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\begin{document}
\title{Ergodic properties of Anosov maps\\ with rectangular holes}
\author{N. Chernov$^{01}$
\\ Department of Mathematics\\
University of Alabama in Birmingham\\
Birmingham, AL 35294, USA\\
E-mail: chernov@vorteb.math.uab.edu; Fax: 1-(205)-934-9025
\and R. Markarian$^{02}$
\\Instituto de Matem\'atica y Estad\'{\i}stica
``Prof. Ing. Rafael Laguardia''\\
Facultad de Ingenier\'{\i}a.
Universidad de la Rep\'ublica\\
C.C. 30, Montevideo, Uruguay\\
E-mail: roma@fing.edu.uy; Fax: (598-2)-715-446
}
\date{ }
\maketitle
.\hfill{\it To the memory of Ricardo Ma\~{n}\'e}\\
\begin{abstract}
We study Anosov diffeomorphisms on manifolds in which some `holes'
are cut. The points that are mapped into those holes disappear and
never return. The holes studied here are rectangles of a Markov
partition. Such maps generalize
Smale's horseshoes and certain open billiards. The set of nonwandering
points of a map of this kind is a Cantor-like set called {\it repeller}.
We construct invariant and conditionally invariant measures on the sets
\footnotetext{$^1$ Partially supported by NSF grant DMS-9401417.}
of nonwandering points. Then we establish ergodic, statistical, and
fractal properties of those measures.
\footnotetext{$^2$ Partially supported by CONICYT (Uruguay).}
\end{abstract}
\centerline{\em AMS classification numbers: 58F12, 58F15, 58F11}
\vspace*{1cm}\noindent {\em Keywords}: repellers, scattering theory, chaotic
dynamics, conditionally invariant measures, Anosov diffeomorphisms.
\newpage
\renewcommand{\baselinestretch}{1.7}
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\section{Introduction and main results}
\label{secI}
Let $T: M'\to M'$ be a topologically transitive Anosov
diffeomorphism of class $C^{1+\alpha}$ on a compact Riemannian
manifold $M'$. Recall that a diffeomorphism $T:M'\to M'$
is said to be Anosov if at every point $x\in M'$ there
is a $DT$-invariant splitting
\be
{\cal T}_x M' = E_x^u \oplus E_x^s
\label{EE}
\ee
such that
\begin{eqnarray}
||DT^{-n} v|| &\leq& C_T\lambda_T^n ||v||
\ \ \ {\rm for}\ {\rm all}\ \ v\in E_x^u\ \ {\rm and}\ \ n>0,\nonumber\\
||DT^n v|| &\leq& C_T\lambda_T^n ||v||
\ \ \ {\rm for}\ {\rm all}\ \ v\in E_x^s\ \ {\rm and}\ \ n>0,
\label{decom}
\end{eqnarray}
for some constants $C_T>0$ and $\lambda_T\in (0,1)$ independent
of $v$ and $x$. The splitting (\ref{EE}) is continuous
in $x$. Topological transitivity of $T$ means that
it has a dense orbit in $M'$.
Sinai \cite{Si68} and Bowen \cite{Bo75} constructed Markov
partitions for transitive Anosov diffeo- morphisms\footnote{Bowen's
construction actually covers larger systems -- Axiom A diffeomorphisms --
which we do not consider here.}. Let ${\cal R}'$ be a Markov partition
of $M'$ into rectangles $R_1,\ldots,R_{I'}$. We assume
that these rectangles are small enough, so that the
symbolic dynamics can be defined \cite{Si68,Bo75}.
Let $I*0$ such that $\mu(T^{-1}A\cap M_+)
=\lambda\mu(A\cap M_+)$ for any Borel set $A\subset M$.
\begin{theorem}
The map $T$ has a unique conditionally invariant probability
measure $\mu_+\in{\cal M}_+^u$. For any other $\mu\in{\cal M}_+^u$
the sequence $T^n_+\mu$ weakly converges, as $n\to\infty$,
to $\mu_+$.
\label{tm2}
\end{theorem}
We also call this unique measure $\mu_+$ the {\it eigenmeasure}
of the map $T$, and the corresponding factor $\lambda_+
=\lambda\in (0,1)$ the {\it eigenvalue} of $T$.
\begin{theorem}
For any smooth measure $\mu$ on $M$ (see a convention below)
the sequence $T^n_+\mu$ weakly converges, as $n\to
\infty$, to the eigenmeasure $\mu_+$. Furthermore, the
sequence $\lambda_+^{-n}\cdot T_\ast^n\mu$ weakly converges,
as $n\to\infty$, to the measure $c[\mu]\cdot\mu_+$, where
$c[\mu]>0$ is a linear functional on smooth measures on $M$.
\label{tm3}
\end{theorem}
{\it Remark}. The conditionally invariant measure $\mu_+$
constructed in this way is very natural according to the
above Pianigiani-Yorke physical motivation \cite{PY}. This
measure coincides with Sinai-Bowen-Ruelle measure in the
case $H=\emptyset$. \medskip
{\it Convention}. We call a measure on $M$ smooth if it is
absolutely continuous with respect to the Riemannian volume
on $M$, and its conditional measures on unstable fibers have
H\"older continuous densities (cf. also the previous convention!). \medskip
This theorem shows that the eigenmeasure $\mu_+$ can be
naturally obtained by iterating smooth measures under
$T$ on $M$.
One can think of an experiment in which we place $N=N(0)$
points (particles) in $M$ at random according to a smooth
probability distribution $\mu$. Then those points are
mapped by successive iterations of $T$. The number
of points that stay in $M$ (do not escape) after $n$
iterations, $N(n)$, is approximately
\be
N(n)\sim N(0)\cdot c[\mu]\cdot e^{-n\ln\lambda_+^{-1}}
\label{NtN0}
\ee
We call $\gamma_+=\ln\lambda_+^{-1}$ the {\it escape
rate}, cf. \cite{GD,GR,GN}.
Next, we show that the eigenmeasure $\mu_+$ can be also
obtained by iterating singular measures supported on
individual unstable fibers.
For any unstable fiber $U\in\cal U$ let $\mu_U^u\in{\cal M}$
be a (canonical) singular probability measure supported on $U$,
which coincides on $U$ with the measure $\nu_+^u$, described
in the remark after Theorem~\ref{tm1}.
\begin{theorem}
For any $U\in\cal U$ and any singular measure $\mu_U\in{\cal M}$
supported on $U$ with a H\"older continuous density with
respect to the Riemannian volume on $U$, the sequence
$T_+^n\mu_U$ weakly converges, as $n\to\infty$, to $\mu_+$.
Furthermore, the sequence of measures $\lambda_+^{-n}
\cdot T_\ast^n\mu_U^u$ weakly converges, as $n\to\infty$,
to a measure supported on $M_+$ and proportional to $\mu_+$.
\label{tm4}
\end{theorem}
\begin{proposition}
The function $e(U)$ on the set of unstable fibers $U\in\cal U$
defined by
\be
\lim_{n\to\infty}\lambda_+^{-n}\cdot T_\ast^n\mu_U^u=e(U)\cdot\mu_+
\label{elim}
\ee
is bounded away from 0 and $\infty$ and its restriction on
the set of fibers $U\in{\cal U}_+$ satisfies the equation
\be
\int_{{\cal U}_+} e(U)\, d\hat{\mu}_+(U)=1
\label{enorm}
\ee
where $\hat{\mu}_+$ is the factor measure of the eigenmeasure
$\mu_+$.
\label{pr1}
\end{proposition}
Next, since the set $M_+$ is invariant under $T^{-1}$, it
makes sense to define the inverse images of $\mu_+$ under
$T_\ast$, i.e. $T_\ast^{-n}\mu_+$ for $n\geq 1$, by
\be
(T^{-n}_\ast\mu_+) (A) = \mu_+(T^n[A\cap M_{-n}])
\label{Tastpri}
\ee
for any Borel set $A\subset M$. In virtue of Theorem~\ref{tm2}
the measure $T_\ast^{-n}\mu_+$, $n\geq 1$, simply
coincides with the conditional measure $\mu_+(\cdot /M_{-n})$
defined by
\be
\mu_+(A/M_{-n})=\mu_+(A\cap M_{-n})/\mu_+(M_{-n})
=\lambda_+^{-n}\cdot \mu_+(A\cap M_{-n})
\label{mupcon}
\ee
\begin{theorem}
The sequence of measures $T^{-n}_\ast\mu_+=\mu_+(\cdot /M_{-n})$
weakly converges, as $n\to\infty$, to a probability measure,
$\eta_+\in{\cal M}$, supported on the set $\Omega=M_+\cap M_-$.
The measure $\eta_+$ is $T$-invariant, i.e.
\be
\eta_+(T^{-1}A)=\eta_+(TA)=\eta_+(A)
\label{etainv}
\ee
for every Borel set $A\subset M$.
\label{tm5}
\end{theorem}
\begin{proposition}
The factor measure $\hat{\eta}_+$ of the measure $\eta_+$
on the set of unstable fibers $U\in {\cal U}_+$ is absolutely
continuous with respect to the factor measure $\hat{\mu}_+$
of the eigenmeasure $\mu_+$, and its Radon-Nikodym derivative
is
\be
\frac{d\hat{\eta}_+}{d\hat{\mu}_+}(U)=e(U)
\label{dde}
\ee
where $e(U)$ is the function introduced in Proposition~\ref{pr1}.
\label{pr2}
\end{proposition}
We call the closed set $\Omega=M_+\cap M_-$ the {\it repeller}
of the map $T$. It is normally a Cantor-like set. The
$T$-invariant measure $\eta_+$ on $\Omega$ can be obtained
naturally by iterating smooth measures on $M$ as follows.
For any probability measure $\mu\in{\cal M}$ and $n,m\geq 1$
we denote by $\mu_{n,m}$ the measure $T_+^n\mu$ conditioned
on $M_{-m}$, i.e.
\be
\mu_{n,m}(A)=T_+^n\mu (A\cap M_{-m})\cdot [T^n_+\mu(M_{-m})]^{-1}
\label{munm}
\ee
for any Borel $A\subset M$.
\begin{theorem}
For any smooth probability measure $\mu$ on $M$
the sequence of measures $\mu_{n,m}$ weakly converges, as $m,n
\to\infty$, to the invariant measure $\eta_+$ on the repeller
$\Omega$. Moreover, the sequence of measures $\mu_{n,m}^\ast$
defined by
\be
\mu_{n,m}^\ast(A)=\lambda_+^{-n-m}\cdot T_\ast^n\mu(A\cap M_{-m})
\label{mustanm}
\ee
weakly converges, as $m,n\to\infty$, to the measure $c[\mu]\cdot
\eta_+$, where $c[\mu]$ is the positive linear functional on
measures, involved in Theorem~\ref{tm3}.
\label{tm6}
\end{theorem}
Next, we establish the ergodic properties of the invariant
measure $\eta_+$ on the repeller $\Omega$.
\begin{theorem}
The measure $\eta_+$ is an equilibrium measure for the H\"older
continuous potential
\be
g_+(x)=-\log J^u(x)
\label{g+}
\ee
on $\Omega$ and the topological pressure
$P(\eta_+)=-\log\lambda_+^{-1}=-\gamma_+$.
Thus, $\eta_+$ is a Gibbs measure.
\label{tm7}
\end{theorem}
\begin{corollary}
The measure $\eta_+$ is ergodic, mixing, K-mixing and Bernoulli.
Its correlations decay exponentially fast and it satisfies the
central limit theorems and its invariance principle.
\label{cr1}
\end{corollary}
{\it Remark}. There are certainly other Gibbs invariant measures
on $\Omega$, see \cite{CMS2}. Some particularly interesting ones
are the measure of maximal entropy and the Hausdorff measure
\cite{Mo}. Our measure $\eta_+$ is the only one generated by
originally smooth measures $\mu$ on $M$, in the sense of
Theorems~\ref{tm3} and \ref{tm6} and the original Pianigiany-Yorke
philosophy \cite{PY}. Let us note that Theorems~\ref{tm3} and
\ref{tm6} cannot be obtained by the study of the symbolic
dynamics on the repeller $\Omega$ alone.
\begin{theorem}
The sum of positive Lyapunov exponents of the map $T$ is
\be
\chi^+_{\eta_+}=\int_{\Omega}\log J^u(x)\, d\eta_+(x)>0\ \ \ \ \ {\rm a.e.}
\label{chi+}
\ee
and the sum of negative Lyapunov exponents of $T$ is
\be
\chi^-_{\eta_+}=\int_{\Omega}\log J^s(x)\, d\eta_+(x)<0\ \ \ \ \ {\rm a.e.}
\label{chi-}
\ee
The variational principle
\be
-\gamma_+=h_{\eta_+}(T)-\int_{\Omega}\log J^u(x)\, d\eta_+(x)
=\sup_{\eta}\{h_\eta(T)-\int_{\Omega}\log J^u(x)\, d\eta(x)\}
\label{vp}
\ee
holds, where $h_\eta(T)$ denotes the Kolmogorov-Sinai entropy of
the measure $\eta$, and the supremum is taken over all
$T$-invariant probability measures on the repeller $\Omega$.
The left equation in (\ref{vp}) is equivalent to
\be
\chi^+_{\eta_+}=h_{\eta_+}(T)+\gamma_+
\label{vp1}
\ee
\label{tm8}
\end{theorem}
The equation (\ref{vp1}) generalizes Pesin's formula for smooth
hyperbolic maps, for which $h=\chi^+$ and $\gamma_+=0$. This equation
can be understood as follows. The exponential rate of separation of
nearby trajectories, characterized by $\chi^+$, contributes to both
the chaoticity of the dynamics on the repeller, measured by $h(T)$,
and the scattering away from the repeller measured by the escape
rate $\gamma_+$.
