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\centerline{\tenbf Nonexistence of SBR Measures for some Diffeomorphisms}
\centerline{\tenbf that are ``Almost Anosov''}
\vskip 0.5 true cm
\centerline{\sl Huyi Hu \quad and \quad Lai-Sang Young\footnote*
{\rm This research is partially supported by the National Science Foundation.}}
\medskip
\centerline{\sl Department of Mathematics, University of Arizona,}
\centerline{\sl Tucson, AZ 85721}
\bigskip
{\it Abstract.} The purpose of the paper is to present some simple examples
that are hyperbolic everywhere except at one point, but which do not admit
SBR measures. Each example has a fixed point at which the larger eigenvalue
is equal to one and the smaller eigenvalue is less than one.
\bigskip
\bigskip
{\bf\S0 Introduction}
\medskip
Let $f:M\to M$ be a $C^2$ Anosov diffeomorphism of a compact connected
Riemannian manifold, and let $m$ denote the Riemannian measure on $M$.
A result of Sinai (see e.g. [S]) says that $f$ admits a unique invariant
Borel probability measure $\mu$ with the property that $\mu$ has
absolutely continuous conditional measures on unstable manifolds.
This is the invariant measure that is observed physically, for if
$\phi: M\to I\!\!R$ is a continuous function, then for $m \ae x\in M$,
$${1\over n}\sum_{i=0}^{n-1}\phi(f^ix)\to\int\phi d\mu$$
as $n\to \infty$. The dynamical system $(f,\mu)$ is ``chaotic''
in the following sense: it has positive Lyapunov exponents; its
metric entropy is equal to the sum of its positive Lyapunov exponents;
$(f,\mu)$ is measure-theoretically isomorphic to a Bernoulli shift;
and it has exponential decay of correlations for H\"older continuous
test functions. These results have been extended
to Axiom A attractors by Bowen, Ruelle, etc. (See e.g. [B].)
In this article we will refer to an invariant measure having
absolutely continuous conditional measures on unstable manifolds as a
{\it Sinai-Bowen-Ruelle measure} or an {\it SBR measure}.
The work of Oseledec, Pesin and others allows us to extend this notion
to a nonuniform setting. (See [P] and [LS].) While some of the properties
of SBR measures carry over (see e.g. [LY], Part I), the question of
{\it existence} of SBR measures in this broader context remains poorly
understood.
We formulate this question more precisely: given a diffeomorphism which
appears to be hyperbolic in a large part of phase space,
can one decide whether or not it admits an SBR measure?
So far there are very few results outside of Axiom A, and these results
involve delicate estimates. See e.g. [BC], [BY] for results on the H\'enon
attractors.
The purpose of the paper is to present some very simple examples
that are hyperbolic everywhere except at one point, but which do not admit
SBR measures. Precise statements of our results are given in \S1. For now
imagine slowly deforming a hyperbolic toral automorphism near the origin $O$
until its derivative has one eigenvalue equal to $1$ and the other
eigenvalue less than $1$. Our theorem says that for the resulting
diffeomorphism, ${1\over n}\sum\limits_{i=0}^{n-1}\delta_{f^ix}\to\delta_O$
for almost every $x$ with respect to Lebesgue measure. ($\delta_x$ denotes
Dirac measure at $x$.) This example can be made to be topologically conjugate
to the original toral automorphism, and so it is topologically ``chaotic''.
>From the statistical point of view, however, it is totally deterministic
in the sense that for almost every initial condition, the trajectory
spends nearly one hundred percent of its time arbitrarily near the origin $O$.
Our result can be thought of as a two dimensional version of the following
result. Let $f:[0,1]\to[0,1]$ be a piecewise $C^2$, piecewise expanding map
of the unit interval with $f'=1$ at a fixed point. Then $f$ cannot
admit a finite absolutely continuous invariant measure. (See [PI].)
The two dimensional situation is, however, not entirely identical to
that in one dimension, for clearly there exist area preserving diffeomorphisms
with positive Lyapunov exponents and nonhyperbolic fixed points.
A more detailed analysis of whether or not systems that are ``almost Anosov''
can admit SBR measures will be given in a forthcoming paper
by the first named author.
\bigskip
\bigskip
{\bf\S1 Assumptions and Statements of Results}
\medskip
Let $M$ be a $C^{\infty}$ two dimensional compact manifold without boundary,
let $m$ denote the Riemannian measure on $M$, and let $f\in \Diff^2(M)$.
We assume throughout this paper that $f$ satisfies
the following two conditions.
\medskip
{\bf Assumption I.}
1. $f$ has a fixed point $p$, i.e. $fp=p$.
