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\begin{document}
\title{Statistical properties of piecewise smooth hyperbolic systems
in high dimensions}
\author{N. Chernov
\\ Department of Mathematics\\
University of Alabama at Birmingham\\
Birmingham, AL 35294, USA\\
E-mail: chernov@vorteb.math.uab.edu
}
\date{\today}
\maketitle
\begin{abstract}
We study smooth hyperbolic systems with singularities and
their SRB measures. Here we assume uniform hyperbolicity, aside
from singularities, and the existence of one-sided derivatives
on singularities. We prove that the SRB measures exist,
are finitely many, and mixing SRB measures enjoy exponential
decay of correlations and a central limit theorem. These
properties have been proved previously only for 2-D
systems.
\end{abstract}
{\em Keywords}: Decay of correlations, Sinai-Ruelle-Bowen measures,
hyperbolic dynamics.
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\section{Introduction}
\label{secI}
\setcounter{equation}{0}
Let $M$ be an open connected domain in a $d$-dimensional
$C^\infty$ Riemannian manifold, such that $\bar{M}$ is compact,
and let $\Gamma\subset \bar{M}$ be a closed subset.
Assume that ${\cal S}:=\Gamma\cup\partial M$
is a finite union of smooth compact submanifolds of
codimension one, possibly with boundary. We denote by
${\cal S}_1,{\cal S}_2,\ldots {\cal S}_r$
the smooth components of $\cal S$.
% ${\cal S}={\cal S}_1\cup\cdots\cup {\cal S}_r$.
% such that each ${\cal S}_i$ is bounded by
% a finite number of submanifolds of codimension two.
We consider a map $T:\, M\setminus {\cal S}\to M$ such that
\medskip
{\bf (H1)} $T$ is a $C^2$ diffeomorphism
of $M\setminus {\cal S}$ onto its image. We also assume that $T$
and $T^{-1}$ are twice differentiable up to the boundaries
of their domains (only one-sided derivatives are required
at the boundary).
\medskip
The set $\cal S$ will be referred to as the singularity set for $T$.
For $n\geq 1$ denote by
$$
{\cal S}^{(n)}={\cal S}\cup T^{-1}{\cal S}\cup\cdots\cup T^{-n+1}{\cal S}
$$
the singularity set for $T^n$. Define
$$
M^+=\{x\in M: T^nx\notin {\cal S},\, n\geq 0\},
\ \ \ \ \ \
M^-=\cap_{n\geq 0}T^n(M\setminus {\cal S}^{(n)})
$$
and
$$
M^{0}=\cap_{n\geq 0}T^n(M^+)=M^+\cap M^-
$$
The sets $M^+$ and $M^-$ consist, respectively,
of points were all the future and past iterations
of $T$ are defined, and $M^{0}$ is the set of points
where all the iterations of $T$ are defined. We denote by
$\rho$ the Riemannian metric in $M$ and by Vol$(\cdot )$
the Lebesgue measure (volume) in $M$.
Three most interesting classes of maps are
\begin{enumerate}
\item {\bf Conservative case:} volume-preserving maps,
or, more generally, maps with an absolutely continuous
invariant measure (a.c.i.m.).
\item {\bf Dissipative case:} no a.c.i.m. exist, yet
$T(M\setminus {\cal S})$ is dense in $M$. In this
case, like in the previous one, $M^{0}$ has full Lebesgue measure.
\item {\bf Attractor case:} when the closure of
$T(M\setminus {\cal S})$ is a proper subset of $M$. In this case
Vol$(M^{0})<$Vol$(M)$, and often Vol$(M^{0})=0$.
The set $\overline{M^{0}}$ is then called an attractor.
\end{enumerate}
Below we list our additional assumptions on $T$. \medskip
{\bf (H2)} $T$ is uniformly hyperbolic, i.e. there exist
two families of cones $C^u_x$ and $C^s_x$ in the tangent
spaces ${\cal T}_xM$, $x\in \bar{M}$, such
that $DT(C^u_x)\subset C^u_{Tx}$ and $DT(C^s_x)\supset
C^s_{Tx}$ whenever $DT$ exists, and
$$
|DT(v)|\geq \Lambda_{\min}|v|\ \ \ \ \forall v\in C^u_x
$$
$$
|DT^{-1}(v)|\geq \Lambda_{\min}|v|\ \ \ \ \forall v\in C^s_x
$$
with some constant $\Lambda_{\min}>1$. These families of cones
are continuous on $\bar{M}$ and the angle between $C^u_x$
and $C^s_x$ has a positive lower bound. \medskip
Technically, the families of cones $C^{u,s}_x$ are specified
by two continuous families of linear subspaces $P^{u,s}_x
\subset{\cal T}_xM$ such that $P^u_x\oplus P^s_x={\cal T}_xM$,
and two continuous functions $\alpha^{u,s}(x)>0$. The cones
$C^{u,s}_x$ are defined by
$$
\angle (v,P^{u,s}_x):=\min_{w\in P^{u,s}_x}\angle (v,w)\leq \alpha^{u,s}(x)
\ \ \ \ \
\forall v\in C^{u,s}_x
$$
The angle between the cones $C^u_x$
and $C^s_x$ is set to $\min\{\angle(v,w):\, v\in C^u_x, w\in C^s_x\}$.
We denote $d_{u,s}={\rm dim}P^{u,s}_x$ (these are independent
of $x$, since $P^{u,s}_x$ are continuous and $M$ is
connected, and $d_u+d_s=d={\rm dim}M$).
Denote $\Lambda_{\max}=\max\{\sup_x ||DT(x)||,\sup_x ||DT^{-1}||\}$.
In plain words, $\Lambda_{\min}$ and $\Lambda_{\max}$ are lower and
upper bounds on the expansion factor of unstable vectors
and contraction factor of stable vectors.
For any submanifold
$W\subset M$ we denote by $\rho_W$ the metric on $W$
induced by the Riemannian metric in $M$, and by
$\nu_W$ the Lebesgue measure on $W$ generated by $\rho_W$.
We call $U$ a u-manifold if it is a smooth $d_u$-dimensional
submanifold in $M$ of finite diameter (in the inner metric $\rho_U$)
and at every $x\in U$ the tangent space
${\cal T}_xU$ lies in $C^u_x$. Any u-manifold is expanded (locally)
by $T$ in every direction by a factor between $\Lambda_{\min}$
and $\Lambda_{\max}$. Similarly, s-manifolds are defined.
\medskip
{\bf (H3)} The angle between $\cal S$ and $C^u$ has a positive
lower bound. \medskip
Technically, the angle between $\cal S$ and $C^u_x$ at $x\in \cal S$
is defined to be $\max\{0,\angle (P^u_x,{\cal T}_x{\cal S})-\alpha^u(x)\}$.
Here $\angle (P^u_x,{\cal T}_x{\cal S})=\max_{v\in P^u_x}
\min_{w\in{\cal T}_x{\cal S}}\angle (v,w)$.
As a consequence of (H3), any u-manifold intersects $\cal S$ transversally,
and the angle between them has a positive lower bound.
It is convenient to assume that for every ${\cal S}_i\subset \Gamma$
we have $\partial{\cal S}_i\subset \cup_{j\neq i}{\rm int}{\cal S}_j
\cup \partial M$,
i.e. every interior singularity manifold with boundary terminates
on some other singularity manifolds or on $\partial M$. This is
not a restrictive assumption, since if this is not the case
for some ${\cal S}_i\subset\Gamma$, we can extend ${\cal S}_i$
until it terminates on other hypersurfaces of ${\cal S}$ or on
the boundary of $M$.
A point $x\in {\cal S}^{(m)}$ of the singularity set
${\cal S}^{(m)}$ of $T^m$ is said to be multiple if it
belongs to $l\geq 2$ smooth components
of ${\cal S}^{(m)}$, and then $l$ is called the multiplicity
of $x$ in ${\cal S}^{(m)}$.
\medskip
{\bf (H4)} There are $K_0\geq 1$ and $m\geq 1$ such that
the multiplicity of any point $x\in {\cal S}^{(m)}$
does not exceed $K_0$, and $K_0<\Lambda_{\min}^m-1$. \medskip
This is a standard assumption which ensures that the
singularity manifolds of ${\cal S}^{(m)}$ do not pile up
too fast anywhere as $m$ grows. The expansion of any u-manifold $U$
under $T^m$ is hereby guaranteed to be stronger than
the cutting (shredding) of $U$ inflicted by ${\cal S}^{(m)}$.
We make this claim precise below in Section~\ref{secU}.
It is also standard to assume that $m=1$ here,
which we do, since we can simply consider $T^m$
instead of $T$. (The assumptions (H1)-(H3) obviously
hold for all $T^m$, $m\geq 1$.)
For any $x\in M^+$ and $y\in M^-$ we set
$$
E^s_x=\cap_{n\geq 0}DT^{-n}(C^s_{T^nx}),\ \ \ \ \ \ \
E^u_y=\cap_{n\geq 0}DT^n(C^u_{T^{-n}y})
$$
respectively. It is standard, see, e.g., \cite{Pes92}, that \\
(a) $E^s_x$, $E^u_x$ are linear subspaces in ${\cal T}_xM$,
dim$E^{u,s}_x=d_{u,s}$, and $E^s_x\oplus E^u_x={\cal T}_xM$
for $x\in M^{0}$;\\
(b) $DT(E^{u,s}_x)=E^{u,s}_{Tx}$, and $DT$ expands vectors
in $E^u_x$ and contracts vectors in $E^s_x$;\\
(c) the subspaces $E^u_x$ and $E^s_x$ are continuous in $x$
(on $M^-$ and $M^+$, respectively), and the angle between
them on $M^{0}$ has a positive lower bound.
As a consequence, there can be no zero Lyapunov exponents
on $M^{0}$. The space $E^u_x$ is spanned by all vectors
with positive Lyapunov exponents, and $E^s_x$ by those
with negative Lyapunov exponents.
We call a submanifold $W^u\subset M$ a local unstable manifold
(LUM),
if $T^{-n}$ is defined and smooth on $W^u$ for all $n\geq 0$,
and $\forall x,y\in W^u$ we have $\rho(T^{-n}x,T^{-n}y)
\to 0$ as $n\to\infty$ exponentially fast. Similarly,
local stable manifolds (LSM), $W^s$, are defined. Obviously,
dim$W^{u,s}=d_{u,s}$, and at any
point $x\in W^{u,s}$ the tangent space
${\cal T}_xW^{u,s}$ coincides with $E^{u,s}_x$.
We denote by $W^u(x)$, $W^s(x)$ local unstable and stable
manifolds containing $x$, respectively. The existence and
abundance of LUM's and LSM's in $M$ is proved in Sect.~\ref{secE}.
We state our main result, with necessary definitions
following it.
\begin{theorem}
Let $T$ satisfy (H1)-(H4). Then
{\rm (a)} Existence: $T$ admits a Sinai-Ruelle-Bowen (SRB) measure $\mu$;
{\rm (b)} Ergodic properties:
any SRB measure $\mu$ has a finite number of ergodic
components, on each of which it is, up to a finite cycle,
mixing and Bernoulli;
{\rm (c)} Statistical properties:
if $(T^n,\mu)$ is ergodic $\forall n\geq 1$, then
$(T,\mu)$ has exponential decay of correlations
and satisfies the central limit theorem for H\"older
continuous functions on $M$.
\label{tmmain}
\end{theorem}
{\bf Definition 1.} A $T$-invariant measure $\mu$ concentrated
on $M^{0}$
is called a Sinai-Ruelle-Bowen (SRB) measure if the conditional
measures of $\mu$ on local unstable manifolds are absolutely
continuous with respect to the Lebesgue measures on those
manifolds. \medskip
The part (b) of the theorem means that $\mu$ has a finite
number of ergodic components $M^{0}_1,\ldots, M^{0}_s$,
and on each $M^{0}_i$ the map $(T,\mu|M^{0}_i)$ either
is mixing and Bernoulli, or else $M^{0}_i$ is further
decomposed into a finite number of subcomponents
$M^{0}_i=M^{0}_{i,1}\cup\cdots\cup M^{0}_{i,s_i}$
which are permuted cyclicly by $T$. In the latter case
the map $(T^{s_i},\mu|M^{0}_{i,j})$ is mixing and
Bernoulli for every $M^{0}_{i,j}$. The part (c) of the
theorem applies to the dynamical system
$(T^{s_i},\mu|M^{0}_{i,j})$ for each $M^{0}_{i,j}$.
It is also standard that {\em any} SRB measure
on $M^{0}$ is a weighted sum of (unique) ergodic
SRB measures concentrated on the components $M^{0}_1,
\ldots,M^{0}_s$. Thus, all the SRB measures for $T$
make an $s$-dimensional simplex, whose vertices are
ergodic SRB measures. \medskip
SRB measures are the only physically observable
invariant measures for smooth or piecewise
smooth hyperbolic dynamical systems. In the conservative
case (Case 1 above), any a.c.i.m. is an SRB measure
automatically, and vice versa. In the
dissipative and attractor cases 2 and 3, SRB measures
are weak Cesaro limits of iterations of smooth measures
on $M$. Furthermore, for any ergodic SRB measure $\mu$ there is a
positive volume set consisting of $\mu$-generic points,
i.e. points $x\in M$ such that
$\frac 1n\sum_{i=0}^{n-1} f\circ T^i(x)\to\int f\, d\mu$
for all continuous functions $f:M\to\IR$, see, e.g., \cite{LSY}.
(This property is sometimes taken as the definition of
SRB measures.)
%Also, a.e. point $x\in M$ is $\mu$-generic for an SRB measure $\mu$.
