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\begin{document}
\title{Anosov maps with rectangular holes. \\ Nonergodic cases.}
\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
\begin{abstract}
We study Anosov diffeomorphisms on manifolds in which some `holes'
are cut. The points that are mapped into our holes will disappear and
never return. We study the case where the holes are rectangles of a Markov
partition. Such maps with holes generalize Smale's horseshoes and certain
open billiards. The set of nonwandering points of our map
is a Cantor-like set we call a {\it repeller}. In our previous
paper, we assumed that the map restricted to the remaining
rectangles of the Markov partition is topologically mixing.
Under this assumption we constructed invariant and conditionally
invariant measures on the sets of nonwandering points.
\footnotetext{$^1$ Partially supported by NSF grant DMS-9401417.}
Here we relax the mixing assumption and extend
our results to nonmixing and nonergodic cases.
\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{\theequation}{\arabic{section}.\arabic{equation}}
\section{Introduction}
\label{secI}
Let $T: M'\to M'$ be a topologically transitive Anosov
diffeomorphism of class $C^{1+\alpha}$ on a compact Riemannian
manifold $M'$. Sinai \cite{Si68} and Bowen \cite{Bo75} constructed
Markov partitions for transitive Anosov diffeomorphisms.
Let ${\cal R}'$ be an arbitrary Markov partition
of $M'$ into rectangles $R_1,\ldots,R_{I'}$. We assume
that these rectangles are small enough, so that the
symbolic dynamics is well defined \cite{Si68,Bo75}.
Let $I*0$ such that
$\mu(T^{-1}A\cap M_+) =\lambda\mu(A\cap M_+)$ for any $A\subset M$.
\begin{theorem}
Assume the mixing condition.
The map $T$ has a unique conditionally invariant probability
measure $\mu_+$ whose conditional measures on unstable fibers
are H\"older continuous. In fact, those conditional measures
are u-SBR measures $\nu^u_U$, $U\in{\cal U}_+$. For any smooth measure $\mu$
on $M$ (see again the above convention)
the sequence $T^n_+\mu$ weakly converges, as $n\to
\infty$, to the measure $\mu_+$. 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{tm1}
\end{theorem}
{\it Remark}. The conditionally invariant measure $\mu_+$
constructed in this way is physically natural according to the
original Pianigiani-Yorke motivation \cite{PY}. This
measure coincides with the Sinai-Bowen-Ruelle measure
in the case $H=\emptyset$. %\medskip
\begin{corollary}
Let $U\in{\cal U}$.
If $\mu$ is a singular measure supported on $U$ with
H\"older continuous density (on $U$),
then the sequence $T^n_+\mu$ weakly converges to $\mu_+$.
\label{cr1}
\end{corollary}
We call $\gamma_+=\ln\lambda_+^{-1}$ the {\it escape
rate}, cf. \cite{GD,GR,GN,CM}.
Next, since the set $M_+$ is invariant under $T^{-1}$,
the measures $T_\ast^{-n}\mu_+$ for $n\geq 1$
are probability measures for all $n\geq 0$.
In virtue of Theorem~\ref{tm1} they coincide with
the conditional measures $\mu_+(\cdot /M_{-n})$
satisfying
\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,
$\eta_+$, 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{tm2}
\end{theorem}
\begin{theorem}
The measure $\eta_+$ is an equilibrium state for the H\"older
continuous potential
\be
g_+(x)=-\log J^u(x)
\label{g+}
\ee
on $\Omega$ and its topological pressure is
$P(\eta_+)=-\log\lambda_+^{-1}=-\gamma_+$.
Thus, $\eta_+$ is a Gibbs measure.
The sum of positive Lyapunov exponents of the map $T$ is
\be
\chi^+_{\eta_+}=\int_{\Omega}\log J^u(x)\, d\eta_+(x)\ \ \ \ \ {\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 $\Omega$.
The left equation in (\ref{vp}) is equivalent to
the following escape rate formula
\be
\chi^+_{\eta_+}=h_{\eta_+}(T)+\gamma_+
\label{vp1}
\ee
\label{tm3}
\end{theorem}
\section{Symbolic dynamics and new results for the mixing case}
\label{secSR}
\setcounter{equation}{0}
We translate Theorems~\ref{tm1}-\ref{tm3} into the
language of symbolic dynamics to obtain new properties
of the measures $\mu_+$ and $\eta_+$ under the mixing
condition.
Define a transition matrix $A'=(A_{ij}')$ of size $I'\times I'$
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
$$
\Sigma_{A'}'=\{\underline{\omega}\in\Sigma':\,
A_{\omega_i\omega_{i+1}}'=1\ \ {\rm for}\ {\rm all}
\ -\infty**0$ such that for all $n\geq 0$
$$
C_1\leq \mu^u(M_n)/e^{n\gamma_+} \leq C_2
$$
Combining the above facts gives the following property
of the measure $\bar{\mu}_+=\Pi^{-1}\mu_+$ on the symbolic
space $\Sigma_+$.
\begin{theorem}
For any admissible cylinder $C=(\omega_{-n},\ldots,\omega_m)
\subset\Sigma_+$ and every symbolic sequence $\underline{\omega}
\in C$ we have
\be
C_3\leq\frac{\bar{\mu}_+(C)}{\exp\left (\sum_{i=-n}^m\bar{g}
(\sigma^i\underline{\omega})+n\gamma_+\right )}\leq C_4
\label{Gibbs}
\ee
where $C_3,C_4>0$ are constants independent of the cylinder
$C$ or the values of $n,m$.
\end{theorem}
Comparing this theorem to Bowen's definition of Gibbs
measures \cite{Bo75} suggests us to call the measure
$\bar{\mu}_+$ on $\Sigma_+$ a `hybrid' Gibbs measure
with the potential function $\bar{g}(\underline{\omega})$.
Unlike Bowen's definition, however, here the `positive'
and `negative' components of the cylinder $C$ have different
`topological pressures', $P_+=0$ and $P_-=-\gamma_+$ respectively.
For any $n\geq 1$ the measure $\sigma_\ast^{-n}\bar{\mu}_+$ is
supported on $\sigma^{-n}\Sigma_+$. This space has the same
cylinders of length $2n+1$, i.e. $(\omega_{-n},\ldots,\omega_n)$,
as the space $\Sigma_A$. It is clear that $\sigma_\ast^{-n}\bar{\mu}_+$
converges, as $n\to\infty$, to the Gibbs measure $\bar{\eta}_+ =
\Pi^{-1}\eta_+$ on $\Sigma_+$ corresponding to the same potential
function $\bar{g}(\underline{\omega})$. This is exactly
what Theorems~\ref{tm2} and \ref{tm3} say.
Lastly, let $C=(\omega_0,\ldots,\omega_k)$ be any admissible
cylinder of length $k+1$ in $\Sigma_{A'}'$, and let $\omega_0\leq I$.
Denote by $\bar{\mu}_{+,C}$ the measure $\bar{\mu}_+$ conditioned
on $\Sigma_+\cap C$. Its inverse images $\sigma_\ast^{-n}(\bar{\mu}_{+,C})$
behave asymptotically, as $n\to\infty$, just like $\sigma_\ast^{-n}
(\bar{\mu}_+)$, because the cylinder $C$ is moved under $\sigma^{-n}$
to the right and eventually its influence vanishes.
Thus, the measure $\sigma_\ast^{-n}(\bar{\mu}_{+,C})$
weakly converges to the same Gibbs measure $\bar{\eta}_+$. Back
on $M$, this last conclusion means the following.
\begin{corollary}
Let $R'$ be any $s$-inscribed subrectangle in any rectangle
$R_i\in{\cal R}$ (i.e., $R'$ is a union of some stable fibers
$S\in{\cal S}, S\subset R_i$). Denote by $\mu_{+,R'}$ the measure
$\mu_+$ conditioned on $R'\cap M_+$. Then the sequence $T^{-n}_\ast
\mu_{+,R'}$ weakly converges, as $n\to\infty$, to the measure $\eta_+$.
