ka_m-a_l|\leq kR$. Hence,
$$
\left |\frac{a_n}{n}-\frac{ka_m}{km+l}-\frac{a_l}{km+l}\right |
\leq \frac{kR}{km+l}\,.
$$
Letting $n\to\infty$ gives
$$
\frac{a_m}{m}-\frac Rm \leq
\liminf_n\frac{a_n}{n}\leq
\limsup_n\frac{a_n}{n}\leq
\frac{a_m}{m}+\frac Rm\,.
$$
Hence, $P:=\lim a_n/n$ exists. Next, assume
that $a_m>Pm+2R$ for some $m$. Then
$a_{2^nm}>2^nmP+(2^n+1)R$
which follows by induction on $n$.
Hence $\lim\sup a_n/n\geq P+R/m$,
a contradiction. A similar contradiction results
from the assumption $a_m0$ that only depends on the Gibbs measure $\mu$.
\label{lmR2}
\end{lemma}
{\em Proof}. The proof of the previous lemma applies with the
following simple adjustments. Note that if $y\in R_1\cap R_2$,
then the orbit of $y$ $\varepsilon_1$-shadows that of $x$ on
$\Lambda_1\cup\Lambda_2$. So, to get an upper bound on
$\mu(R_1\cap R_2)$, we can take into account all $n$-periodic
orbits that $\alpha$-shadow the orbit of $x$ on
$\Lambda_1\cup\Lambda_2$ with $\alpha=\varepsilon_1$. Now, if
$\Lambda_1$ and $\Lambda_2$ overlap, the argument is exactly like
in the proof of the previous lemma. Let $\Lambda_1$ and
$\Lambda_2$ be disjoint with dist$(\Lambda_1,\Lambda_2)=J$. If
$J\leq 2K$, we can simply disregard such a small gap and apply the
previous argument. If $J>2K$, we replace the part of the orbit of
$y\in\,$Fix$(T^n,X)$ of length $J$ between $\Lambda_1$ and
$\Lambda_2$ by periodic orbits of period $J-2K$. To conclude the
argument, we now need an obvious extension of (\ref{3B}) from two
to four pseudoorbits with gaps of length $K$ in between. This
extension is straightforward. $\Box$\medskip
\begin{lemma}
There is a constant $\delta_0>0$ such that for all $n\geq 1$
and $x\in\,{\rm Fix}(T^n,X)$ we have
$$
\varphi(x)+\varphi(Tx)+\cdots+\varphi(T^{n-1}x)\leq -\delta_0n\,.
$$
\end{lemma}
{\em Proof}. Let $\delta_x$ be the delta measure concentrated
at $x$. The measure
$$
\delta_{x,n}=\frac 1n (\delta_x+\cdots+\delta_{T^{n-1}x})
$$
is $T$-invariant, so by the variational principle we have
$$
\delta_{x,n}(\varphi)=\frac 1n \Big(
\varphi(x)+\cdots+\varphi(T^{n-1}x)\Big )\leq 0\,.
$$
We now need to prove that
$$
\sup_{n\geq 1}\sup_{x\in\,{\rm Fix}(T^n,X)}\delta_{x,n}(\varphi)<0\,.
$$
If this is not true,
then there is a sequence of periodic points $x_k\in\,$Fix$(T^{n_k},X)$
such that $\delta_{x_k,n_k}(\varphi)\to 0$. We take any limit point of the
sequence of measures $\delta_{x_k,n_k}$ in the weak topology,
it will be a $T$-invariant
measure, call it $\nu$. We have $\nu(\varphi)=0$, so by the uniqueness
part of the variational principle $\nu=\mu$, so $\mu(\varphi)=0$ and
hence $h_{\mu}(T)=0$.
But it is known that $h_{\mu}(T)>0$ for any Gibbs measure,
a contradiction. $\Box$\medskip
Combining this lemma with the specification property and (\ref{3B}) gives
\begin{corollary}
There is a constant $B_0>0$ such that for all
$n\geq 1$ and $x\in X$
$$
-\Delta_0n\leq
\varphi(x)+\varphi(Tx)+\cdots+\varphi(T^{n-1}x)\leq B_0-\delta_0n
$$
with $\Delta_0=||\varphi||_{\infty}$.