In the dissipative case 2, $\mu$ is
typically singular, i.e. Vol$(M^{0})=0$, but the support
of $\mu$ can coincide with $\bar{M}$, i.e. the
$\mu$-measure of every open set may be positive. That
happens to typical transitive Anosov diffeomorphisms
\cite{Si72} and some nonequilibrium stationary distributions
studied in modern statistical physics \cite{GC}. In the attractor
case (3), the support of $\mu$ normally has zero volume,
the best studied examples here being Lorenz, Lozi and Belykh
attractors \cite{ACS}. \medskip
Now, let ${\cal H}_\eta$ be the class of H\"older continuous
functions on $M$ with exponent $\eta >0$:
$$
{\cal H}_\eta=\{f:\, M\to \IR\, |\,
\exists C>0:\, |f(x)-f(y)|\leq C\rho(x,y)^\eta ,\ \forall x,y\in M\}
$$
{\bf Definition 2.} We say that
$(T,\mu)$ has exponential decay of correlations for H\"older
continuous functions if $\forall\eta>0$ $\exists\gamma
=\gamma(\eta)\in (0,1)$ such that $\forall f,g\in{\cal H}_\eta$
$\exists C=C(f,g)>0$ such that
$$
\left |
\int_M (f\circ T^n)g\, d\mu-
\int_M f\, d\mu\int_M g\, d\mu
\right |
\leq C\gamma^{|n|}\ \ \ \ \ \forall n\in\ZZ
$$
{\bf Definition 3.} We say that $(T,\mu)$ satisfies
a central limit theorem (CLT) for H\"older
continuous functions if $\forall \eta>0,f\in{\cal H}_\eta$,
with $\int f\,d\mu=0$, $\exists \sigma_f\geq 0$ such that
$$
\frac{1}{\sqrt{n}}\sum_{i=0}^{n-1}f\circ T^i
\stackrel{\rm distr}{\longrightarrow}{\cal N}(0,\sigma_f^2)
$$
Furthermore, $\sigma_f=0$ iff $f=g\circ T-g$ for some $g\in L^2(\mu)$
\medskip
{\em History}. The existence of SRB measures and their ergodic
properties have been
obtained for the so called generalized hyperbolic attractors
by Pesin \cite{Pes92} and sharpened by Sataev \cite{Sat92}.
Pesin and Sataev assumed much less of the geometry of $M$ and the
smoothness of $T$ than we do. On the other hand, they assumed
special bounds on the volume of $\varepsilon$-neighbohoods of
the singularity set $\cal S$, see (H5)-(H7) in our Sect.~\ref{secE}.
For the systems like ours the assumptions (H5)-(H7) have
been verified only in the 2-D case (first in ref. \cite{Pes92}
under a more restrictive assumption than our (H4) and then in
ref. \cite{ACS} under the one equivalent to (H4)). We do that
verification in any dimension in Sect.~\ref{secE}.
The statistical properties of the SRB measures
have been studied, again only in 2-D case, by Liverani
\cite{L} and Young \cite{LSY}.
Earlier, a subexponential or ``stretched'' exponential
bound on correlations and a CLT in that same 2-D case were
obtained in ref. \cite{ACS}. In higher dimensions,
only a class of dispersing billiards has been studied
in ref. \cite{Ch94}, where
a stretched exponential decay of correlations and
a CLT were proved. Here we combine the recent general results
by Young \cite{LSY} and some techniques of ref. \cite{Ch94}
to obtain statistical properties of SRB measures
in any dimension.
\section{Expansion and filtration of u-manifolds}
\label{secU}
\setcounter{equation}{0}
Here we study u-manifolds. The general theme will be showing
that the expansion of u-manifolds by $T$ is ``stronger''
(in many senses) than cutting by singularity manifolds $\cal S$.
More precisely, we will show that the images of small
u-manifolds under $T^n$, $n\geq 0$, grow in size
exponentially in $n$ `on the average', until they
reach a certain `fixed' size ($\delta_1$ below).
{\em Notations}. Let $U$ be a u-manifold. We denote by
diam$U$ the diameter of $U$ in the $\rho_U$ metric.
For any point $x\in U\setminus {\cal S}$ denote by
$J^u(x)=|{\rm det}(DT|{\cal T}_xU)|$ the jacobian of the map
$T$ restricted to $U$ at $x$, i.e. the factor of the
volume expansion on $U$ at the point $x$.
For $n\geq 1$ the connected components of
$T^n(U\setminus {\cal S}^{(n)})$ are called {\em components}
of $T^nU$.
The following is standard, and we omit the proofs:\medskip
(a) {\em Curvature}. $\exists B'>B''>0$ such that if
the sectional curvature of a u-manifold $U$ is $\leq B''$,
then all the components of $T^nU$, $n\geq 1$, have sectional
curvature $\leq B'$. As a result, sectional curvature
of any LUM $W^u$ is bounded above by $B'$. We will always
assume that sectional curvature of our u-manifolds
is bounded above by $B'$. \medskip
(b) {\em Distorsions}.
Let $x,y\in U\setminus {\cal S}^{(n-1)}$ and
$T^nx,T^ny$ belong in one component of $T^nU$, denote it by $V$. Then
\be
\log\prod_{i=0}^{n-1}\frac{J^u(T^ix)}{J^u(T^iy)}
\leq C'\rho_{V}(T^nx,T^ny)
\label{distor1}
\ee
with some $C'=C'(T)>0$. \medskip
(c) {\em Absolute continuity}.
Let $U_1,U_2$ be two sufficiently small u-manifolds,
so that any local stable manifold $W^s$ intersects each of $U_1$
and $U_2$ in at most one point. Let $U_1'=\{x\in U_1:\,
W^s(x)\cap U_2\neq\emptyset\}$. Then we define a map $h:U_1'\to U_2$
by sliding along stable manifolds. This map is often
called a holonomy map. It is absolutely continuous with
respect to the Lebesgue measures $\nu_{U_1}$ and $\nu_{U_2}$,
and its jacobian (at any point of density of $U_1'$)
is bounded, i.e.
\be
1/C''\leq\frac{\nu_{U_2}(h(U_1'))}{\nu_{U_1}(U_1')}\leq C''
\label{ac}
\ee
with some $C''=C''(T)>0$. \medskip
Our assumption (H4) implies that $\exists\bar{\delta}>0$
such that any $\bar{\delta}$-ball in $M$ intersects
at most $K_0$ smooth components of $\cal S$.
We now fix a $\delta_0\ll\bar{\delta}$ and will assume
that it is small enough for all our future needs. \medskip
{\bf Definition}.
We say that a connected u-manifold $U$ is {\em admissible} if \\
(a) its sectional curvature is $\leq B'$ everywhere;\\
(b) diam$U\leq \delta_0$;\\
(c) its boundary $\partial U$ is piecewise
smooth, i.e. it is a finite union of smooth compact submanifolds
of dimension dimension $d_u-1$, possibly with boundary. \medskip
%In what follows we always consider admissible u-manifolds
%without specifying that. \medskip
{\bf Key Remark}.
Let $U$ be an admissible u-manifold. Since $\delta_0$ is very small,
the tangent spaces ${\cal T}_xU$ are almost parallel
at all points $x\in U$. If $n\geq 1$ and $U'\subset T^nU$
is another admissible u-manifold, then
$T^n_{\ast}\nu_U|U'$ (the $n$th iterate of the Lebesgue
measure on $U$ conditioned on $U'$) has an almost constant
density with respect to $\nu_{U'}$, due to (\ref{distor1}).
These important observations will allow us to approximate
any admissible u-manifold
by a $d_u$-dimensional flat domain in $\IR^d$, i.e. a domain
on a $d_u$-dimensional linear subspace of $\IR^d$
with piecewise smooth boundary.
In addition, we assume that $\delta_0\ll$ the minimum
radius of curvature of singularity manifolds ${\cal S}^{(i)}\subset \cal S$.
Thus, if a u-manifold $U$ intersects a singularity manifold
${\cal S}^{(i)}$, all the tangent spaces to ${\cal S}^{(i)}$ at the points
of ${\cal S}^{(i)}\cap U$ are almost parallel. Hence, the manifold
${\cal S}^{(i)}$ is almost flat on the `microscopic' scale of
diam$U\leq\delta_0$. \medskip
Let $U$ be an admissible u-manifold. The components of
its iterates, $T^nU$, $n\geq 1$, may not be admissible,
since they grow in size. We will partition them into
smaller, admissible u-manifolds. \medskip
{\bf Definition}. Let $U$ be an admissible u-manifold,
and $V\subset U$ an open subset with piecewise smooth
boundary (i.e., $\partial V$ consist of a finite number of
smooth compact $(d_u-1)$-dimensional submanifolds in
$\bar{M}$). $\forall x\in V$ we denote by $V(x)$ the connected
component of $V$ that contains $x$. We say that $V$ is
$n$-{\em admissible}, $n\geq 0$,
if $T^n$ is smooth on $V$ and $\forall x\in V$
the u-manifold $T^nV(x)$ is admissible. \medskip
Observe that $V$ need not be connected, in fact, it
almost never is in our arguments. Observe also that
for an $n$-admissible open set $V$ we have
$V\subset U\setminus {\cal S}^{(n)}$.
Let $U$ be an admissible u-manifold, and $V\subset U$
an $n$-admissible open subset. Let
\be
r_{V,n}(x)=\rho_{T^nV(x)}(T^nx,\partial T^nV(x))
\label{rVn}
\ee
be the distance from $T^nx$ to the boundary of the connected
component of $T^nV$ where this point belongs. (The distance
is measured in the induced Riemannian metric on that component.)
We put
\be
Z[U,V,n]=
\sup_{\varepsilon>0}\frac{\nu_U(x\in V:\, r_{V,n}(x)<\varepsilon)}
{\varepsilon\cdot\nu_U(U)}
\label{ZUVn}
\ee
This supremum is finite because $\partial T^nV(x)$ in
(\ref{rVn}) is piecewise smooth $\forall x\in V$.
In the case $\nu_U(U\setminus V)=0$,
the value of $Z[U,V,n]$ characterizes, in a certain way,
the `average size' of the components of $T^nV$ --
the larger they are the smaller $Z[U,V,n]$.
In particular,
the value of $Z[U,U,0]$ characterizes the size of $U$
in a way illustrated by the following examples:
\medskip
{\em Examples}. Let $U$ be a ball of radius $r$, then
$Z[U,U,0]\sim r^{-1}$. Let $U$ be a cylinder
whose base is a ball of radius $r$ and height $h\gg r$,
then again $Z[U,U,0]\sim r^{-1}$. Let $U$ be a rectangular
box with dimensions $l_1\times l_2\times\cdots\times l_{d_u}$,
then $Z[U,U,0]\sim 1/\min\{l_1,\ldots, l_{d_u}\}$. \medskip
{\bf Definition} Let $U$ be an admissible u-manifold.
A decreasing sequence of open subsets $U=U_0\supset U_1
\supset U_2\supset\cdots$ is called a {\em u-filtration} of $U$ if \\
(a) $\forall n\geq 0$ the set $U_n$ is $n$-admissible;\\
(b) $\forall n\geq 0$ the set $U_n$ is dense in $U$,
i.e. $\bar U_n=\bar{U}$.\\
We also put $U_{\infty}=\cap_{n\geq 0}U_n$\medskip
Observe that all $U_n$ and $U_{\infty}$ have full $\nu_U$-measure.
On the other hand, $U_{\infty}$ has to be totally disconnected.
Let $\{U_n\}$ be a u-filtration of an admissible
u-manifold $U$. We then put for brevity $r_n=r_{U_n,n}$
a function on $U_n$ defined by (\ref{rVn}) and
$Z_n=Z[U,U_n,n]$ for all $n\geq 0$. The value of
$Z_n$ characterizes the `average size' of the
connected components of $T^nU_n$.
\begin{theorem}
There are $\alpha=\alpha(T)\in (0,1)$ and
$\beta=\beta(T)>0$ such that for any admissible
u-manifold $U$ there is a u-filtration $\{U_n\}$
such that \\
{\rm (i)} we have
\be
Z_1\leq \alpha Z_0+\beta\delta_0^{-1}
\label{exp1}
\ee
and $\forall n\geq 2$
\be
Z_n\leq \alpha^n Z_0+\beta\delta_0^{-1}(1+\alpha+\cdots +\alpha^{n-1})
\label{expn}
\ee
{\rm (ii)} let $\bar{\beta}=2\beta/(1-\alpha)$: then
$Z_n\leq\max\{Z_0,\bar{\beta}/\delta_0\}$ for all $n\geq 0$;\\
{\rm (iii)} $Z_n\leq\bar{\beta}/\delta_0$ for all $n\geq a\ln Z_0+b$.\\
Here $a=-(\ln\alpha)^{-1}$ and $b=-\ln(\delta_0(1-\alpha)/\beta)
/\ln\alpha$ are independent of $U$.
\label{tmexp}
\end{theorem}
{\em Remark}.
Effectively, the theorem asserts that if a u-manifold $U$
is small or thin, so that $Z_0$ is
very large, then the connected components of $T^nU$
grow larger, on the average, so that $Z_n$ decreases exponentially
in $n$ until it becomes small enough, $\leq\bar{\beta}/\delta_0$.
This is our exact version of the well-known concept `small unstable
manifolds grow exponentially in size' in the context
of high dimensions. \medskip
{\em Proof of Theorem~\ref{tmexp}}. We start with a construction
of an open dense 1-admissible subset $U_1\subset U$ that satisfies
(\ref{exp1}). For brevity, we will write $\nu$ instead of $\nu_U$
and $\rho$ instead of $\rho_U$.
Step 1. Assume first that $U\cap {\cal S}=\emptyset$ and diam$U\leq
\delta_0\Lambda_{\max}^{-1}$. Then $TU$ is an admissible
u-manifold, and we set $U_1=U$. Then $r_1(x)\geq \Lambda_{\min}r_0(x)$
for any $x\in U$, and so
\be
\nu(r_1<\varepsilon)\leq
\nu(r_0<\varepsilon/\Lambda_{\min})\leq
Z_0\Lambda_{\min}^{-1}\nu(U)\cdot\varepsilon
\label{C1smooth}
\ee
Step 2. Assume that diam$U>\delta_0\Lambda_{\max}^{-1}$.
Then we will define an open dense subset $U_1'\subset U$ whose
connected components will have diameter $<\delta_0\Lambda_{\max}^{-1}$.