2. There exist a constant $\kappa^s< 1$, a continuous function
$\kappa^u$ with
$$\kappa^u(x)\cases{=1, \qquad &at $x=p$\ ,\cr
>1, &elsewhere,\cr}$$
and a decomposition of the tangent space $T_xM$ at every $x\in M$ into
$$T_xM=E_x^u\oplus E_x^s$$
such that
$$\|Df_xv\|\le\kappa^s\|v\|,\quad\qquad\ \forall v\in E_x^s,$$
$$\|Df_xv\|\ge\kappa^u(x)\|v\|,\qquad \forall v\in E_x^u,$$
and
$$\|Df_pv\|=\|v\|,\qquad \qquad \forall v\in E_p^u.$$
\medskip
{\bf Assumption II.}\ $f$ is topologically transitive on $M$.
\medskip
{\bf Definition 1.1.}\ {\sl
An $f-$invariant Borel probability measure $\mu$ on $M$ is called
an {\it SBR measure} for $f:M\to M$ if
i)\ \ $(f,\mu)$ has positive Lyapunov exponents almost everywhere;
ii)\ $\mu$ has absolutely continuous conditional measures
on unstable manifolds.}
\goodbreak
\medskip
We give the precise meaning of the second condition above.
Let $\xi$ be a
measurable partition of a measure space $X$, and let $\nu$
be a probability measure on $X$. Then there is a family of probability
measures $\{\nu_x^\xi: x\in X\}$ with $\nu_x^\xi\bigl(\xi(x)\bigr)=1$
such that for every measurable set $B\subset X$, $x\to \nu_x^{\xi}(B)$
is measurable and
$$\nu B=\int_X\nu_x^{\xi}(B)d\nu(x). \eqno (1.1)$$
The family $\{\nu_x^\xi\}$ is called a {\it canonical system of conditional
measures} for $\nu$ and $\xi$. (For a reference, see e.g. [R].)
Suppose now that $f: (M,\mu)\to (M,\mu)$ has positive Lyapunov exponents
almost everywhere. Then for a.e. $x$, the unstable manifold $W^u(x)$ exists
and is an immersed submanifold of $M$ (see [P]). A measurable partition
$\xi$ of $M$ is said to be {\it subordinate to unstable manifolds} if for
$\mu\ae x$, $\xi(x)\subset W^u(x)$ and contains an open neighborhood of $x$
in $W^u(x)$. Let $m^u_x$ denote the Riemannian measure induced on $W^u(x)$.
We say that $\mu$ has {\it absolutely continuous conditional measures
on unstable manifolds} if for every measurable partition $\xi$ that is
subordinate to unstable manifolds, $\mu_x^\xi$ is absolutely continuous
with respect to $m^u_x$ (written $\mu_x^\xi<\!< m^u_x$) for $\mu\ae x\in M$.
(For more details, see e.g. [LS]).
It is easy to verify that if $\mu_x^\xi<\!< m^u_x$ for one measurable partition
$\xi$ subordinate to unstable manifolds, then the same property holds
for all other measurable partitions subordinate to unstable manifolds.
\medskip
We now state our results. Let $f$ and $M$ be as in the beginning of this
section.
\medskip
{\bf Theorem A.}\ {\sl
$f$ does not admit SBR measures.}
\medskip
{\bf Theorem B.}\ {\sl
For $m\ae x\in M$,
$$\lim_{n\to\infty}{1\over n}\sum_{i=0}^{n-1}\delta_{f^ix}=\delta_p$$
where $\delta_z$ is the Dirac measure at $z$, and the above convergence
is in the weak~${}^*$ topology.}
\medskip
As a by-product of our proofs for Theorem A and Theorem B, we obtain the
following.
{\bf Theorem C.}\ {\sl $f$ has an infinite invariant measure $\bar \mu$
with the following properties:
\ (i)\ if $U$ is any open neighborhood of $p$ in $M$, then
$\bar \mu(M\backslash U)< \infty$;
(ii)\ $\bar \mu$ has absolutely continuous conditional measures
on weak unstable manifolds.}
\medskip
{\bf Remark 1.2.}\
Weak unstable manifolds are defined in Proposition 2.2 (2).
Note that the definition of absolutely continuous conditional measures
on unstable manifolds makes sense even though $\bar \mu$ is
a $\sigma-$finite measure.
\medskip
One could think of $\bar \mu$ as an infinite SBR measure.
In this paper, however, the term ``SBR measure'' without any qualifications
will always be reserved for probability measures.