\label{cr2}
\end{corollary}
This corollary is dual to Corollary~\ref{cr1}, for it shows that
the measure $\eta_+$ can be obtained by backward iterations of
a measure supported on just one stable fiber, $S\cap M_+$,
the latter measure is $\mu_+$ conditioned on $S\cap M_+$. This
corollary was missing in Ref.~\cite{CM}, and we need it in
this paper.
\section{Topologically transitive case}
\label{secTT}
\setcounter{equation}{0}
Here we replace the mixing condition by the following weaker one. \medskip
{\bf Transitivity condition}. The symbolic dynamics generated by
the partition ${\cal R}=\{R_1,\ldots,R_I\}$ of $M$ is
a topologically transitive subshift of finite type. Equivalently,
for any $R_i,R_j\in{\cal R}$ there is a $k_{ij}\geq 1$ such that
int$R_i\cap T^{k_{ij}}(R_j\cap M_{-k_{ij}}) \neq\emptyset$. \medskip
Under this condition the subshift is either topologically mixing
(i.e. $T$ satisfies the mixing condition) or periodic.
The latter means that there is a finite $p\geq 2$ (period) and a
partition of $\cal R$ into $p$ subgroups ${\cal R}_1,\ldots,
{\cal R}_p$ cyclically permuted by the shift. Precisely,
int$R_i\cap T(R_j\cap M_{-1}) \neq\emptyset$
if and only if $R_i\in{\cal R}_l$ and $R_j\in{\cal R}_{l+1}$
for some $l$ (here and on $l$ is a cyclic index, i.e. $l=p+1$ is
identified with $l=1$).
Besides, the map $T^p$ restricted to $M^{(l)}=\cup_{R\in{\cal R}_l}R$
for any $l$ satisfies the mixing assumption.
The map $T^p$ restricted to $M^{(l)}$ has
all the properties listed in the previous section. In particular,
there are conditionally invariant measures $\mu_+^{(l)}$ on
$M^{(l)}_+=M_+\cap M^{(l)}$ and $T^p$-invariant measures
$\eta_+^{(l)}$ on the sets $\Omega^{(l)}=\Omega\cap M^{(l)}$.
We call these {\em basic measures}. These measures satisfy
Theorems~\ref{tm1}-\ref{tm3} with $T$ replaced by $T^p$
and $M$ by $M^{(l)}$.
It is standard in the ergodic theory to reduce transitive but
nonmixing subshifts to mixing ones by replacing $T$ with its
appropriate iterate, $T^p$. It is interesting, however, to
extend Theorems~\ref{tm1}-\ref{tm3} directly to the nonmixing
map $T$, the task we accomplish in this section.
According to Theorem~\ref{tm1}, every basic measure $\mu_+^{(l)}$
is a weak limit of $c[\mu]\cdot \left [\lambda_+^{(l)}\right ]^{-n}
T_\ast^{pn}\mu$, as $n\to\infty$, for any smooth measure $\mu$
on $M^{(l)}$.
It is then clear that the eigenvalues of the measures $\mu_+^{(l)}$
under $T^p$ coincide, i.e. $\lambda_+^{(l)}=\bar{\lambda}_+$ for all
$l$. Also, for any $l$ the measure $T_\ast\mu^{(l)}_+$ is
proportional to $\mu_+^{(l+1)}$, i.e. $T_\ast\mu_+^{(l)}
=\lambda_l \mu_+^{(l+1)}$ with some $\lambda_l\in (0,1]$.
Then $\bar{\lambda}_+=\lambda_1\cdots\lambda_p$. From
these remarks and the cyclic character of the map $T$ we
derive the following.
\begin{theorem}
There is a unique conditionally invariant measure $\mu_+$
for the map $T$, whose conditional measures on unstable fibers
are smooth. These are, in fact, the u-SBR measures $\nu_U^u$.
The eigenvalue of $\mu_+$ is $\lambda_+=\left (
\bar{\lambda}_+\right )^{1/p}$. The measure $\mu_+$ is a weighted
sum of basic measures
$$
\mu_+=w_1\mu_+^{(1)}+\cdots +w_p\mu_+^{(p)}
$$
where the weights $w_l>0$ are uniquely determined by the equations
$w_l\lambda_l=w_{l+1}\lambda_+$ for all $l$ and $w_1+\cdots
+w_p=1$.
\end{theorem}
{\em Example}. Let $p=2$, and $\lambda_1=1$, $\lambda_2=1/4$.
Then the eigenvalue of the measure $\mu_+$ is $1/2$ and the
weights are $w_1=1/3$ and $w_2=2/3$. \medskip
However, the images $T_+^n\mu$ of an arbitrary
smooth measure $\mu$ on $M$ generally need not converge, as
$n\to\infty$, to the measure $\mu_+$. Normally, the sequence
$T_+^n\mu$ periodically approaches a finite number ($\leq p$)
of limit measures, all of them being some weighted sums
of the basic measures $\mu_+^{(1)},\ldots,\mu_+^{(p)}$
(in particular, they are all equivalent to $\mu_+$).
The Cesaro limit of the sequence $T_+^n\mu$ always exists
and does not depend on $\mu$. But it is an equidistributed sum
of the basic measures
$$
\mu_+^0=\frac 1p \left (\mu_+^{(1)}+\cdots +\mu_+^{(p)}\right )
$$
Even though this measure is equivalent to $\mu_+$, it is generally
different from $\mu_+$.
\begin{theorem}
{\rm (i)} The equidistributed sum of basic measures $\eta_+^{(p)}$,
$$
\eta_+=\frac 1p \left (\eta_+^{(1)}+\cdots +\eta_+^{(p)}\right )
$$
is a $T$-invariant measure on $\Omega$. It is the only
$T$-invariant measure equivalent to $\eta_+^{(l)}$ on
$M^{(l)}$ for every $l$. \\
{\rm (ii)} The weak Cesaro limit of the sequence $T_\ast^{-n}\mu_+$,
as $n\to\infty$, is $\eta_+$.
\label{tm2t}
\end{theorem}
{\em Proof}. The basic measures $\eta_+^{(l)}$ on $\Omega^{(l)}$
satisfy the invariance property $T_\ast\eta_+^{(l)}=
\eta_+^{(l+1)}$. This follows from Theorem~\ref{tm2}
and Corollary~\ref{cr2}. Then the part (i) is immediate.
It is easy to see that
the measure $\eta_+$ is the weak limit of the sequence
$T^{-n}_\ast\mu_+^0$ as $n\to\infty$. However, the sequence
$T_\ast^{-n}\mu_+=\mu_+(\cdot/M_{-n})$ generally does
not converge to any measure on $\Omega$. Instead, it
periodically approaches a finite number ($\leq p$)
of measures on $\Omega$ that will be weighted sums of the
basic measures $\eta_+^{(l)}$.
In the above example, the two limit measures for
the sequence $T^{-n}_\ast\mu_+$ have weight distributions
$(1/3,2/3)$ and $(2/3,1/3)$. It is now clear that
the part (ii) of Theorem~\ref{tm2t} holds.
\begin{theorem}
{\rm (i)} The measure $\eta_+$ is a Gibbs measure with potential function
$g(x)=-\ln J^u(x)$ and topological pressure $P=\ln\lambda_+$. \\
{\rm (ii)} It satisfies the equation (\ref{vp1}).
\label{tm3t}
\end{theorem}
{\em Proof}.