According to our Key Remark, the manifold $U$ is almost
flat. We first assume that $U$ is exactly
a flat $d_u$-dimensional surface in $\IR^d$ with piecewise
smooth boundary. We choose a coordinate system in $\IR^d$
so that $U$ is parallel
to the first $d_u$ coordinate axes, i.e. $x_{d_u+1}=\cdots=x_d=0$
on $U$. Also, we assume that $\nu$ is the $d_u$-dimensional
volume on $U$.
For each $i=1,\ldots,d_u$ we take an array of parallel
hyperplanes $\{x_i=a_i+m\delta'\}$, $m\in\ZZ$, where
$\delta'=\delta_0\Lambda_{\max}^{-1}/\sqrt{2d_u}$,
and with some fixed $a_i\in [0,\delta')$. All these
hyperplanes together `shred' (or `dice') the domain $U$ into
cubic pieces of diameter $\delta_0\Lambda_{\max}^{-1}
/\sqrt{2}< \delta_0\Lambda_{\max}^{-1}$. Then the set
\be
U_1':=U\setminus \left (\cup_{i,m} \{x_i=a_i+m\delta'\}\right )
\label{U1'}
\ee
is open, dense in $U$, and completely determined by the vector
$(a_1,\ldots,a_{d_u})$, which will be fixed shortly.
For each $i=1,\ldots,d_u$ and $m\in\ZZ$ put
$D_{m,a_i}=U\cap\{x_i=a_i+m\delta'\}$. Observe
that $\partial U_1'=\partial U\cup(\cup_{i,m}D_{m,a_i})$.
For $\varepsilon>0$ put ${\cal U}_{\varepsilon}^0=\{ x\in U:\,
\rho(x,\partial U)<\varepsilon\}$ and
${\cal U}_{\varepsilon}'=\{ x\in U:\, \rho(x,\partial U_1')
<\varepsilon\}$.
Now we will optimize the parameters $a_1,\ldots,a_{d_u}$
so that $\nu({\cal U}_{\varepsilon}')$
will be small enough, $\forall\varepsilon>0$. For every
$D_{m,a_i}$ denote by
$$
{\cal C}_{m,a_i}(\varepsilon)=D_{m,a_i}\times [a_i+m\delta'-\varepsilon
\leq x_i\leq a_i+m\delta'+\varepsilon]
$$
the solid cylinder in $\IR^{d_u}$ of height $2\varepsilon$ whose
middle cross-section is $D_{m,a_i}$. Observe that for any
point $x\in {\cal U}_{\varepsilon}'\setminus {\cal U}_{\varepsilon}^0$
the $d_u$-dimensional ball in $U$ of radius $\rho(x,\partial U_1')$
centered at $x$ is touching one of the $(d_u-1)$-dimensional
domains $D_{m,a_i}$. Therefore, the region
${\cal U}_{\varepsilon}'\setminus {\cal U}_{\varepsilon}^0$ is covered
by the union of the cylinders ${\cal C}_{m,a_i}(\varepsilon)$.
Therefore
\be
\nu({\cal U}_{\varepsilon}'\setminus {\cal U}_{\varepsilon}^0)
\leq
2\varepsilon \sum_{i=1}^{d_u}S_{a_i}
\label{r'}
\ee
where $S_{a_i}$ is the total $(d_u-1)$ dimensional volume of
the domains $D_{m,a_i}$, $m\in\ZZ$.
We now fix $a_i\in [0,\delta')$ so that $S_{a_i}$ takes its
minimum value. In particular, this fixes our subset $U_1'$
defined by (\ref{U1'})!
Obviously, for each $i=1,\ldots,d_u$ we have
$$
\nu(U)=\int_0^{\delta'}S_{a_i}\, da_i
$$
so that $\min_{a_i}\,S_{a_i}\leq \nu(U)/\delta'$.
Therefore
\be
\nu({\cal U}_{\varepsilon}'\setminus {\cal U}_{\varepsilon}^0)
\leq 2\varepsilon d_u\nu(U)/\delta'<
4d_u^{3/2}\delta_0^{-1}\Lambda_{\max}\nu(U)\cdot\varepsilon
\label{nur'}
\ee
Step 3. Assume now that $U\cap{\cal S}\neq\emptyset$.
Since $\delta_0\ll\bar{\delta}$, then, according to (H4),
$U$ intersects at most $K_0$ singularity manifolds ${\cal S}_j\subset \cal S$.
We again assume that $U$ is a flat
$d_u$-dimensional surface in $\IR^d$ with piecewise
smooth boundary. Besides, we assume that each singularity
manifold ${\cal S}_j$ intersecting $U$ is a hyperplane in $\IR^d$,
cf. Key Remark. It may happen that some ${\cal S}_j$ terminates
inside $U$, then it must terminate on some other ${\cal S}_{j'}$,
see Introduction. In that case we treat ${\cal S}_j$ as
a hyperplane cutting one part of $U$ after $U$ was previously cut
into two parts by the hyperplane ${\cal S}_{j'}$.
% In the case $U\cap\partial {\cal S}_j\neq\emptyset$
% we additionally extend ${\cal S}_j$ so that it cuts completely across $U$.
The set $U_1''=U\setminus \cal S$ is open and dense in $U$.
It is obtained by cutting
the domain $U$ by $k\leq K_0$ hyperplanes in $\IR^d$.
Unlike Step 2, however, we no longer
can control the position of the new cutting hyperplanes.
So, we need the following lemma:
\begin{lemma}
Let $\Sigma$ be an arbitrary hyperplane cutting $U$.
For any $\varepsilon>0$ put ${\cal U}^0_{\varepsilon}=\{ x\in U:\,
\rho(x,\partial U)<\varepsilon\}$
and ${\cal U}_{\varepsilon,1}''=\{ x\in U:\,
\rho(x,\Sigma )<\varepsilon\}$. Then
$\nu({\cal U}_{\varepsilon,1}''\setminus {\cal U}^0_{\varepsilon})
\leq\nu({\cal U}^0_{\varepsilon})$.
\label{lmSigma}
\end{lemma}
{\em Proof}. If $x\in {\cal U}_{\varepsilon,1}''\setminus
{\cal U}^0_{\varepsilon}$,
then the $d_u$-dimensional ball in $U$ of radius
$\rho(x,\Sigma\cap U)$ centered at $x$
is touching the $(d_u-1)$-dimensional region $\Sigma\cap U$.
Therefore, the set ${\cal U}_{\varepsilon,1}''\setminus
{\cal U}^0_{\varepsilon}$ is foliated by segments in $U$
orthogonal to $\Sigma\cap U$ in such a way that each segment
crosses $\Sigma\cap U$ and sticks out by $\leq\varepsilon$
on each side of $\Sigma\cap U$. On the other hand, the line containing
any of those segments intersects ${\cal U}^0_{\varepsilon}$ by two
segments of length $\geq\varepsilon$ each. Hence the lemma. $\Box$\medskip
{\em Remark}. We will later need the following modification of
Lemma~\ref{lmSigma}. Let $B\subset U$ be some $d_u$-dimensional ball,
and ${\cal U}_{\varepsilon}'''=\{x\in U\setminus B:\,
\rho(x,B)<\varepsilon\}$. Then $\nu({\cal U}_{\varepsilon}'''
\setminus{\cal U}_{\varepsilon}^0)\leq\nu({\cal U}_{\varepsilon}^0)$,
$\forall\varepsilon>0$. The proof of this is similar
to that of Lemma~\ref{lmSigma} if one uses the foliation of
${\cal U}_{\varepsilon}'''$ by segments of rays emanating
from the center of $B$. \medskip
Lemma~\ref{lmSigma} asserts that cutting $U$ by any
hyperplane effectively adds at most as much volume to the
$\varepsilon$-neighborhood of the boundary as there was originally.
\begin{corollary}
Let $\Sigma_1,\ldots,\Sigma_k$ be arbitrary hyperplanes crossing $U$.
For any $\varepsilon>0$ put
${\cal U}_{\varepsilon}''=\{ x\in U:\,
\rho(x,\cup_i\Sigma_i)<\varepsilon\}$.
Then $\nu({\cal U}_{\varepsilon}''\setminus {\cal U}^0_{\varepsilon})
\leq k\cdot \nu({\cal U}^0_{\varepsilon})$.
\label{crSigma}
\end{corollary}
Applying Corollary~\ref{crSigma} to the $k\leq K_0$ singularity
hyperplanes ${\cal S}_j$ that cut $U$ gives
\be
\nu({\cal U}_{\varepsilon}''\setminus {\cal U}^0_{\varepsilon})
\leq K_0\cdot \nu({\cal U}^0_{\varepsilon})
\label{nur''}
\ee
Step 4. We put $U_1=U_1'\cap U_1''$.
Observe that $\{x\in U:\, \rho(x,\partial U_1)
<\varepsilon\}={\cal U}_{\varepsilon}^0\cup
{\cal U}_{\varepsilon}'\cup {\cal U}_{\varepsilon}''$.
Combining (\ref{nur'}) and (\ref{nur''}) gives
$$
\nu(x\in U:\,\rho(x,\partial U_1)<\varepsilon)
\leq (K_0+1)\cdot \nu({\cal U}_{\varepsilon}^0)
+4d_u^{3/2}\delta_0^{-1}\Lambda_{\max}\nu(U)\cdot\varepsilon
$$
This last estimate combined with (\ref{C1smooth}) gives
(formally) the bound (\ref{exp1}) with $\alpha=(K_0+1)/\Lambda_{\min}$ and
$\beta=4d_u^{3/2}\Lambda_{\max}/\Lambda_{\min}>0$.
Note that $\alpha<1$ due to (H4).
Due to the actual nonflatness of both $U$ and $\cal S$
we have to slightly (depending on $\delta_0$)
increase the above values of $\alpha$ and $\beta$, and we can
keep $\alpha$ below 1 assuming $\delta_0$ be small enough.
Step 5. Next, (\ref{expn}) follows from (\ref{exp1}) by
induction on $n$. To define $U_n$ inductively, assume
that $U_{n-1}$ is defined. Every connected component
$V$ of $T^{n-1}U_{n-1}$ is an admissible u-manifold.
Applying the proof of (\ref{exp1}) to $V$ defines
an open dense subset $V_1\subset V$. Then $U_n$ is
defined to be the union of
$T^{-n+1}V_1$ over all $V\subset T^{n-1}U_{n-1}$. Lastly,
the measure $T^n_{\ast}\nu_U$ conditioned on any
admissible u-manifold $V\subset T^nU_n$ is almost
uniform (depending on $\delta_0$) with respect to
$\nu_V$, cf. Key Remark. Its actual nonuniformity,
however, requires an additional slight increase of $\alpha$
and $\beta$ in the above calculations, which we can
afford assuming that $\delta_0$ is small enough.
The clauses (ii) and (iii) trivially follow from (i).
Theorem~\ref{tmexp} is proved. $\Box$\medskip
{\em Remark}. The choice of the vector $(a_1,\ldots,a_{d_u})$
made in Step 2 defines the subset $U_1$. Applying
this choice to every connected component of $T^{n-1}U_{n-1}$
defines the subset $U_n$. Thus, the entire u-filtration $\{U_n\}$
is defined. We say that the u-filtration so defined is
{\em admissible}. Admissible u-filtrations always satisfy
(i)-(iii) of the above theorem. \medskip
{\em Remark}. Our estimates (\ref{C1smooth}), (\ref{nur'})
and (\ref{nur''}) yield a little more than the part (i)
of the theorem. In fact, for any admissible u-manifold $U$,
an admissible u-filtration $\{U_n\}$ of $U$, and $\forall
\varepsilon >0$ we have
\be
\nu_U(r_1<\varepsilon)\leq \alpha\Lambda_{\min}\cdot\nu_U
(r_0<\varepsilon/\Lambda_{\min})+\varepsilon\beta\delta_0^{-1}\cdot\nu_U(U)
\label{exp1nu}
\ee
and hence $\forall n\geq 2$
\be
\nu_U(r_n<\varepsilon)\leq (\alpha\Lambda_{\min})^n\cdot
\nu_U(r_0<\varepsilon/\Lambda_{\min}^n)+\varepsilon
\beta\delta_0^{-1}(1+\alpha+\cdots +\alpha^{n-1})\cdot\nu_U(U)
\label{expnnu}
\ee
Let $\delta_1=\delta_0/(2\bar{\beta})$. According to the part
(iii) of Theorem~\ref{tmexp}, $Z_n\leq (2\delta_1)^{-1}$
for all $n\geq a\ln Z_0+b$. Hence,
\be
\nu_U(r_{U_n,n}(x)>\delta_1)>\nu_U(U)/2
\label{50}
\ee
In other words, at least 50\% of the points in
$T^nU$ (with respect to the measure induced by $\nu_U$)
lie a distance $\geq\delta_1$ away from the boundaries of $T^nU$.
\section{Existence and ergodic properties of SRB measures}
\label{secE}
\setcounter{equation}{0}
Here we prove the parts (a) and (b) of our main theorem \ref{tmmain}.
In \cite{Pes92}, Pesin proved the existence and ergodic
properties of SRB measures for a wide class of hyperbolic
maps with singularities (he called them generalized
hyperbolic attractors), covering the class we study here,
under two extra assumptions, which in our notation are: \medskip
{\bf (H5)} $\exists C>0,q>0$ such that $\forall\varepsilon>0,n\geq 1$
$$
{\rm Vol}(T^{-n}{\cal U}_{\varepsilon}({\cal S}))\leq C\varepsilon^q
$$
{\bf (H6)} $\exists z\in M^{0}$ with a local unstable
manifold\footnote{$W^u(z)$ is defined \cite{Pes92} via
a function $\phi^u: B^u_z\to E^s_z$, where $B^u_z$ is a ball
in $E^u_z$ centered at $z$, whose graph is then mapped
onto $M$ by the exponential map. Observe that such $W^u(z)$ has
smooth boundary, so it is admissible.}
$W^u(z)$ and $C>0,q>0$ such that $\forall \varepsilon>0,n\geq 1$
$$
\nu_{W^u(z)}(W^u(z)\cap T^{-n}{\cal U}_{\varepsilon}({\cal S}))
\leq C\varepsilon^q
$$
Here ${\cal U}_{\varepsilon}(\cdot)$ stands for $\varepsilon$-neighborhood
in the $\rho$ metric.