\label{crh}
\end{corollary}
We can now prove analogues of Facts~1 and 2 of
Section~\ref{secPTMC} for Anosov diffeomorphisms. Our constants,
such as $c_i,\theta_i$, will only depend on the Gibbs measure
$\mu$ and the values of $\varepsilon_0,\varepsilon_1$ in the
definition of EQR rectangles. We use notation of Lemmas~\ref{lmR1}
and \ref{lmR2}.
\begin{lemma}
Let $R$ be a EQR rectangle and $k:=k^+- k^-$. Then
$$
c_5\theta_5^{k}\leq\mu(R)\leq c_6\theta_6^{k}
$$
with some $c_5,c_6>0$ and $\theta_5,\theta_6\in (0,1)$.
\label{lmF1}
\end{lemma}
\begin{lemma}
Let $R_1,R_2$ be EQR rectangles and the intervals
$\Lambda_1=[k^-_1,k^+_1]$ and $\Lambda_2=[k^-_2,k^+_2]$ overlap.
Let $k=|\Lambda_2\setminus\Lambda_1|$. Then
$$
\mu(R_1\cap R_2)\leq c_7\theta_7^{k}\mu(R_1)
$$
with some $c_7>0$ and $\theta_7\in (0,1)$.
\label{lmF2}
\end{lemma}
{\em Proof}. Lemmas~\ref{lmF1} and \ref{lmF2}
follow from Lemmas~\ref{lmR1} and \ref{lmR2}
and Corollary~\ref{crh}. $\Box$\medskip
Note that so far we only used the property (i) of the
quasiround rectangles, we did not use (\ref{lne}).
\begin{lemma}
Let $R_1,R_2$ be EQR rectangles and the intervals
$\Lambda_1=[k^-_1,k^+_1]$ and $\Lambda_2=[k^-_2,k^+_2]$ be
disjoint. Let $k=\,{\rm dist}(\Lambda_1,\Lambda_2)$. Then
$$
|\mu(R_1\cap R_2)-\mu(R_1)\mu(R_2)|\leq
c_8\frac{\mu(R_1)+\mu(R_2)}{|ak+b|^{1+\gamma}}
$$
with some constants $c_8>0$, $a>0$ and $b$.
\label{lmF3}
\end{lemma}
{\em Proof}. Our proof uses Markov partitions and symbolic
dynamics. Let ${\cal R}$ be a Markov partition and $\Sigma$ the
corresponding symbolic space, a topological Markov chain. We now
partition the rectangles $R_1$ and $R_2$ into subrectangles
generated by the Markov partition $\cal R$ as follows. Let
$C\subset\Sigma$ be a cylinder defined on an interval
$\Lambda\subset\ZZ$. We say that its projection $\pi(C)$ is {\em
properly inside} $R_i$, $i=1,2$, if \\ (i) $\pi(C)\subset R_i$,
and\\ (ii) for any larger cylinder $C'\supset C$ its projection
$\pi(C')$ is not a subset of $R_i$.\\ Denote by ${\cal C}_i$ the
collection (in general, countable) of cylinders that are properly
inside $R_i$. Since $R_i$ is a rectangle, one can easily check
that all the cylinders in ${\cal C}_i$ are disjoint. Next, it
follows from the assumption (\ref{lne}) that $\mu(\partial
R_i)=0$, hence $$
\mu\big (R_i\setminus\cup_{C\in{\cal C}_i}\pi(C)\big )=0\,
$$
i.e.~the rectangles $\pi(C)$, $C\in{\cal C}_i$,
make a (mod 0) partition of $R_i$.
Now consider the collection ${\cal C}_1$ and an arbitrary cylinder
$C\in{\cal C}_1$ defined on an interval $\Lambda=[k^-,k^+]$.