\bigskip
\bigskip
{\bf\S2. Preliminaries}
\medskip
{\bf Lemma 2.1.}\ {\sl The maps $x\to\{E_x^u\}$ and $x\to\{E_x^s\}$
are continuous.}
This is an easy consequence of the ``gap'' between $\kappa^s$ and
$1=\inf\{\kappa^u(x):\ x\in M\}$. The proof is left to the reader.
\medskip
We will use the following notation: for $\beta>0$,
$E^u_x(\beta)=\{v\in E^u_x:\ |v|\le\beta\}$,
$E^s_x(\beta)=\{v\in E^s_x:\ |v|\le\beta\}$, and
$E_x(\beta)=E^u_x(\beta)\times E^s_x(\beta)$.
\medskip
{\bf Proposition 2.2.}\ {\sl
There exist two continuous foliations ${\cal F}^u$ and ${\cal F}^s$ on $M$
tangent to $E^u$ and $E^s$ respectively for which the following hold.
(1)\ The leaf of ${\cal F}^s$ through $x$, denoted by ${\cal F}^s(x)$,
is the stable manifold at $x$, i.e.
$${\cal F}^s(x)=W^s(x)=\{y\in M:\ \exists C=C_y, \
\hbox{s.t.}\ d(f^nx,f^ny)\le C(\kappa^s)^n\ \forall n\ge 0\}.$$
(2)\ The leaf of ${\cal F}^u$ through $x$, denoted by ${\cal F}^u(x)$,
is the unstable or ``weak unstable'' manifold at $x$, i.e.
$${\cal F}^u(x)=\{y\in M:\ \lim_{n\to\infty}d(f^{-n}x,f^{-n}y)=0\}.$$
(3)\ There exist constants $\beta>0$ and $D>0$ such that for all $x\in M$, if
${\cal F}^u_\beta(x)$ is the component of ${\cal F}^u(x)\cap\exp_xE_x(\beta)$
containing $x$, then $\exp_x^{-1}{\cal F}^u_\beta(x)$ is the graph of
a function $\phi^u_x:\ E^u_x(\beta)\to E^s_x(\beta)$ with $\phi^u_x(0)=0$
and $\|\phi^u_x\|_{C^2}\le D$.
The analogous statement holds for ${\cal F}^s_\beta(x)$.}
{\it Proof:} \quad
These results follow from Theorem 5.5 and Theorem 5A.1 in [HPS].
We indicate how ${\cal F}^u$ is obtained.
Let $x\in M$ be fixed. If $\phi: E^u_x(\beta)\to E^s_x(\beta)$ is a function
with $\phi(0)=0$ and $\hbox{Lip}(\phi)\le 1$, we let
${\cal G}_x\phi: E^u_{fx}(\beta)\to E^s_{fx}(\beta)$ be the function defined by
$$\graph({\cal G}_x\phi)= \bigl(\exp_{fx}^{-1}\circ f\circ
\exp_x(\graph\phi)\bigr)\cap E_{fx}(\beta).$$
Note that ${\cal G}_x\phi$ is always well defined, even though
$f$ is not uniformly hyperbolic. The function $\phi^u_x$ in assertion (3)
is then obtained as the pointwise limit as $n\to\infty$ of
$${\cal G}_{f^{-1}x}\circ\cdots\circ{\cal G}_{f^{-n+1}x}\circ
{\cal G}_{f^{-n}x}(0),$$
where $0$ represents the $0$ function from $E^u_{f^{-n}x}(\beta)$ to
$E^s_{f^{-n}x}(\beta)$, and ${\cal F}^u(x)$ is defined to be
$$\bigcup_{n\ge 0}f^n\bigl({\cal F}^u_\beta(f^{-n}x)\bigr). \eqno \qed $$
\medskip
{\bf Remark 2.3.}\
For convenience we will write $W^u(x)={\cal F}^u(x)$,
$W^u_\beta(x)={\cal F}^u_\beta(x)$ etc. and refer to $W^u(x)$ and
$W^u_\beta(x)$ as the ``unstable manifold'' and ``local unstable manifold''
respectively at $x$, even though points on these manifolds may not
be contracted exponentially in backwards time.
\medskip
The Lipschitzness of the $W^s-$foliation will be very important
for us later on. We give the form of the definition that will be used.