One should note that,
under these assumptions, Pesin \cite{Pes92} proved that any SRB measure
has at most countable number of ergodic components -- a weaker
statement than we claim in Theorem~\ref{tmmain}. Sataev \cite{Sat92}
showed that the number of ergodic components is finite under
the above (H5) and the following: \medskip
{\bf (H7)} $\exists C>0,q>0$ such that
for any ball-like u-manifold $U$ (i.e., a u-manifold
$U$ that is a ball in
the $\rho_U$ metric) there are $n_U>0$ and $B_U>0$
such that $\forall\varepsilon>0$
(a) $\nu_U(U\cap T^{-n}{\cal U}_{\varepsilon}({\cal S}))
\leq \nu_U(U)\cdot C\varepsilon^q\ \ \ \ \forall n>n_U$
(b) $\nu_U(U\cap T^{-n}{\cal U}_{\varepsilon}({\cal S}))
\leq \nu_U(U)\cdot B_U\varepsilon^q\ \ \ \ \forall n>0$
\begin{proposition}
If the map $T$ satisfies (H1)-(H4), then it satisfies
(H5)-(H7).
\label{pr567}
\end{proposition}
{\em Proof}.
The property (H5) with $q=1$ is obvious for $n=0$,
cf. \cite{Fed}. To prove it for $n\geq 1$,
we foliate $M$ by admissible u-manifolds
${\cal U}^f=\{U\}$ in a smooth way. Let $\nu^f_U$
be the conditional measures on $U\in {\cal U}^f$
induced by the Lebesgue volume in $M$, and
$d\mu^f(U)$ the factor measure on ${\cal U}^f$.
If the foliation is smooth enough and $\delta_0$
small enough, every $\nu_U^f$ will have almost
uniform density with respect to the Lebesgue
measure $\nu_U$ on $U$. For every $U\in{\cal U}^f$
let $\{U_n\}$ be an admissible u-filtration
of $U$, and $r_{U_n,n}(x)$ be defined by (\ref{rVn}).
Observe that
\be
r_{U_{n+1},n+1}(x)1$ and $\varepsilon>0$. Define
$$
M^{\pm}_{\Lambda,\varepsilon}=\{x\in M^{\pm}:\,
\rho(T^{\pm n}x,{\cal S})>\varepsilon\Lambda^{-n}\ \ \ \
\forall n\geq 0\}
$$
and
$$
M_{\Lambda}^{\pm}=\cup_{\varepsilon>0}M_{\Lambda,\varepsilon}^{\pm}
\ \ \ \ \ \ \ \
M_{\Lambda}^0=M_{\Lambda}^+\cap M_{\Lambda}^-
$$
The following is standard \cite{Pes92,LSY}:\medskip
{\bf Fact}. Let $1<\Lambda<\Lambda_{\min}$ and $\varepsilon>0$.
Then $\forall x\in
M^-_{\Lambda,\varepsilon}$ there is a LUM $W^u(x)$ such that
$\rho (x,\partial W^u(x))\geq\varepsilon$. Similarly,
$\forall x\in M^+_{\Lambda,\varepsilon}$ there is an
LSM $W^s(x)$ such that $\rho (x,\partial W^s(x))\geq\varepsilon$. \medskip
Therefore, stable manifolds exist everywhere on
$M_{\Lambda}^-$, and unstable ones everywhere on $M_{\Lambda}^+$.
For $x\in M_{\Lambda,\varepsilon}^-$ we denote by $W^u_{\varepsilon}(x)$
the $\varepsilon$-ball in $W^u(x)$ centered at $x$, in the
$\rho_{W^u(x)}$ metric. It is, indeed, a ball, since
$\rho_{W^u(x)}(x,\partial W^u(x))\geq\varepsilon$.
Similarly, $W^s_{\varepsilon}(x)$
is defined $\forall x\in M_{\Lambda}^+$. We will call
$W^s_{\varepsilon}(x)$ and $W^u_{\varepsilon}(x)$
stable and unstable {\em disks} of radius $\varepsilon$
through $x$, respectively.
\begin{lemma}
$\forall\Lambda>1$ we have
$M^0_{\Lambda}\neq\emptyset$.
\label{lmMpm}
\end{lemma}
Pesin \cite{Pes92} proved this lemma under the assumption (H5),
that we already proved,
for a larger class of hyperbolic systems than we study here.
On the other hand, Young \cite{LSY} provided a
direct argument for 2-D case, which we will extend below to
our systems.
Let $U$ be an admissible u-manifold, and let
$$
\bar{\nu}_N=\frac {1}{N}\sum_{i=0}^{N-1}T^i_{\ast}\nu_U
$$
This is a pre-compact sequence of Borel measures on $\bar{M}$.
Any limit point $\hat{\mu}$ of this sequence, normalized, is a
$T$-invariant probability measure concentrated on $M^0$.
Theorem~\ref{tmexp}, see also the above proof of (H7),
ensures that $\exists C>0$ such that
$\hat{\mu}({\cal U}_{\varepsilon}({\cal S}))\leq C\varepsilon$, $\forall
\varepsilon>0$. Then the standard application of
Borel-Cantelli lemma \cite{LSY} proves Lemma~\ref{lmMpm}.
$\Box$\medskip
Proposition~\ref{pr567} is proved.$\Box$\medskip
This concludes the proof of parts (a) and (b) of
Theorem~\ref{tmmain}. $\Box$
\begin{corollary}[\cite{Pes92,LSY}]
For any SRB measure $\mu$ and any $\Lambda>1$ we have
$\mu(M_{\Lambda}^0)=1$,
i.e., LUM's and LSM's exist a.e. with respect to any SRB measure.
\end{corollary}
{\em Remark}. Applying Borel-Cantelli lemma to $\nu_U$
rather than $\hat{\mu}$ yields that for any u-manifold
$U$ we have $\nu_U(U\setminus M_{\Lambda}^+)=0$,
i.e. an LSM $W_x^s$ exists for $\nu_U$-a.e. point $x\in U$. \medskip
Having proved the first two parts of Theorem~\ref{tmmain},
we conclude that all the SRB measures satisfying the
assumptions of the part (c) are actually mixing and Bernoulli.
\section{Refined filtration of u-manifolds}
\label{secF}
\setcounter{equation}{0}
The techniques of Section~\ref{secU} are not enough
to obtain the statistical properties of $T$. We will be
dealing with rectangles defined later in Sect.~\ref{secR}.
Those are made of points whose both stable and unstable
manifolds are large enough. The results of Sect.~\ref{secU}
allow us only to control the sizes of unstable manifolds
and their iterates. In order to locate points on a given
unstable manifold with large
enough stable manifolds, we have to, according to the
Fact given in the previous section, discard the points
whose orbits come too close to the singularity manifold $\cal S$.
Technically, we again consider the iterations of an admissible
u-manifold $W$ under $T^n$, $n\geq 0$. The admissible
u-filtration $\{W_n\}$ constructed in Sect.~\ref{secU}
will be refined here, so that points in $W$ whose images come
too close to the singularity manifolds will be set
apart and no longer iterated under $T$. This will create
countably many gaps in $W$ in which stable manifolds
fail to be long enough.
We start with a technical construction around the singularity manifold $\cal S$.
For every $\delta'\ll\delta_0$ we define
two parallel hypersurfaces at distance $\delta'$ from
every singularity manifold ${\cal S}_j$
(located on both sides of ${\cal S}_j$).
They are obtained by moving every point $x\in{\cal S}_j$
the distance $\delta'$ from ${\cal S}_j$ along the normal
vectors to ${\cal S}_j$ at $x$ in both directions from ${\cal S}_j$.
Since $\delta_0$ is less than the minimum radius of curvature of
${\cal S}_j$, the resulting hypersurfaces will be smooth
$\forall\delta'<\delta_0$. We also make sure that those hypersurfaces
terminate on the same components of $\cal S$ as the original
manifold ${\cal S}_i$. We denote by $\hat{\cal S}^{\delta'}_j$
the {\em union} of these two hypersurfaces, $\forall j=1,\ldots,r$.
Now, fix a $\Lambda\in (1,\Lambda_{\min})$ and let $\delta_2\ll\delta_1
=\delta_0/(2\bar{\beta})$.
The two parameters $\Lambda$ and $\delta_2$
will govern all our further constructions in this
section.
Let $W$ be an admissible u-manifold and
$\{W_n\}$ its admissible u-filtration
defined in Sect.~\ref{secU}.
For any $n\geq 0$ we define an open
subset $W_n'\subset W_n$ by
\be
W_n'=\left\{x\in W_n:\,
0<\rho (T^nx,{\cal S})
< \delta_2\Lambda^{-n}\right\}
\label{Wn'}
\ee
This is a set of points whose $n$-th iterates come too close
to the singularity manifolds.
Observe that the sets $W_n'$, $n\geq 0$, may overlap.
So, we define $W_n^0=W_n'\setminus (\overline{W_0'}\cup
\cdots\cup\overline{W_{n-1}'})$. This set consists of
points whose trajectories come too close to $\cal S$
at time $n$, not earlier. Hence, $W_n^0$ is
a gap in $W$ created at the $n$-th iteration.
We then set $W_0^1=W$ and $W_n^1=W\setminus
(\overline{W_0'}\cup\cdots\cup \overline{W_{n-1}'})$
for $n\geq 1$. Thus, $W^1_n$ is the part of $W$ that survives
$n$ iteration of $T$ without coming too close to ${\cal S}$.
All $W_n^0$ and $W_n^1$ are $n$-admissible open subsets of $W$.
We call the two collections $\{W_n^1\}$ and $\{W_n^0\}$
the {\em refinement of the u-filtration} $\{W_n\}$,
or a refined u-filtration.
We denote it by $(\{W_n\},\{W_n^1\},\{W_n^0\})$.
We put $W_{\infty}^1=\cap_{n\geq 0}W_n^1$.
Observe that $W_{\infty}^1\subset M_{\Lambda,\delta_2}^+$, and so
a stable disk $W^s_{\delta_2}(x)$ of radius $\delta_2$
exists at every point $x\in W_{\infty}^1$.
Next, we characterize the `sizes' of the u-manifolds
$T^nW^1_n$ and $T^nW^0_n$, $n\geq 0$, in the manner
similar to that of Sect.~\ref{secU}. Define $\forall n\geq 0$
$$
Z_n^1=Z[W,W_n^1,n]
\ \ \ \ \ {\rm and}\ \ \ \ \
Z_n^0=Z[W,W_n^0,n]
$$
based on (\ref{ZUVn}). In the case $W_n^1=\emptyset$ we have,
of course, $Z_n^1=0$, but this will never actually happen in our
further constructions.
Observe that $Z_0^1=Z_0$, where $Z_0$ was defined in
Theorem~\ref{tmexp}. Put also
$$
w_n^1=\nu_W(W_n^1)/\nu_W(W)
\ \ \ \ \ {\rm and}\ \ \ \ \
w_n^0=\nu_W(W_n^0)/\nu_W(W)
$$
Observe that $w_n^1=1-w_0^0-\cdots -w_{n-1}^0$ and
$w_n^1\searrow w_{\infty}^1
\stackrel{\rm def}{=}\nu_W(W^1_{\infty})/\nu_W(W)$ as $n\to\infty$.
\begin{theorem}
Let $W$ be an admissible u-manifold and $\{W_n\}$
its admissible u-filtration. Let $1<\Lambda<\Lambda_{\min}$
and $\delta_2\ll\delta_1$. Then the refinement
$(\{W_n\},\{W_n^1\},\{W_n^0\})$ of the u-filtration
$\{ W_n\}$ satisfies the following bounds:\\
{\rm (i)} we have
\be
Z_1^1 \leq \alpha Z_0^1+\beta\delta_0^{-1}
\label{expw}
\ee
with the same $\alpha\in (0,1)$ and $\beta>0$ as in Theorem~\ref{tmexp},
and for any $n\geq 2$
\be
Z_n^1\leq \alpha^n Z_0^1+\beta\delta_0^{-1} (1+\alpha+\cdots +\alpha^{n-1})
\label{expnw}
\ee
{\rm (ii)} for any $n\geq 0$ we have $Z_n^0 \leq (3K_0+1)Z_n^1$;\\
{\rm (iii)} for any $n\geq 0$ we have $w_n^0\leq Z_n^0C'\delta_2\Lambda^{-n}$
with some constant $C'=C'(T)>0$.
\label{tmexpa}
\end{theorem}
{\em Proof}. To prove (\ref{expw}), we only need to modify
Step 3 of the proof of Theorem~\ref{tmexp}. According to
(\ref{Wn'}), the sets $W^0_0$ and $W^1_1$
are made by cutting $W$ with the hypersurfaces
${\cal S}_{j_i}$ and
$\hat{\cal S}_{j_i}^{\delta_2}$, $1\leq i\leq k$. Thus,
in addition to $k\leq K_0$ singularity hyperplanes
$\Sigma_j$, $1\leq j\leq k$, in the notations used
in the proof of Theorem~\ref{tmexp},
we now have $k$ pairs of their parallel copies,
which we shall call $\Sigma_j'$ and $\Sigma_j''$,
$1\leq j\leq k$.
Lemma~\ref{lmSigma} admits the following easy modification:
\begin{lemma}
Let $W$ be a domain in $\IR^{d_u}\subset\IR^d$ with
piecewise smooth boundary, and $\Sigma',\Sigma''$ two
parallel hyperplanes in $\IR^d$. Denote by $B$ the layer
in $\IR^d$ between $\Sigma'$ and $\Sigma''$.
For any $\varepsilon>0$ put ${\cal U}^0_{\varepsilon}=\{ x\in W:\,
\rho_W(x,\partial W)<\varepsilon\}$
and
${\cal U}_{\varepsilon,1}''=\{ x\in W\setminus B:\,
\rho_W(x,\Sigma'\cup\Sigma'')<\varepsilon\}$.
Then
$\nu_W({\cal U}_{\varepsilon,1}''\setminus {\cal U}^0_{\varepsilon})
\leq\nu_W({\cal U}^0_{\varepsilon})$.
