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 BODY
\documentstyle {amsppt}
\magnification \magstep1
\openup3\jot
\NoBlackBoxes
\pageno=1
\hsize 6 truein
%\hoffset -.5 truein
%\voffset -.5 truein
 
%here are the needed definitions

\font\ttt=cmtt10 scaled \magstep1
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%here I redifine "\enddemo"
 
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\predefine\reall{\Re}

\redefine\Re{\Bbb R}
\def\N{\Bbb N}
\def\Z{\Bbb Z}
\def\To{\Bbb T}
\def\P{\Bbb P}
\def\ve{\varepsilon}
\def\Id{\text{{\rm 1}\!\! \cmss 1}}
\def\Idfoot{\text{{\rm 1}\!\! \cmssfoot 1}}
\def\d{\hbox{dist}}
\def\M{{\hbox{\ttt M}}}
\def\Mp{{\hbox{\tt M}}}
\def\F{{\Cal F}}

\def\e #1{{\hbox{{\rm Exp}$\left[#1\right]$}}}
\def\eu #1{{\hbox{{\rm Exp}$_u\left[#1\right]$}}}
\def\es #1{{\hbox{{\rm Exp}$_s\left[#1\right]$}}}
\def\missingstuff{\vskip1cm\centerline{\bf MISSING STUFF}\vskip1cm}

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   January\or February\or March\or April\or May\or June\or July\or August\or
   September\or October\or November\or December\fi
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%here starts the document

\topmatter
 
\title
FLOWS, RANDOM PERTURBATIONS AND RATE OF MIXING
\endtitle
\author
Carlangelo Liverani
\endauthor
\affil University of Rome {\sl Tor Vergata}
\endaffil
\address
Liverani Carlangelo,
Mathematics Department,
University of Rome II, Tor Vergata,
00133 Rome, Italy.
\endaddress
\email
liverani@mat.utovrm.it
\endemail
\date
Dicember 6, 1996
\enddate

\abstract
A new approach to the study of the rate of mixing in Anosov flows, recently
proposed by N. Chernov, is simplified and generalized to the higher dimensional case.
\endabstract
\thanks
\bf I whish to thank Viviane Baladi, Nikolai Chernov and Thomas Spencer for
stimulating my interest in the subject. I am indebted to Francois Ledrappier for many
extremely helpful and enjoyable discussions and for his continuing encouragement during
this project. Finally, I acknowledge the hospitality of the Ervin Shr\"odinger
Institute, Vien, of IMPA, Rio de Janeiro, the support of the grant CHRX-CT94-0460 of
the Commission of the European Community and the agreement CNPq-CNR.
\endthanks
\endtopmatter
 


\vskip -.5cm
\centerline{\bf CONTENT}
%\vskip -.5cm

\newdimen\riga
\newdimen\rigat
\riga=\baselineskip
\rigat=\lineskip
\baselineskip=.5\baselineskip
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\roster

\item"0."  Introduction\dotfill p. \ \ 2
\item"1."  Preliminaries\dotfill p. \ \ 3
\item"2."  Random perturbations\dotfill p. \ \ 5
\item"3."  Estimating the Kernel\dotfill p. \ \ 8
\item"4."  Decay of correlations \dotfill p. 10 
\item" "   Appendix I (Averages)\dotfill p. 13
\item" "   Appendix II (Balls)\dotfill p. 16
\item" "   Appendix III (C-Frames)\dotfill p. 22
\item" "   Appendix IV (Product Sets)\dotfill p. 25
\item" "   Appendix V (A change of coordinates)\dotfill p. 26
\item" "   References\dotfill p. 28

\endroster

\baselineskip=\riga
\lineskip=\rigat


\document
        
\vskip1cm
\subhead \S 0 Introduction 
\endsubhead

The problem of studying the rate of mixing in dynamical systems has
been the subject of many investigations in the last decades.
Many different techniques have been developed allowing to gain a
good understanding of the situation as far as uniformly hyperbolic maps are 
concerned. For this systems it is possible to show
that H\"older continuous observables enjoy an exponential decay of correlation
\cite{Bo}. This turns out to be true even if the system is only piecewise smooth,
provided the singularities are not too wild \cite{Li1}, \cite{Yo} or when the
mechanism that produces the hyperbolicity is not a straightforward one \cite{BY}. 