{\bf Definition 2.4.}\ {\sl
Let $\Sigma_1$ and $\Sigma_2$ be two $W^u-$leaves, and let
$\theta: \Sigma_1\to\Sigma_2$ be a continuous map defined
by sliding along the $W^s-$leaves, i.e. for $x\in\Sigma_1$,
$\theta(x)\in\Sigma_2\cap W^s(x)$. We say $W^s$ is {\it Lipschitz}
if $\theta$ is Lipschitz for every $(\Sigma_1, \Sigma_2; \theta)$.}
\medskip
For $y\in W^s(x)$, let $d^s(x,y)$ denote the distance between $x$ and $y$
measured along $W^s(x)$, and for $z\in W^u(x)$, let $d^u(x,z)$ be defined
similarly.
\medskip
{\bf Proposition 2.5.}\ {\sl The $W^s-$foliation is Lipschitz. In fact,
given $D_1>0$, there exists $L_1>0$ such that for every
$(\Sigma_1, \Sigma_2; \theta)$ with $d^s(x,\theta(x))< D_1$
$\forall x\in\Sigma_1$, the Lipschitz constant of $\theta$ is less than
or equal to $L_1$.}
{\it Proof:} \quad
This result follows from the stronger statement that the map $x\to E^s_x$ is
$C^1$, which can be obtained using the same ideas in the proof of
Theorem 6.3 in [HP]. We sketch a more direct proof here for
the convenience of the reader.
Let $x_1\in\Sigma_1$, and let $\gamma$ be an arbitrarily short segment in
$\Sigma_1$ containing $x_1$. We will argue that
$l(\gamma)\approx l(\theta\gamma)$, where $l$ denotes length and ``$\approx$''
means ``up to a constant''.
By taking a suitably large iterate of $f$, we may assume that $f^n\gamma$ and
$f^n(\theta\gamma)$ are very near each other, and
$l(f^n\gamma)\approx d^s(f^nx,f^n(\theta x))$ for $x\in\gamma$.
Notice that $\forall x\in\gamma$,
$d^s(f^n(x),f^n(\theta x))\le D_1(\kappa^s)^n$ $\forall n>0$.
Also, $\bigl|Df_x^n|_{E^u_x}\bigr|\ge 1$ $\forall n\ge 0$.
Observe the following:
(1)\quad $l(f^n\gamma)\approx l(\theta (f^n\gamma))$;
(2)\quad $\bigl|Df_x^n|_{E^u_x}\bigr|\approx
\bigl|Df_{\theta x}^n|_{E^u_{\theta x}}\bigr|$\ $\forall x\in\gamma$;
(3)\quad $\forall y_1, y_2\in \gamma$,
$$\eqalign{
{\bigl|Df_{y_1}^n|_{E^u_{y_1}}\bigr|\over\bigl|Df_{y_2}^n|_{E^u_{y_2}}\bigr|}
\approx & \prod_{i=0}^{n-1}\bigl(1\pm \hbox{const}\cdot d(f^iy_1,f^iy_2)\bigr)
\le \bigl(1\pm \hbox{const}\cdot l(f^n\gamma)\bigr)^n \cr
\approx & \bigl(1\pm \hbox{const}\cdot D_1(\kappa^s)^n \bigr)^n
\approx\hbox{const}. \cr
}$$
\noindent (The proof of (3) uses the boundness of the $C^2$ norms
of $\phi^u_x$. See Proposition 2.2.3).)
Combining (1)-(3), we get $l(\gamma)\approx l(\theta\gamma)$.
\qed
\medskip
Lemma 2.1 and Proposition 2.2 imply that $f$ has a local product structure,
i.e. there exist constants $0<\epsilon<\beta$ such that $\forall y, z\in M$
with $d(y, z)<\epsilon$, $[y, z]:=W^u_\beta(y)\cap W^s_\beta(z)$ and
$[z, y]:=W^u_\beta(z)\cap W^s_\beta(y)$ each contains exactly one point.
A rectangle $R$ is a set in $M$ such that $y, z\in R$ implies
$[y,z], [z,y] \in R$. If $\gamma^u$, $\gamma^s$ are segments of $W^u-$
and $W^s-$leaves respectively, then $[\gamma^u, \gamma^s]$ denotes
the rectangle $\{[y,z]:\ y\in \gamma^u, z\in\gamma^s\}$ provided
that everything makes sense. If $R$ is a rectangle and $x\in R$, we let
$W^u(x,R)=W^u_\beta(x)\cap R$ and $W^s(x,R)=W^s_\beta(x)\cap R$.
If $Q$ and $R$ are two rectangles, we say that $f^nQ$ {\it $u-$crosses}
$R$ if $\forall x\in Q$ with $f^nx\in R$, $f^nW^u(x,Q)\cap R=W^u(f^nx,R)$.
We record a simple fact that will be used in \S4.