\label{lmSigma2}
\end{lemma}
We apply Lemma~\ref{lmSigma2} to each pair of hyperplanes
$\Sigma_j',\Sigma_j''$, $1\leq j\leq k$ and then sum
over $j=1,\ldots,k$ as we did in Corollary~\ref{crSigma}.
This proves (\ref{expw}).
The bound (\ref{expnw}) follows from (\ref{expw}) by
induction on $n$, as in Step 5 of the proof of Theorem~\ref{tmexp}.
We simply apply the bound (\ref{expw}) to every connected
component of $T^nW_n^1$, which is an admissible u-manifold.
To prove (ii) for $n=0$, we apply Corollary~\ref{crSigma} to
the entire collection of $3k$ hyperplanes $\Sigma_j, \Sigma_j',\Sigma_j''$,
$1\leq j\leq k$, and then get
$$
\nu_W(x\in W_0^0:\, \rho(x,\partial W_0^0)<\varepsilon)\leq
(3K_0+1)Z_0^1\,\varepsilon\nu_W(W)
$$
To prove (ii) for $n\geq 1$, we apply the above argument
to every connected component of $T^nW_n^1$.
To prove (iii), observe that, in the notations of
(\ref{Wn'}), $\forall x\in W_n^0$ we have
$$
\rho_{T^nW^0_n(x)}\left (T^nx,\partial T^nW^0_n(x)\right )\leq
\rho_{T^nW^0_n(x)}\left (T^nx,{\cal S}\right )
\leq C'\delta_2\Lambda^{-n}
$$
where $C'>0$ depends on the minimum angle between
${\cal S}$ and the unstable cone family. We apply
(\ref{rVn}) and (\ref{ZUVn}) with $U=W$, $V=W_n^0$,
$\varepsilon=C'\delta_2\Lambda^{-n}$ and observe that then
$$
\nu_W(x\in W_n^0:\, r_{W_n^0,n}(x)0}\frac{\nu_W(x\in V:\, r_{V,n}(x)<\varepsilon)}
{\varepsilon\cdot\nu_W(V)}
=Z[W,V,n]\times\frac{\nu_W(W)}{\nu_W(V)}
\label{ZVn}
\ee
This value depends on $V$ but not on $W$.
It characterizes the average size of the components of $T^nV$
just like $Z[U,V,n]$ did in Section~\ref{secU} for subsets
$V\subset U$ of full measure. Accordingly, the values of
$$
Z[W_n^1,n]=Z_n^1/w_n^1
\ \ \ \ \ {\rm and}\ \ \ \ \
Z[W_n^0,n]=Z_n^0/w_n^0
$$
characterize the average size of the components of
$T^nW^1_n$ or $T^nW^0_n$, respectively.
In our further constructions, the set $W^1_{\infty}$ will be
often very dense in $W$, so that $w^1_{\infty}>0.9$. We call
this a special case, and Corollary~\ref{crwzn} then implies:\medskip
{\em Special case}. If $w^1_{\infty}>0.9$, then for all
$n\geq a\ln Z_0+b$ we have $Z[W^1_n,n]\leq 0.6/\delta_1$.
We will say then that the components of $T^nW^1_n$ are
large enough, on the average.\medskip
{\em Remark}. The values of $Z[U,V,n]$ in (\ref{ZUVn}) and
the values of $Z^{1,0}_n$, $w^{1,0}_n$ in this section will certainly
not change if we replace the Lebesgue measures, $\nu_U$ in (\ref{ZUVn})
and $\nu_W$ here, by any measures proportional to those. It is also
straightforward that all the results of Sections~\ref{secU}
and \ref{secF} extend to countable disjoint unions
of admissible u-manifolds with finite measures that are
linear combinations of the Lebesgue measures on individual components.
Precisely, let $U=\cup_k U^{(k)}$ be a countable union of pairwise
disjoint admissible u-manifolds and $\hat{\nu}_U=\sum_k u_k\nu_{U^{(k)}}$,
with some $u_k>0$, a finite measure on $U$. Then $Z[U,V,n]$ is
still defined by (\ref{ZUVn}), with $\nu_U$ replaced by $\hat{\nu}_U$,
for any set $V=\cup_k V^{(k)}$,
where $V^{(k)}$ are some $n$-admissible open subsets of $U^{(k)}$.
The definition of u-filtration and the proof of Theorem~\ref{tmexp}
go through with only minor obvious changes.
Likewise, the definitions and results of this
section apply to any countable union $W=\cup W^{(k)}$ of admissible
u-manifolds with any finite measure $\hat{\nu}_W=\sum_ku_k\nu_{W^{(k)}}$,
provided we use the same parameters $\Lambda$ and
$\delta_2$ for all $W^{(k)}$. \medskip
{\em Final Remark}. Let $W'$ be an admissible u-manifold, $k\geq 1$,
and $V'\subset W'$ a $k$-admissible open subset. Then $W=T^kV'$ is a finite
or countable union of admissible u-manifolds. The measure
$\tilde{\nu}_W:=T^k_{\ast}\nu_{W'}|W$ on $W$ is almost uniform
(proportional to the Lebesgue measure $\nu_W$) on each component of $W$,
according to Key Remark of Sect.~\ref{secU}. All the results of
Sections~\ref{secU} and \ref{secF} will then apply to $(W,\tilde{\nu}_W)$,
instead of $(W,\nu_W)$, but the slight nonuniformity of
the measure $\tilde{\nu}_W$ with respect to $\nu_W$ might slightly
affect the values of the constants, such as $\alpha,\beta,a,b$.
The smaller $\delta_0$, the smaller changes in the constants will
be inflicted. In all that follows we assume that the constants
are adjusted accordingly, so that the results of Sections~\ref{secU}
and \ref{secF} apply to pairs $(W,\tilde{\nu}_W)$ as above. \medskip
Lastly, we generalize the above special case:
\begin{proposition}
Let $(\{W_n\},\{W_n^1\},\{W_n^0\})$ be a refined u-filtration of
an admissible u-manifold $W$, such that
$w^1_{\infty}=p>0$. Then for all $n\geq a_1(\ln Z_0-\ln p)
+b_1$ we have $\nu_W(W^1_{\infty})/\nu_W(W^1_n)\geq 0.9$
and $Z[W^1_n,n]\leq 0.6/\delta_1$, i.e. the components of
$T^nW^1_n$ will be large enough, on the average.
Here $a_1=a+(\ln\Lambda)^{-1}$ and $b_1$ is another constant
determined by $\alpha,\beta,\delta_0,\Lambda,C''$.
\label{prwp}
\end{proposition}
{\em Proof}. Due to the part (ii) of Corollary~\ref{crwzn},
we have $Z_n^1\leq (2\delta_1)^{-1}$, and hence
$Z[W^1_n,n]\leq (2\delta_1p)^{-1}$, for all $n\geq n':=
a\ln Z_0+b$. Due to the part (iii) of the same corollary,
we have $\sum_{i=n}^{\infty}
w_i^0\leq p/20$ for all $n\geq n'':=\log_{\Lambda}
[20\,C''\bar{Z}_0p^{-1}/(1-\Lambda^{-1})]$. Now let $k=n'+n''$.
The set $\tilde{W}:=T^kW^1_k$ is a finite or countable union of
admissible u-manifolds. It carries the measure $\tilde{\nu}_{\tilde{W}}
:=T^k_{\ast}\nu_W|\tilde{W}$, so that the results of this section apply
to $(\tilde{W},\tilde{\nu}_{\tilde{W}})$, according to Final Remark.
The subsets $T^kW^1_{m}\subset\tilde{W}$, $m\geq k$, correspond
to a refined u-filtration
$(\{\tilde{W}_n\},\{\tilde{W}^1_n\},\{\tilde{W}^0_n\})$
of $\tilde{W}$ with $\delta_2$ replaced by $\delta_2\Lambda^{-k}$,
so that $T^kW^1_m=\tilde{W}^1_{m-k}$, $\forall m\geq k$.
Since $k\geq n'$, we have $Z[\tilde{W},\tilde{W},0]=Z[W^1_k,k]
\leq (2\delta_1p)^{-1}$. Since $k\geq n''$, we have
$$
\tilde{w}^1_{\infty}
=\tilde{\nu}_{\tilde{W}}(\tilde{W}^1_{\infty})/
\tilde{\nu}_{\tilde{W}}(\tilde{W})
=\nu_W(W^1_{\infty})/\nu_W(W^1_k)\geq 0.9
$$
Thus, the refined u-filtration
$(\{\tilde{W}_n\},\{\tilde{W}^1_n\},\{\tilde{W}^0_n\})$
of $\tilde{W}$ falls in the above special
case. Hence, $Z[\tilde{W}^1_n,n]\leq 0.6/\delta_1$ for
all $n\geq n''':=-a\ln (2\delta_1p)+b$. Therefore,
for the original refined u-filtration of $W$, we have
$Z[W^1_n,n]\leq 0.6/\delta_1$ for all $n\geq n'+n''+n'''$.
It is then an easy calculation that $n'+n''+n'''\leq
a_1(\ln Z_0-\ln p)+$const. $\Box$. \medskip
{\em Final Remark (Part 2)}. The above proposition also applies
to any pair $(W,\tilde{\nu}_W)$ described in Final Remark before the
proposition. Likewise, some further results stated and proved
for admissible u-manifolds, $W$, with Lebesgue measures $\nu_W$,
will also apply to measures $\tilde{\nu}_W=T^k_{\ast}\nu_{T^{-k}W}$
on $W$ for any $k\geq 1$ such that $T^{-k}$ is defined on $W$.
We will assume this without any more reminders.
\section{Rectangles}
\label{secR}
\setcounter{equation}{0}
The key instrument in Young's proofs \cite{LSY} of statistical properties
of hyperbolic dynamical systems is a set with hyperbolic
product structure. Its full definition
is quite long, but for uniformly hyperbolic maps studied here
such a set is just a rectangle or parallelogram in Sinai-Bowen sense,
cf. \cite{Si68,Bo75}. \medskip
{\bf Definition}. A subset $R\subset M^0$ is called a
{\em rectangle} if
$\exists\varepsilon>0$ such that for any $x,y\in R$
there is an LSM $W^s(x)$ and an LUM $W^u(y)$,
both of diameter $\leq\varepsilon$, that
meet in exactly one point, which also belongs
in $R$. We denote that point by $[x,y]=W^s(x)\cap W^u(y)$.
\medskip
In all our rectangles, we will have $\varepsilon<\delta_0$.
A subrectangle $R'\subset R$ is called a u-subrectangle if
$W^u(x)\cap R=W^u(x)\cap R'$
for all $x\in R'$. Similarly, s-subrectangles are defined.
We say that a rectangle $R'$ u-crosses another rectangle
$R$ if $R'\cap R$ is a u-subrectangle in $R$ and an
s-subrectangle in $R'$.
We introduce some more notation.
Let $x\in M$ and $r\in (0,\delta_0)$. We denote by $S_r(x)$
any s-manifold that is a ball of radius $r$ centered at $x$ in
its own metric, $\rho_{S_r(x)}$. By that we mean $\rho_{S_r(x)}
(x,y)=r$, $\forall y\in\partial S_r(x)$.
We call such $S_r(x)$ an {\em s-disk}.
In order to define s-disks also around points close to $\partial M$
we extend the cone families $C^u$ and $C^s$ continuously
beyond the boundaries of $M$ into
the $\delta_0$-neighborhood of $M$.
Then s-disks $S_r(x)$ exist $\forall x\in M, \forall r\in (0,\delta_0)$.
Note that $S_r(x)$ is by no means uniquely determined by $x$ and $r$.
Let $U$ be a u-manifold of diameter $<\delta_0$, and $x\in M$.
Clearly, any s-disk $S_{\delta_0}(x)$ can meet $U$ in at most
one point (as we always require $\delta_0$ be small enough).
We call
$$
H_x(U)=\{y\in U:\, y=S_{\delta_0}(x)\cap U\ \
{\rm for}\ \ {\rm some}\ \ S_{\delta_0}(x)\}
$$
the {\em s-shadow} of $x$ on $U$.
We say that a point $x\in M$ is overshadowed by a u-manifold
$U$ if $\forall S_{\delta_0}(x)$ we have $S_{\delta_0}(x)
\cap U\neq\emptyset$. Note that in this case, of course,
$\rho(x,U)\leq\delta_0$. We call
$$
\rho^s(x,U)=\sup_{S_{\delta_0}(x)}\rho_{S_{\delta_0}(x)}
(x,S_{\delta_0}(x)\cap U)
$$
the {\em s-distance} from $x$ to $U$ (this one is also $\leq\delta_0$
whenever defined).
Let $U,U'$ be two u-manifolds of diameters $<\delta_0$. We
call
$$
H_U(U')=\cup_{x\in U}H_x(U')
$$
the s-shadow of $U$ on $U'$. We say that $U'$ overshadows
$U$ if it overshadows every point $x\in U$. In this case
we define
$$
\rho^s(U,U')=\sup_{x\in U}\rho^s(x,U')
$$
the s-distance from $U$ to $U'$. It is not symmetric,
since no two u-manifolds can simultaneously overshadow each other:
geometrically, $U'$ overshadows $U$ if $U$ is close to $U'$
and $U'$ stretches all the way along $U$ and a little beyond it.
Let $\Lambda\in (1,\Lambda_{\min})$ be the one fixed in
Sect.~\ref{secF}. We assume that $\delta_0$, and hence
$\delta_1$, are small
enough, so that $M_{\Lambda,\delta_1}^-\neq\emptyset$.
Therefore,
$$
A_{\delta_1}\stackrel{\rm def}{=}\{x\in M:\, {\rm the}\
{\rm unstable}\ {\rm disk}\ \
W^u_{\delta_1}(x)\ \ {\rm exists}\}\neq\emptyset
$$
(recall, cf. Sect.~\ref{secE}, that $W^u_{\varepsilon}(x)$
is the ball of radius $\epsilon$ centered at $x$ in the local
unstable manifold $W^u(x)$).