On the contrary, very little it is known in the non-uniformly hyperbolic case;
this still stands as a challenge. Nevertheless, recently Chernov \cite{Ch} 
has made
a decisive progress concerning Anosov flows. On the one hand, hyperbolic flows
can be considered to be an intermediate situation between uniform hyperbolicity
and non-uniform hyperbolicity due to the zero Lyapunov exponent in the
flow direction.  On the other hand, the study of flows bears a clear interest
both in itself and for its physical implications. The study of correlation 
for flows is open since the seventies \cite{BR}
but very little progress has been made since; apart from few results for the
geodesic flows on constant negative curvature \cite{CEG}, \cite{Mo},
\cite{Po1} and \cite{Ra}.

Chernov has been able to show that for Anosov flows on a three dimensional
manifold satisfying some uniform non-integrability condition\footnote{For 
contact flows Chernov conditions turns out to be essentially equivalent to requiring
that the stable and unstable foliations are Lipschitz \cite{H}, see appendix III for
more details.} and, for H\"older continuous observables, the correlations decay at
least as $e^{-\sqrt{t}}$. In this paper I present Chernov idea in a nutshell,
avoiding any reference to Markov partitions and therefore greatly simplifying
Chernov's approach. As a byproduct of the above mentioned simplification I am able to
extend the applicability of Chernov's method to the higher dimensional situation. 

In order to further simplify the presentation of the technique I deal only
with contact flows but the generalization to Anosov should present no difficulties.
The results of the paper consist in the following.

\proclaim{Theorem A} Given a uniformly hyperbolic contact flow $\phi_t$, with
Lipschitz stable and unstable foliation, on a $2d+1$-dimensional  manifold $\M$ there
exists a constant $\gamma_0>0$ such that for each  two H\"older continuous functions
$f,\,g$ 
$$
|\mu(fg\circ\phi_t)-\mu(f)\mu(g)|\leq K e^{-\gamma_0\sqrt{t}}
$$
for some constant $K$.
\endproclaim

\proclaim{Theorem B} Given a uniformly hyperbolic contact flow $\phi_t$  on a 
$2d+1$-dimensional 
manifold $\M$ there exists a constant $\gamma_0>0$ such that for each 
two H\"older continuous functions $f,\,g$ and each
$\theta<\frac{2\alpha-1}{1-\alpha}$
$$
|\mu(fg\circ\phi_t)-\mu(f)\mu(g)|\leq 
\frac K{1+(1+|t|)^\theta}
$$
where $\alpha<1$ is the order of H\"older continuity of the stable and unstable 
foliations.
\endproclaim

Theorem B yields a polynomial decay of correlations for
$\alpha>\frac{1}2$, with the degree of the polynomial going to infinity as
$\alpha\to 1$. In particular, it follows that, for $\alpha>\frac 23$,
$$
\int_0^\infty dt\int_{\Mp}d\mu g\circ\phi_t g <\infty ,
$$
 for each $g\in C^{(1)}(\M)$, $\int_{\Mp}g=0$. The relevance of the above fact is
that the integrability of the self-correlation is a basic ingredient to obtain the
Central Limit Theorem for the function $g$ \cite{Li2}.

Theorem B shows that Chernov ideas can
be pushed beyond the realm of ``uniform non-integrability" which is not satisfied if
the foliation is only H\"older continuous (see appendix III for more details). I would
like to remark that theorem B is not optimal and that a look at the proof suggests
that there may be several ways to improve it, whereby obtaining faster decays and
results for less regular foliations. It is instead unclear if the strategy of the
proof of theorem A can be modified to yield a sharper bound.