In a particular case, where dim$\, M'=2$, let $\delta^u_+$
and $\delta^s_-$ be the Hausdorff dimensions of the invariant
measure $\eta_+$ on unstable fibers $U\subset M_+$ and on
stable fibers $U\subset M_-$, respectively.
\begin{theorem}
Let ${\rm dim}\, M=2$. According to Manning's formula \cite{Ma}, we have
\be
h_{\eta_+}(T)=\delta_+^u\chi_{\eta_+}^+
=-\delta_+^s\chi_{\eta_+}^-
\ee
This agrees with Young's formula \cite{LSY} for the Hausdorff
dimension of the measure $\eta_+$:
\be
HD(\eta_+)=h_{\eta_+}(T)\left (\frac{1}{\chi_{\eta_+}^+}
-\frac{1}{\chi_{\eta_+}^-}\right ) = \delta^u_++\delta^s_+
\ee
\label{tm9}
\end{theorem}
By reversing the time, we can define the eigenmeasure $\mu_-$
on $M_-$ for the map $T^{-1}$, whose eigenvalue is $\lambda_-
\in (0,1)$. We then can define the corresponding invariant measure
$\eta_-$ on the repeller $\Omega$. These also have all the properties
described in the above theorems. The measure
$\eta_-$ and the values of $\lambda_-$ and $\chi_{\eta_-}^{\pm}$
are, generally, different from the previously described
measure $\eta_+$ and the quantities $\lambda _+$
and $\chi_{\eta_+}^{\pm}$, see some examples in
\cite{C86}. However, there are remarkable exceptions. \medskip
{\it Definition}. We say that the repeller $\Omega$ is
time-symmetric if $\eta_+=\eta_-$, $\lambda_+=\lambda_-$,
$\chi_{\eta_+}^+=\chi_{\eta_-}^+=|\chi_{\eta_+}^-|=|\chi_{\eta_-}^-|$.
\begin{theorem}
The measures $\eta_+$ and $\eta_-$ on the repeller $\Omega$
coincide if and only if there is a constant $Z>0$ such that
for every periodic point $x\in\Omega$, $T^kx=x$, we have
$$
{\rm det}\, DT^k(x)=J^u_k(x)\cdot J^s_k(x)=Z^k
$$
Moreover, the repeller $\Omega$ is time-symmetric if and
only if $Z=1$.
\label{tm10}
\end{theorem}
\begin{corollary}
If the original Anosov diffeomorphism $T:M'\to M'$ preserves
an absolutely continuous invariant measure on $M'$, then
the repeller $\Omega$ is time-symmetric.
\label{cr2}
\end{corollary}
The history of the subject goes back to 1979, when
Pianigiani and Yorke \cite{PY} constructed conditionally
invariant measures for expanding (noninvertible) maps.
Their results are analogous to our Theorems\ref{tm2} and
\ref{tm3}. In 1981-86 \v{C}encova \cite{C81,C86} undertook
a detailed study of both invariant and conditionally
invariant measures for smooth Smale's horseshoes
(her results are a particular case of our Theorems
\ref{tm1}-\ref{tm6}). In 1994, Collet, Martinez and
Schmitt \cite{CMS1} constructed invariant measures on
the sets of nonwandering points for Pianigiani-Yorke
transformations (their results are similar to our
Theorems \ref{tm5}-\ref{tm7}). In a later manuscript
\cite{CMS2} the same authors constructed
conditionally invariant measures for some symbolic
subshifts of finite type. Smooth hyperbolic systems
other than horseshoes were first considered in
this context by Lopes and Markarian recently
\cite{LM}. They studied an open billiard
system -- a particle bouncing off three circular
scatterers placed sufficiently far apart. Their results
are a particular case of our Theorems \ref{tm2},
\ref{tm3}, \ref{tm5}, and \ref{tm7}-\ref{tm9}.
Theorem~\ref{tm10} applies to open billiards,
answering a question posed in \cite{LM}.
Let us also point out physical papers by
Gaspard et. al. \cite{GB,GD,GN,GR,LS} in which the dynamics
on repellers was discussed and some equations, like
our (\ref{NtN0}) and (\ref{vp1}), were conjectured
and their connections with other equations in statistical
physics established.
>From measure-theoretic point of view, our systems
resemble probabilistic Markov chains with absorbing states.
For such chains, conditionally invariant distributions
(called quasi-stationary distributions) have been studied
in \cite{FKMP,MV}.
The purpose of the present paper is threefold. First,
we cover much larger classes of smooth hyperbolic systems
with `holes' than the previous papers did. Second, we collect
all the existing results in this direction scattered in other
papers, add some new ones (e.g., \ref{tm10} and \ref{cr2}),
and present the complete
(up-to-date) program for studying smooth hyperbolic repellers.
Third, we simplify and improve the matrix techniques for
the construction of conditionally invariant measures
used by \v{C}encova \cite{C86}. The matrix method
she used goes back to Sinai \cite{Si72}, but its
realizations are sometimes lengthy and heavy, as it
unfortunately happened to \cite{C86}. In our framework,
this method works quite effectively and easily. Moreover,
at present it is nearly the only workable method in the
context of systems with countable Markov partitions,
like billiards with `holes', open Lorentz gases
\cite{GN,GR} and other models of physical interest. We
sharpen the matrix method preparing it for an attack on
billiards, but such an attack is beyond the scopes of this paper.
The paper is organized as follows. Section~\ref{secBAD} provides
necessary results on Markov partitions and symbolic dynamics
for Anosov diffeomorphisms. Section~\ref{secCIMUF} contains a
proof of Theorem~\ref{tm1} and other properties of conditional
measures on unstable fibers. In Section~\ref{secCIM} we describe,
in general terms, the matrix techniques for constructing invariant
measures. Then we construct the conditionally invariant measure
$\mu_+$ proving Theorem~\ref{tm2}. In Section~\ref{secLTmu+}
we prove the limit theorems~\ref{tm3} and \ref{tm4} along with
Proposition~\ref{pr1}. In Section~\ref{secIMR} we construct
the invariant measure $\eta_+$ and prove statements~\ref{tm5}-\ref{tm6}.
In Section~\ref{secEPM} we prove
the ergodic and fractal properties of the measure $\eta_+$
described by the statements~\ref{tm7}-\ref{cr2}. In Section~\ref{secGOP}
we discuss possible generalizations of our main results and related
open problems. Appendix provides necessary techniques from the theory
of positive matrices. \medskip
{\bf Acknowledgements}. R.M. is indebted to S.~Mart\'{\i}nez
for introducing him to the subject and stimulating discussions.
This work was initiated during
the authors' visits at Princeton university, for which
we are grateful to Ya.~Sinai and J.~Mather. Special
thanks go to Ya.~Sinai who mentioned to us \v{C}encova's
papers. This work was essentially completed when
N.Ch. visited IMERL, Facultad de Ingenier\'{\i}a,
Uruguay, for which he is the most indebted.
N.Ch. acknowledges the support of
NSF grant DMS-9401417.
\section{Background on Anosov diffeomorphisms}
\label{secBAD}
\setcounter{equation}{0}
This section provides necessary tools from the
theory of Anosov diffeomorphisms.
It is known that Anosov diffeomorphisms enjoy strong ergodic
properties if they are of class $C^{1+\alpha}$, not just $C^1$, i.e.
$$
|| DT(x) - DT(y) || \leq C_\alpha\cdot [d(x,y)]^{\alpha}
$$
for some $C_{\alpha}>0$,
where $d(x,y)$ is the distance in the Riemannian metric. The
constant $\alpha\in(0,1]$ will be fixed throughout the paper.
The local unstable manifolds $W^u_\varepsilon(x)$,
$x\in M'$, are defined by
$$
W^u_\varepsilon(x)=\{y\in M':\,
d(T^nx,T^ny)\leq\varepsilon\ \ \forall n\leq 0\}
$$
for small $\varepsilon>0$. Similarly, local stable manifolds
$W^s_\varepsilon(x)$ are defined taking positive $n$.
It is known that these manifolds are `as smooth as the map'
$T$, see \cite{AS}. Precisely, they are of class $C^{1+\alpha}$,
i.e. the tangent space $E^u_x$ is H\"older continuous along
each $W^u$, with the H\"older exponent $\alpha$, and the same is
true for $E^s_x$ along stable manifolds. The tangent bundles
$E^u_x$ and $E^s_x$ over the whole of $M'$ are also H\"older
continuous \cite{AS,Mane}, but the exponent may be different from
$\alpha$.
Therefore, the Jacobians $J^u(x)$ and $J^s(x)$ are H\"older
continuous function on $M'$. Moreover, the restrictions of
$\log J^u(x)$ on unstable manifolds are H\"older continuous
with the exponent $\alpha$:
\be
|\log J^u(x)-\log J^u(y)|\leq C_J\cdot [d_u(x,y)]^{\alpha}
\label{JHc}
\ee
with some $C_J>0$,
for all $x,y\in U$, $U\in{\cal U}'$ (the same is true for
$J^s$, of course). Here and elsewhere $d_u$ and $d_s$ are
intrinsic metrics on unstable and stable manifolds, respectively,
induced by the Riemannian metric on $M'$.
For any $x,y\in M'$ we put
$$
[x,y]= W^s_\varepsilon(x)\cap W^u_\varepsilon(y)
$$
There is a $\delta>0$ such that if $d(x,y)<\delta$, then
$[x,y]$ consists of a single point. A subset $R \subset M'$
is called a rectangle if diam$\, R<\delta$ and $[x,y] \in R$
whenever $x, y \in R$. A rectangle $R$ is called {\em proper}
if $R = \overline{{\rm int} R}$ and for any point $x\in R$
the sets $W^u_\varepsilon(x)\cap\partial R$ and
$W^u_\varepsilon(x)\cap\partial R$ have zero Riemannian
volumes in the manifolds $W^u_\varepsilon(x)$ and
$W^s_\varepsilon(x)$, respectively. For $x \in R$ we put
$$
W^{u,s}(x,R) = W^{u,s}_{\varepsilon}(x)\cap R
$$
Recall \cite{Bo75} that $R'\subset R$ is called a $u$-subrectangle
in a rectangle $R$ if $W^u(R,x)\subset R'$ for all $x\in R'$.
Similarly, $R'\subset R$ is an $s$-subrectangle in $R$ if
$W^s(R,x)\subset R'$ for all $x\in R'$.
A Markov partition of $M'$ is a finite covering
${\cal R'} = \{ R_1, R_2, \ldots, R_{I'}\} $ of $M'$ by proper
rectangles such that\\
(i) ${\rm int} R_i \cap {\rm int} R_j = \emptyset$ for $i \neq j$;\\
(ii) if $x \in {\rm int} R_i$ and $Tx \in {\rm int} R_j$, then
$T W^u(x,R_i) \supset W^u(Tx,R_j)$ and $T W^s(x,R_i) \subset W^s(Tx,R_j)$
Equivalently, for any $R_i,R_j$ and $n\geq 1$ such that
int$(T^nR_i\cap R_j)\neq\emptyset$
the set $T^nR_i\cap R_j$ is a $u$-subrectangle in $R_j$
and $R_i\cap T^{-n}R_j$ is an $s$-subrectangle in $R_i$.
Every topologically transitive Anosov diffeomorphism $T:M'\to M'$
has Markov partitions of arbitrary small diameter.
We work with a fixed Markov partition ${\cal R}'$ of
a sufficiently small diameter.
For every $z \in R_i$ we define the projection $h_z^s : R_i
\to W^u(z,R_i)$ by $h_z^s(x) = [x,z]$.
For every $x\in R_i$ this is a one-to-one map from $W^u(x,R_i)$
to $W^u(z,R_i)$, which is called canonical isomorphism or
holonomy map. This map is absolutely continuous in the
sense that its Jacobian with respect to Riemannian volume
on unstable fibers is bounded and positive. Moreover,
the Jacobian $Dh_z^s(x)$ of the map $h_z^s: W^u(x,R_i)\to
W^u(z,R_i)$ satisfies the Anosov-Sinai formula \cite{AS}
$$
Dh_z^s(x) = \lim_{n\to\infty} J^u_n(x)/J^u_n(h_z^s(x))
$$
The Jacobian of the holonomy map is H\"older continuous
in the following sense: for any $x,y\in W^u(x,R_i)$ we have
\be
|Dh_z^s(x) - Dh_z^s(y)| \leq C'\cdot [d_u(x,y)]^{\alpha'}
\label{ac1}
\ee
and
\be
|Dh_z^s(x)| \leq \exp\left ( C'\cdot [d_s(x,h_z^s(x))]^{\alpha'}\right )
\label{ac2}
\ee
for some constants $C'>0$, $\alpha'>0$. (For proofs of these results,
see for example, the book by Ma\~n\'e \cite{Mane}, Chapter 3, Lemmas
2.7 and 3.2).
We now recall the basic definitions of symbolic dynamics.