Observe that if $t:=k^+-k_1^+>0$, then, using the notation of
(\ref{Ru}), we have $\pi(C)\subset R^u_{1,z}(\varepsilon)$ with
$\varepsilon=c\theta^t$ for any $z\in R_1$. Here $c>0$ and
$\theta\in (0,1)$ are constants determined by the hyperbolicity
properties of $T$ and the sizes of rectangles of the Markov
partition $\cal R$. Similarly, if $C\in{\cal C}_2$ is defined on
an interval $\Lambda=[k^-,k^+]$ and $t=k_1^--k^->0$, then
$\pi(C)\subset R^s_{2,z}(\varepsilon)$ with
$\varepsilon=c\theta^t$.
Now define subcollections ${\cal C}_i'\subset {\cal C}_i$ for
$i=1,2$ that contain all cylinders $C$ defined on intervals
$\Lambda=[k^-,k^+]$ satisfying $k^+-k_1^+>k/3$ for $i=1$ and
$k_1^--k^->k/3$ for $i=2$ (recall that $k=\,{\rm
dist}(\Lambda_1,\Lambda_2)$). By the assumption (\ref{lne})
$$
\mu\left (\cup_{C\in{\cal C}_i'} \pi(C)\right )
\leq C_0\frac{\mu(R_i)}{|ak+b|^{1+\gamma}}
$$
with constants $a=-\ln\theta^{1/3}>0$ and $b=-\ln c$. So, the
parts $\pi(C)$, $C\in{\cal C}_i'$, can be removed from $R_i$ with
no harm. Denote by
$$
\tilde{R}_i=R_i\setminus\left (\cup_{C\in{\cal C}_i'} \pi(C)\right )
$$
the remaining parts of $R_i$.
Note that $\tilde{R}_1$ and $\tilde{R}_2$ consist (mod 0) of
cylinders $C'\in{\cal C}_1\setminus{\cal C}_1'$ and $C''\in{\cal
C}_2\setminus{\cal C}_2'$, respectively, and the gap between the
intervals on which $C'$ and $C''$ are defined is always $\geq
k/3$. Hence we can use the subadditivity of the correlation
function and Fact~3 of Section~\ref{secPTMC} to get
\begin{eqnarray*}
|{\rm Cov}(\tilde{R}_1,\tilde{R}_2)| &:=&
\big|\mu\big((\cup\pi(C')\big)\cap \big(\cup\pi(C'')\big)
-\mu\big(\cup\pi(C')\big)\mu\big(\cup\pi(C'')\big)\big|\\
&\leq &
\sum_{C'}\sum_{C''}\big|\mu\big(\pi(C')\cap \pi(C'')\big)
-\mu\big(\pi(C')\big)\mu(\pi(C'')\big)\big|
\\ &\leq &
\sum_{C'}\sum_{C''}
c_3\theta_3^{k/3}\mu\big(\pi(C')\big)\mu\big(\pi(C'')\big)
\\ &\leq&
c_3\theta_3^{k/3}\mu(\tilde{R}_1)\mu(\tilde{R}_2)\,.
\end{eqnarray*}
This completes the proof of Lemma~\ref{lmF3}. $\Box$
\begin{lemma}
Let $R_1,R_2$ be EQR rectangles with the corresponding intervals
$\Lambda_1=[k^-_1,k^+_1]$ and $\Lambda_2=[k^-_2,k^+_2]$. If
$k_1^+\leq k_2^+$, then
$$
|\mu(R_1\cap R_2)-\mu(R_1)\mu(R_2)|\leq
c_9\frac{\mu(R_1)+\mu(R_2)}{|a_1k+b|^{1+\gamma}}\,,
$$
where $k=k_2^+-k_1^+$. The same holds if $k_2^-\leq k_1^-$,
but then $k=k_1^--k_2^-$. Here $a_1=a/2$, and $c_9>0$ is a constant.
\label{lmRR}
\end{lemma}
{\em Proof}. We prove the first claim, the second is symmetric to
the first. Then $k=k_2^+-k_1^+$. If the intervals $\Lambda_1$ and
$\Lambda_2$ overlap, or if at least $k_2^+-k_2^-\geq k/2$, the
claim follows from Lemmas~\ref{lmF1} and Lemma~\ref{lmF2}, one can
even drop the term $\mu(R_2)$. If $k_2^+-k_2^-0$ is a constant.