The idea of the proof essentially consists in introducing a random 
perturbation  of the flow. This provides us with the missing ``hyperbolicity"
in the flow direction. Consequently, it is possible to compute the rate
of decay for the random perturbation. The cornerstone of the approach is the
possibility to remove the random perturbation while still keeping a control
on the rate of decay of the correlations, whereby obtaining information on
the rate of mixing for the deterministic flow.

It seems to me that this approach has good potentiality of yielding results
also in other ``non-uniform" situations. In addition, given to the possibility
of avoiding Markov partitions, it is conceivable that one can apply similar
ideas to non-smooth flows (e.g. billiards).


The content of the paper is as follows.
Section one contains few preliminaries concerning contact flows  and hyperbolicity. 
In section two a special random perturbation of the
flow is introduced and studied assuming a key estimate (theorem 2.1), that it is
proven in section three. Section four shows how to use the knowledge gained in the
previous sections to study the decay of correlations for the deterministic flow.
Finally, in appendix I it is proven that random perturbations with the
properties required in section two exist. Appendix II contains some
measure--geometrical estimates used in appendix I; while in appendix III we recall the
idea and properties of the C-frames, which are the essential geometrical tool used in
section three.\footnote{These are the H-frames introduced by Chernov.}   
Appendix IV contains more measure-geometric estimates used in section three,
while appendix V concludes the paper with some considerations on a change of
coordinates used in section four.

\vskip1cm
\subhead \S1 Preliminaries
\endsubhead

We will consider contact flows on a $2d+1$ connected compact Riemannian manifold 
$\M$.\footnote{For the reader convenience here is an almost verbatim quote concerning
contact flows, taken from \cite{KB}: ``A contact form on $\Mp$ is a $C^1$
differential 1-form $\omega$ such that the $(2d+1)$-form $\omega\wedge (d\omega)^d$
is non-zero at every point. The kernel of $\omega$ is a codimension 1 distribution on
$\Mp$. The restriction of the 2-form $d\omega$ to Ker $\omega$ determines a
symplectic structure there. There is a unique vector field $X$ on $\Mp$ such that
$d\omega(X,\,Y)=0$ for  all vector fields $Y$ and $\omega(X)=1$. The flow $\phi_t$
defined by $X$ is called the {\it contact flow on $\Mp$}. It preserves the contact
form $\omega$. Conversely, any flow on $\Mp$ that preserves $\omega$ is a constant
reparametrization of
$\phi_t$. The contact flow preserves the distribution Ker $\omega$, the symplectic
structure there and the measure $\mu$ on $\Mp$ determined by the volume form
$\omega\wedge(d\omega)^d$."}

For simplicity we assume that the Riemannian volume $\mu$ and the contact one
coincide,  moreover, if $X$ is the vector field generating the flow, we assume
$\|X\|=1$ (if this is not the case, one can always change the Riemannian structure to
obtain such properties).

We assume that the flow is uniformly hyperbolic, namely  at every point $p\in\M$, the 
tangent space $\Cal T_p\M$ can be written as $E_p^0\oplus E_p^s\oplus E_p^u$,
$\hbox{dim}E_p^0=1$, $\hbox{dim}E_p^s=\hbox{dim}E_p^u$. In addition, there exists
$\lambda>0$, $c>1$ such that, for each $p\in\M$, $E_p^0$ is the flow direction; for
each $t>0$, $d_p\phi_t E_p^s=E_{\phi_tp}^s$, $d_p\phi_t E_p^u=E_{\phi_tp}^u$ and, for
each $v\in E_p^s$, $w\in E_p^u$,\footnote{By $d_p\phi_t$ is meant the differential of
$\phi_t$ at the point $p$, some time I will write only $d\phi_t$ if no confusion
arises.}
$$
\aligned
&\|d\phi_t v\|\leq ce^{-\lambda t}\|v\|,\\
&\|d\phi_t w\|\geq c^{-1} e^{\lambda t}\|w\|.
\endaligned
$$

The above hypotheses imply that the contact flow is Bernoulli (hence, mixing and 
ergodic) \cite{KB}. Contact flows arise naturally in many contexts, e.g.
Hamiltonian dynamics \cite{Arn} and differential geometry. In particular, it is well
known that the geodesic flow on a $d$ dimensional Riemannian manifold
$\Cal M$ can be viewed as a contact flow on the unitary tangent bundle $\M$
\cite{Arn}. If the manifold $\Cal M$ has strictly negative curvature then the
geodesic flow is uniformly hyperbolic \cite{An}, \cite{Po2}. 