Let $z\in A_{\delta_1}$. Consider $W(z):=W^u_{\delta_1/3}(z)$,
the `central part' of the existing unstable disk $W^u_{\delta_1}(z)$.
It is an admissible u-manifold, and a perfect ball in its own
metric. It is an easy exercise that for a perfect ball $W$ of
radius $\delta$ in $\IR^{d_u}$ one has $Z[W,W,0]=d_u/\delta$.
Since the manifolds $W(z)$, $z\in A_{\delta_1}$, actually have
some (bounded) sectional curvature, $Z[W(z),W(z),0]$ might
be larger than $3d_u/\delta_1$, but if $\delta_1$ is small enough,
that difference is not big, and we will have
\be
Z[W(z),W(z),0]\leq 4d_u/\delta_1
\label{Z1}
\ee
for all $z\in A_{\delta_1}$.
Let $\delta_2\ll\delta_1$ to be specified below, and
$(\{W_n(z)\},\{W_n^1(z)\},\{W_n^0(z)\})$ the
refined u-filtration of $W(z)$ defined in Sect.~\ref{secF}
and governed by the two parameters $\Lambda$ and $\delta_2$.
\begin{lemma}
If $\delta_2/\delta_1$ is small enough, then $\forall z\in A_{\delta_1}$
we have $\nu_{W(z)}(W^1_{\infty}(z))\geq 0.9\cdot \nu_{W(z)}(W(z))$.
\label{lm0.9}
\end{lemma}
This follows from (\ref{Z1}) and the part (v)
of Corollary~\ref{crwzn}, provided
\be
\frac{\delta_2}{\delta_1}\leq
\frac{1-\Lambda^{-1}}{40\, C''d_u}
\label{d12}
\ee
{\em Convention}. We will treat our small parameters $\delta_i$,
$i\geq 0$, in the following way. On the one hand, all of them
are assumed to be small, and on the other hand, the ratios
$\delta_{i+1}/\delta_i$, $i\geq 0$, are also small.
Moreover, we will fix their ratios $\delta_{i+1}/\delta_i$,
$i\geq 1$, at some points below, but still allow them to vary
altogether with their ratios fixed.
\medskip
In fact, the ratio $\delta_1/\delta_0=(2\bar{\beta})^{-1}$
is already fixed in Section~\ref{secU}.
We now fix $\delta_2/\delta_1$ that satisfies (\ref{d12}).
Recall that $\forall x\in W^1_{\infty}(z)$ a stable disk
$W^s_{\delta_2}(x)$ exists, cf. Sect.~\ref{secF}.
\begin{lemma}
Let $z\in A_{\delta_1}$, and consider a refined u-filtration
$(\{W_n(z)\},\{W_n^1(z)\},\{W_n^0(z)\})$ of the
unstable disk $W(z)=W^u_{\delta_1/3}(z)$. Then
$\forall n\geq n_0':=-\ln(16d_u)/\ln\alpha+1$
we have \\
{\rm (i)} $Z_n^1<(2\delta_1)^{-1}$ and $Z[W_n^1(z),n]<0.6/\delta_1$;\\
{\rm (ii)} $
\nu_{W(z)}(x\in W_n^1(z):\, r_{W_n^1(z),n}(x)>\delta_1)
> 0.4\cdot \nu_{W(z)}(W_n^1(z))
> 0.4\cdot \nu_{W(z)}(W_{\infty}^1(z))$.\\
In other words, (ii) means that at least 40\% of the points in
$T^nW^1_n(z)$ (with respect to the measure induced by $\nu_{W(z)}$)
lie a distance $\geq\delta_1$ away from the boundaries of $T^nW^1_n(z)$.
\label{lm40}
\end{lemma}
{\em Proof.} This follows from the part (ii) of Corollary~\ref{crwzn},
Lemma~\ref{lm0.9} and (\ref{Z1}), recall also a similar
bound (\ref{50}). $\Box$ \medskip
{\em Remark}. Let $z\in A_{\delta_1}$. For a moment, let
$W(z)=W_{\varepsilon}^u(z)$ be the stable disk of any
radius $\varepsilon\in (\delta_1/3,\delta_1)$.
That disk $W(z)$ is larger than $W^u_{\delta_1/3}(z)$,
and so (\ref{Z1}) still holds. Therefore, the statements
(i) and (ii) of the above lemma hold as well. Furthermore,
if, again for a moment, we decrease $\delta_2$ thus making
the ratio $\delta_2/\delta_1$ smaller than the one fixed above,
then Lemma~\ref{lm0.9} will still hold, and then so will (i)
and (ii) of Lemma~\ref{lm40}.
\medskip
Let $\delta_3\ll\delta_2$, to be specified later.
\begin{proposition}
Let $W$ be an admissible u-manifold, and $W'$ another
u-manifold that overshadows $W$ and $\rho^s(W,W')\leq\delta_3$.
Let $(\{W_n\},\{W_n^1\},\{W_n^0\})$ be a refined
u-filtration of $W$. Then $\forall n\geq 1$ and any connected
component $V$ of $W^1_n$ there is a connected domain $V'\subset W'
\setminus {\cal S}^{(n)}$ such that the u-manifold $T^nV'$
overshadows the admissible u-manifold $T^nV$,
and $\rho^s(T^nV,T^nV')\leq\delta_3\Lambda_{\min}^{-n}$.
\label{pruu}
\end{proposition}
{\em Proof}. The proof easily goes by induction on $n$, so that
it suffices to prove the proposition for $n=1$. Put $n=1$, and
let $V$ be a connected component of $W^1_0$. Then
$\forall x\in V$ we have $\rho(x,{\cal S})>\delta_2$ by (\ref{Wn'}),
so that any s-disk $S_{2\delta_3}(x)$ will cross $W'$ but not
${\cal S}$, provided $\delta_3/\delta_2$ is small enough.
Hence, the s-shadow $H(V,W')$
belongs in one connected component of $W' \setminus
{\cal S}$. It is then easy to see by direct inspection
that its image under $T$ overshadows $TV$ and the s-distance
from $TV$ to that image is $\leq\delta_3\Lambda_{\min}^{-1}$. $\Box$\medskip
For any $z\in A_{\delta_1}$ we define a `canonical'
rectangle $R(z)$ as follows:
$y\in R(z)$ iff $y= W^s_{\delta_2}(x)\cap W^u$ for some
$x\in W^1_{\infty}(z)$ and for some LUM $W^u$ that
overshadows $W(z)=W^u_{\delta_1/3}(z)$, and such that
$\rho^s(W(z),W^u)\leq\delta_3$.
Observe that if $\delta_3/\delta_20$ is
determined by the minimum angle between the stable and
unstable cone families, then every $W^u$ that overshadows
$W(z)$ and is $\delta_3$-close to it in the
above sense will meet all stable disks $W^s_{\delta_2}(x)$,
$x\in W^1_{\infty}(z)$. In that case $R(z)$ will be a rectangle, indeed.
We fix the ratio $\delta_3/\delta_2$ now as follows:
\be
\delta_3/\delta_2=\min\{c',1-\Lambda^{-1}, 1/3\}
\label{d21}
\ee
For any connected subdomain $V\subset W(z)$ the set
$R_V(z):=\{y\in R(z):\, W^s(y)\cap V\neq\emptyset\}$ is
an s-subrectangle in $R(z)$ ``based on $V$''.
Let $n\geq 1$. The partition of $W^1_n(z)$ into connected
components, $\{V\}$, induces a partition of $R(z)$ into
s-subrectangles $\{R_V(z)\}$ that are based on those components.
Let $R_V(z)$ be one of those s-subrectangles.
It follows from Proposition~\ref{pruu}
that $T^nR_V(z)$ is a rectangle. We call every rectangle
$T^nR_V(z)$ a component of the set $T^nR(z)$, note that
the entire set $T^nR_V(z)$ does not have to be a rectangle
itself. We next consider the intersections of $T^nR_V(z)$
with $R(z')$ for $z'\in A_{\delta_1}$.
\begin{lemma}
There is a $c_1>0$ such that $\forall z,z'\in A_{\delta_1}$
such that $\rho(z,z')\delta_3/2
\label{assx}
\ee
Observe that $\forall m\geq 0$, the map $T^{-m}$ is defined and
smooth on both $W^u_{\delta_1/2}(z)$ and $W^u_{\delta_1/2}(z')$.
The distance between the inverse images $W_m:=T^{-m}
(W^u_{\delta_1/2}(z))$ and $W_m':=T^{-m}(W^u_{\delta_1/2}(z'))$
grows with $m$, and eventually these images may be separated
by a singularity manifold.
Let $m\geq 1$ be the largest integer that satisfies two conditions: \\
(i) $T^{-m}$ is smooth on a connected domain in $M$ that contains
both $W^u_{\delta_1}(z)$ and $W^u_{\delta_1}(z')$;\\
(ii) $\rho(T^{-m}z,W_m')\leq\delta_1$.\\
Observe that $\rho^s(z,W^u_{\delta_1/2}(z'))\leq C'c_1\delta_3$
for some $C'$ determined by the minimum angle between the
stable and unstable cone families. It is easy to see that
$\rho^s(T^{-m}z,W_m')\leq C''c_1\delta_3\Lambda_{\max}^m$,
where $C''$ is another constant determined by the minimum angle
between the stable and unstable cone families. On the other hand,
we have the following lower bound on the inner radius of the manifold $W_m$:
$\rho_{W_m}(T^{-m}z,\partial W_m)\geq \delta_1\Lambda_{\max}^{-m}/2$.
It follows from (H3) that the LUM's $W_m$ and $W_m'$ cannot be separated
by singularities as long as the distance between them is $\ll$
the inner radius of $W_m$. Therefore, $2c_1\delta_3\delta_1^{-1}
\Lambda_{\max}^{2m} \geq c''/C''$, where $c''$ is some constant
determined by the minimum angle between the unstable cone
family and $\cal S$. Recall that $\delta_3/\delta_2$ and
$\delta_2/\delta_1$ are fixed, so that $\delta_3=\tilde{c}\delta_1$
with some $\tilde{c}=$const$>0$. Hence, $m\geq\ln [(2\tilde{c}
c_1)^{-1}c''/C'']/\ln\Lambda_{\max}^2$, so that $m$ can be made arbitrarily
large by choosing $c_1$ in the lemma very small. On the other hand,
assuming (\ref{assx}) we arrive at
$$
\rho^s(T^{-m}x,W_m')\geq
\delta_3\Lambda_{\min}^m/2=\tilde{c}\delta_1\Lambda_{\min}^m/2
$$
When $m$ is large enough, the right hand side will be
$\gg\delta_1$, which contradicts the requirement (ii) above
(note that both manifolds $W_m$ and $W_m'$ will be
tiny, of diameter $\ll\delta_1$).
This proves the lemma. $\Box$\medskip
We set $\delta_4=c_1\delta_3$. We also fix $n_0''=
\min\{n\geq 1:\, \Lambda_{\min}^n>2\}$
\begin{proposition}
Let $z\in A_{\delta_1}$ and $n\geq n_0''$. Let $V$ be a connected component
of $W^1_n(z)$ and $x\in V$ such that $r_{V,n}(x)>\delta_1$
and $\rho(T^nx,z')<\delta_4$ for some $z'\in A_{\delta_1}$.
Then the rectangle $T^nR_V(z)$ u-crosses the rectangle $R(z')$,
i.e. $T^nR_V(z)\cap R(z')$ is {\rm (i)} a u-subrectangle in
$R(z')$ and {\rm (ii)} an s-subrectangle in $T^nR_V(z)$.
\label{prret}
\end{proposition}
{\em Proof}. According to Lemma~\ref{lmzz'}, the LUM $T^nV$
overshadows $W(z')$, and $\rho^s(W(z'),T^nV)\leq\delta_3/2$.
According to our choice of $n_0''$ and Proposition~\ref{pruu},
the LUM $W^u(y)$ for every $y\in T^nR_V(z)$ overshadows
$W(z')$, and $\rho^s(W(z'),W^u(y))\leq\delta_3$. This
implies (ii).
To prove (i), we need to show that $\forall x'\in W^1_{\infty}(z')$
the point $y'=W^s_{\delta_3}(x')\cap T^nV$ belongs in $T^nR_V(z)$.
It is enough to show that the point $y=T^{-n}y'\in W^1_{\infty}(z)$.
Firstly, $y\in V\subset W^1_n(z)$, so we only need
to prove that $y\in W^1_{n+1+k}(z)$, $\forall k\geq 0$.
Observe that $\rho(T^kx',{\cal S})>\delta_2\Lambda^{-k}$,
$\forall k\geq 0$, since $x'\in W^1_{\infty}(z')$. Next, observe that
$\rho(T^kx',T^ky')<\delta_3\Lambda^{-k}$, $\forall k\geq 0$, by
virtue of Proposition~\ref{pruu}. Therefore, $\rho(T^ky',
{\cal S})>(\delta_2-\delta_3)\Lambda^{-k}>\delta_2\Lambda^{-n-k}$,
the last inequality follows from (\ref{d21}).
Thus, $y\in W^1_{n+1+k}(z)$, $\forall k\geq 0$. $\Box$. \medskip
\section{Rectangular structure and return times}
\label{secS}
\setcounter{equation}{0}
The scheme of our proof of the part (c) of Theorem~\ref{tmmain}
is the following. Let $\mu$ be an SRB measure such that
$(T^n,\mu)$ is ergodic $\forall n\geq 1$. According to
the last remark of Sect.~\ref{secE}, $\mu$ is also mixing
and Bernoulli. Clearly, there is a $\delta_0>0$ and a
$z_1\in A_{\delta_1}$ (remember that $\delta_1=\delta_0/(2\bar{\beta})$)
such that $\mu(R(z_1))>0$.
Note also that for any other ergodic SRB measure $\mu'\neq\mu$
we have $\mu'(R)=0$. We fix such a $\delta_0$ and one such
$z_1\in A_{\delta_1}$. We then denote, for brevity,
$R=R(z_1)$, $W=W(z_1)$, $W^1_{\infty}=W^1_{\infty}(z_1)$, etc.