According to Theorem~\ref{tm3} the basic measures $\eta_+^{(l)}$
are Gibbs with potential $g_l(x)=-\ln (J^u(x)\cdots J^u(T^{p-1}x))$,
$x\in\Omega^{(l)}$, and topological pressure $P_l=\ln\bar{\lambda}_+$.
Then the part (i) easily follows.
The equation (\ref{vp1}) holds for the measure
$\eta_+^{(l)}$ and the map $T^p$ on $\Omega^{(l)}$.
It is easy to check that every quantity involved in
(\ref{vp1}) decreases by a factor of $p$ if we replace
$T^p:\Omega^{(l)}\to\Omega^{(l)}$ by $T:\Omega\to\Omega$
and $\eta_+^{(l)}$ by $\eta_+$. This gives the part (ii). \medskip
Summarizing, we find that Theorems~\ref{tm1}-~\ref{tm3}
still hold under the transitivity assumption, with two
exceptions. First, the images $T^n_+\mu$ of an arbitrary
smooth measure $\mu$ on $M$ do not exactly converge to
$\mu_+$. They approach a finite number of limit measures
on $M_+$, all equivalent to $\mu_+$. The same goes to the
sequence $T^{-n}_\ast\mu_+$ and the limit measure $\eta_+$.
In the latter case, however, the Cesaro limit of $T^{-n}_\ast
\mu_+$ is always $\eta_+$. Corollaries~\ref{cr1} and \ref{cr2}
cannot be extended to nonmixing cases.
\section{Nonrecurrent rectangles}
\label{secNR}
\setcounter{equation}{0}
One can classify the rectangles $R\in{\cal R}$ just like states
of Markov chains are classified in probability theory.
We call a rectangle $R\in{\cal R}$ recurrent
if its interior points come back to $R$ under $T$,
i.e. int$R\cap T^n(R\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
all empty, and the phase space $M$ `escapes' entirely.
The recurrent rectangles can be grouped, within each group points
from any rectangle are eventually mapped into any other rectangle.
The symbolic dynamics within every group is a topologically
transitive (TT) subshift of finite type.
In this section we still assume that there is just one transitive
group of recurrent rectangles $R_1,\cdots,R_{I_0}$, but we allow
some nonrecurrent rectangles $R_{I_0+1},\ldots,R_I$ as well.
Put $M^{(1)}=R_1\cup\ldots \cup R_{I_0}$.
Nonrecurrent rectangles are further subdivided into three groups: \\
(i) incoming: such that int$T^n(R_i\cap M_{-n})\cap M^{(1)}
\neq\emptyset$ for some $n\geq 1$;\\
(ii) outgoing: such that int$T^{-n}(R_i\cap M_{n})\cap M^{(1)}
\neq\emptyset$ for some $n\geq 1$;\\
(iii) isolated: such that int$T^n(R_i\cap M_{-n})\cap M^{(1)}
=\emptyset$ for all $n\in\ZZ$.
The set of nonwandering points $\Omega$ obviously belongs
in $M^{(1)}$. The restriction of the map $T$ to $M^{(1)}$ satisfies
the transitivity assumption in the previous section.
Thus, there is a conditionally invariant measure $\mu_+^{(1)}$
on $M_+^{(1)}=M_+\cap M^{(1)}$ with eigenvalue $\lambda_+^{(1)}$,
and the corresponding $T$-invariant measure $\eta_+^{(1)}$ on $\Omega$.
The isolated rectangles escape to $H$ altogether
in a finite time and have no influence on the measures
$\mu_+, \eta_+$ whatsoever.
The incoming rectangles are absorbed into $M^{(1)}$ in
a finite time, so their presence (or absence) cannot affect
the properties of the measures $\mu_+$ or $\eta_+$ either.
The set $M_+$ intersects only recurrent and outgoing
rectangles. The measures $T^n_\ast\mu_+^{(1)}$ (the images
of $\mu_+^{(1)}$ under the maps $T^n$ on $M$) will be
supported on $M_+$, and their restrictions to $M_+^{(1)}$
will be always proportional to $\mu_+^{(1)}$. It is then
easy to check the following.
\begin{theorem}
Under the above conditions,
there is a unique conditionally invariant measure
$\mu_+$ for the map $T$ supported on $M_+$ with
absolutely continuous conditional measures on unstable
fibers. They are, in fact, the u-SBR measures $\nu_U^u$.
The measure $\mu_+$ is proportional to $\mu_+^{(1)}$
on the set $M_+^{(1)}$. These two measures have the same
eigenvalue $\lambda_+=\lambda_+^{(1)}$.
\label{tmNRRmu}
\end{theorem}
If the transitive group of rectangles is topologically mixing,
the sequence $T^n_+\mu$ converges, as $n\to\infty$, to $\mu_+$
for any smooth measure $\mu$ on $M$. In nonmixing cases the
situation is equivalent to the one in the previous section.
\begin{theorem}
{\rm (i)} The $T$-invariant measure $\eta_+$ on $\Omega$ simply coincides
with the measure $\eta_+^{(1)}$. Thus, it enjoys all the properties
established by Theorems~\ref{tm3} and \ref{tm3t}. \\
{\rm (ii)} The measure $\eta_+$ is the weak Cesaro limit of
$T^{-n}_\ast\mu_+$ as $n\to\infty$ (in the mixing case,
it is just the weak limit).
\end{theorem}
{\em Proof}. Only the part (ii) needs a proof. According
to Theorem~\ref{tm2t}, the weak Cesaro limit of the
sequence $T^{-n}_\ast\mu_+^{(1)}$ is $\eta_+$.
Consider the measure $\mu_+$ conditioned on
outgoing rectangles. It will be transferred under $T^{-n}_\ast$
into measures supported on some $s$-inscribed subrectangles
in some rectangles $R_i\subset M^{(1)}$ and on those subrectangles
those measures will be proportional to $\mu_+^{(1)}$.
Due to Corollary~\ref{cr2}, such measures converge to $\eta_+$,
as $n\to\infty$, in the same way as the sequence $T^{-n}\mu_+^{(1)}$.
The theorem is proved.
\section{Two TT groups of rectangles}
\label{secTT2}
\setcounter{equation}{0}
As it was remarked in the previous section, recurrent rectangles can be
divided into topologically transitive (TT) groups
so that in each group points from any rectangle can be
mapped into any other rectangle. In this Section we will assume that
there are just two TT groups of rectangles:
$M^{(1)}=R_1\cup \cdots \cup R_{I_1}$,
$M^{(2)}=R_{I_1+1}\cup\cdots\cup R_{I_2}$,
and some non-recurrent rectangles $R_{I_2+1}, \ldots ,R_I$. If there is
no connection between these groups, i.e. int$(T^nM^{(1)}\cap T^mM^{(2)})
=\emptyset$ for all $m,n\in\ZZ$, then we have two trivially
independent repellers, one in $M^{(1)}$ and another in $M^{(2)}$.
There may be, however, a one-way route from $M^{(1)}$ to $M^{(2)}$,
i.e. int$(T^nM^{(1)}\cap M^{(2)})\neq\emptyset$ for some $n\geq 1$.
In this case the picture gets more intricate.
The rate of escape from $M^{(1)}$ is still
the same as for the map $T|_{M^{(1)}}$, as if $M^{(2)}$ did not
exist. The escape from $M^{(2)}$, however, is combined with
the influx from $M^{(1)}$. The resulting escape rate from
$M$ and conditionally invariant measures will be then determined by
three factors: the escape of mass from $M^{(1)}$, $M^{(2)}$ and
the transfer of mass from $M^{(1)}$ to $M^{(2)}$.