A transition matrix $A'=(A_{ij}')$ of size $I'\times I'$
is defined by
$$
A_{ij}'=\left \{\begin{array}{ll}
1 & {\rm if}\ \ {\rm int}\, R_i\cap T^{-1}
({\rm int}\, R_j)\neq\emptyset\\
0 & {\rm otherwise}
\end{array}\right .
$$
In the space $\Sigma'=\{1,2,\ldots,I'\}^{\ZZ}$ of doubly infinite
sequences $\underline{\omega}=\{\omega_i\}_{-\infty}^{\infty}$
with the product topology we consider a closed subset
$$
\Sigma_{A'}'=\{\underline{\omega}\in\Sigma':\,
A_{\omega_i\omega_{i+1}}'=1\ \ {\rm for}\ {\rm all}
\ -\infty**0$, the class of measures $\mu\in{\cal M}$
such that their conditional measures $\mu_U$ on unstable fibers
$U\in{\cal U}$ are absolutely continuous with respect to the Riemannian
volume $m_U$ with densities $f_{\mu}(x)$ whose logarithms are
H\"older continuous with the exponent $\alpha$ and constant $G>0$:
\be
|\log f_\mu (x)-\log f_\mu (y)| \leq G\cdot [d_u(x,y)]^{\alpha}
\label{ffG}
\ee
for all $x,y\in U$ and $U\in{\cal U}$.
\section{Conditionally invariant measures on unstable fibers}
\label{secCIMUF}
\setcounter{equation}{0}
In this Section we prove Theorem~\ref{tm1} and some lemmas on the
evolution of measures under $T_\ast$, that will be used in the
forthcoming sections.\medskip
{\it Proof of Theorem~\ref{tm1}.} Our Theorem~\ref{tm1} is in fact an adapted
version of a result by Sinai for ordinary Anosov systems
(without holes). In our notations, his result reads \medskip
{\bf Fact} \cite[Lemma 2.3]{Si68}. Let $T$ be a $C^2$ transitive
Anosov diffeomorphism. Then there exists a unique family
of conditionally invariant probability measures $\nu_U^u$ on
unstable fibers $U\in{\cal U}'$ satisfying (\ref{rhoci}) with
Lipschitz continuous densities $\rho^u_U(x)=d\nu^u_U/dm_U(x)$.
\medskip
{\it Remarks}. Actually, Sinai constructed measures on
stable fibers, but this does not matter because one can
take $T^{-1}$ instead of $T$. Our map $T$ need not be $C^2$,
it may be less regular than Sinai's. This is why our
densities are only H\"older continuous. \medskip
We now start the proof. Let $\mu\in{\cal H}(G)$. Our proof works
for measures defined on $M'$ with (\ref{ffG}) valid on all $U\in
{\cal U}'$. Take a fiber $U\in{\cal U}'$. The measure $\mu_n=
T^n_{\ast}\mu$ on $M'$ conditioned on $U$ has a density $f_n(x)
=d\mu_{n,U}/dm_U(x)$. Due to (\ref{ffJ}), we have
\be
\frac {f_n(x)}{f_n(y)} = \frac {J^u(T^{-1}y)\cdots J^u(T^{-n}y)}
{J^u(T^{-1}x)\cdots J^u(T^{-n}x)}
\cdot\frac{f_{\mu}(T^{-n}x)}{f_{\mu}(T^{-n}y)}
\label{fnxy}
\ee
for every $x,y \in U$.
Note that
$$
d_u(T^{-n}x,T^{-n}y) \leq C_T\lambda_T^n\cdot d_u(x,y)
$$
Since both $J^u$ and $f_{\mu}$ are H\"older continuous
on unstable fibers, see (\ref{JHc}) and (\ref{ffG}), we have
\be
|\log J^u(T^{-n}y) - \log J^u(T^{-n}x)| \leq
C_J\cdot C_T^{\alpha}\lambda_T^{\alpha n}[d_u(x,y)]^{\alpha}
\label{JJnxy}
\ee
and
\be
|\log f_{\mu}(T^{-n}x) - \log f_{\mu}(T^{-n}y)| \leq
G\cdot C_T^{\alpha}\lambda_T^{\alpha n}[d_u(x,y)]^{\alpha}
\label{ffnxy}
\ee
Hence, the ratio in (\ref{fnxy}) converges, as $n\to\infty$, to
$$
r(x,y)=\lim_{n\to\infty} f_n(x)/f_n(y)
=\lim_{n\to\infty} J^u_n(T^{-n}y)/J^u_n(T^{-n}x)
$$
Moreover, this convergence is uniformly exponential in $n$:
$$
\left |\log[f_n(x)/f_n(y)] - \log r(x,y)\right |
\leq (c_1+c_2G)\lambda_T^{\alpha n}
$$
with some $c_1,c_2$ independent of $x,y,U,\mu$ and $G$.
We define a function $\rho^u_U(x)$ on $U$ by
$$
\rho_U^u(x) = \left (\int_U r(x,x_0)\, dm_U(x)\right )^{-1}
r(x,x_0)
$$
for any $x_0\in U$. The function $\rho^u_U(x)$ so defined
does not depend on $x_0$ and is a density of a probability
measure, $\nu_U^u$, on $U$. It is a direct calculation that
\be
\rho_U^u(x) = \lim_{n\to\infty} f_n(x)
\label{rholimf}
\ee
and
\be
|\log f_n(x) - \log \rho_U^u(x)|\leq (c_3+c_4G)\lambda_T^{\alpha n}
\label{logfrho}
\ee
with some $c_3,c_4$ independent of $x,y,U,\mu$ and $G$.
Obviously, the function $\rho_U^u(x)$ is bounded away from
zero and infinity, and it is H\"older continuous with the
exponent $\alpha$:
\be
|\log \rho_U^u(x)-\log \rho_U^u(y)|
\leq G_{\ast}\cdot [d_u(x,y)]^{\alpha}
\label{rhoHG}
\ee
with
$$
G_{\ast} = C_J\cdot C_T^{\alpha}\cdot
\sum_{n=0}^{\infty}\lambda_T^{\alpha n}
$$
which is independent of $U$.
The conditional invariance (\ref{rhoci}) now follows from the fact that
$$
f_n(x) = \mu_{n,U}(T^{-1} U') \cdot J^u(x) \cdot f_{n+1} (Tx)
$$
for all $x\in U$, $Tx\in U'$, which is just a particular
case of (\ref{ffJ}). Taking the limit as $n\to\infty$ yields
(\ref{rhoci}).
The uniqueness of the conditionally invariant family of measures
follows from the convergence to it of any other family of measures
with H\"older continuous densities on unstable fibers, under
the iterates of $T_\ast$, due to (\ref{rholimf}).
The restriction of the conditionally invariant family of measures
$\nu^u_U$ to ${\cal U}_+$ will then satisfy Theorem~\ref{tm1}.
Theorem~\ref{tm1} is now proved. \medskip
We now establish a few useful lemmas.
\begin{lemma}
There is a constant $G_0>0$, and for any $G>0$ there is
an integer $n_G\geq 1$ such that if $\mu\in{\cal H}(G)$,
then $T^n_\ast\mu\in {\cal H}(G_0)$ for all $n\geq n_G$.
\label{lmGG0}
\end{lemma}
{\it Proof}. It follows from (\ref{fnxy})-(\ref{ffnxy})
that if $\mu\in{\cal H}(G)$, then $T_\ast^n\mu\in{\cal H}
(G_n)$ with
$$
G_n\leq G\cdot C_T^{\alpha}\lambda_T^{\alpha n} + G_{\ast}
$$
Lemma~\ref{lmGG0} is then established for any $G_0> G_{\ast}$.
Lemma~\ref{lmGG0} means the following. If the densities of
the conditional measures $\mu_U$ on $U\in{\cal U}$ oscillate
wildly ($G$ is big), then the map $T$ stretching unstable
fibers will quickly `smooth out' those densities. In fact,
the H\"older constant $G_n$ decreases basically like a
geometric progression as $n$ grows. There is a natural
bound, $G_{\ast}$, however, under which the values of $G_n$ will
not drop.
\begin{lemma}
The function $\rho_U^u(x)$ and its logarithm are H\"older
continuous (with some exponent $\alpha'>0$) on every Markov
rectangle $R_i\in{\cal R}'$.
\label{lmHrho}
\end{lemma}
{\it Proof}. The H\"older continuity of $\rho_U^u(x)$ along
every unstable fiber $U\in{\cal U}'$ (with the exponent
$\alpha$) was established by (\ref{rhoHG}).
Its H\"older continuity along stable fibers (with some
positive exponent) follows from the H\"older continuity
of $J^u(x)$ along stable manifolds and the H\"older
continuity of the holonomy map (\ref{ac2}). \medskip
Let $\mu\in{\cal H}(G_0)$. For any $n\geq 0$ and $U\in {\cal U}_n$
denote by $\mu_{n,U}$ the measure $\mu_n=T^n_\ast\mu$
conditioned on $U$.
\begin{lemma}
For any $U\in{\cal U}_n$ the above measure $\mu_{n,U}$ is equivalent
to $\nu^u_U$ and
$$
e^{-c\lambda^n}\leq \frac{d\mu_{n,U}}{d\nu_U^u} \leq e^{c\lambda^n}
$$
where $c>0$ and $\lambda\in (0,1)$ are independent of $U,n,\mu$.
\label{lmmunun}
\end{lemma}
{\it Proof}. This follows from (\ref{logfrho}) with $\lambda=
\lambda_T^{\alpha}$ and $c=c_3+c_4G_0$. \medskip
In the notations of the previous lemma, let $m\geq 0$ and
$B\in{\cal R}_m$ be an atom of the partition ${\cal R}_m$
of the set $M_m$, and $U,U'\subset B$ two unstable
fibers. Let $A\subset U$
and $A'\subset U'$ be two canonically isomorphic Borel
subsets, i.e. $A'=h_z^s(A)$ for any $z\in U'$.
\begin{lemma}
For any $n\geq m$ we have
\be
e^{-c\lambda^m}\leq \frac{\nu_U^u(A)}{\nu_{U'}^u(A')} \leq e^{c\lambda^m}
\label{AA'+}
\ee
and
\be
e^{-c\lambda^m}\leq \frac{\mu_{n,U}(A)}{\mu_{n,U'}(A')} \leq e^{c\lambda^m}
\label{AA'n}
\ee
with some $c>0$ and $\lambda\in (0,1)$ independent of $U,n,\mu$.
\label{lmAA}
\end{lemma}
{\it Proof}. First, note that
$$
d_{\cal U}(U,U')\leq D_sC_T\lambda_T^m
$$
where $D_s$ is the maximum diameter of stable fibers $S\in{\cal S}'$.
The bound (\ref{AA'+}) now follows from Lemma~\ref{lmHrho}
and the H\"older continuity of the Jacobian of the holonomy map
(\ref{ac2}). The bound (\ref{AA'n}) follows from (\ref{AA'+})
and the previous lemma. \medskip
{\it Convention}. Without loss of generality, we can assume that
the values of $c$ and $\lambda$ are the same in both lemmas. \medskip
The next three statements involve the mixing power $k_0$ of
the transition matrix A.
\begin{lemma}
There is a constant $\beta > 0$ such that for any
$\mu \in {\cal H}(G_0)$ and $R_j\in{\cal R}$ we have
\be
\inf_{U \in {\cal U}} \mu_U (T^{-k_0}
(R_j \cap M_{k_0})\cap U) \geq \beta
\label{beta1}
\ee
\end{lemma}
{\it Proof.} In virtue of the mixing assumption,
for any $U \in {\cal U}$ and any $R_j \in {\cal R}$ we have
$\nu_U^u(T^{-k_0}(R_j \cap M_{k_0}) \cap U) > 0.$ For every
$i=1,\ldots,I$ we pick an arbitrary `representative' fiber
$\tilde{U}_i \subset R_i$ and from Lemmas~\ref{lmmunun} and
\ref{lmAA} it follows that for any other $ U \subset R_i$ we have
$$
\mu_U (T^{-k_0}(R_j \cap M_{k_0}) \cap U) \geq
e^{-2c} \nu_{\tilde{U}_i}^u (T^{-k_0}(R_j \cap M_{k_0}) \cap \tilde{U}_i)
$$
The bound (\ref{beta1}) follows with
$$
\beta = e^{-2c} \cdot \min_{j,i}
\nu_{\tilde{U}_i}^u (T^{-k_0}(R_j \cap M_{k_0}) \cap \tilde{U}_i) > 0
$$
\begin{lemma}
There is a $\beta > 0$ such that for any
$\mu \in {\cal H}(G_0)$ and $R_j\in{\cal R}$
and all $k\geq k_0$ we have
\be
\inf _{U \in {\cal U}} \mu_U (T^{-k}
(R_j \cap M_k)\cap U) \geq \beta\cdot \sup _{U \in {\cal U}}
\mu_U (T^{-k}(R_j \cap M_k) \cap U)
\ee
\end{lemma}
{\it Proof.}
Put $m = k - k_0.$ For $U\in {\cal U}$, let $T^{k_0}(U\cap M_{-k_0}) =
U_1 \cup \cdots \cup U_L$ for some fibers $U_l\in {\cal U}_{k_0}$. From
(\ref{mucond}) we obtain
\begin{eqnarray*}
\mu_U(T^{-k}(R_j \cap M_k)\cap U) &=& \sum_{l=1}^L
\mu_U(T^{-k_0}U_l) \cdot (T^{k_0}_\ast\mu)_{U_l}
(T^{-m}(R_j \cap M_m) \cap U_l) \\
&=& \sum_{i=1}^I \sum _{l:U_l \subset R_i}
\mu_U(T^{-k_0}U_l) \cdot
(T^{k_0}_\ast\mu)_{U_l} (T^{-m}(R_j \cap M_{m}) \cap U_l)
\end{eqnarray*}
Using once again `representatives' $\tilde{U}_i \subset R_i$ and
Lemmas~\ref{lmmunun} and \ref{lmAA}, we get an upper bound,
\begin{eqnarray*}
\mu_U(T^{-k}(R_j \cap M_k) \cap U) &\leq&
e^{2c} \cdot \sum _{i=1}^I
\mu_U(T^{-k_0}(R_i \cap M_{k_0}) \cap U)
\cdot \nu_{\tilde{U}_i}^u(T^{-m}(R_j \cap M_{m})
\cap \tilde{U}_i) \\
&\leq& e^{2c} \cdot \sum _{i=1}^I
\nu_{\tilde{U}_i}^u (T^{-m}(R_j \cap M_{m}) \cap \tilde{U}_i)
\end{eqnarray*}
By invoking (\ref{beta1}), we get a lower bound,
\begin{eqnarray*}
\mu_U(T^{-k}(R_j \cap M_k) \cap U) &\geq&
e^{-2c} \cdot \sum _{i=1}^I
\mu_U(T^{-k_0}(R_i \cap M_{k_0}) \cap U)
\cdot \nu_{\tilde{U}_i}^u (T^{-m}(R_j \cap M_{m})
\cap \tilde{U}_i) \\
&\geq& e^{-2c} \cdot \beta \cdot \sum _{i=1}^I
\nu_{\tilde{U}_i}^u(T^{-m}(R_j \cap M_{m}) \cap \tilde{U}_i)
\end{eqnarray*}
Then we decrease the value of $\beta$ by a factor of $e^{-4c}$
and complete the proof.