Let ${\cal Z}=\{z_1,z_2,\ldots,z_p\}$ be a finite $\delta_4$-dense
subset of $A_{\delta_1}$ containing the above point $z_1$.
We call ${\cal R}=\cup_i R(z_i)$ the rectangular
structure. It is a finite union of rectangles that most
likely overlap and do not cover $M$ or even the support
of $\mu$.
We will partition the set $W_{\infty}^1$ into a countable
collection of subsets $W_{\infty,k}^1$, $k\geq 0$,
such that for every $k\geq 1$ there is an integer $r_k\geq 1$ such that
for the s-subrectangle $R_k\subset R$ based on $W^1_{\infty,k}$
the set $T^{r_k}(R_k)$ will be a u-subrectangle in some $R(z_i)$,
$z_i\in {\cal Z}$. By the s-subrectangle $R_k\subset R$
based on $W^1_{\infty,k}$ we mean
the set $R_k=\{x\in R: W^s(x)\cap W^1_{\infty}\in W^1_{\infty,k}\}$.
Topologically, $W_{\infty,k}^1$, $k\geq 1$, are $d_u$-dimensional
Cantor sets for systems with singularities. We will
call them {\em gaskets}.
We consider the fact that $T^{r_k}(R_k)$ is a u-subrectangle
in some $R(z_i)$ as a {\em proper return} (of $R_k$ into ${\cal R}$).
We define a function $r(x)$ on $W^1_{\infty}$ by
$r(x)=r_k$ for $x\in W^1_{\infty,k}$, $k\geq 1$, and
$r(x)=\infty$ for $x\in W^1_{\infty,0}$. We call $r(x)$
the return time.
L.-S. Young proved \cite{LSY} the following:
\begin{theorem}
If $\int_{W_{\infty}^1} r(x)\, d\nu_W<\infty$,
then there is an SRB measure $\mu_R$ concentrated on
$\cup_{n\geq 0} T^nR$. That measure is ergodic, thus unique.
\label{tmY1}
\end{theorem}
\begin{theorem}
If $\nu_W\{r(x)>n\}\leq C\theta^n$, $\forall n\geq 1$,
for some $C>0$, $\theta\in (0,1)$, then the system
$(T,\mu_R)$ enjoys an exponential decay of correlations
and a central limit theorem.
\label{tmY2}
\end{theorem}
We state these theorems here in a slightly wider version than
Young did in \cite{LSY}. One gets her original theorems
if one sets $p=1$, i.e. when the rectangular structure
contains just one rectangle. However, Young worked with
finite unions of (overlapping) rectangles in Section~7
of \cite{LSY}, and showed that it was equivalent to
working with one rectangle.
Alternatively, one can define the returns of $R$ to itself
rather than to $\cup_i R(z_i)$ and then apply the original
Young's theorems (with $p=1$) directly. This can be done
by using the mixing property of the measure $\mu$, along
the lines of \cite{BSC91,Ch92}, but this is not necessary
in view of the above.
The uniqueness of $\mu_R$ in Theorem~\ref{tmY1} implies
$\mu_R=\mu$. Note also that if $T^{r_k}(R_k)
\subset R(z_i)$, then $\mu(R(z_i))>0$, so there are no
possible returns to rectangles $R(z_i)\subset {\cal R}$
of zero $\mu$-measure, i.e. they can be
simply ignored. In summary, it remains to define the
function $r(x)$ and prove an exponential tail bound:
\be
\nu_W\{r(x)>n\}\leq C\theta^n
\label{tail}
\ee
for some $C>0$, $\theta\in (0,1)$, and all $n\geq 1$.
In the rest of this section, we define the partition
$W_{\infty}^1=\cup_k W_{\infty,k}^1$ and the return
time $r(x)$. Our definition consists in several steps. \medskip
{\bf Initial growth.}
First, we take $n_1=\max\{n_0',n_0''\}$.
According to Lemma~\ref{lm40}, we have \\
(a) $Z_{n_1}^1<(2\delta_1)^{-1}$ and $Z[W_{n_1}^1,n_1]<0.6/\delta_1$,
i.e. the components of $T^{n_1}W^1_{n_1}$ are large enough,
on the average, and \\
(b) $\nu_{W}\{x\in W^1_{n_1}:\,
r_{W^1_n,n}(x)\geq \delta_1\} \geq 0.4\, \nu_W(W^1_{n_1})$,
i.e. at least 40\% of the points in $T^{n_1}W^1_{n_1}$ (with respect to
the measure induced by $\nu_W$) lie a distance $\geq\delta_1$
away from $\partial T^{n_1}W^1_{n_1}$.\\
(Recall that (b) actually follows from (a), cf. Lemma~\ref{lm40}.)
Denote $W^g=T^{n_1}W^1_{n_1}$, and $\tilde{\nu}_{W^g}=T^{n_1}_{\ast}
\nu_W|W^g$ the induced measure on $W^g$.
For every connected component $V\subset W^g$ such that
$\exists x_V\in V:\, \rho_V(x_V,\partial V)\geq\delta_1$ we arbitrarily
fix one such point $x_V$. Then $x_V\in A_{\delta_1}$, and
$\exists z_V\in {\cal Z}$ such that $\rho(x_V,z_V)<\delta_4$.
We fix one such $z_V$, too.
Then we label the set $T^{-n_1}(V\cap R(z_V))$ as one of our
gaskets $W^1_{\infty,k}$, and we define $r_k=n_1$ on it. According
to Proposition~\ref{prret}, $T^{r_k}(R_k)$ is a u-subrectangle
in $R(z_V)$, indeed. Note that we are defining at most one gasket
in each component $V$ of $W^g$. We will sometimes slightly abuse
the terminology and call the set $V\cap R(z_V)$ a gasket, too.
\begin{lemma}
There is a $q=q(T)>0$ such that, independently of the choice
of the points $x_V$ and $z_V$ in the components
$V\subset W^g$, the just defined gaskets
$W^1_{\infty,k}$ satisfy
$$
\nu_W\left (\cup W^1_{\infty,k}\right )\geq q\, \nu_W(W^1_{n_1})
$$
\label{lmq}
\end{lemma}
{\em Proof}. The lemma follows from Lemmas~\ref{lm0.9}
and \ref{lm40}, along with the absolute continuity (\ref{ac}). $\Box$\medskip
Thus, a certain fraction ($\geq q$) of $W^g$ returns
at the $n_1$-th iteration. This is the earliest return in
our construction.
Further returns are harder to define, and we first explain why.
Let $n>n_1$ and $\exists x\in V:\, \rho_V(x,\partial V)>\delta_1$ for
some connected component $V$ of $T^nW^1_n$. If we arbitrarily
pick some points $x_V$ and $z_V$ as before, then the set
$T^{-n}(V\cap R(z_V))$ may overlap with some previously defined gaskets
$W^1_{\infty,k}$, so we cannot label it as another gasket.
To avoid possible overlaps, we perform the following construction. \medskip
{\bf Capture}.
Every connected component $V$ of $W^g$ where a point
$x_V$ is picked is now subdivided
into two connected sets: $V^c:=W^u_{\delta_1/2}(x_V)$
and $V^f:=V\setminus V^c$. Observe that $V^c$ overshadows
$W(z_V)$, according to Lemma~\ref{lmzz'}, and so the gasket
$V\cap R(z_V)$ defined above lies wholly
in $V^c$. We say that $V^c$ is `captured' at the
$n_1$-th iteration. The rest of $V$, i.e. the set
$V^f$, is `free to move'. The captured parts of $W^g$
are taken out of circulation, for the moment, and the rest of
$W^g$, let us call it $W^{f}$, is mapped further under $T$, it
contains no points of the previously defined gaskets.
Denote $W^f_n=W^f\cap T^{n_1}W^1_{n_1+n}$ for $n\geq 0$.
Observe that the manifolds $W^f_n$, $n\geq 0$, correspond to a refined
u-filtration $\{W^f_n,W^{f,1}_n,W^{f,0}_n\}$ of
the u-manifold $W^f$
in the sense of Sect.~\ref{secF} with $\delta_2$ replaced
by $\delta_2\Lambda^{-n_1}$, so that $W^f_n=W^{f,1}_n$,
$\forall n\geq 0$.
We would like to see, first of all, that the components of
$T^nW^f_n$ for some $n\geq 0$ are large, on the average,
precisely that $Z[W^f_n,n]<0.6/\delta_1$. This may not
be the case for $n=0$, for the following reasons.
The removal of the captured parts from $W^g$
will create more boundary in the remaining part, $W^f$,
and also reduce its measure. As a result, $Z[W^{f},W^f,0]
>Z[W^g,W^g,0]=Z[W^1_{n_1},n_1]$. However, the $\varepsilon$-neighborhood
of the boundary increases at most twice $\forall\varepsilon>0$,
cf. the remark after Lemma~\ref{lmSigma}. It is also
clear that $\tilde{\nu}_{W^g}(W^f)>\tilde{\nu}_{W^g}(W^g)/2$.
Therefore, $Z[W^{f},W^f,0]<4\cdot Z[W^1_{n_1},n_1]<
2.4/\delta_1$. Applying then the part (ii) of Corollary~\ref{crwzn}
to the manifold $W^f$, which is
possible according to Final Remark
of Section~\ref{secF}, with $\delta_2$
replaced by $\delta_2\Lambda^{-n_1}$ yields
$Z[W^{f},W^f_n,n]<(2\delta_1)^{-1}$ for all
$n\geq n_2$, where $n_2:=[-\ln 9.6/\ln\alpha]+1$.
Also, the part (iv) of the same corollary, along
with (\ref{d12}), yields $\tilde{\nu}_{W^g}(W^f_n)
>(1-0.06\,\Lambda^{-n_1}/d_u)\,\tilde{\nu}_{W^g}
(W^f)>0.9\,\tilde{\nu}_{W^g}(W^f)$ for all $n\geq 0$.
Therefore, due to (\ref{ZVn}), we have
$Z[W^f_n,n]=Z[W^f,W^f_n,n]\, \tilde{\nu}_{W^g}(W^f)
/\tilde{\nu}_{W^g}(W^f_n)<0.6/\delta_1$ for all $n\geq n_2$,
as desired.
In other words, it takes a fixed number of iterations,
$n_2$, to recover the lost average size of the freely
moving manifold, $T^nW^{f}_n$, $n\geq 0$,
after the removal of the
captured parts from $W^g$. As soon as this is done,
i.e. at the iteration $n=n_2$,
at least 40\% of the image $T^{n}W^f_n$,
will lie a distance $\geq\delta_1$ from its boundary,
just as in the claim (b) above.
Now we inductively repeat the above procedure of
picking points $x_V,z_V$ in the large components $V$
of the freely moving manifold, defining new gaskets
$V\cap R(z_V)$, capturing disks covering the
newly defined gaskets, moving
the remaining manifold another $n_2$
iterations under $T$ until its components grow large
enough, on the average, etc. According to Lemma~\ref{lmq},
the points of the freely moving manifold are being
captured at an exponential rate: at least a fraction
$q>0$ of them is captured every $n_2$ iterations of $T$.
Let $t_0(x)$, $x\in W^1_{\infty}$, be the number
of iterations it takes to capture the image of the point $x$.
Observe that $t_0(x)=n_1+kn_2$ for some $k=0,1,\ldots$.
Lemma~\ref{lmq} implies that
\be
\nu_W(t_0(x)>n)/\nu_W(W^1_{\infty})\leq C_0\theta_0^n
\label{t0}
\ee
with $\theta_0=q^{1/n_2}<1$ and some $C_0>0$. In
particular, $t_0(x)<\infty$ for a.e. $x\in W^1_{\infty}$. \medskip
{\bf Release}.
Next, we take care of the captured parts of the manifolds $T^nW_n^1$,
$n\geq 1$. They are all very similar.
Let $B^c\subset T^{n_c}W^1_{n_c}$ is a part captured
at the $n_c$-th iteration of $T$, $n_c\geq n_1$. Then
$B^c$ is a perfect ball of radius $\delta_1/2$ in
some connected component of $T^{n_c}W^1_{n_c}$.
It carries the measure $\tilde\nu_{B^c}=T^{n_c}_{\ast}\nu_W|B^c$.
The center $x_c$ of the disk $B^c$ belongs in $A_{\delta_1}$,
and there is a point $z_c\in {\cal Z}$ such that $\rho(x_c,z_c)
<\delta_4$ and such that the set $B^c_R:=B^c\cap R(z_c)$ makes a new
gasket at the moment of capture. The points of the gasket
successfully return to $R(z_c)$, i.e. $r(x)=n_c$ for
$x\in T^{-n_c}B^c_R$. Denote also $B^c_{\infty}=
B^c\cap T^{n_c}W^1_{\infty}$. We now have to take care of
the subset $B^c_{\infty}\setminus B^c_R$.
Denote $B^c_n=B^c\cap T^{n_c}W^1_{n_c+n}$ for $n\geq 0$.
According to the remark after Lemma~\ref{lm40},
we have
\be
Z[B^c,B^c,0]\leq 4d_u/\delta_1
\ \ \ \ \ {\rm and}\ \ \ \ \
Z[B^c_n,n]<0.6/\delta_1,\ \forall n\geq n_0'
\label{ZBc}
\ee
In other words, it takes $n_0'$ iterations of $T$ to make
the components of $T^nB^c_n$ large enough, on the average.
In order to define a new gasket in any large component $V$
of $T^nB^c_n$ and avoid possible overlaps with the image
$T^nB_R^c$ of the gasket $B^c_R$, we will make sure that
$V$ contains no points of $T^nB_R^c$.
To get a control of that, we define
a `release time' (we will call it also
`point release time'), $f(x)$, for points
$x\in B^c_{\infty}\setminus B^c_R$. A point $x$
will be `released' if $T^{f(x)}(x)$ is sufficiently
far from $T^{f(x)}B_R^c$, so that for all $n\geq f(x)$
the component of $T^nB^c_n$ containing $T^nx$ will
contain no points of $T^nB_R^c$.