\medskip
{\bf Proposition 2.6.}\ {\sl $W^u(p)$ and $W^s(p)$ are both dense in $M$.}
{\it Proof:} \quad
We only prove the proposition for $W^s(p)$.
Let $P$ be a rectangle containing $p$ and let $R$ be any other rectangle,
both with nonempty interiors. Let $\hat R$ be a strictly smaller rectangle
lying in the interior of $R$. By the topological transitivity of $f$,
$\exists n>0$ such that $f^n{\hat R}\cap P\ne\emptyset$. For $n$ sufficiently
large, $f^nR$ is considerably longer than $f^n{\hat R}$ in the $u-$direction.
We may therefore assume that $f^nR$ $u-$crosses $P$. This implies that
$f^{-n}W^s(p,P)\cap R\ne\emptyset$.
\qed
\bigskip
\bigskip\goodbreak
{\bf\S3. Distortion Estimates}
\medskip
The goal of this section is to prove the following.
{\bf Proposition 3.1.}\ {\sl
Given any small rectangle $P$ containing $p$ in its interior, there exist
constants $\delta>0$ and $J>1$ such that if $\gamma$ is a $W^u-$segment with
$l(\gamma)\le\delta$ and $\gamma\cap P=\emptyset$,
then $\forall y,z\in \gamma$ and $n>0$,
$$J^{-1}\le {\bigl| Df^{-n}_z|_{E^u_z}\bigr|\over
\bigl|Df^{-n}_y|_{E^u_y}\bigr|}\le J.$$}
\medskip
We write $y_i=f^{-i}y$, $z_i=f^{-i}z$ and $\gamma_i=f^{-i}\gamma$.
If $\Gamma$ is a $W^u-$segment in $P$,
let $\tilde \Gamma=\{[p,x]:\ x\in\Gamma\}$.
In what follows, the letter $C$ will be used to denote a generic constant,
which is allowed to depend only on $f$.
\medskip
{\bf Lemma 3.2.}\ {\sl
Let $P^+$ be one of the components of $fP\backslash P$, and let
$\gamma=W^u(x,P^+)$ for some $x\in P^+$.
Assume that $\gamma_i\subset P$ for $i=1,\cdots n-1$.
Then for any $y,z\in \gamma$,
$$\log{\bigl| Df^{-n}_z|_{E^u_z}\bigr|\over
\bigl|Df^{-n}_y|_{E^u_y}\bigr|}
\le C{d^u(y,z)\over l(\gamma)}.$$}
{\it Proof:} \quad
First, we have for $j\le n$,
$$\log{\bigl| Df^{-j}_z|_{E^u_z}\bigr|\over
\bigl|Df^{-j}_y|_{E^u_y}\bigr|}=
\log \prod_{i=0}^{j-1}\Bigl(1+
{\bigl| Df^{-1}_{z_i}|_{E^u_{z_i}} -Df^{-1}_{y_i}|_{E^u_{y_i}}\bigr|
\over \bigl|Df^{-1}_{y_i}|_{E^u_{y_i}}\bigr|}\Bigr)
\le C\sum_{i=0}^{j-1}\bigl| Df^{-1}_{z_i}|_{E^u_{z_i}}
-Df^{-1}_{y_i}|_{E^u_{y_i}}\bigr|.$$
Using Proposition 2.2 (3) and the Lipschitzness of $W^s$ (Proposition 2.5),
we see that
$$\bigl| Df^{-1}_{z_i}|_{E^u_{z_i}}-Df^{-1}_{y_i}|_{E^u_{y_i}}\bigr|
\le Cd^u(z_i,y_i) \le Cl(\gamma_i)\le Cl(\tilde \gamma_i).$$
Since the $\tilde \gamma_i$ are pairwise disjoint, we have
$$\sum_{i=1}^{j-1}l(\tilde \gamma_i)\le l(W^u(p,P)).$$
The arguments above tell us that $\forall j\le n$,
$${d^u(z_j,y_j)\over l(\gamma_j)}\le C{d^u(z,y)\over l(\gamma)}.$$
We conclude that
$$\log{\bigl| Df^{-n}_z|_{E^u_z}\bigr|\over \bigl|Df^{-n}_y|_{E^u_y}\bigr|}
\le C\sum_{j=0}^{n-1}d^u(z_j,y_j)
\le C {d^u(z,y)\over l(\gamma)}. \eqno \qed$$
\goodbreak
\medskip
{\it Proof of Proposition 3.1:}\
Let $0=n_0=n_0+k_0\le n_1From the Taylor expansion of $h$, we know that for $x>0$,
$$hx\le x+ Lx^2 \eqno (*)$$
for some $L$. Increasing $L$ if necessary, we may assume that
$a_1\ge{1\over L}$. We will show that $a_i\ge {1\over Li}$ implies that
$a_{i+1}\ge {1\over L(i+1)}$. Suppose this is not true. Then
$$a_{i+1}+La_{i+1}^2< {1\over L(i+1)}\Bigl(1+{1\over i+1}\Bigr)
={1\over Li}\Bigl(1-{1\over i+1}\Bigr)\Bigl(1+{1\over i+1}\Bigr)
<{1\over Li}\le a_i,$$
contradicting $(*)$ with $x=a_{i+1}$.