\begin{corollary}
There is a $\beta > 0$ such that for any
$\mu \in {\cal H}(G_0)$ and any s-subrectangle
$D \in R_i$ (in particular for any atom $D \in
{\cal R}_{-m}$, $m \geq 0$) and all $k\geq k_0$ we have
\be
\inf_{U \in {\cal U}} \mu_U (T^{-k}
(D \cap M_k) \cap U) \geq \beta\cdot \sup _{U \in {\cal U}}
\mu_U (T^{-k}(D \cap M_k) \cap U)\; ,
\ee
\label{cormu}
\end{corollary}
Without loss of generality, the values of $\beta\in (0,1)$ are
assumed to be the same in these statements.
\section{Conditionally invariant measure $\mu_+$ on $M_+$}
\label{secCIM}
\setcounter{equation}{0}
In this section we prove Theorem~\ref{tm2}. First, we describe
the concepts on which our proofs in this and the following
sections are based.
We invoke the Perron-Frobenius theorem for positive matrices
and related techniques developed by Sinai and \v{C}encova.
One can think of the matrices we will work with as
finite-dimensional approximations to the usual
Perron-Frobenius operator on (infinite-dimensional) space
of measures. To clarify this connection, let us sketch
how these matrix techniques work for an arbitrary
measurable transformation $T: M\to M$.
The adjoint operator, $T_\ast$, on the space of measures
on $M$ acts by $T_\ast\mu(A)=\mu(T^{-1}A)$ for any measurable
subset $A\subset M$. Constructions of the invariant measures
and studies of their statistical properties usually rely
on the convergence of the sequence of measures $\mu_n=T_\ast^n\mu$,
as $n\to\infty$, to a $T$-invariant measure $\mu_0$ on $M$. To study this
convergence, one can take an increasing sequence of finite
partitions $\xi_1<\xi_2<\cdots$ of $M$,
where $\xi_m=\{A_1^{(m)},\ldots,A_{k_m}^{(m)}\}$, that converges
to a partition into single points. Then one can represent
any measure $\mu$ on $M$ by a sequence of (row) vectors
${\bf p}_m(\mu)$ with components $({\bf p}_m(\mu))_i=
\mu(A_i^{(m)})$, $1\leq i\leq k_m$. A probability measure
$\mu$ is represented by unit vectors, $|{\bf p}_m(\mu)|=1$,
the norm $|\cdot|$ for (row) vectors being defined below. Then,
under certain regularity conditions that we leave out here,
the weak convergence of a sequence of measures $\mu_n$,
as $n\to\infty$, to a measure $\mu_0$ is equivalent to the
componentwise convergence of the sequence of vectors
${\bf p}_m(\mu_n)$, as $n\to\infty$, to the vector
${\bf p}_m(\mu_0)$ for every $m\geq 1$.
For a fixed $m\geq 1$ and a measure $\mu$, the vectors
${\bf p}_m(\mu)$ and ${\bf p}_m(T_\ast\mu)$ are related by
\be
{\bf p}_m(T_\ast\mu)={\bf p}_m(\mu)\Pi_m(\mu)
\label{ppPi0}
\ee
where $\Pi_m(\mu)$ is a $k_m\times k_m$ matrix with components
$(\Pi_m(\mu))_{ij}=\mu(T^{-1}A_j^{(m)}\cap A_i^{(m)})/
\mu(A_i^{(m)})$ (we assume $\mu(A_i^{(m)})\neq 0$ for all
$m,i$). Therefore, we have
\be
{\bf p}_m(T_\ast^n\mu)={\bf p}_m(\mu)\Pi_m(\mu)
\cdot\Pi_m(T_\ast\mu)\cdots\Pi_m(T_\ast^{n-1}\mu)
\label{Pin0}
\ee
If the partitions $\xi_m$ have nice geometric properties
(e.g., they are Markov partitions or alike), then the
matrices in (\ref{Pin0}) are very close to each other,
and so one can replace their product by $\tilde{\Pi}_m^n$ with
some matrix $\tilde{\Pi}_m$ close to all of the matrices in
(\ref{Pin0}). All these matrices have nonnegative
entries, and usually some power, $\tilde{\Pi}_m^{n_m}$, $n_m\geq 1$,
has all positive entries. In that case Perron-Frobenius theorem
for positive matrices, see Appendix, applies. It provides
a (unique) positive unit eigenvector, $\tilde{\bf p}_m$, for the
matrix $\tilde{\Pi}_m$, corresponding to its largest eigenvalue
$\tilde{\lambda}_m>0$ (of multiplicity one). We call $\tilde{\bf p}_m$
the Perron eigenvector and $\tilde{\lambda}_m$ the Perron eigenvalue.
Moreover, for any other positive unit vector ${\bf q}_m$ the
sequence of vectors ${\bf q}\tilde{\Pi}_m^n$ converges, as $n\to\infty$,
to $\tilde{\bf p}_m$ (exponentially fast in $L=n/n_m$). These facts
can be used to prove that for some suitable probability measures
$\mu$ the vectors ${\bf p}_m(T_\ast^n\mu)$ will be close to
the Perron eigenvector ${\bf p}_m$ for large enough $n$.
Now, the limit of the Perron eigenvectors $\tilde{\bf p}_m$,
as $m\to\infty$, defines a measure $\mu_0$ on $M$, which
will be the weak limit of $T^n_\ast\mu$, as $n\to\infty$.
The details of this scheme depend on the specific dynamical
system and specific sequence of partitions $\xi_m$. Various
versions of this matrix method work well for systems with
sufficiently strong hyperbolic or expanding properties.
We prefer this matrix machinery to the Perron-Frobenius
functional operator techniques for two reasons. First,
it allows us to compute some characteristics of limit
invariant measures which are not readily available otherwise,
like the ones in our Propositions~\ref{pr1} and \ref{pr2}.
Second, this machinery looks flexible enough to work well
for nonuniformly hyperbolic systems, in particular billiards,
where other techniques fail.
We now make a few conventions. As it is already clear, we
will study vectors ${\bf p}$ whose components correspond
to atoms $A\in\xi$ of some finite partitions $\xi$ of $M$.
We will not enumerate or even order those atoms, so our
`vectors' will be just collections of numbers, denoted by
${\bf p}_A$, $A\in\xi$. Likewise, we will work with
`matrices' $\Pi$ whose entries correspond to (ordered)
pairs $A,B$ of atoms of the partition $\xi$, and we
denote them by $\Pi_{A,B}$. Despite the lack of order,
we think of our vectors as row vectors, and the product
${\bf q}={\bf p}\Pi$ is naturally defined to be another
(row) vector with components
$$
{\bf q}_B = \sum_{A\in\xi}{\bf p}_A\Pi_{A,B}
$$
Next, for any (row) vector ${\bf p}$ we define its norm by
$$
|{\bf p}|=\sum_{A\in\xi}|{\bf p}_A|
$$
and we call a positive vector $\bf p$ a unit vector if
$|{\bf p}|=1$. For a positive matrix $\Pi$ the ratio of
rows, $P$, is defined by
$$
P=\max_{A',A'',B\in\xi} \Pi_{A',B}/\Pi_{A'',B}
$$
Any two positive matrices, $\Pi$ and $\Pi'$, are said
to be close with the constant of proximity $R\geq 1$
if for all $A,B\in\xi$ we have
$$
R^{-1}\leq\Pi_{A,B}/\Pi_{A,B}'\leq R
$$
We now begin the proof of Theorem~\ref{tm2}. Recall that
any measure $\mu\in{\cal M}_+^u$ is supported on $M_+$,
has conditional measures $\nu_U^u$ on fibers $U\in{\cal U}_+$
and is then completely defined by its factor measure $\hat{\mu}$
on ${\cal U}_+$. Due to Theorem~\ref{tm1} the operator $T_\ast$
and the transformation $T_+$ leave ${\cal M}_+^u$ invariant.
The conditionally invariant measures $\mu\in{\cal M}_+^u$
are fixed points of the transformation $T_+$.
Consider the increasing sequence of partitions
${\cal R}_1^+ < {\cal R}_2^+ <\cdots$ of $M_+$
defined in Sect.~\ref{secBAD}
Any measure $\mu\in{\cal M}_+^u$ can be represented by
a sequence of (row) vectors
$$
{\bf p}_m(\mu)=\{\mu(B):\, B\in{\cal R}_m^+\}
$$
The weak convergence of a sequence of measures,
$\mu_n\to\mu$, in ${\cal M}_+^u$, is equivalent to
the componentwise convergence ${\bf p}_m(\mu_n)\to
{\bf p}_m(\mu)$, as $n\to\infty$, for every $m\geq 1$.
According to (\ref{ppPi0}), for any $\mu\in{\cal M}_+^u$
and $k\geq 1$ we have
\be
{\bf p}_m(T_\ast^k\mu)={\bf p}_m(\mu)\Pi_m^{(k)}(\mu)
\label{ppPi}
\ee
where $\Pi_m^{(k)}(\mu)$ is a matrix with components
\be
\{\mu(T^{-k}[B''\cap M_k]\cap B')/\mu(B'):\, B',B''\in{\cal R}_m^+\}
\label{Pimk}
\ee
(here $B'$ is the `row number' and $B''$ is the `column number').
Note that if $\mu'$ is proportional to $\mu$, $\mu'=a\cdot\mu$
with some constant $a>0$, then $\Pi_m^{(k)}(\mu')=\Pi_m^{(k)}(\mu)$
for all $m,k\geq 1$. \medskip
{\it Remark}. Some entries of $\Pi_m^{(k)}(\mu)$
may not be defined by (\ref{Pimk})
if $\mu(B')=0$. In that case we can define them arbitrarily
without doing any harm to the equation (\ref{ppPi}). We simply
pick a $U\subset B'$ and set the component (\ref{Pimk}) to
$\nu_U^u(T^{-k}[B''\cap M_k]\cap U)$. \medskip
Next, the equation (\ref{ppPi}) directly implies that
\be
{\bf p}_m(T_+^k\mu)=
\frac{{\bf p}_m(\mu)\Pi_m^{(k)}(\mu)}
{|{\bf p}_m(\mu)\Pi_m^{(k)}(\mu)|}
\label{T+k}
\ee
\begin{lemma}
For any $m\geq 1$
and $k\geq k_0$ the matrices $\Pi_m^{(m+k)}(\mu)$, $\mu\in
{\cal M}_+^u$, satisfy two conditions:\\
{\rm (i)} the ratio of its rows is bounded by $P=\beta^{-1}$:
\be
\beta \leq
\frac{\mu(T^{-m-k}[B''\cap M_{m+k}]\cap B_1')/\mu(B_1')}
{\mu(T^{-m-k}[B''\cap M_{m+k}]\cap B_2')/\mu(B_2')}
\leq \beta^{-1}
\label{rr}
\ee
for all $B_1',B_2',B''\in{\cal R}_m^+$;\\
{\rm (ii)} the matrices $\Pi_m^{(m+k)}(\mu_1)$ and
$\Pi_m^{(m+k)}(\mu_2)$, for any $\mu_1,\mu_2\in{\cal M}_+^u$,
are close to each other with the constant of proximity
$R=\exp(c\lambda^m)$, i.e.
\be
e^{-c\lambda^m} \leq
\frac{\mu_1(T^{-m-k}[B''\cap M_{m+k}]\cap B')/\mu_1(B')}
{\mu_2(T^{-m-k}[B''\cap M_{m+k}]\cap B')/\mu_2(B')}
\leq e^{c\lambda^m}
\label{cp}
\ee
for all $B',B''\in{\cal R}_m^+$.
\label{lm2+}
\end{lemma}
{\it Proof}. Put ${\cal U}_{+,B}=\{U\in{\cal U}_+:\, U\subset B\}$
for $B\in{\cal R}_m^+$. Then the components of the matrix
$\Pi_m^{(m+k)}(\mu)$ can be expressed by
\be
\frac{\mu(T^{-m-k}[B''\cap M_{m+k}]\cap B')}{\mu(B')}=
\frac{1}{\hat{\mu}({\cal U}_{B',+})}
\int_{{\cal U}_{B',+}}\nu_U^u(T^{-m-k}[B''\cap M_{m+k}]\cap U)\, d\hat{\mu}(U)
\label{Pint}
\ee
For any $B''\in{\cal R}_m^+$ there is an atom $D\in{\cal R}_{-m}$
such that $T^{-m}B''=M_+\cap D$. So, for any $k\geq 0$
we have $T^{-m-k}[B''\cap M_{m+k}]=T^{-k}[D\cap M_k]\cap M_+$.