Nonrecurrent rectangles are now subdivided into four groups:
\smallskip
\noindent
(i) incoming: such that int$T^n(R_i\cap M_{-n})\cap (M^{(1)}
\cup M^{(2)})\neq\emptyset$ for some $n\geq 1$ but
int$T^{-n}(R_i\cap M_{n})\cap (M^{(1)}
\cup M^{(2)}) = \emptyset$ for all $n\geq 1$;
\smallskip
\noindent
(ii) outgoing: such that int$T^{-n}(R_i\cap M_{n})\cap (M^{(1)}
\cup M^{(2)})\neq\emptyset$ for some $n\geq 1$ but
int$T^{n}(R_i\cap M_{-n})\cap (M^{(1)}
\cup M^{(2)}) = \emptyset$ for all $n\geq 1$;
\smallskip
\noindent
(iii) isolated: such that int$T^n(R_i\cap M_{-n})\cap (M^{(1)}
\cup M^{(2)})=\emptyset$ for all $n\in\ZZ$;
\smallskip
\noindent
(iv) transmitting: such that int$\, T^n(R_i\cap M_{-n})\cap M^{(2)}
\neq\emptyset$ for some $n\geq 1$ and\\ int$\, T^{-m}(R_i\cap M_{m})
\cap M^{(1)}\neq \emptyset$ for some $m\geq 1$.
\smallskip
Slightly abusing the language, we will say that incoming rectangles
have one-way connections {\em to} $M^{(1)}\cup M^{(2)}$, and
the outgoing rectangles have one-way connections {\em from}
$M^{(1)}\cup M^{(2)}$. We may also say that transmitting rectangles
are connected {\em from} $M^{(1)}$ and {\em to} $M^{(2)}$.
%We denote by $M^{(t)}$ the union of transmitting rectangles.
%We will also assume that outgoing rectangles from $M^1$ and $M^2$
%are not dynamically related; i.e.
%int$T^{-n}(R_i\cap M_n)\cap R_j = \emptyset$
%for every $n \geq 1$ , $R_i $ outgoing rectangle from $M^1 ~(M^2)$, and
%$R_j$ outgoing rectangle from $M^2 (M^1)$. All these assumptions mean
%that there is no "flow" from $M^2$ to $M^1$ and between outgoing
%rectangles from different transitive groups.
For the map $T$ restricted to $M^{(i)}$, $i=1,2$, we denote by
$M^{(i)}_{\pm}$ and $\Omega^{(i)}$ the corresponding sets defined as
in Introduction, and by $\mu^{(i)}_{\pm}$ and $\eta^{(i)}_{\pm}$
their conditionally invariant and invariant measures, respectively.
We denote by $\lambda_{\pm}^{(i)}$, $i=1,2$, the corresponding
eigenvalues.
It is clear that $M_+=\cup_{n\geq 0}T^n(M_+^{(1)}\cap M_+^{(2)})$
and $M_-=\cup_{n\geq 0}T^{-n}(M_-^{(1)}\cap M_-^{(2)})$. In the
present case, the set $M_+$ consists, in addition to $M_+^{(1)}\cup
M_+^{(2)}$, of some unstable fibers in outgoing and transmitting
rectangles, as well as some unstable fibers in $M^{(2)}$ not
included in $M^{(2)}_+$. These fibers are images of $M^{(1)}_+$
under $T^n$, $n\geq 1$, and they are getting closer and closer
to $M^{(2)}_+$ as $n\to\infty$. Symmetric statements can be
made of the set $M_-$.
The set of nonwandering points $\Omega=M_+\cap M_-$ includes
$\Omega^{(1)}$ and $\Omega^{(2)}$, but is not limited to them.
It also contains (i) points of intersection of stable fibers
of $M^{(2)}_-$ and unstable fibers of $(T^nM^{(1)}_+)\cap M^{(2)}$,
as well as (ii) similar points in $M^{(1)}$, and (iii) the points in
transmitting rectangles which belong in $T^nM^{(1)}_+\cap
T^{-m}M^{(2)}_-$, $n,m\geq 1$. \medskip
{\em Standing assumption for all theorems in
Sections~\ref{secTT2} and \ref{secTT3}}.
The map $T$ restricted to every TT component is not only
topologically transitive, but also topologically mixing. \medskip
This is assumed for simplicity only. Nonmixing maps require the
modifications to our results completely described in Section~\ref{secTT}.
We now construct the conditionally invariant measure $\mu_+$
for $T$ on $M$. Obviously, isolated and incoming rectangles
do not affect the measure $\mu_+$. Outgoing and transmitting
rectangles can capture some fraction of this measure as we
described in the previous section. A new twist here is
a flow of mass into $M^{(2)}$ from the transmitting
rectangles or directly from $M^{(1)}$. The flowing mass then
evolves in $M^{(2)}$ and approaches $M^{(2)}_+$
competing with the measure $\mu^{(2)}_+$.
This flow is characterized by parameters described below.
Denote by $M^{(1+)}$ the union of $M^{(1)}$ with all
outgoing rectangles connected {\it from} $M^{(1)}$ and
all transmitting rectangles. Consider the restriction of
the map $T$ to $M^{(1+)}$. This restriction has one TT
group of rectangles (it is $M^{(1)}$) and others act
just like outgoing rectangles in the previous section.
We proved there that $T$ on $M^{(1+)}$ has a conditionally
invariant measure $\mu^{(1+)}_+$ with the same eigenvalue
$\lambda_+^{(1)}$ as the measure $\mu^{(1)}_+$. Similarly,
denote by $M^{(2+)}$ the union of $M^{(2)}$ with all
outgoing rectangles connected {\it from} $M^{(2)}$.
Let $\mu_+^{(2+)}$ be the conditional invariant measure
for the restriction of $T$ to $M^{(2+)}$. Note that, in a
peculiar way, the sets $M^{(1+)}$ and $M^{(2+)}$ may have
common outgoing rectangles. But even in this case the
measures $\mu_+^{(1+)}$ and $\mu_+^{(2+)}$ are supported
on disjoint closed sets.
Now, let $q_1^{(12)} > 0$ be the fraction of $\mu_+^{(1+)}$ transmitted
to $M^{(2+)}$ under the action of $T$, i.e. $q_1^{(12)}=T_\ast\mu_+^{(1+)}
(M^{(2+)})$. Denote by $\mu_1^{(12)}$ the measure
$T_\ast\mu_+^{(1+)}$ conditioned on $M^{(2+)}$. For any $k\geq 2$
let $q_k^{(12)} > 0$ be the fraction of $\mu_+^{(1+)}$ transmitted to
$M^{(2+)}$ and surviving $k-1$ iterations of $T$ within $M^{(2+)}$,
i.e. $q_k^{(12)}=q_1^{(12)} T_\ast^{k-1}\mu_1^{(12)}(M^{(2+)})$.
For any $k\geq 2$ let $\mu_k^{(12)}=T_+^{k-1}\mu_1^{(12)}$.
The measure $\mu_1^{(12)}$
is supported on some unstable fibers in $M^{(2+)}$. Its
further evolution under $T_+^k$, $k\geq 1$, within $M^{(2+)}$
will satisfy Theorem~\ref{tmNRRmu}. According to that theorem,
$\mu_k^{(12)}$ will weakly converge to $\mu_+^{(2+)}$
as $k\to\infty$, and $q_k^{(12)}\sim [\lambda_+^{(2)}]^k$, i.e.
$q_k^{(12)} [\lambda_+^{(2)}]^{-k}\to{\rm const}>0$ as $k\to\infty$.
\begin{theorem}
Assume that the two TT groups of rectangles are topologically mixing.
{\rm (i)} If $\lambda^{(1)}_+ > \lambda^{(2)}_+$, then there are
two conditionally invariant probability measures for $T$ whose
conditional measures on unstable fibers are u-SBR measures. One
coincides with $\mu_+^{(2+)}$ and has eigenvalue $\lambda_+^{(2)}$.