With $a_i\ge {1\over Li}$ $\forall i\ge 1$, the desired conclusion follows.
\qed
\medskip
Before giving the proof of Theorem A, we recall some facts from general
nonuniform hyperbolic theory. Let $f$ be an arbitrary $C^{1+\alpha}$
diffeomorphism (not having anything to do with the situation in this paper),
and suppose that $f$ preserves an SBR measure $\mu$. Let $\xi$ be a partition
subordinate to $W^u$. Then it is proved in [L] that for $\mu \ae x$,
the density $\rho_x$ of $\mu^{\xi}_x$ with respect to $m_x$ satisfies
$${\rho_x(z)\over \rho_x(y)}=
{\prod\limits_{i=0}^{\infty}\bigl| Df^{-1}_{z_i}|_{E^u_{z_i}}\bigr| \over
\prod\limits_{i=0}^{\infty}\bigl| Df^{-1}_{y_i}|_{E^u_{y_i}}\bigr|}$$
for all $y,z\in\xi(x)$. (In particular the quotient on the right makes
sense.) The following is also known to be true.
Let $\{\Lambda_l\}$ be the ``Pesin sets'', i.e. sets on which
$f^n$ has uniform estimates. Then for every $l$, $\exists \delta_l>0$
such that $W^u_{\delta_l}(x)$ exists for every $x\in\Lambda_l$.
Also, $\exists C_l>0$ such that $\forall x\in \Lambda_l$,
$$C_l^{-1}\le{\rho_x(z)\over \rho_x(y)}\le C_l \qquad
\forall y,z\in W^u_{\delta_l}(x).$$
(See e.g. [P].)
\medskip
We now return to the situation considered in this paper, i.e. $f$ is again
assumed to satisfy Assumptions I and II.
\medskip
{\it Proof of Theorem A:} \quad
Suppose, to derive a contradiction, that $f$ admits an SBR measure $\mu$.
Then there is a rectangle $R\subset M$ such that $\mu(R\cap\Lambda_l)>0$
for some $l$, and $W^u(x,R)\subset W^u_{\delta_l}(x)$
$\forall x\in R\cap\Lambda_l$.
We fix a rectangle of the form $P=[W^u_\delta(p), W^s_\delta(p)]$ and
let $Q=f^{-1}P\backslash P$. Let $\xi$ be the partition of $Q$ given by
$\xi(x)=W^u(x,Q)$. An argument similar to that in Proposition 2.6
shows that $f^nR$ $u-$crosses $Q$ for some $n>0$. It follows
from our discussion above that there is a set ${\hat Q}\subset Q\cap f^nR$
with $\mu{\hat Q}>0$ such that
(i)\ $x\in {\hat Q}$ $\Longrightarrow$ $\xi(x)\subset{\hat Q}$; \ and
(ii) $\exists C_0>0$ such that $\forall x\in {\hat Q}$,
$$C_0^{-1}\le{\rho_x(z)\over \rho_x(y)}\le C_0 \qquad
\forall y,z\in W^u(x,Q).$$
Let
$$Q^{(i)}=\{y\in {\hat Q}:\ f^jy\in P\ \hbox{for}\ j=1,\cdots,i\},$$
and let $\tilde\gamma^{(i)}$ be the projection of $Q^{(i)}$ onto $W^u(p, P)$
by sliding along $W^s$. Then the density estimate above together with
the Lipschitzness of the $W^s-$foliation gives
$$\mu Q^{(i)}\ge Cl(\tilde\gamma^{(i)}).$$
Using the facts that $f^iQ^{(i)}$, $i=1,2,\cdots$, are pairwise
disjoint subsets of $P$, and $\mu$ is an invariant measure, we have
$$\mu P \ge\sum_{i=1}^\infty\mu (f^iQ^{(i)}) =\sum_{i=1}^\infty\mu Q^{(i)}
\ge C\sum_{i=1}^\infty l(\tilde\gamma^{(i)}).$$
Lemma 4.1 applied to $f|_{W^u_\beta}(p)$ tells us that this sum diverges,
contradicting $\mu(M)=1$.