Now, the estimate (\ref{rr}) follows from (\ref{Pint}) and
Corollary~\ref{cormu}.
To prove (\ref{cp}), notice that the set $T^{-k}[D\cap M_k]$
is a finite union of $s$-subrectangles (some atoms of
${\cal R}_{-m-k}$). Thus, for any two unstable fibers
$U,U'\subset B'$ the sets $T^{-k}[D\cap M_k]\cap U$ and
$T^{-k}[D\cap M_k]\cap U'$ are canonically isomorphic, and
Lemma~\ref{lmAA} implies
\be
e^{-c\lambda^m}\leq
\frac{\nu_U^u(T^{-k}[D\cap M_k]\cap U)}
{\nu_{U'}^u(T^{-k}[D\cap M_k]\cap U')}
\leq e^{c\lambda^m}
\label{nuUU'}
\ee
This and (\ref{Pint}) prove (\ref{cp}). Lemma~\ref{lm2+} is proved.
We continue the proof of Theorem~\ref{tm2}. For any $m\geq 1$
and $B\in{\cal R}_m^+$ we pick an arbitrary `representative'
unstable fiber $\tilde{U}_B\subset B$. For any $m\geq 1$, $k\geq 1$,
denote by $\tilde{\Pi}_m^{(k)}$ the matrix with components
\be
\{\nu_{\tilde{U}_{B'}}^u(T^{-k}[B''\cap M_k]\cap \tilde{U}_{B'}):\,
B',B''\in{\cal R}_m^+\}
\label{Pitil}
\ee
Note that $\tilde{\Pi}_m^{(k)}=\Pi_m^{(k)}(\tilde{\mu})$ for any
measure $\tilde{\mu}\in{\cal M}_+^u$ supported on the union of
representative fibers $\tilde{U}_B$, $B\in{\cal R}_m^+$, and
such that $\tilde{\mu}(\tilde{U}_B)>0$ for all $B\in{\cal R}_m^+$.
Thus, the matrix $\tilde{\Pi}_m^{(m+k)}$, $k\geq k_0$, satisfies
the bound (\ref{rr}) on the ratio of rows and is close to any
$\Pi_m^{(m+k)}(\mu)$, $\mu\in{\cal M}_+^u$, with the constant
of proximity $P_m$, see (\ref{cp}).
According to the Perron-Frobenius theorem, provided in Appendix,
the matrix $\tilde{\Pi}_m^{(m+k_0)}$ has a positive unit (row)
eigenvector, $\tilde{\bf p}_m$, corresponding to its largest
eigenvalue.
We put
$$
\gamma=\max\{\lambda,1-\beta/2\}
$$
and fix an $m_0$ such that
$$
(1-\beta)e^{2c\lambda^{m_0}}<\gamma
$$
\begin{proposition}
There is a constant $C_1>0$ such that for all $m\geq m_0$, $m_1=m+k_0$,
and $n\geq m$ we have
$$
|{\bf p}_m(T_+^n\mu)-\tilde{\bf p}_m|
\leq C_1(\gamma^{[n/m_1]} + \lambda^m)
$$
\label{prpp}
\end{proposition}
{\it Proof}. Put $L=[n/m_1]$ and $l=n-m_1L$, so that $n=m_1L+l$,
$0\leq l l\geq 1$ and any vector ${\bf p}_m$ whose
components correspond to atoms $B\in{\cal R}_m^+$,
we denote by ${\bf p}_{m\downarrow l}$ the vector with
components
$$
({\bf p}_{m\downarrow l})_{B'}=\sum_{B\subset B'} ({\bf p}_m)_B
$$
corresponding to atoms $B'\in{\cal R}_l^+$.
\begin{proposition}
For any $l\geq 1$ there exists a limit
$$
{\bf r}_l=\lim_{m\to\infty}\tilde{\bf p}_{m\downarrow l}
$$
The sequence of vectors ${\bf r}_l$ satisfies the equations
\be
|{\bf r}_l|=1\ \ \ \ {\rm and}
\ \ \ \ {\bf r}_{l\downarrow k}={\bf r}_k
\label{rrll}
\ee
for all $l>k\geq 1$. Moreover, for all $m\geq l$ we have
\be
|\tilde{\bf p}_{m\downarrow l}-{\bf r}_l|\leq 4C_1\gamma^m
\label{prl}
\ee
\label{prrl}
\end{proposition}
{\it Proof}. Let $\mu\in {\cal M}_+^u$, $l\geq 1$ and
$n>m (\geq l)$ be large enough. For any $s\geq n(n+k_0)$
Proposition~\ref{prpp} yields
$$
|{\bf p}_m(T^s_+\mu)-\tilde{\bf p}_m|\leq 2C_1\gamma^m
$$
and
$$
|{\bf p}_n(T^s_+\mu)-\tilde{\bf p}_n|\leq 2C_1\gamma^n
$$
By using an obvious fact that ${\bf p}_{n\downarrow m}(T_+^s\mu)
={\bf p}_{m}(T_+^s\mu)$, we get
\be
|\tilde{\bf p}_{m\downarrow l}
-\tilde{\bf p}_{n\downarrow l}| \leq
|\tilde{\bf p}_{m}
-\tilde{\bf p}_{n\downarrow m}|
\leq 4C_1\gamma^m
\label{pmnl}
\ee
Thus, for any $l\geq 1$ the sequence of vectors
$\tilde{\bf p}_{n\downarrow l}$, $n\geq 1$, is a
Cauchy sequence, so it converges to a vector that
we denote by ${\bf r}_l$. Now (\ref{prl}) follows
from (\ref{pmnl}). It, in turn, readily implies (\ref{rrll}).
Proposition~\ref{prrl} is proved.
Due to (\ref{rrll}), the sequence of vectors ${\bf r}_l$,
$l\geq 1$, specifies a probability measure $\mu_+\in{\cal M}_+^u$
such that ${\bf p}_l(\mu_+)={\bf r}_l$ for all $l\geq 1$.
\begin{corollary}
For any measure $\mu\in{\cal M}_+^u$ the sequence $\{T_+^n\mu\}$
weakly converges, as $n\to\infty$, to $\mu_+$. Moreover, for
all $l\geq 1$ and $n>\max\{m_0^2,l^2\}$ we have
$$
|{\bf p}_l(T^n_+\mu)-{\bf p}_l(\mu_+)|\leq C_2\gamma^{\sqrt{n}},
$$
with some constant $C_2>0$.
\end{corollary}
Clearly, $T_+\mu_+=\mu_+$ and
$$
T_\ast\mu_+=\lambda_+\mu_+\ \ \ \ {\rm with}
\ \ \ \ \lambda_+=\mu_+(M_{-1})
$$
Theorem~\ref{tm2} is now proved.
\section{Limit theorems for the measure $\mu_+$}
\label{secLTmu+}
\setcounter{equation}{0}
Here we prove Theorems~\ref{tm3} and \ref{tm4}. The proofs
require the extension of the previous analysis from the
class of measures ${\cal M}_+^u$ to the larger classes ${\cal M}_n$.
For any measure $\mu\in{\cal M}_n$ we denote by $\mu_U$ its
conditional measures on unstable fibers $U\subset M_n$,
and by $\hat{\mu}$ its factor measure on ${\cal U}_n$.
For any measure $\mu\in{\cal M}_n$ we can consider a
finite sequence of vectors,
$$
{\bf p}_m(\mu)=\{\mu(B):\, B\in{\cal R}_m\}
$$
for $1\leq m\leq n$. Note that if we have a sequence of
measures $\mu_n\in{\cal M}_n$, for which the sequence
of factor measures
$\hat{\mu}_n$ weakly converges, then its limit is a
factor measure $\hat{\mu}$ of some $\mu\in{\cal M}_+^u$.
This is equivalent to a componentwise convergence
${\bf p}_m(\mu_n)\to{\bf p}_m(\mu)$, as $n\to\infty$,
for every $m\geq 1$.
According to (\ref{ppPi0}), for any $n\geq m\geq 1$, $k\geq 1$
and $\mu\in{\cal M}_n$ we have $T_\ast^k\mu\in
{\cal M}_{n+k}$ and
$$
{\bf p}_m(T_\ast^k\mu)={\bf p}_m(\mu)\cdot\Pi_m^{(k)}(\mu)
$$
where $\Pi_m^{(k)}(\mu)$ is the matrix with components
$$
\{\mu(T^{-k}[B''\cap M_k]\cap B')/\mu(B'):\, B',B''\in{\cal R}_m\}
$$
(here, as in (\ref{Pimk}), $B'$ is the `row number' and $B''$
is the `column number'). The equation (\ref{T+k}) holds
without changes. The remark before Lemma~\ref{lm2+} also applies,
but now $B',B''$ are atoms of ${\cal R}_m$ instead of
${\cal R}_m^+$. The following lemma is an analog of
Lemma~\ref{lm2+}:
\begin{lemma}
Let $\mu\in{\cal H}(G)$ with some $G>0$. Then for any $m\geq 1$,
$k\geq k_0$ and $n\geq m+ n_G$
the matrix $\Pi_m^{(m+k)}(\mu_n)$ for the measure
$\mu_n=T_\ast^n\mu\in {\cal M}_n$ satisfies two
conditions:\\
{\rm (i)} the ratio of its rows is bounded by $\beta^{-1}$:
\be
\beta \leq
\frac{\mu_n(T^{-m-k}[B''\cap M_{m+k}]\cap B_1')/\mu_n(B_1')}
{\mu_n(T^{-m-k}[B''\cap M_{m+k}]\cap B_2')/\mu_n(B_2')}
\leq \beta^{-1}
\label{rr1}
\ee
for all $B_1',B_2',B''\in{\cal R}_m$;\\
{\rm (ii)} for all $B',B''\in{\cal R}_m$ we have
\be
e^{-2c\lambda^m} \leq
\frac{\mu_n(T^{-m-k}[B''\cap M_{m+k}]\cap B')/\mu_n(B')}
{\mu_+(T^{-m-k}[B''\cap M_{m+k}]\cap B')/\mu_+(B')}
\leq e^{2c\lambda^m}
\label{cp1}
\ee
\label{lm2n}
\end{lemma}
{\it Proof}. Note that $\mu_n\in{\cal H}(G_0)$ due to
Lemma~\ref{lmGG0}.
for $B\in{\cal R}_m$. Then the components of the matrix
$\Pi_m^{(m+k)}(\mu_n)$ can be expressed by
\be
\frac{\mu_n(T^{-m-k}[B''\cap M_{m+k}]\cap B')}{\mu_n(B')}=
\frac{1}{\hat{\mu}_n({\cal U}_{B'})}
\int_{{\cal U}_{B'}}\mu_{n,U}(T^{-m-k}[B''\cap M_{m+k}]\cap U)\,
d\hat{\mu}_n(U)
\label{Pint1}
\ee
For any $B''\in{\cal R}_m$ the set $D=T^{-m}B''$
is an atom of ${\cal R}_{-m}$. So, for any $k\geq 0$
we have $T^{-m-k}[B''\cap M_{m+k}]=T^{-k}[D\cap M_k]$.
Now, the estimate (\ref{rr1}) follows from (\ref{Pint1})
and Corollary~\ref{cormu}.
The first part of the proof of (\ref{cp1}) repeats word by word
that of (\ref{cp}), but then (\ref{nuUU'}) must be combined
with Lemma~\ref{lmmunun}. This gives
$$
e^{-2c\lambda^m}\leq
\frac{\mu_{n,U}(T^{-k}[D\cap M_k]\cap U)}
{\nu_{U'}^u(T^{-k}[D\cap M_k]\cap U')}
\leq e^{2c\lambda^m}
$$
for all $U,U'\subset B'$.
This and (\ref{Pint}) with (\ref{Pint1}) prove (\ref{cp1}).
Lemma~\ref{lm2n} is proved.
The following proposition is an analog of Proposition~\ref{prpp}:
\begin{proposition}
There is a constant $C_3>0$ such that for all $m\geq m_0$, $m_1=m+k_0$,
$n\geq m$ and any $G>0$, $\mu\in{\cal H}(G)$ we have
$$
|{\bf p}_m(T_+^{n+n_G}\mu)-\tilde{\bf p}_m|
\leq C_3(\gamma^{[n/m_1]} + \lambda^m)
$$
\label{prpp1}
\end{proposition}
It is enough to prove this for $\mu\in{\cal H}(G_0)$ and $n_G=0$.
The proof then repeats that of Proposition~\ref{prpp} word by word.
Combining Propositions~\ref{prpp1} and \ref{prrl} gives
\begin{corollary}
Let $G>0$ and $\mu\in{\cal H}(G)$.
For every $l\geq 1$ the sequence of vectors ${\bf p}_l
(T_+^n\mu)$ converges to ${\bf r}_l={\bf p}_l(\mu_+)$.
Moreover, for all $n\geq\max\{m_0^2,l^2\}$ we have
$$
|{\bf p}_l(T^{n+n_G}_+\mu)-{\bf p}_l(\mu_+)|
\leq C_4\gamma^{\sqrt{n}},
$$
with some constant $C_4>0$.