The other has eigenvalue $\lambda_+^{(1)}$, it is a weighted sum
\be
\mu_+=Q^{-1}\cdot\left (
\mu_+^{(1+)}+\sum_{k=1}^\infty q_k^{(12)}[\lambda_+^{(1)}]^{-k}\mu_k^{(12)}
\right )
\label{www}
\ee
where $Q^{-1}$ is the normalization factor:
$$
Q = 1+\sum_{k=1}^\infty q_k^{(12)} [\lambda_+^{(1)}]^{-k}
$$
In particular, $\mu_+(M^{(2+)})=1-Q^{-1}$ and $\mu_+(M_+^{(2)})=0$.
{\rm (ii)} If $\lambda^{(1)}_+ \leq \lambda^{(2)}_+$, then
the only conditionally invariant probability measure for $T$ with
u-SBR conditional distributions on unstable fibers is
$\mu_+^{(2+)}$ with eigenvalue $\lambda_+^{(2)}$.
For any smooth measure $\mu$ on $M$ the sequence
$T_+^n\mu$ weakly converges, as $n\to\infty$, to one
of the above conditionally invariant measures. In the case
(i) this limit measure is the one from (\ref{www}) (rather than
$\mu_+^{(2+)}$) if and only if $\mu(M^{(1-)})\neq 0$, where
$M^{(1-)}$ is the union of $M^{(1)}$ and all incoming
rectangles connected {\it to} $M^{(1)}$.
\label{tmTT2mu}
\end{theorem}
{\em Proof}. It is enough to investigate the evolution under
$T_\ast$ of the measure $\mu_0=x\mu_+^{(1+)}+y\mu_+^{(2+)}$
with arbitrary $x,y\geq 0$, $x+y=1$. Its image, $T_\ast\mu_0$, is
$$
x\lambda_+^{(1)}\mu_+^{(1+)}+xq_1^{(12)}\mu_1^{(12)}
+y\lambda_+^{(2)}\mu_+^{(2+)}
$$
Its $k$-th image, $T_\ast^k\mu_0$, is
\be
x[\lambda_+^{(1)}]^k\mu_+^{(1+)}+x\sum_{i=1}^k q_i^{(12)}[\lambda_+^{(1)}]^{k-i}
\mu_i^{(12)}+y[\lambda_+^{(2)}]^k\mu_+^{(2+)}
\label{three}
\ee
The norm of the second term in (\ref{three}) is
$$
x[\lambda_+^{(1)}]^k\sum_{i=1}^k c_i[\lambda_+^{(2)}/\lambda_+^{(1)}]^{i}
$$
with some $c_i=O(1)$, i.e. $c_i$ are bounded away from 0 and $\infty$.
This series converges iff $\lambda_+^{(1)}>\lambda_+^{(2)}$. In this
case the asymptotics of $T_\ast^k\mu_0$ will be determined by the
first two terms in (\ref{three}) provided $x\neq 0$ and by the third term
alone otherwise. In the case $\lambda_+^{(1)}\leq\lambda_+^{(2)}$
we use the convergence of $\mu_k^{(12)}$ to $\mu_+^{(2+)}$.
Renormalizing and taking limit as $k\to\infty$ prove the theorem.
%{\em Remark}. Outgoing rectangles add very little to the above
%theorem: they will simply capture a fraction of $\mu_+$ in the
%way described in the previous section. Note that, in a peculiar
%way, there can be some outgoing rectangles to which transitions
%from both components $M^{(1)}$ and $M^{(2)}$ are possible. In that case,
%however, the measures coming from $M^{(1)}$ and $M^{(2)}$ to that
%rectangle will be supported on disjoint closed subsets of it.
\begin{theorem}
{\rm (i)} If $\lambda^{(1)}_+ > \lambda^{(2)}_+$, then the
measure $\eta_+=\lim T_\ast^{-n}\mu_+$ is either $\eta_+^{(1)}$
or $\eta_+^{(2)}$ depending on $\mu_+$ being defined by (\ref{www})
or being equal to $\mu_+^{(2+)}$;
{\rm (ii)} If $\lambda^{(1)}_+ \leq \lambda^{(2)}_+$, then $\eta_+=
\eta_+^{(2)}$.
\noindent In either case $\eta_+$ is a $T$-invariant Gibbs measure with
potential function $g(x)=-\ln J^u(x)$ and topological pressure
$P=\ln\lambda_+$. It satisfies the equation (\ref{vp1}).
\label{tmTT2eta}
\end{theorem}
This theorem readily follows from the previous one, in view
of Theorems~\ref{tm2} and \ref{tm3}. \medskip
In the case (i) of Theorems~\ref{tmTT2mu} and \ref{tmTT2eta},
the options $\mu_+=\mu_+^{(2+)}$ and $\eta_+=\eta_+^{(2)}$ can be
regarded as quite singular. Indeed, these measures are generated by
initially smooth measures $\mu$ such that $\mu(M^{(1-)})=0$.
>From now on, we will rule out such degenerate measures: \medskip
{\em Definition}. The measures $\mu_+$ and $\eta_+$ are said to be
regular if they are generated by smooth measures on $M$ that are
positive on every open set. \medskip
In particular, $\mu(M^{(1-)})>0$, so that in each case in
Theorems~\ref{tmTT2mu} and \ref{tmTT2eta} the regular measures
are unique. We will restrict ourselves to regular measures.
Then these theorems can be summarized as follows. \medskip
{\bf Rule 1}.
The eigenvalue $\lambda_+$ of the map $T$ on $M$ equals
the largest of the eigenvalues of $T$ restricted to TT components.
The conditionally invariant measure $\mu_+$ is determined by
that of the component with the largest eigenvalue.
The other components that have one-way connections {\it to}
the one with the largest eigenvalue, play no role.
The other components that have one-way connections {\it from}
the one with the largest eigenvalue, play
the same role as outgoing rectangles, capturing a
fraction of $\mu_+$. The invariant measure $\eta_+$
coincides with the one on the TT component with
the largest eigenvalue, as if the others did not exist. \medskip
Motivated by this rule, we will call the TT components
with the largest eigenvalue (i.e., the smallest escape rate)
the {\it dominating} components.
We will see later that the above rule holds for maps with
any number of TT components, provided the dominating component
is unique. Necessary corrections in the case of several
dominating components will be made below.
\section{Three TT groups of rectangles}
\label{secTT3}
\setcounter{equation}{0}
The description of measures $\mu_+$ and $\eta_+$
gets more complicated in the case of more than two TT groups of
rectangles. However, the entire picture is still
determined by the rates of escape of mass from
every transitive component and by the rates of transfer of
mass between components.
It is clear that there can be only one-way routes between components.
These routes make an oriented graph in which transitive components
are vertices. Moreover, there can be no (oriented) loops in this graph,
so that it is actually a tree. We can assume that it is a connected
tree, otherwise it decomposes into two or more trivially independent
trees.
In this section we study maps with three transitive components.
Again, for simplicity we assume that all these components are
topologically mixing. Let
us denote them by $M^{(1)}=R_1\cup \ldots \cup R_{I_1}$ ,
$M^{(2)}=R_{I_1+1}\cup\ldots\cup R_{I_2}$ and
$M^{(3)}=R_{I_2+1}\cup\ldots\cup R_{I_3}$. In addition, there
may be some non-recurrent rectangles $R_{I_3+1}, \ldots ,R_I$.
For the map $T$ restricted to $M^{(i)}$, $i=1,2,3$, we will
use the notations $M_+^{(i)}$, $\lambda_{\pm}^{(i)}$, $\mu_{\pm}^{(i)}$
and $\eta_{\pm}^{(i)}$ introduced in the previous section.
There are four nonisomorphic connected oriented trees with three vertices.