\qed
\bigskip
\bigskip
{\bf \S5 Proofs of Theorems B and C}
\medskip
We first construct some neighborhoods of $p$ that are convenient to work with.
For a closed rectangle $R$, let
$\partial^sR=\{x\in R: x\notin\intset\, W^u(x,R)\}$.
\medskip
{\bf Lemma 5.1.}\ {\sl
There exist rectangles $P$ of the form $P=[\gamma^u, \gamma^s]$ satisfying
the follows:
(1)\ $\gamma^u$ and $\gamma^s$ are segments of $W^u(p)$ and $W^s(p)$
respectively such that $p$ lies in the interiors of $\gamma^u$ and $\gamma^s$;
(2)\ there is a compact segment $\hat W^s\subset W^s(p)$ with
$f\hat W^s\subset\hat W^s$ such that $\partial^sP\subset \hat W^s$ and
$\bigl(\hat W^s\backslash W^s(p, P)\bigr)\bigcap \intset P=\emptyset$;
(3)\ there is a compact segment $\hat W^u\subset W^u(p)$ with analogous
properties.
\noindent Moreover, the diameter of $P$ can be chosen arbitrarily small.}
{\it Proof:} \quad
Use the fact that $W^u(p)$ and $W^s(p)$ are dense in $M$. See Lemma 2.6.
\qed
\medskip
We fix $P$ as above, and consider the first return map
$g: M\backslash P\to M\backslash P$. That is, for $x\in M\backslash P$,
if $\tau(x)$ is the smallest positive integer with
$f^{\tau(x)}x\in M\backslash P$, then $gx=f^{\tau(x)}x$. Note that $g$
is not defined on a set of Lebesgue measure $0$ on $M\backslash P$,
but this will not concern us.
\medskip
{\bf Lemma 5.2.}\ {\sl
There exists a $g-$invariant Borel probability measure $\mu$ with the
property that $\mu$ has absolutely continuous conditional measures
on the unstable manifolds of $f$.}
{\it Proof:} \quad
Let $L=W^u(x,P^+)$, where $P^+$ is one of the components of $fP\backslash P$
and $x\in P^+$. Let $m_L$ be the Lebesgue measure on $L$ and let
$g^n_*m_L$ be the push-forward of $m_L$, i.e., $(g^n_*m_L)(E)=m_L(g^{-n}E)$.
We may take $\mu$ to be any accumulation point in the weak~${}^*$ topology of
${1\over n}\sum\limits_{i=0}^{n-1}g^i_*m_L$.
The $g-$invariance of $\mu$ is clear. To show that it has
absolutely continuous conditional measures, it suffices to consider
rectangles $R$ in $M\backslash P$ with small diameters. We assume also
that $\intset R\cap{\hat W^s}=\emptyset$, where $\hat W^s$
is as in Lemma 5.1. The significance of the second condition is as follows:
$g^iL$ is the disjoint union of $W^u-$segments, each one of which begins
and ends at some point in $g^j{\hat W^s}$ for some $j\ge 0$.
Since $f{\hat W^s}\subset {\hat W^s}$, each component of $g^iL$ that
intersects $R$ $u-$crosses $R$. Let $\rho_i$ denote the density of $g^i_*m_L$
with respect to Lebesgue measure on $g^iL$. Then by Proposition 3.1,
$\exists J>0$ (independent of $i$) such that for all $x$, $y$
in the same component of $g^iL\cap R$,
$$J^{-1}\le {\rho_i(x)\over \rho_i(y)}\le J. $$
This bound on densities is passed on to the limit measure $\mu$.
\qed
\medskip
{\it Proof of Theorem C:} \quad
Let $Q_i=\{x\in M\backslash P:\ \tau(x)=i\}$, where $\tau$ is the return time
to $M\backslash P$. Define
$$\bar\mu=\sum_{i=1}^{\infty}\sum_{j=0}^{i-1}f^j_*(\mu|_{Q_i}).$$
Then $\bar\mu$ is clearly $f-$invariant.
Let $U$ be a neighborhood of $p$. Then $M\backslash U$ is contained in
$M\backslash \bigl(\bigcap\limits_{i=-n}^nf^iP\bigr)$ for some large $n$,
and this latter set
clearly has finite $\bar\mu-$measure. This proves that $\bar\mu$ is at most
$\sigma-$finite. It cannot be finite because it has absolutely continuous
conditional measures on unstable manifolds, and Theorem A says that
$f$ does not admit $SBR$ measures.