\label{crppmu}
\end{corollary}
\begin{corollary}
Let $G>0$ and $\mu\in{\cal H}(G)$.
The sequence of factor measures $\hat{\mu}_n$, where
$\mu_n=T_+^n\mu$, weakly converges, as $n\to\infty$,
to the factor measure $\hat{\mu}_+$ on ${\cal U}_+$.
\end{corollary}
We now begin the proofs of Theorems~\ref{tm3} and \ref{tm4}.
\begin{proposition}
Let $G>0$ and $\mu\in{\cal H}(G)$. The sequence of
measures $\mu_n=T_+^n\mu$, $n\geq 1$, weakly converges
to the measure $\mu_+$.
\label{prlim+}
\end{proposition}
{\it Proof}. Since $T^{n_G}_+\mu\in{\cal H}(G_0)$, we may
assume that $\mu\in{\cal H}(G_0)$. It is enough to show that
for every $l\geq 0$, $k\geq 0$, every atom $B\in{\cal R}_l$
and every atom $D\in{\cal R}_{-k}$ we have a convergence
\be
\mu_n(B\cap D)\to\mu_+(B\cap D)\ \ \ {\rm as}\ \ n\to\infty
\label{muBD}
\ee
In the following, $B$ and $D$ may be also unions of some
atoms of ${\cal R}_l$ and ${\cal R}_{-k}$, respectively, in
one Markov rectangle $R_i\in\cal R$.
Let $n\geq\max\{ m_0^2,l^2\}$. Put $m=[\sqrt{n}]$.
Then $B$ is the union of some atoms of ${\cal R}_m$,
let us denote them by $B_1,\ldots,B_L$. In every $B_i$ we pick a
`representative' fiber $\tilde{U}_i\subset B_i$. Note that
$$
\mu_n(B\cap D) = \int_{{\cal U}_B}\mu_{n,U}(D\cap U)
\, d\hat{\mu}_n(U)
$$
where $\mu_{n,U}$ is $\mu_n$ conditioned on the fiber $U$
and $\hat{\mu}_n$ is its factor measure on ${\cal U}_n$.
Due to Lemmas~\ref{lmmunun} and \ref{lmAA} we have
\begin{eqnarray*}
\mu_n(B\cap D) &\leq& e^{c\lambda^m}\sum_{i=1}^L
\mu_{n,\tilde{U}_i}(D)\cdot\mu_n(B_i) \\
&\leq& e^{2c\lambda^m}\sum_{i=1}^L
\mu^u_{\tilde{U}_i}(D)\cdot\mu_n(B_i)
\end{eqnarray*}
The corresponding estimate from below with negative exponents
also holds. In the same way Lemma~\ref{lmAA} yields
$$
\mu_+(B\cap D) \leq
e^{c\lambda^m}\sum_{i=1}^L
\mu^u_{\tilde{U}_i}(D)\cdot\mu_+(B_i)
$$
and the corresponding lower bound with the negative exponent.
Now, Corollary~\ref{crppmu}, in which we can set $l=m$, implies
\begin{eqnarray}
\mu_n(B\cap D) &\leq& e^{3c\lambda^m}
\mu_+(B\cap D) + e^{3c}\cdot\sup_{U\in{\cal U}_B}\mu_U^u(D)
\cdot |{\bf p}_m(\mu_n) - {\bf p}_m(\mu_+)|\nonumber\\
&\leq& e^{3c\lambda^m}
\mu_+(B\cap D) + e^{3c}\cdot\sup_{U\in{\cal U}_B}\mu_U^u(D)
\cdot C_4\gamma^m
\label{mu+mu1}
\end{eqnarray}
and, respectively,
\be
\mu_n(B\cap D) \geq e^{-3c\lambda^m}
\mu_+(B\cap D) - e^{3c}\cdot\sup_{U\in{\cal U}_B}\mu_U^u(D)
\cdot C_4\gamma^m
\label{mu+mu2}
\ee
These two bounds readily imply (\ref{muBD}).
Proposition~\ref{prlim+} is proved.
The first statement of Theorem~\ref{tm3} now
follows immediately. To prove the second, it is
enough to establish the following:
\begin{proposition}
For any $G>0$ and $\mu\in{\cal H}(G)$ the limit
\be
c[\mu ]=\lim_{n\to\infty} \lambda_+^{-n}||T_\ast^n\mu ||
\label{cmu}
\ee
exists and $c[\mu]>0$.
\label{prcmu}
\end{proposition}
{\it Proof}. Clearly,
\be
||T_\ast^n\mu||=\prod_{i=0}^{n-1}||T_\ast(T^i_+\mu)||
=\prod_{i=0}^{n-1}(T^i_+\mu)(M_{-1})
\label{Tnmup}
\ee
Let $\mu_n=T^n_+\mu$ for $n\geq 0$.
It is enough to show that the series
$$
\sum_{n=0}^{\infty}\log\frac{\mu_n(M_{-1})}{\lambda_+}=
\sum_{n=0}^{\infty}\log\frac{\mu_n(M_{-1})}{\mu_+(M_{-1})}
$$
converges. Note that $\mu_{n_G}\in
{\cal H}(G_0)$, so we may again assume that $\mu\in{\cal H}(G_0)$.
Now, let $R_i\in{\cal R}$ and $D=R_i\cap M_{-1}$.
Let $P_0^{-1}=\min_j \mu_+(R_j\cap M_{-1})$.
Then the bounds (\ref{mu+mu1}) and (\ref{mu+mu2})
combined with (\ref{AA'+}) imply that
$$
e^{-3c\lambda^m} - P_0\cdot C_4\gamma^m \leq
\mu_n(D)/\mu_+(D) \leq e^{3c\lambda^m} + P_0\cdot C_4\gamma^m
$$
for all $n>m_0^2$ with $m=[\sqrt{n}]$. Therefore,
$$
\left | \log\frac{\mu_n(M_{-1})}{\mu_+(M_{-1})}\right |
\leq C_5\gamma^{\sqrt{n}}
$$
with some constant $C_5>0$. Proposition~\ref{prcmu} is proved.
Theorem~\ref{tm3} is then proved also. \medskip
{\it Remark}.
For every $G>0$ the convergence in (\ref{cmu}) is uniform in
$\mu\in{\cal H}(G)$. In particular, if $\mu\in{\cal H}(G_0)$,
then for all $m\geq m_0^2$
\be
|\log c[\mu]-\log(\lambda_+^{-m}||T^m_\ast\mu||)|
\leq C_5\cdot\sum_{n=m}^{\infty}\gamma^{\sqrt{n}}
\label{rm4uni}
\ee
We conclude this section with proofs of Theorem~\ref{tm4}
and Proposition~\ref{pr1}. The first part of Theorem~\ref{tm4}
is a particular case of Proposition~\ref{prlim+}. Next, since
$\mu_U^u\in {\cal H}(G_0)$, Proposition~\ref{prcmu}
applies and ensures the second part of Theorem~\ref{tm4} with
\be
e(U) = c[\mu_U^u] = \lim_{n\to\infty}\lambda_+^{-n}\mu_U^u(M_{-n})
\label{eUlim}
\ee
In virtue of Corollary~\ref{cormu}, the function $e(U)$ is positive
and bounded:
\be
\sup_{U\in{\cal U}}e(U) \leq \beta^{-1}\inf_{U\in{\cal U}}e(U)
\label{ebound}
\ee
This bound and (\ref{rm4uni}) imply the following:
\begin{corollary}
There is $C_6>1$ such that for any $m\geq 0$ and any
$U\in{\cal U}$ we have
$$
C_6^{-1}\leq \lambda_+^{-m}\mu^u_U(M_{-m})\leq C_6
$$
\label{cor4C}
\end{corollary}
Due to (\ref{rm4uni}), the normalization (\ref{enorm})
will follow if we show that for all $n\geq 0$
$$
\lambda_+^{-n}\int_{{\cal U}_+}||T^n_\ast\mu_U^u||\,
d\hat{\mu}_+(U) = 1
$$
This equation is verified as follows:
$$
\int_{{\cal U}_+}||T^n_\ast\mu_U^u||\, d\hat{\mu}_+(U) =
\int_{{\cal U}_+}\mu_U^u(M_{-n})\, d\hat{\mu}_+(U) =
\mu_+(M_{-n}) = \lambda_+^n
$$
Proposition~\ref{pr1} is proved. \medskip
{\it Remark}. There is an alternative proof of Theorem~\ref{tm4},
along the lines of \cite{C86}, based on the following
observation. Recall that the matrix
$\Pi_m^{m+k_0}(\mu_+)$, cf. Sect.~\ref{secCIM}, has the largest
eigenvalue $\lambda_+^{m+k_0}$ and the Perron row eigenvector
${\bf p}_m(\mu_+)$. According to the Perron-Frobenius theorem,
see Appendix, it also has a positive column eigenvector,
${\bf p}^\ast_m(\mu_+)$, such that
$$
\Pi_m^{m+k_0}(\mu_+){\bf p}^\ast_m(\mu_+)
=\lambda_+^{m+k_0}{\bf p}^\ast_m(\mu_+)
$$
The sequence of vectors ${\bf p}^\ast_m(\mu_+)$ `converges', as
$m\to\infty$, to the function $e(U)$ on ${\cal U}_+$ in the
following sense: for any $U\in{\cal U}_+$ the numerical sequence
$$
\{({\bf p}^\ast_m(\mu_+))_B,\ \ {\rm where}\ \ B\in{\cal R}_m^+
\ \ {\rm is}\ {\rm such}\ {\rm that}\ \ B\supset U\}
$$
converges, as $m\to\infty$, to $e(U)$ exponentially fast
in $m$. We do not elaborate this proof here, it is given
in full detail in \cite{C86} for the case where $T$ is a
smooth horseshoe.
\section{Invariant measure $\eta_+$ on the repeller $\Omega$}
\label{secIMR}
\setcounter{equation}{0}
Here we prove Theorems~\ref{tm5} and \ref{tm6}.
For any $n\geq 1$ the measure $\mu_+^{(n)}=T^{-n}_\ast\mu_+$
defined by (\ref{Tastpri}) is supported on $M_+\cap M_{-n}$.
Its conditional measures on $U\cap M_{-n}$, $U\subset M_+$,
i.e. $\nu^u_U(\cdot /M_{-n})$, they are absolutely continuous
with respect to the Riemannian volume on $U$ with densities
$$
\rho^u_{n,U}(x)=[\nu_U^u(U\cap M_{-n})]^{-1}\rho^u_U(x),\ \ \
x\in U\cap M_{-n}
$$
Its factor measure, $\hat{\mu}_+^{(n)}$, on ${\cal U}_+$
is absolutely continuous with respect to $\hat{\mu}_+$,
and its Radon-Nikodym derivative is
$$
\frac{d\hat{\mu}_+^{(n)}}{d\hat{\mu}_+}(U)=
\lambda_+^{-n}\cdot \nu_U^u(U\cap M_{-n})
$$
for $U\subset {\cal U}_+$, in virtue of (\ref{mupcon}).
Due to (\ref{eUlim}), we have
$$
\lim_{n\to\infty}\frac{d\hat{\mu}_+^{(n)}}{d\hat{\mu}_+}(U)=e(U)
$$
for any $U\in {\cal U}_+$.
\begin{corollary}
The sequence of measures $\hat{\mu}^{(n)}_+$ weakly converges
to the measure $\hat{\mu}_0$ on ${\cal U}_+$ defined by
$$
d\hat{\mu}_0(U) = e(U) d\hat{\mu}_+(U)
$$
\label{cor5e}
\end{corollary}
We now complete the proof of Theorem~\ref{tm5}. Let $k\geq 1$,
$l\geq 1$, and consider two arbitrary atoms $B\in{\cal R}_l$
and $D\in{\cal R}_{-k}$. For all $n\geq k$ we have
\begin{eqnarray*}
\mu_+^{(n)}(B\cap D)&=&\mu_+(T^n[(B\cap D)\cap M_{-n}]) \\
&=&\mu_+(T^{n-k}[T^k(B\cap D)\cap M_{-n+k}])
= \mu_+^{(n-k)}(T^k(B\cap D))
\end{eqnarray*}
The set $T^k(B\cap D)$ is an atom of ${\cal R}_{k+l}$.
Due to Corollary~\ref{cor5e} we have
$$
\lim_{n\to\infty}\mu_+^{(n)}(B\cap D)=
\lim_{n\to\infty}\mu_+^{(n-k)}(T^k(B\cap D))=
\hat{\mu}_0\{U\in{\cal U}_+:\, U\subset T^k(B\cap D)\}
$$
Hence, the sequence of measures $\mu_+^{(n)}=T^{-n}_\ast\mu_+$
weakly converges, as $n\to\infty$, to a measure $\eta_+$,
which is supported on the closed set $M_+\cap(\cap_{n\geq 1}
M_{-n})=\Omega$. The invariance of $\eta_+$ under $T$ follows
from two equations:
$$
\mu_+^{(n)}(T(B\cap D))=\mu_+^{(n-k+1)}(T^k(B\cap D))
$$
and
$$
\mu_+^{(n)}(T^{-1}(B\cap D))=\mu_+^{(n-k-1)}(T^k(B\cap D))
$$
($B$ and $D$ are the same as above). By taking the limit
as $n\to\infty$, we obtain (\ref{etainv}). Theorem~\ref{tm5}
is proved.
Proposition~\ref{pr2} follows from Corollary~\ref{cor5e}.