They are (up to renumbering of vertices):\\
(I) $M^{(1)}\to M^{(2)}\to M^{(3)}$,\\
(II) $M^{(1)}\to M^{(2)}$ and $M^{(1)}\to M^{(3)}$, \\
(III) $M^{(1)}\to M^{(2)}\to M^{(3)}$ and $M^{(1)}\to M^{(3)}$,\\
(IV) $M^{(1)}\to M^{(3)}$ and $M^{(2)}\to M^{(3)}$.
For every of these configurations, the nonrecurrent rectangles
can be classified and the sets $M_{\pm}$ and $\Omega$
can be described in a way similar to the one we gave in the
previous section. We do not dwell on this, since it will
not be essential to our analysis. We turn to the study of the
measures $\mu_+$ and $\eta_+$ for the map $T$ on $M$.
The first configuration logically reduces to the study of
two TT groups if we consider first the subgroup $M^{(2)}
\to M^{(3)}$ independently of $M^{(1)}$ and then
the pair $M^{(1)}$ and $M^{(2)}\cup M^{(3)}$. The
results will then perfectly fit Rule 1 at the end
of the previous section. We leave out the details
and turn to the more interesting configurations (II)-(IV).
The configurations (II) and (III) are characterized by
flows of mass from $M^{(1)}$ into $M^{(2)}$ and $M^{(3)}$
(directly or via transmitting nonrecurrent rectangles).
The flowing mass then evolves in both $M^{(2)}$ and $M^{(3)}$
approaching the sets $M^{(2)}_+$ and $M^{(3)}_+$ respectively.
In the configuration (III) there is also a flow of mass from
$M^{(2)}$ to $M^{(3)}$. These flows are characterized by
parameters described below.
For simplicity, we assume that there are no outgoing
or transmitting rectangles in the system. If there are any,
one has to take unions of $M^{(i)}$ with outgoing and transmitting
rectangles connected {\em from} $M^{(i)}$ like we did in the
previous section. This amounts to somewhat heavier notations
but makes little difference in our arguments.
For any pair of components $M^{(i)}$ and $M^{(j)}$ we introduce
a sequence of numbers $\{q^{(ij)}_k\}$ similarly to the
sequence $\{q_k^{(12)}\}$ in the previous section. Let $q^{(ij)}_1 > 0$ be
the fraction of $\mu_+^{(i)}$ transmitted to $M^{(j)}$ under the
action of $T$, i.e. $q^{(ij)}_1=T_\ast\mu_+^{(i)}(M^{(j)})$.
Denote by $\mu_1^{(ij)}$ the measure
$T_\ast\mu_+^{(i)}$ conditioned on $M^{(j)}$. For any $k\geq 2$
let $q^{(ij)}_k > 0$ be the fraction of $\mu_+^{(i)}$ transmitted to
$M^{(j)}$ and surviving $k-1$ iterations of $T$ restricted to $M^{(j)}$,
i.e. $q^{(ij)}_k=q^{(ij)}_1 T_\ast^{k-1}\mu_1^{(ij)}(M^{(j)})$.
For any $k\geq 2$ let $\mu_k^{(ij)}$ be the measure
$T_\ast^{k-1}\mu_1^{(ij)}$ conditioned on $M^{(j)}$.
The measure $\mu_1^{(ij)}$
is supported on some unstable fibers in $M^{(j)}$. Its
further evolution under the restriction of $T^k$, $k\geq 1$, to $M^{(j)}$
will satisfy Theorem~\ref{tm1}. According to that theorem,
$\mu_k^{(ij)}$ will weakly converge to $\mu_+^{(j)}$
as $k\to\infty$, and
$q^{(ij)}_k\sim [\lambda_+^{(j)}]^k$, i.e. $q^{(ij)}_k
[\lambda_+^{(j)}]^{-k}\to{\rm const}>0$ as $k\to\infty$.
\begin{theorem} Assume the configuration (II), the mixing
condition within every TT group of rectangles, and the absence
of outgoing and transmitting rectangles in the system.
{\rm (i)} If $\lambda_+^{(1)} > \max\{\lambda_+^{(2)},\lambda_+^{(3)}\}$,
then the unique regular conditionally invariant measure $\mu_+$
has eigenvalue $\lambda^{(1)}_+$ and is a weighted sum
\be
\mu_+=Q^{-1}\cdot\left (\mu_+^{(1)}
+\sum_{k=1}^\infty
q^{(12)}_k[\lambda_+^{(1)}]^{-k}\mu_k^{(12)}
+\sum_{k=1}^\infty
q^{(13)}_k[\lambda_+^{(1)}]^{-k}\mu_k^{(13)}\right )
\label{wwww}
\ee
where $Q^{-1}$ is the normalization factor:
$$
Q = 1 + \sum_{k=1}^\infty q^{(12)}_k[\lambda_+^{(1)}]^{-k}
+ \sum_{k=1}^\infty q^{(13)}_k[\lambda_+^{(1)}]^{-k}
$$
In particular, $\mu_+(M_+^{(2)}\cup M_+^{(3)})=0$.
{\rm (ii)} Let $\lambda_+^{(1)} \leq \max\{\lambda_+^{(2)},\lambda_+^{(3)}\}$
and $\lambda_+^{(2)}\neq\lambda_+^{(3)}$. Without loss of generality,
assume that $\lambda_+^{(2)} > \lambda_+^{(3)}$. Then the only regular
conditionally invariant measure $\mu_+$ coincides with $\mu_+^{(2)}$
and has eigenvalue $\lambda_+^{(2)}$.
{\rm (iii)} If $\lambda_+^{(1)} \leq \max\{\lambda_+^{(2)},\lambda_+^{(3)}\}$
and $\lambda_+^{(2)} = \lambda_+^{(3)}$, then any weighted sum
of the measures $\mu_+^{(2)}$ and $\mu_+^{(3)}$ is a regular
conditionally invariant measure for $T$. Its eigenvalue is
$\lambda_+^{(2)} = \lambda_+^{(3)}$.
For any smooth measure $\mu$ on $M$ positive on every open set
the sequence $T_+^n\mu$ weakly converges, as $n\to\infty$,
to a regular conditionally invariant measure $\mu_+$. In the
case (iii) the resulting measure $\mu_+$ is determined by the
initial distribution of $\mu$ between the TT components.
\label{tmTT3mu}
\end{theorem}
{\em Proof}. As in the proof of Theorem~\ref{tmTT2mu},
it is enough to investigate the evolution under $T_\ast$
of the measure $\mu_0=x\mu_+^{(1)}+y\mu_+^{(2)}+z\mu_+^{(3)}$
with arbitrary $x,y,z\geq 0$ such that $x+y+z=1$.
Its image, $T_\ast\mu_0$, is
\begin{eqnarray*}
x\lambda_+^{(1)}\mu_+^{(1)}&+&xq^{(12)}_1\mu_1^{(12)}+y\lambda_+^{(2)}\mu_+^{(2)}\\
&+&xq^{(13)}_1\mu_1^{(13)}+z\lambda_+^{(3)}\mu_+^{(3)}
\end{eqnarray*}
Its $k$-th image, $T_\ast^k\mu_0$, is
\begin{eqnarray}
x[\lambda_+^{(1)}]^k\mu_+^{(1)}&+&x\sum_{i=1}^k q^{(12)}_i[\lambda_+^{(1)}]^{k-i}
\mu_i^{(12)}+y[\lambda_+^{(2)}]^k\mu_+^{(2)}\nonumber\\
&+&x\sum_{i=1}^k q^{(13)}_i[\lambda_+^{(1)}]^{k-i}
\mu_i^{(13)}+z[\lambda_+^{(3)}]^k\mu_+^{(3)}
\label{five}
\end{eqnarray}
The rest of the proof goes like that of Theorem~\ref{tmTT2mu}.
\begin{theorem}
Under the conditions of the previous theorem we have
{\rm (i)} If $\lambda_+^{(1)} > \max\{\lambda_+^{(2)},\lambda_+^{(3)}\}$,
then the measure $\eta_+=\lim T_\ast^{-n}\mu_+$ coincides with
$\eta_+^{(1)}$.