\qed
\medskip
Recall that $m$ denotes the Lebesgue measure on $M$.
We now study the asymptotic behavior of trajectories starting at $x$ for
$m \ae x$.
\medskip
{\bf Lemma 5.3.}\ {\sl
Let $g$ and $\mu$ be as in Lemma 5.2. Then $(g,\mu)$ is ergodic.}
{\it Proof:} \quad
We will follow the two standard steps in the proof of ergodicity of SBR
measures for hyperbolic systems without discontinuities.
The first step is to use Hopf's argument to show that given a rectangle $R$,
$m\ae x\in R$ is ``future-generic'' with respect to some ergodic
measure $\mu_R$, with $\mu_R$ possibly depending on $R$.
(``Future-genericity'' means that ${1\over n}\sum\limits_{i=0}^{n-1}
\phi\circ g^i(x)\to\int\phi d\mu_R$ as $n\to +\infty$ for every continuous
function $\phi: M\to R$. ``Past-genericity'' is defined similarly.)
The second step is to show that $\mu_R=\mu$ for all $R$.
Let $R\subset M\backslash P$ be a rectangle. Note that when we use the word
``rectangle'' or the symbol ``$W^u(x,R)$'' in this paper, we are always
referring to the stable and unstable manifolds of $f$ --- which are not
necessarily stable and unstable manifolds of $g$! First we need to argue
that for suitable $R$, $W^u(x,R)$ {\it is} indeed a local unstable manifold
of $g$, in the sense that $\forall y\in W^u(x,R)$, $d(g^{-n}x,g^{-n}y)\to 0$
as $n\to\infty$. This is true if $\intset R\cap {\hat W^s}=\emptyset$,
for this condition will guarantee that $\forall n\ge 0$, $f^{-n}W^u(x,R)$
is either entirely in $P$ or it does not intersect $P$. Similarly,
$W^s(x,R)$ is a local stable manifold of $g$ if
$R\cap \hat W^u=\emptyset$.
We recall Hopf's argument (see, e.g. [A]) for a rectangle $R$ with
the properties in the last paragraph. Since the conditional measures
of $\mu$ are absolutely continuous on unstable manifolds (Lemma 5.2),
there exists $L=W^u(x,R)$ such that $m^u_x\ae y\in L$ is generic
(both future and past) with respect to some ergodic measure $\nu_y$.
All the $\nu_y$'s are in fact identical because as $n\to +\infty$,
$d(g^{-n}y, g^{-n}z)\to 0$ \ $\forall y,z\in L$.
We call this common measure $\mu_R$.
Now if $y$ is future generic with respect to $\mu_R$,
then $z$ is future generic with respect to $\mu_R$ $\forall z\in W^s(y,R)$.
It then follows from the Lipschitzness of the $W^s-$foliation
(Proposition 2.5) that $m\ae z\in R$ is future generic with respect to $\mu_R$.
To carry out the second step, it suffices to observe that if $R_1$ and $R_2$
are rectangles with the properties above, then $\exists n>0$ such that
$g^nR_1\cap R_2\ne \emptyset$.
\qed
\medskip
{\it Proof of Theorem B:} \quad
We will show that given arbitrarily small numbers $\alpha>0$ and $\epsilon>0$,
there exist neighborhoods $P_2\subset P_1$ of $p$ with
$\diam P_1\le\alpha$ such that for $m\ae x\in M\backslash P_2$,
$${\#\{0\le k\le n:\ f^nx\in M\backslash P_1\}\over
\#\{0\le k\le n:\ f^nx\in P_1\backslash P_2\}}<\epsilon$$
for all sufficient large $n$.
To see this, let $P_1$ be a rectangle of the type in Lemma 5.1.
Let $g_1: M\backslash P_1\to M\backslash P_1$ and $\mu_1$ be as
in Lemma 5.2, and let $\bar \mu$ be the infinite measure in the proof
of Theorem C. Let $P_2$ be chosen small enough that
$\bar \mu(M\backslash P_1)\le \epsilon {\bar \mu}(P_1\backslash P_2)$.
Then $\mu_2:=\bar \mu|_{M\backslash P_2}$ is invariant under the first
return map $g_2: M\backslash P_2\to M\backslash P_2$, and $(g_2, \mu_2)$
is ergodic. The Birkhoff Ergodic Theorem applied to $(g_2, \mu_2)$
completes the proof.
\qed
\bigskip
\bigskip\goodbreak
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\vskip 0.5 true cm
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\goodbreak
\end