We now prove Theorem~\ref{tm6}.
\begin{proposition}
For any $G>0$ and any measure $\mu\in{\cal H}(G)$ the
sequence of measures $\mu_{n,m}$ defined by (\ref{munm})
weakly converges, as $m,n\to\infty$, to $\eta_+$.
\label{pr5mn}
\end{proposition}
{\it Proof}. Since $T^{n_G}_+\mu\in{\cal H}(G_0)$, we may
assume that $\mu\in{\cal H}(G_0)$.
It is enough to show that for every $k,l\geq 0$,
every atom $B\in{\cal R}_l$ and every atom $D\in{\cal R}_{-k}$
we have
$$
\lim_{m,n\to\infty}\mu_{n,m}(B\cap D)=\eta_+(B\cap D)
$$
Let $m>k$ and $n>l$. Note that $\eta_+(B\cap D) = \eta_+
(T^k(B\cap D))$ and $\mu_{n,m}(B\cap D) = \mu_{n+k,m-k}
(T^k(B\cap D))$, and $T^k(B\cap D)$ is an atom of ${\cal R}_{k+l}$.
Thus, it is enough to show that
\be
\lim_{m,n\to\infty}\mu_{n,m}(B)=\eta_+(B)
\label{mnB}
\ee
Recall that $\mu_+^{(n)}(B)\to \eta_+(B)$ as $n\to\infty$
by Theorem~\ref{tm5}. Thus, (\ref{mnB}) is equivalent to
the following:
\be
\lim_{m,n\to\infty}\frac{\mu_{n,m}(B)}{\mu_+^{(m)}(B)}=1
\label{munm+}
\ee
Let $\mu_n=T^n_+\mu$.
The ratio of the above measures can be rewritten as
$$
\frac{\mu_{n,m}(B)}{\mu_+^{(m)}(B)}=
\frac{\mu_n(B\cap M_{-m})}{\mu_+(B\cap M_{-m})}\cdot
\frac{\mu_+(M_{-m})}{\mu_n(M_{-m})}
$$
First, we will show that
\be
\lim_{m,n\to\infty}
\frac{\mu_n(B\cap M_{-m})}{\mu_+(B\cap M_{-m})}=1
\label{mulim1}
\ee
A direct application of bounds (\ref{mu+mu1}),
(\ref{mu+mu2}) and Corollary~\ref{cor4C} gives
$$
\mu_n(B\cap M_{-m})\leq e^{3c\lambda^{\sqrt{n}}}
\mu_+(B\cap M_{-m}) + e^{4c}\cdot C_6\lambda_+^m\cdot C_4\gamma^{\sqrt{n}}
$$
and
$$
\mu_n(B\cap M_{-m})\geq e^{-3c\lambda^{\sqrt{n}}}
\mu_+(B\cap M_{-m}) - e^{4c}\cdot C_6\lambda_+^m\cdot C_4\gamma^{\sqrt{n}}
$$
for all $n\geq\max\{m_0^2,l^2\}$ and $m\geq m_0^2$.
Due to Corollary~\ref{cor4C} we have
$$
C_6^{-1}\mu_+(B)
\leq \frac{\mu_+(B\cap M_{-m})}{\lambda_+^m}\leq
C_6\mu_+(B)
$$
Combining all the previous bounds yields (\ref{mulim1}).
The equation (\ref{mulim1}) holds, in particular, for $l=0$
and $B=R_i$, $1\leq i\leq I$. Since $M_{-m}=\cup_{i=1}^I
(R_i\cap M_{-m})$, we immediately obtain
$$
\lim_{m,n\to\infty}
\frac{\mu_+(M_{-m})}{\mu_n(M_{-m})}=1
$$
thus completing the proof of (\ref{munm+}) and Proposition~\ref{pr5mn}.
The first part of Theorem~\ref{tm6} is then established.
\begin{proposition}
For any $G>0$ and $\mu\in{\cal H}(G)$ the limit (see (\ref{mustanm}))
$$
c[\mu]=\lim_{m,n\to\infty}
||\mu_{n,m}^{\ast}||
$$
exists, and $c[\mu]$ is the same as in Proposition~\ref{prcmu}.
\label{prcmu1}
\end{proposition}
{\it Proof}. We have
\begin{eqnarray*}
\lim_{m,n\to\infty}
||\mu_{n,m}^{\ast}|| &=&
\lim_{m,n\to\infty}\lambda_+^{-n-m}
(T^n_\ast\mu)(M_{-m}) \\
&=& \lim_{m,n\to\infty}\lambda_+^{-n-m}
\mu(M_{-n-m})
\end{eqnarray*}
which is equal to $c[\mu]$ due to Proposition~\ref{prcmu}.
Proposition~\ref{prcmu1} is proved.
The proof of Theorem~\ref{tm6} is completed.
\section{Ergodic properties of the measure $\eta_+$}
\label{secEPM}
\setcounter{equation}{0}
Here we prove the ergodic and fractal properties of the
invariant measure $\eta_+$ on the repeller, given by
statements~\ref{tm7}-\ref{cr2}.
Let $k,l\geq 0$, and take arbitrary atoms $B\in{\cal R}_l$ and
$D\in{\cal R}_{-k}$. Assume that int$(B\cap D)\neq\emptyset$ and
pick a point $x\in B\cap D$.
\begin{lemma}
There is a constant $C_7>1$ independent of $x,B,D,k,l$ such that
$$
C_7^{-1}\leq \lambda_+^{k+l} J^u_{k+l}(T^{-l}x)\cdot \eta_+(B\cap D)\leq C_7
$$
\label{lmetaBD}
\end{lemma}
{\it Proof}. The set $E=T^k(B\cap D)$ is an atom of ${\cal R}_{k+l}$, and
due to (\ref{dde})
$$
\eta_+(B\cap D)=\eta_+(T^k(B\cap D))=\int_{{\cal U}_E}e(U)\, d\hat{\mu}_+(U)
$$
In virtue of (\ref{enorm}) and (\ref{ebound}) we have
$$
\beta\leq \inf_{U\in{\cal U}} e(U)\leq \sup_{U\in{\cal U}}
e(U)\leq \beta^{-1}
$$
so that
$$
\beta\leq \frac{\eta_+(B\cap D)}{\mu_+(E)}\leq \beta^{-1}
$$
Next, the conditional invariance of $\mu_+$ implies that
$$
\mu_+(E)=\lambda_+^{-k-l}\mu_+(F)=\lambda_+^{-k-l}
\int_{{\cal U}(F)}\nu_U^u(F\cap U)\,\hat{\mu}_+(U)
$$
where $F=T^{-l}(B\cap D)$ is an atom of ${\cal R}_{-k-l}$ and
${\cal U}(F)=\{U\in{\cal U}:\, U\cap F\neq\emptyset\}$.
To estimate this last integral, recall that the measures $\nu_U^u$
on unstable fibers $U\in{\cal U}$ have densities uniformly bounded away
from zero and infinity, and note that for all $U\subset{\cal U}(F)$
$$
0<{\rm const}\leq
m_U(F\cap U)\cdot J^u_{k+l}(T^{-l}x)
\leq{\rm const}<\infty
$$
which follows from the absolute continuity of stable and
unstable foliations, Sect.~\ref{secBAD}.
This completes the proof of Lemma~\ref{lmetaBD}.
This lemma immediately implies that $\eta_+$ is a Gibbs measure
with the potential $g_+(x)=-\log J^u(x)$ and the topological
pressure $P(\eta_+)=\log\lambda_+=-\gamma_+$, see \cite{Bo75}.
Theorems~\ref{tm7} and \ref{tm8} are now proved.
Corollary~\ref{cr1} mostly follows from \cite{Bo75},
for more advanced limit theorems than the central
limit theorem see, e.g., \cite{GH}.
Theorem~\ref{tm9} is self-evident.
We now turn to Theorem~\ref{tm10}.
The measure $\eta_-$ is also a Gibbs measure, with potential
$$
g_-(x) = \log J^s(T^{-1}x)
$$
and topological pressure $P(\eta_-)=-\log\lambda_-^{-1}=-\gamma_-$.
The next lemma is a direct consequence of \cite[Proposition 4.5]{Bo75}.
\begin{lemma}
The following three conditions are equivalent:\\
{\rm (i)} $\eta_+=\eta_-$;\\
{\rm (ii)} there is a constant $Z>0$ such that for any periodic
point $x\in\Omega$, $T^kx=x$, we have $J^u_k(x)\cdot J^s_k(x)=Z^k$;\\
{\rm (iii)} the functions $g_+(x)$ and $g_-(x)$ are cohomologous,
i.e. there is a constant $R$ and a H\"older continuous
function $u(x)$ such that $g_+(x)-g_-(x)=R+u(Tx)-u(x)$.\\
If those conditions are satisfied, then
$$
-\ln Z=R=P(\eta_+)-P(\eta_-)=\gamma_- - \gamma_+
$$
\end{lemma}
Theorem~\ref{tm10} now follows immediately. This theorem, combined
with Proposition~4.14 from \cite{Bo75}, gives Corollary~\ref{cr2}.
Possible applications of Theorem~\ref{tm10} and Corollary~\ref{cr2}
cover hyperbolic repellers constructed on the base of Hamiltonian
systems (those preserve Liouville measures that are absolutely continuous).
In particular, these include billiard systems, like the open billiard
with three circular scatterers studied in \cite{LM}, where repellers
are thus always time-symmetric.
Another interesting class of repellers are {\it linear} repellers.
Let the rectangles $R_1,\ldots,R_I$ be subset in $\IR^d$ and all
$E^u_x$ and all $E^s_x$ be parallel. Let the map $T$ be linear
on each $R_i$, with the constant derivative $DT=$const on $M$,
so that the functions $J^u(x)=J^u$ and $J^s(x)=J^s$ are constant
on $M$. In this case the measures $\eta_+$ and $\eta_-$ always
coincide, and both coincide with the measure of maximal entropy
on $\Omega$, see \cite{Bo75} for definitions and details. In
this case the repeller $\Omega$ is, however, time symmetric if
and only if $\det DT=J^u\cdot J^s=1$, i.e. if $T$ preserves
the Lebesgue measure in $\IR^d$.
\section{Generalizations and open problems}
\label{secGOP}
\setcounter{equation}{0}
In our arguments, we never essentially relied on the fact that $T$
was a diffeomorphism of a connected manifold, in fact the
action of $T$ on $H=M'\setminus M$ never came into play.
All our results hold true under the following, more
general assumptions:
Let $M$ be a finite union of disjoint closed domains $R_1,
\ldots,R_I$ in a smooth Riemannian manifold $\cal M$. Let
$T:M\to{\cal M}$ be a diffeomorphism of $M$ onto its image,
which is $C^{1+\alpha}$ up to the boundary $\partial M$.
We assume the Anosov splitting (\ref{EE}) at every $x\in M$,
and require (\ref{decom}) if the corresponding
iterations of $T$ are defined. Let the bundles $E^{u,s}_x$
be H\"older continuous and
integrable over every $R_i$, so that $R_i$ is foliated by
H\"older continuous families of $C^{1+\alpha}$ submanifolds
$W_x^{u,s}$ such that ${\cal T}W^{u,s}_x=E^{u,s}_x$ at
every $x\in R_i$. Assume that every $R_i$ is a rectangle
and $\{R_1\ldots,R_I\}$ is a Markov partition of $M$ in
the sense of Section~\ref{secBAD}.
Under these assumptions our results remain true. The above
setting is very convenient for horseshoe-like maps, studied
in \cite{C86,Nit}.
We now discuss what happens if we relax the mixing assumption
in Section~\ref{secBAD}. First, we can classify the rectangles
like one does states of Markov chains. We call a rectangle
$R_i$ recurrent if its points come back to itself under $T$,
i.e. int$R_i\cap T^n(R_i\cap M_{-n})\neq\emptyset$ for some
$n\geq 1$. In the trivial case, where all the rectangles are
nonrecurrent (transient), the sets $M_+,M_-$ and $\Omega$ are
empty, and the phase space $M$ `escapes' entirely.
The recurrent
rectangles can be grouped, in each group points from any rectangle
can be mapped into any other rectangle, so that the symbolic
dynamics within every group is transitive.
Let us assume first that there is only one transitive group
of rectangles $R_1,\cdots,R_{I_0}$, and put $M_0=R_1\cup\ldots
\cup R_{I_0}$. This group is periodic if there is a $k\geq 1$
such that the periods of all the periodic points in $M_0$
are multiples of $k$. In that case this group can be divided
into $k$ subgroups cyclicly permuted by $T$, and the restriction
of $T^k$ to any subgroup is topologically mixing. The study of
the map $T$ admits a standard reduction to that of $T^k$, well
known in the theory of Axiom~A diffeomorphisms
\cite{Bo75}, so that we can restrict ourselves to the case
$k=1$. Then the repeller $\Omega$ belongs in $M_0$.
The nonrecurrent rectangles $R_i$, $i>I_0$, can be of three
types: isolated (such that int$T^nR_i\cap M_0=\emptyset$
for all $n\in\ZZ$), incoming (such that int$T^nR_i\cap M_0\neq\emptyset$
for some $n>0$) and outgoing (such that int$T^nR_i\cap M_0\neq\emptyset$
for some $n<0$). The set $M_+$ intersects only recurrent and outgoing
rectangles, $M_-$ only recurrent and incoming ones. The measures
$\mu_{\pm}$ conditioned on $M_0$ coincide with the corresponding
measures for the restriction of $T$ to $M_0$. The measures
$\eta_{\pm}$ and the escape rates $\gamma_{\pm}$ will be the
same for $T|_M$ and $T|_{M_0}$. So, nonrecurrent rectangles
do not really affect the properties of the repeller $\Omega$
studied in this paper, they only may enlarge the sets $M_{\pm}$
and `stretch' the measures $\mu_{\pm}$ accordingly.