The definition of the release time requires a
classification of points $x\in B^c_{\infty}\setminus B^c_R$. \medskip
{\em Type I points} are such that there is an LSM
$W^s(x)$ meeting the manifold $W^u_{\delta_1}(z_c)$
in one point, call it $h(x)$. Then $h(x)\notin
W^1_{\infty}(z_c)$, otherwise $x$ would belong in $B^c_R$.
Hence, either $h(x)\in W^u_{\delta_1}(z_c)\setminus
W^u_{\delta_1/3}(z_c)$ or $h(x)\in W^0_m(z_c)$ for some
$m=m(x)\geq 0$.
In the former case, we set $m(x)=0$ and $\varepsilon(x)=
\rho(h(x),W^u_{\delta_1/3}(z_c))$. In the latter case
we set $\varepsilon(x)=\rho(T^mh(x),\partial T^mW^0_m(z_c))$.
We now define the release time to be $f(x)=m(x)+\log_{\Lambda}
(\delta_0/\varepsilon(x))$, one formula for both cases. \medskip
{\em Type II points} have no local stable manifolds that
extend to $W^u_{\delta_1}(z_c)$. Let $x\in B^c_{\infty}$
be such a point. According to the second
statement in Lemma~\ref{lmzz'}, $\rho^s(x,W^u_{\delta_1}(z_c))
\leq\delta_3/2$. Hence, no local stable manifold $W^s(x)$
contains a stable disk of radius $\delta_3/2$
around $x$. Therefore, $x\notin M^+_{\Lambda,\delta_3/2}$,
in virtue of the Fact of Section~\ref{secE}. Let then $m=m(x)
=\min\{m'>0:\, \rho(T^{m'}x,{\cal S})\leq\delta_3\Lambda^{-m'}/2\}$.
We claim that, on the component of $T^{m}B^c_{m}$ containing
$T^{m}x$, there are no points of $T^{m}B_R$ in the
$(\delta_2\Lambda^{-m}/2)$-neighborhood of $T^{m}x$. Indeed, if some point
$y\in T^{m}B_R$ were there, its local stable fiber $W^s(y)$ would
contain a point $y'\in T^{m}W^1_{\infty}(z_c)$, which is at distance
$\leq \delta_3\Lambda^{-m}$ from $y$. Then $\rho(y',{\cal S})
\leq \delta_2\Lambda^{-m}$, since $\delta_3\leq\delta_2/3$,
cf. (\ref{d21}). This, however, contradicts the definition
of $W^1_{\infty}(z_c)$, cf. (\ref{Wn'}). We now define
the release time to be $f(x)=2m(x)+\log_{\Lambda}(2\delta_0
/\delta_2)$. \medskip
It is clear that for any point $x\in B^c_{\infty}\setminus B^c_R$
of either type and any $n\geq f(x)$ the point $T^nx$ should be
at least the distance $\delta_0$ from $T^nB_R^c$ (measured
along $T^nB^c_n$), so that, in fact, the component of
$T^nB^c_n$ containing $T^nx$ does not intersect $T^nB_R^c$ at all.
Therefore, we are free to define new gaskets
and capture new disks on any component $V\subset T^nB^c_n$ that
contains at least one released point, i.e. such that $\exists x\in T^{-n}V:
\, f(x)\leq n$. We can only define a gasket, however, if $\exists x\in V:\,
\rho_V(x,\partial V)\geq\delta_1$, i.e. if $V$ is large enough.
Hence the next step in our construction.\medskip
{\bf Growth}.
To get a control on the size of the components of $T^nB^c_n$, we
gather, for every $n\geq 0$, the components $V\subset T^nB^c_n$
released at the $n$-th iteration. We say that $V$ is released
at the $n$-th iteration if at least one point of $V$ is
released at this iteration, and none of the points of
the component of $T^{i}B_i^c$ that contains $T^{-(n-i)}V$
is released at the $i$-th iteration for any $i=0,\ldots,n-1$.
In that case we define another function, $s(x)=n$, on $B^c_{\infty}
\cap T^{-n}V$. We call $s(x)$ the `component release time' (as
opposed to the point release time $f(x)$ defined earlier).
Observe that $s(x)$ is defined for each $x\in B^c_{\infty}
\setminus B_R^c$ and $s(x)\leq f(x)$.
Fix an $s\geq 0$ (the `component release time') and let
\be
\tilde{W}=\tilde{W}(s)=\cup\{V\subset T^sB^c_s:\,
s(x)=s\ \forall x\in B^c_{\infty}\cap T^{-s}V\}
\label{Ws}
\ee
be the union of the components
of $T^{s}B^c_{s}$ released exactly at the $s$-th iteration.
The manifold $\tilde{W}$ carries the measure $\tilde{\nu}_{\tilde{W}}=
T^{s}_{\ast}\tilde{\nu}_{B^c}|\tilde{W}$. Observe that the sets
$\tilde{W}\cap T^{s}B^c_{s+n}$, $n\geq 0$, correspond to
a refined u-filtration $\{\tilde{W}_n,\tilde{W}^1_n,\tilde{W}^0_n\}$
of $\tilde{W}$ in the sense of Sect.~\ref{secF}
with $\delta_2$ replaced by $\delta_2\Lambda^{-n_c-s}$,
so that $\tilde{W}\cap T^sB^c_{s+n}=\tilde{W}^1_n$. Denote
\be
p(s)=
\tilde{\nu}_{\tilde{W}}(\tilde{W}^1_{\infty})/
\tilde{\nu}_{\tilde{W}}(\tilde{W})=
\tilde{\nu}_{\tilde{W}}(\tilde{W}\cap T^{s}B^c_{\infty})/
\tilde{\nu}_{\tilde{W}}(\tilde{W})
\label{ps}
\ee
In a trivial case, when $p(s)=0$, there is nothing in $\tilde{W}$
to worry about, and we disregard such a release time $s$. If
$p(s)>0$, then
Proposition~\ref{prwp} applies to $(\tilde{W},\tilde{\nu}_{\tilde{W}})$,
according to Final Remark (Part 2). Hence, $\exists n\geq 1$
such that $Z[\tilde{W}^1_n,n]\leq 0.6/\delta_1$, i.e. the
components of $T^n\tilde{W}^1_n$ are large enough, on the average.
Let $g$ be the minimum of such $n$'s. We call $g$ the `growth time' and
define another function, $g(x)=g$ on $B^c_{\infty}\cap T^{-s}\tilde{W}$
(note that $g(x)$ is a constant function on $B^c_{\infty}\cap
T^{-s}\tilde{W}$, and it only depends on $s$, so we will
also write it as $g(s)$).
Consider now the manifold $\hat{W}=T^{g}\tilde{W}^1_{g}$
and the measure $\tilde{\nu}_{\hat{W}}=
T_{\ast}^{g}\tilde{\nu}_{\tilde{W}}|\hat{W}$
on it. Denote $\hat{W}^1_{\infty}=
T^g(\tilde{W}^1_{\infty})=T^g(\tilde{W}\cap T^sB^c_{\infty})$
the subset of $\hat{W}$ we will keep track of.
According to Proposition~\ref{prwp}, we have \\
(c) $\tilde{\nu}_{\hat{W}}(\hat{W}^1_{\infty})>
0.9\, \tilde{\nu}_{\hat{W}}(\hat{W})$, and \\
(d) $Z[\hat{W},\hat{W},0]\leq 0.6/\delta_1$, so that
at least 40\% of the points in $\hat{W}$ (with respect
to the measure $\tilde{\nu}_{\hat{W}}$) lie a distance
$\geq\delta_1$ away from $\partial\hat{W}$. \\
Next, we define new gaskets and capture disks covering them
on the large components of $\hat{W}$, as we did to $W^g$
early in this section. Then we move the remaining parts
of $\hat{W}$ under $T^{n_2}$, again define new gaskets and
capture new disks, etc., exactly repeating the procedure
applied to $W^g$ during the period of initial growth.
Let $t(x)$ be the `capture time' for $x\in \hat{W}^1_{\infty}$,
i.e. the minimum of $t\geq 0$ such that $T^tx$ belongs in a
captured disk. Note that $T^tx$ might be luckily covered by a
gasket, then it returns to $\cal R$, or else it has to be
iterated further under $T$.
\begin{lemma}
We have $\tilde{\nu}_{\hat{W}}(t(x)>n)/\tilde{\nu}_{\hat{W}}
(\hat{W}^1_{\infty})\leq C_0\theta_0^n$ with the same
constants as in (\ref{t0}).
\label{lmt}
\end{lemma}
{\em Proof}. The lemma follows from the properties (c)
and (d) of the manifold $\hat{W}$ just like Lemma~\ref{lmq}
and (\ref{t0}) followed from the similar properties of
the manifold $W^g$. $\Box$\medskip
{\bf Summary}.
We summarize the ideas of our constructions. For every release time
$s\geq 0$ we take the union $\tilde{W}$ of the components
of $T^{s}B^c_{s}$ released exactly at the $s$-th iteration,
iterate them further $g$ times without capturing or
defining gaskets, then they become large enough, on the average.
Then our construction repeats inductively. We define new gaskets
and capture new disks on the components of $T^t\tilde{W}$, $t\geq g$,
the gaskets make successful return to $\cal R$ at the time they
are defined, the captured points around gaskets are iterated
further and eventually released, the released components grow
in size until they become large enough, on the average, then
new gaskets are defined, etc. For a.e. point $x\in W^1_{\infty}$,
the cycle `growth$\to$capture$\to$release$\to$growth$\ldots$' repeats
until the point returns to $\cal R$ at a moment of capture.
If it never returns, however, we put it in $W^1_{\infty,0}$ and set
$r(x)=\infty$. This concludes our definition of the partition
$W^1_{\infty}=\cup_k W^1_{\infty,k}$ and the return time $r(x)$.
\section{Exponential tail bound}
\label{secT}
\setcounter{equation}{0}
In this section we prove the exponential tail bound (\ref{tail}).
First, we show that the points of any captured disk $B^c$
are being released at an exponential rate.
\begin{lemma}
There are $C_1>0$ and $\theta_1\in (0,1)$ such that
for every captured disk $B^c$ we have $\tilde{\nu}_{B^c}
(f(x)>n)/\tilde{\nu}_{B^c}(B^c)0$.
In view of the absolute continuity (\ref{ac}), it is enough
to estimate the measure $\nu_{W^u_{\delta_1}(z_c)}
\{h(x):\, f(x)>n\}$.
The measure of the set $\{h(x):\, m(x)>n/2\}$ is
exponentially small in $n$ due to the part (iii)
of Corollary~\ref{crwzn} and (\ref{Z1}).
Next, for every $0\leq m\leq n/2$,
the measure of the set $\{h(x):
m(x)=m\ \&\ \varepsilon(x)<\delta_2\Lambda^{-n/2}\}$
is exponentially small in $n$, uniformly in $m$, in view of
the definition of $Z_m^0$ and the part (i) of Corollary~\ref{crwzn}
and (\ref{Z1}). Thus, the points of type I obey our claim.
For any point $x$ of type II with $m(x)=m$, observe that
the point $\rho_{V}(T^mx,\partial V\cup {\cal S})0$ is a constant depending
on the minimum angle between unstable cones and $\cal S$.
Hence, $\rho_{V'}(T^{m+1}x,\partial V')\leq
C'\Lambda_{\max}\delta_3\Lambda^{-m}/2$, where $V'$ is the
component of $T^{m+1}B^c$ containing $T^{m+1}x$. The
measure of the set of such points is then exponentially small
in $m$ in view of the definition of $Z_{m+1}$ in Sect.~\ref{secU}
as applied to the u-manifold $U=B^c$, combined with the part (ii)
of Theorem~\ref{tmexp} and (\ref{ZBc}). $\Box$.\medskip
For brevity, we normalize the measure $\tilde{\nu}_{B^c}$,
so that $\tilde{\nu}_{B^c}(B^c)=1$.
The next lemma shows that the released components in the images
of any captured disk $B^c$ grow at an exponential rate:
\begin{lemma}
There are $C_2>0$ and $\theta_2\in (0,1)$
such that for every captured disk $B^c$ we have
$\tilde{\nu}_{B^c}(s(x)+g(x)>n)
n)\leq
(1-q)^n$ for all $n\geq 0$.
Now an exponential tail bound on $r(x)$ can be obtained
by a standard argument developed in \cite{Ch92}
(pp. 129--130) and used in \cite{LSY} (Sublemma 6 in
Section 7).
Instead of repeating that argument, we present a different
one here, of a completely probabilistic nature. Its
relevance to our previous discussion
will be quite clear. Let $\xi_n$, $n\geq 1$, be a
sequence of independent identically distributed
random variables taking positive integral
values and satisfying an exponential tail bound
$P(\xi_i=n)\leq c_1\lambda_1^n$ for some
$c_1>0$, $\lambda_1\in (0,1)$. Let also $N$ be a random
variable independent from all $\xi_i$'s,
taking positive integral values, and satisfying
an exponential tail bound
$P(N=n)\leq c_2\lambda_2^n$ for some $c_2>0$
and $\lambda_2\in (0,1)$. Let $S_N=\sum_{i=1}^N\xi_i$.
\begin{proposition}
The random variable $S_N$ satisfies an exponential
tail bound $P(S_N=n)\leq c\lambda^n$ with some
$c>0$ and $\lambda\in (0,1)$.
\end{proposition}
{\em Proof}. The generating function
$$
G_{\xi}(z)=\sum_{n=1}^{\infty}P(\xi_i=n)\, z^n
$$
is analytic in the open disk $|z|<\lambda_1^{-1}$.
The generating function of $S_N$ is
$$
G_{S_N}(z)=\sum_{n=1}^{\infty}P(N=n)G_{\xi}^n(z)
$$
Since $|G_{\xi}(z)|\leq 1$ on the closed unit disk $|z|\leq 1$,
then for any $10$.
Then $G_{S_N}(z)$ is an analytic function in the
open disk $|z|<1+\varepsilon_A$. This implies
$P(S_N=n)\leq {\rm const}\cdot (1+\varepsilon')^{-n}$
for $\varepsilon'<\varepsilon_A$. $\Box$
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\end{document}
\end