{\rm (ii)} Let $\lambda_+^{(1)} \leq \max\{\lambda_+^{(2)},\lambda_+^{(3)}\}$
and $\lambda_+^{(2)} > \lambda_+^{(3)}$ as before. Then
$\eta_+=\eta_+^{(2)}$.
{\rm (iii)} If $\lambda_+^{(1)} \leq \max\{\lambda_+^{(2)},\lambda_+^{(3)}\}$
and $\lambda_+^{(2)} = \lambda_+^{(3)}$, then $\eta_+$ is a
weighted sum of $\eta_+^{(2)}$ and $\eta_+^{(3)}$ with the
same weights as in the case (iii) of the previous theorem.
In every case $\eta_+$ is a $T$-invariant Gibbs measure with
potential function $g(x)=-\ln J^u(x)$ and topological pressure
$P=\ln\lambda_+$. It is ergodic in the cases (i) and (ii), and
has two ergodic components in the case (iii).
The measure $\eta_+$ satisfies the equation (\ref{vp1}).
\label{tmTT3eta}
\end{theorem}
We now turn to the configuration (III). A new twist here is a
secondary flow of mass from $M^{(1)}$ to $M^{(3)}$ via $M^{(2)}$.
For any $m,n\geq 1$ denote by $M^{(1)}_{m,n}$ the set of points
of $M^{(1)}$ whose first $m$ images land in $M^{(2)}$ and the
following $n$ images land in $M^{(3)}$, i.e.
\begin{eqnarray*}
M^{(1)}_{m,n}=\{x\in M^{(1)}:\, T^ix\in M^{(2)}\ \ &{\rm for}&\
1\leq i\leq m\ \ {\rm and}\\ T^jx\in M^{(3)}\ \ &{\rm for}&\
m+1\leq j\leq m+n\}
\end{eqnarray*}
Let $r^{(123)}_{m,n}=\mu_+^{(1)}(M^{(1)}_{m,n})$ and $\mu_{m,n}^{(123)}=
T_\ast^{m+n}(\mu_+^{(1)}|M^{(1)}_{m,n})$, note that $\mu_{m,n}^{(123)}$ is
a probability measure. Obviously, $T_\ast^{-1}\mu_{m,1}^{(123)}$
is the measure $\mu_m^{(12)}$ conditioned on $M^{(2)}\cap T^{-1}M^{(3)}$.
Therefore, the measure $T_\ast^{-1}\mu_{m,1}^{(123)}$ converges,
as $m\to\infty$, to $\mu_+^{(2)}$ conditioned on $M^{(2)}\cap T^{-1}M^{(3)}$.
According to Theorem~\ref{tm1}, the measure $\mu_{m,n}^{(123)}$ then
converges, as $n\to\infty$, to $\mu_+^{(3)}$, and this convergence
is uniform in $m$. Also, $r^{(123)}_{m,n}\sim [\lambda_+^{(2)}]^m
[\lambda_+^{(3)}]^n$, i.e. $r^{(123)}_{m,n}[\lambda_+^{(2)}]^{-m}
[\lambda_+^{(3)}]^{-n}\to{\rm const}>0$ as $m,n\to\infty$, and the
values of $r^{(123)}_{m,n}[\lambda_+^{(2)}]^{-m}[\lambda_+^{(3)}]^{-n}$
are bounded away from 0 and $\infty$.
\begin{theorem} Assume the configuration (III), the mixing
condition within every TT group of rectangles, and the absence
of outgoing and transmitting rectangles in the system.
{\rm (i)} If $\lambda_+^{(1)} > \max\{\lambda_+^{(2)},\lambda_+^{(3)}\}$,
then the unique regular conditionally invariant measure $\mu_+$
has eigenvalue $\lambda^{(1)}_+$, and it is a weighted sum
\begin{eqnarray}
\mu_+=Q^{-1}\cdot\Big (\mu_+^{(1)}
&+&\sum_{k=1}^\infty
q^{(12)}_k[\lambda_+^{(1)}]^{-k}\mu_k^{(12)}
+ \sum_{k=1}^\infty
q^{(13)}_k[\lambda_+^{(1)}]^{-k}\mu_k^{(13)}\nonumber\\
&+&\sum_{m,n=1}^\infty
r^{(123)}_{m,n}[\lambda_+^{(1)}]^{-m-n}\mu_{m,n}^{(123)}
\Big )
\label{wwwww}
\end{eqnarray}
where $Q^{-1}$ is the normalization factor:
$$
Q = 1 + \sum_{k=1}^\infty q^{(12)}_k[\lambda_+^{(1)}]^{-k}
+ \sum_{k=1}^\infty q^{(13)}_k[\lambda_+^{(1)}]^{-k}
+ \sum_{m,n=1}^\infty r^{(123)}_{m,n}[\lambda_+^{(1)}]^{-m-n}
$$
In particular, $\mu_+(M_+^{(2)}\cup M_+^{(3)})=0$.
{\rm (ii)} Let $\lambda_+^{(1)} \leq \lambda_+^{(2)}$
and $\lambda_+^{(2)}>\lambda_+^{(3)}$. Then the only regular
conditionally invariant measure $\mu_+$
has eigenvalue $\lambda^{(2)}_+$ and is a weighted sum
\be
\mu_+=Q^{-1}\cdot\left (\mu_+^{(2)}
+\sum_{k=1}^\infty
q^{(23)}_k[\lambda_+^{(2)}]^{-k}\mu_k^{(23)}\right )
\label{www'}
\ee
where $Q^{-1}$ is the normalization factor:
$$
Q = 1 + \sum_{k=1}^\infty q^{(23)}_k[\lambda_+^{(2)}]^{-k}
$$
In particular, $\mu_+(M_+^{(1)}\cup M_+^{(3)})=0$.
{\rm (iii)} Let $\lambda_+^{(3)} \geq \max\{\lambda_+^{(1)},
\lambda_+^{(2)}\}$. Then the only regular
conditionally invariant measure $\mu_+$ coincides with $\mu_+^{(3)}$
and has eigenvalue $\lambda_+^{(3)}$.
For any smooth measure $\mu$ on $M$ positive on every open set
the sequence $T_+^n\mu$ weakly converges, as $n\to\infty$,
to a regular conditionally invariant measure $\mu_+$.
\label{tmTT3mu1}
\end{theorem}
{\em Proof}. As in the proofs of Theorems~\ref{tmTT2mu} and
\ref{tmTT3mu}, it is enough to investigate the evolution under
$T_\ast$ of the measure $\mu_0=x\mu_+^{(1)}+y\mu_+^{(2)}+z\mu_+^{(3)}$
with arbitrary $x,y,z\geq 0$ such that $x+y+z=1$.