A more involved situation occurs when there are two or more
groups of recurrent rectangles. For simplicity, consider two groups,
$R_1',\ldots,R_{I_0}'$ and $R_1'',\ldots,R_{J_0}''$, and put
$M_0'=\cup R_i'$ and $M_0''=\cup R_j''$. If there is no
connection between these groups, i.e. int$(T^nM_0'\cap T^mM_0'')
=\emptyset$ for all $m,n\in\ZZ$, then we have two trivially
independent repellers in $M_0'$ and $M_0''$, respectively. On the
contrary, if there is a route from $M_0'$ to $M_0''$, i.e. int$(T^nM_0'
\cap M_0'')\neq\emptyset$ for some $n\geq 1$, then the picture
gets intricate. The rate of escape from $M_0'$ is still
the same as for the map $T|_{M_0'}$, as if $M_0''$ did not
exist. The escape from $M_0''$, however, is combined with
the influx of points from $M_0'$. The resulting escape rate from
$M$ will be than influenced by three factors: the escape
rates from $M_0'$, $M_0''$ and by the fraction of
$M_0'$ transmitted to $M_0''$ after escaping from $M_0'$.
We did not investigate here these interesting phenomena.
Another natural extension would be to study Axiom A
diffeomorphisms rather than Anosov ones. Let $T:M'\to M'$
be an Axiom~A diffeomorphism with the basic set $\Omega$.
Let $M\subset M'$ be a proper closed subdomain such that
$\Omega=\cap_{-\infty}^{\infty}T^nM$. Then it might
be possible to construct conditionally invariant
measures on $M_+=\cap_0^{\infty}T^nM$ and invariant
measures on $\Omega$ in the same way as we did for
Anosov diffeomorphisms. We leave this for future researches.
Lastly, there are
nonuniformly hyperbolic diffeomorphisms and hyperbolic
maps with singularities, like billiards, which have countable
Markov partitions and the derivatives growing to infinity
at singularities. Extension of our results to those
models is the most challenging problem at present.
% \appendix
\section*{Appendix}
\setcounter{section}{1}
\setcounter{theorem}{0}
\renewcommand{\thetheorem}{\Alph{section}.\arabic{theorem}}
\renewcommand{\theequation}{\Alph{section}.\arabic{equation}}
\setcounter{equation}{0}
This appendix contains the Perron-Frobenius theorem on positive
matrices and related results. Most of these results are taken
from \cite{C86}.
Let $V_m$ be the space of row $m$-vectors, and $V^\ast_m$ the space
of column $m$-vectors. We equip them with norms
\be
|{\bf a}| = \sum_{i=1}^m |a_i|,\ \ \ \
|{\bf b}^\ast| = \max_{1 \leq i\leq m} |b_i|
\label{eqa1}
\ee
and scalar product
$$
({\bf a, \;b^\ast}) = a_1b_1+\cdots +a_mb_m
$$
for all ${\bf a} \in V_m$ and ${\bf b}^\ast \in V^\ast_m$.
We call vectors ${\bf a}\in V_m$ and ${\bf b}^\ast\in V^\ast_m$
positive if their components are all positive.
Note that $|({\bf a, b^\ast})| \leq |{\bf a}| |{\bf b^\ast}|.$
Let ${\cal P}_m$ be the set of $m \times m$ matrices with positive entries:
$$
{\bf A}=(A_{ij}) \in {\cal P}_m \ \ \ {\rm if }\ \ A_{ij} > 0
\ \ \ \forall\ 1\leq i,j \leq m
$$
Stochastic ($\sum _j A_{ij} = 1$) and substochastic ($\sum _j A_{ij} \leq 1$)
matrices are the best studied classes of matrices in ${\cal P}_m$.
\begin{theorem}[Perron-Frobenius theorem]
Every positive matrix ${\bf A}\in{\cal P}_m$ has a positive row
eigenvector ${\bf p}$ and a positive column eigenvector ${\bf p}^\ast$:
$$
{\bf p A} = \lambda {\bf p}\ \ \ \ {\rm and}
\ \ \ \ {\bf A p}^\ast = \lambda {\bf p}^\ast
$$
where $\lambda>0$ is the largest (in absolute value) eigenvalue
of the matrix $\bf A$. These vectors are unique up to a scalar multiple,
i.e. the multiplicity of $\lambda$ is one.
\end{theorem}
We put ${\bf p}$ and ${\bf p}^\ast$ for the Perron eigenvectors
of $\bf A$ normalized so that
\be
\sum_{i=1}^m p_i = |{\bf p}| = 1\ \ \ \ {\rm and}
\ \ \ \ \sum_{i=1}^m p_ip^\ast_i = ({\bf p}^\ast,{\bf p}) =1
\label {eqa2}
\ee
For a fixed matrix ${\bf A}\in{\cal P}_m$, we introduce other norms
in $V_m$ and $V^\ast_m$ by
\be
|| {\bf a} ||_r = \sum _{i=1}^m |a_i|p_i^\ast,
\ \ \ \ \
||{\bf b}^\ast ||_c = \max _{1 \leq j \leq m}
(|b_j^\ast|/p_j^\ast)
\label{eqa3}
\ee
If the components of ${\bf a}$ are non-negative, then
$$
|| {\bf a}||_r = ({\bf a,p^\ast})\ \ \ \ {\rm and}
\ \ \ \ ||{\bf aA} ||_r = \lambda || {\bf a} ||_r
$$
Note that $|({\bf a},{\bf b}^\ast )| \leq || {\bf a}||_r
||{\bf b^\ast} ||_c.$
We will say that $P \geq 1$ is an {\em estimate of the ratio
of rows} of ${\bf A} \in {\cal P}_m$ if
$$
P^{-1} \leq A_{ij}/A_{kj} \leq P \ \ \ \ \forall\ 1 \leq i,j,k \leq m
$$
If $\bf A$ satisfies this estimate, we write ${\bf A} \in {\cal P}_m(P).$
If ${\bf A} \in {\cal P}_m(P)$, then the components of its Perron
eigenvectors satisfy
$$
P^{-1}\leq p^\ast_i/p^\ast_j \leq P,\ \ \
\lambda P^{-1}\leq A_{ij}/p_j \leq \lambda P,\ \ \
P^{-1} \leq p^\ast_i \leq P
$$
for all $1\leq i,j \leq m$. The norms defined by (\ref{eqa1})
and (\ref{eqa3}) are then equivalent:
$$
P^{-1}|{\bf a}| \leq ||{\bf a} ||_r \leq P|{\bf a}|
\ \ \ \ {\rm and}\ \ \ \
P^{-1}|{\bf b}^\ast| \leq ||{\bf b^\ast}||_c \leq P|{\bf b}|
$$
for all ${\bf a}\in V_m$ and ${\bf b}^\ast\in V^\ast_m$.
The following estimate on the so called coefficients of
ergodicity is also satisfied if ${\bf A} \in {\cal P}_m(P)$:
$$
\sum_{j=1}^m p_j^\ast\cdot \inf_{1 \leq i \leq m}
(A_{ij}/p_i^\ast) \geq \lambda P^{-1}
$$
Denote by $L_m$ and $L_m^\ast$ the orthogonal complements
to the Perron eigenvectors:
$$
L_m = \{ {\bf a} \in V_m :\, ({\bf a},{\bf p}^\ast) = 0\}
\ \ \ {\rm and}\ \ \
L_m^\ast =\{ {\bf b}^\ast \in V^\ast_m :\, ({\bf p},{\bf b}^\ast)=0 \}
$$
Then we have the decompositions
$$
{\bf a} = ({\bf a},{\bf p}^\ast){\bf p}+{\bf a}_0
\ \ \ {\rm with} \ \ \
{\bf a}_0 \in L_m
$$
and
$$
{\bf b}^\ast = ({\bf p},{\bf b}^\ast) {\bf p}^\ast +{\bf b}^\ast_0
\ \ \ {\rm with} \ \ \
{\bf b}_0^\ast \in L_m^\ast
$$
\begin{lemma}
If ${\bf A}\in {\cal P}_m(P)$, then for any ${\bf a} \in L_m$ we have
$$
||{\bf aA} ||_r \leq \lambda(1-P^{-1}) ||{\bf a}||_r
$$
and for any ${\bf b}^\ast \in L_m^\ast$ we have
$$
||{\bf Ab}^\ast ||_c \leq \lambda (1-P^{-2}) ||{\bf b}^\ast||_c
$$
\end{lemma}
If $\lambda < 1 $ (this is the case if ${\bf A}$ is a proper
substochastic matrix), then this lemma says that the
contraction in the orthogonal subspaces $L_m$ and
$L_m^\ast$ is stronger than that in the eigenspaces
spanned by the Perron eigenvectors.
\begin{corollary}
If ${\bf A} \in {\cal P}_m(P)$ and $\theta=1-P^{-1}$, then
$$
\lim_{n \to\infty} \lambda^{-n}{\bf A}^n =
{\bf p}^\ast \otimes {\bf p}
$$
where $({\bf p}^\ast \otimes {\bf p})_{ij} = p_i^\ast p_j$ is
the tensor product of ${\bf p}^\ast$ and ${\bf p}$. Moreover,
$$
||(\lambda^{-n}{\bf A}^n - {\bf p^\ast} \otimes {\bf p})_k ||_r
\leq 2 \theta^n p_k^\ast
$$
where $B_k$, for a matrix $\bf B$, means the $k$-th row.
\end{corollary}
{\it Remark}. If $\bf A$ is a stochastic matrix ($\sum_j A_{ij} = 1$),
then $\lambda=1$ and $p_j^{\ast}=1$ for all $1\leq j\leq n$, and we
recover a well known ergodic theorem for finite Markov chains. \medskip
We now compare the action on positive row vectors by two
positive matrices which are close to each other. We say
that ${\bf B}\in {\cal P}_m$ is close to ${\bf A}\in {\cal P}_m$,
with the {\it constant of proximity} $R\geq 1$ if
\be
R^{-1} \leq B_{ij}/A_{ij} \leq R
\ \ \ \ \forall\ 1 \leq i,j \leq m
\label{eqa11}
\ee
In the following statements, ${\bf A}\in {\cal P}_m(P)$ is a fixed
matrix, ${\bf p}$ is its Perron row eigenvector normalized by
(\ref{eqa2}) and $\lambda$ is the corresponding eigenvalue. We
also set $\theta = 1 - P^{-1}$.
\begin{lemma}
Let ${\bf q}$ be an arbitrary positive row vector such that
$||{\bf q}||_r = 1$. Let ${\bf B} \in {\cal P}_m$ be another matrix
close to $\bf A$ with the constant of proximity $R \geq 1$.
Then
$$
R^{-1} ||{\bf pA}||_r \leq ||{\bf qB}||_r \leq R ||{\bf pA}||_r
$$
and
$$
||{\bf qB}-{\bf pA}||_r \leq \lambda\theta
||{\bf q-p}||_r + \lambda (R-1)
$$
\label{lmA}
\end{lemma}
\begin{lemma}
Let ${\bf B}\in{\cal P}_m$ be as in Lemma~\ref{lmA}. For any positive
row vector ${\bf q}\in V_m$ we have
$$
\left |\left | \frac{{\bf qB}}{||{\bf qB}||_r}
-\frac{{\bf pA}}{||{\bf pA}||_r} \right |\right |_r
\leq \theta R ||{\bf q-p}||_r + 2R(R-1)
$$
\end{lemma}
%\begin{lemma}
%Let ${\bf B}_1, {\bf B}_2, \ldots ,{\bf B}_n\in {\cal P}_m$ be matrices, all close to
%$\bf A$ with the same constant of proximity $R \geq 1$. For any positive
%row vector ${\bf p}\in V_m$ we put ${\bf q}_n={\bf q}{\bf B}_1\cdots {\bf B}_n$.
%Then we have
%$$
% R^{-n} \lambda^n = R^{-n} ||{\bf p}{\bf A}^n ||_r
% \leq ||{\bf q}_n ||_r \leq
% R^n ||{\bf p}{\bf A}^n ||_r = R^n \lambda^n
%$$
%and
%$$
% ||{\bf q}_n - {\bf p}{\bf A}^n ||_r \leq
% \lambda^n \theta^n ||{\bf q-p} ||_r +
% \lambda^n \frac{R-1}{1-\theta } +
% \lambda^n R^k (R^k-1) \frac{R+\theta}{R-\theta}
%$$
%\end{lemma}
\begin{theorem}
Let ${\bf B}_1, {\bf B}_2, \ldots , {\bf B}_n\in {\cal P}_m$
be matrices, all close to $\bf A$ with the same constant of
proximity $R \geq 1$. For any positive row vector ${\bf q}\in V_m$
we put ${\bf q}_n={\bf q}{\bf B}_1\cdots {\bf B}_n$.
In addition, assume that $\theta R <1$. Then we have
$$
\left |\left |\frac{{\bf q}_n}{||{\bf q}_n||_r} -
{\bf p} \right |\right |_r \leq 2\theta^nR^n +
2R(R-1)(1-\theta R)^{-1}
$$
and
$$
\left | \frac{{\bf q}_n}{|{\bf q}_n|} -
{\bf p} \right | \leq 4 P\theta^nR^n +
4PR(R-1)(1-\theta R)^{-1}
$$
\label{tmA}
\end{theorem}
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