\label{lmRRast}
\end{lemma}
{\em Proof}. This follows from standard distortion bounds. $\Box$ \medskip
{\em Proof of Theorem~\ref{tm5}}.
Denote by $k_n^{\pm}$ the integers characterizing
the quasiroundness of $R_n$. We may assume that all $R_n$
are small enough, and then the uniform boundedness of their
aspect ratio ensures that $k_n^+\geq 0$ and $k_n^-\leq 0$
for all $n\geq 1$.
We estimate the quantity
$R_{mn}=\mu(R_m\cap T^{m-n}R_n)-\mu(R_n)\mu(R_m)$.
The set $R_{\ast}=T^{m-n}R_n$ is a connected rectangle
whose quasiroundness is characterized by the integers
$k^{\pm}_{\ast}:=k^{\pm}_n+(n-m)$.
Without loss of generality, assume that $d^u(R_m)\geq d^u(R_n)$.
We consider three cases:
Case 1. Assume that either (i) $n-m\geq 2k_m^+$ or (ii)
$n-m\leq 2k_m^-$. In the case (i) we have
$$
k^+_{\ast}-k^+_m\geq n-m-k_m^+\geq |n-m|/2\,,
$$
and in the case (ii) we have
$$
k^-_m-k^-_{\ast}\geq k_m^--(n-m)\geq |n-m|/2\,.
$$
In either case we apply Lemma~\ref{lmRR} and obtain
$$
|R_{mn}|\leq c_9\frac{\mu(R_m)+\mu(R_n)}{|a_2|n-m|+b|^{1+\gamma}}
$$
with $a_2=a_1/2>0$.
Case 2. Assume that $2k_m^-\leq n-m\leq 2k_m^+$ and $R_{\ast}\cap
R_m\neq\emptyset$.
If $n>m$, then $d^u(R_{\ast})\leq \theta^{n-m} d^u(R_m)$, and if
$n\leq m$, then $d^s(R_{\ast})\leq \theta^{m-n} d^s(R_m)$
for some constant $\theta<1$, due to the uniform hyperbolicity
of $T$. Hence, Lemma~\ref{lmRRast} implies that if $n>m$, then
$$
k^+_{\ast}-k^+_m\geq c_{11}|n-m|\,
$$
and if $n\leq m$, then
$$
k^-_m-k^-_{\ast}\geq c_{11}|n-m|
$$
with some constant $c_{11}>0$. Again, we use Lemma~\ref{lmRR} and obtain
$$
|R_{mn}|\leq c_9\frac{\mu(R_m)+\mu(R_n)}{|a_3|n-m|+b|^{1+\gamma}}
$$
with $a_3=c_{11}a_1$.
Case 3. Assume that $2k_m^-\leq n-m\leq 2k_m^+$ and $R_{\ast}\cap
R_m=\emptyset$. Then $R_{mn}=\mu(R_m)\mu(R_n)$.
It follows from Lemma~\ref{lmF1} that
$$
a_4|\ln\mu(R_m)|+b_4\leq k^+_m-k^-_m\leq a_5|\ln\mu(R_m)|+b_5
$$
with some $a_4,a_5>0$ and $-\infty0$,
due to uniform bounds on expansion and contraction rates of $T$.
Therefore,
$$
\mu(R_n)\leq c_{12}[\mu(R_m)]^{\kappa}
$$
with some constants $c_{12}>0$ and $\kappa>0$. Holding
$m$ fixed and summing over all $n$ that satisfy the conditions
of Case 3 gives
$$
\sum_n R_{mn}\leq 2c_{12}[\mu(R_m)]^{1+\kappa}(a_5|\ln\mu(R_m)|+b_5)
\leq c_{13}\mu(R_m)
$$
with some constant $c_{13}>0$.
Lastly, summing up over all $m,n=M,\ldots,N$ proves (SP).
$\Box$\bigskip
\bigskip
{\bf Acknowledgements}. The authors want to thank Vitaly
Bergelson, Jean-Pierre Conze, Dmitry Dolgopyat, Yves Guivarc'h,
Andres del Junco and Albert Raugi for helpful discussions.
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