Its $k$-th image, $T_\ast^k\mu_0$, is
\begin{eqnarray*}
x[\lambda_+^{(1)}]^k\mu_+^{(1)}&+&
x\sum_{i=1}^k q^{(12)}_i[\lambda_+^{(1)}]^{k-i}\mu_i^{(12)}
+y[\lambda_+^{(2)}]^k\mu_+^{(2)}\\
&+&
x\sum_{i=1}^k q^{(13)}_i[\lambda_+^{(1)}]^{k-i}\mu_i^{(13)}
+z[\lambda_+^{(3)}]^k\mu_+^{(3)}\\
&+&
y\sum_{i=1}^k q^{(23)}_i[\lambda_+^{(2)}]^{k-i}\mu_i^{(23)}+
x\sum_{i+j\leq k} r_{i,j}^{(123)}[\lambda_+^{(1)}]^{k-i-j}
\mu_{i,j}^{(123)}
\end{eqnarray*}
The rest of the proof goes basically like that of Theorems~\ref{tmTT2mu}
and \ref{tmTT3mu}. In the analysis of the case (ii) the measures
$\mu_{i,j}^{(123)}$ play some role. The necessary result follows
from two facts: (i) the measure $T_\ast^{-1}\mu_{i,1}^{(123)}$ converges,
as $i\to\infty$, to $\mu_+^{(2)}$ conditioned on $M^{(2)}\cap T^{-1}M^{(3)}$,
and (ii) for any $j_1,j_2\geq 1$ we have
$$
\lim_{i\to\infty} r_{i,j_1}^{(123)}/r_{i,j_1}^{(123)} =
q_{j_1}^{(23)}/q_{j_2}^{(23)}
$$
\begin{theorem}
Under the conditions of the previous theorem we have
{\rm (i)} If $\lambda_+^{(1)} > \max\{\lambda_+^{(2)},\lambda_+^{(3)}\}$,
then the measure $\eta_+=\lim T_\ast^{-n}\mu_+$ coincides with
$\eta_+^{(1)}$.
{\rm (ii)} Let $\lambda_+^{(1)} \leq \lambda_+^{(2)}$
and $\lambda_+^{(2)} > \lambda_+^{(3)}$. Then
$\eta_+=\eta_+^{(2)}$.
{\rm (iii)} If $\lambda_+^{(3)} \geq \max\{\lambda_+^{(1)},\lambda_+^{(2)}\}$
then $\eta_+=\eta_+^{(3)}$.
In every case $\eta_+$ is an ergodic $T$-invariant Gibbs measure with
potential function $g(x)=-\ln J^u(x)$ and topological pressure
$P=\ln\lambda_+$. It satisfies the equation (\ref{vp1}).
\label{tmTT3eta1}
\end{theorem}
The last configuration, IV, can be reduced to III by eliminating
the flow of mass from $M^{(1)}$ to $M^{(2)}$ together with the
secondary flow from $M^{(1)}$ to $M^{(3)}$ via $M^{(2)}$.
In the previous two theorems this forces $q^{(12)}_k=0$ and
$r_{m,n}^{(123)}=0$ for all $k,m,n$. Then the results of
those theorems apply to the configuration IV in the cases
(i) and (iii). The case (ii) goes through under an additional
assumption that $\lambda_+^{(1)}<\lambda_+^{(2)}$. The possibility
$\lambda_+^{(1)}=\lambda_+^{(2)}>\lambda_+^{(3)}$ is treated
separately in the following theorem.
\begin{theorem}
Assume the configuration (IV), the mixing condition
within every TT group of rectangles, and the absence
of outgoing and transmitting rectangles in the system.
Let $\lambda_+^{(1)}=\lambda_+^{(2)}>\lambda_+^{(3)}$.
Then any regular conditionally invariant measure for $T$
is a weighted sum $\mu_+=w_1\mu_{+,1}+w_2\mu_{+,2}$, where
$$
\mu_{+,i}=
\mu_+^{(i)}+\sum_{k=1}^\infty q_k^{(i3)}
[\lambda_+^{(i)}]^{-k}\mu_k^{(i3)}
$$
for $i=1,2$. The measures $\mu_{+,i}$ are singular with respect to each other.
The eigenvalue of any such $\mu_+$ is $\lambda_+^{(1)} = \lambda_+^{(2)}$.
For any smooth measure $\mu$ on $M$ positive on every open set
the sequence $T_+^n\mu$ weakly converges, as $n\to\infty$,
to some regular conditionally invariant measure $\mu_+$,
whose weights are determined by the initial distribution
of $\mu$ between the TT components.
The $T$-invariant measure $\eta_+=\lim T_\ast^{-k}\mu_+$ is a
weighted sum of $\eta_+^{(1)}$ and $\eta_+^{(2)}$.
It is a Gibbs measure with potential function $g(x)=-\ln J^u(x)$
and topological pressure $P=\ln\lambda_+$. It
satisfies the equation (\ref{vp1}) and has two ergodic components.
\end{theorem}
\section{General conclusions}
\label{secGC}
\setcounter{equation}{0}
Here we generalize (leaving out some technical details and
exact proofs) the theorems obtained in the previous two sections
to systems with arbitrary number of TT components.
First of all, if the system has an only dominating TT component
(the one with the largest eigenvalue), then Rule 1 describes
the properties of measures $\mu_+$ and $\eta_+$.
If the system has more than one dominating TT components, then we
subdivide them into essential and nonessential ones
as follows. Any dominating component $M^{(i)}$ that has a one-way
connection {\em to} another dominating component (possibly, via
some transmitting rectangles and/or other TT components) is
said to be {\em nonessential}. The remaining dominating components
are {\em essential}. \medskip
{\bf Rule 2}.
The measures $\mu_+$ and $\eta_+$ always exist. They
are determined by essential dominating (ED) components only.
If the system has just one ED component, the measures
$\mu_+$ and $\eta_+$ are unique and determined by that
component according to the Rule~1, as if all the other
TT components were not even dominating.\medskip
{\bf Rule 3}. If the system has two or more ED components,
the measures $\mu_+$ and $\eta_+$ are not unique.
Every ED component $M^{(i)}$ determines measures $\mu_{+,i}$
and $\eta_{+,i}$ according to the Rule~2. The measures
$\mu_{+,i}$ are singular with respect to each other.
The set of regular conditionally invariant measures
$\mu_+$ for $T$ is the convex hull of the measures $\mu_{+,i}$.
They have the same eigenvalue, which is
the common eigenvalue of all ED components.
The set of regular invariant measures $\eta_+$ for $T$
is the convex hull of the measures $\eta_{+,i}$.
Every $\eta_+$ is a Gibbs measure with potential
function $g(x)=-\ln J^u(x)$ and topological pressure
$P=\ln\lambda_+$, and it satisfies the
equation (\ref{vp1}). The number of its ergodic components equals
the number of ED components in the system. \medskip
We do not prove Rules 2 and 3 in the general case,
since they directly generalize our theorems proved
in two previous sections. We also leave out detailed description
of the measures $\mu_{+,i}$ that was provided in the previous sections.
We could have given such a description along the lines
developed in the case (i) of Theorems~\ref{tmTT2mu}, \ref{tmTT3mu} and
\ref{tmTT3mu1}, but it would involve unpleasantly heavy, though
conceptually simple, calculations. So, we restricted ourselves to
the detailed analysis of two and three TT groups.
Finally, let us emphasize that Rules 2 and 3 do not require
the topological mixing condition within TT components. This
condition only affects the way the iterations of smooth measures
converge to $\mu_+$, and the way the iterations of $T^{-n}_\ast\mu_+$
converge to $\eta_+$. If the mixing condition within ED components
fails, then the Cesaro limit of $T^n_\ast\mu$, for any smooth
measure $\mu$, is a measure $\mu_+^0$ equivalent to
$\mu_+$, cf. Sect.~\ref{secTT}. Also, the Cesaro limit of
$T^{-n}_\ast\mu_+$ (and the limit of $T^{-n}_\ast\mu_+^0$)
is $\eta_+$. \medskip
{\bf Acknowledgements}. This work was started when R.M. visited
University of Alabama at Birmingham, for which he is the most
indebted. Both authors thank S.~Troubetzkoy for fruitful
discussion and numerous remarks on the paper.
N.Ch. acknowledges the support of NSF grant DMS-9622547,
and R.M. the support of CONICYT (Uruguay).
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\end
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