It is a general result of hyperbolic theory \cite{Pe} that $E_p^u$, $E_p^s$ are 
H\"older continuous distributions.\footnote{The result states that the distributions
are H\"older continuous of some order $\alpha$ depending on the rate between the
minimal and the maximal expansion in the unstable directions and the minimal and
maximal contraction in the stable directions. In fact, there are several H\"older
structures besides the stable and unstable distributions (e.g., the holonomy map and
its Jacobian), here, and in the following, by ``$\alpha$" we will mean the smallest of
all the relevant H\"older exponents (actually, the holonomy map and the
distributions have the same regularity, crf. \cite{Has}, \cite{SS}, but here we are
not concerned with the optimal estimate for $\alpha$: more work in this direction
may be needed).} In addition, they are integrable giving rise to the stable 
($W^s(p)$), the weak-stable ($W^{0s}(p)$),
the unstable ($W^u(p)$) and the weak-unstable ($W^{0u}(p)$)foliations. On the contrary the distribution
$E^s_p\oplus E^u_p$ is not integrable (due to the contact structure,
\cite{KB}) and, in fact, this is a key ingredient in the proof that ergodic contact
flows are Bernoulli.

The distribution may be more regular in special cases. The geodesic 
flow in strictly negative curvature for example. In this case
the distributions $E^s_p$ and $E^u_p$ turns out to be $C^{(1)}$ for two dimensional
manifolds \cite{HK}, while in higher dimensions the distributions are $C^{(1)}$ if
the metric satisfies the so called $\frac 14$-pinching conditions \cite{HP} (see
\cite{Has} for a review on regularity results on Anosov splittings). In principle it
could be possible to take advantage of this to improve the results presented here but
no attempt is done in this direction in the present work.\footnote{At the moment there
exists a claim of D.Dolgopiat along such lines
\cite{D1}, \cite{D2}. He proposes a different approach to prove that, if the
foliation is $C^{(1)}$, then the decay of correlations is exponential. His technique
seems to be extremely sensitive to the regularity of the foliation: for Lipschitz
(but not $C^{(1)}$) foliations he obtains only a decay faster than any polynomial,
which it is worst than Chernov result.}

In the following, by ``random perturbation" of the flow we will not mean a random 
flows. Instead, we will consider the map $T$ defined by the flow at some time $t$ and
construct a random process that is a small perturbation of $T$. Typically, moving
the points by $T$ and then by spreading them around according to some distribution
having a small support. 

Nevertheless, I think that it would be interesting to explore other possibilities.
For  example, in the case of geodesic flows it would seem very natural to consider the
generator $\Cal L_0$ of the flow (simply the vector field viewed as an operator on
$L^2(\M,\,\mu)$) and use, as a perturbation, the stochastic process defined 
by the generator
$$
\Cal L_\ve=\Cal L_0+\ve\Delta,
$$
where $\Delta$ is the Laplace-Beltrami operator on $\M$. In analogy with what is done 
here it should be possible to study the rate of decay for the process generated by
$\Cal L_\ve$ and to obtain informations on the rate of decay of correlations for the
deterministic flow by sending $\ve$ to zero. In my opinion the development of the
techniques necessary to carry out such a program would have an interest in itself.

\vskip1cm
\subhead \S2 Random Perturbations
\endsubhead

Here we will define a suitable random perturbation of a contact flow and
derive some results on the decay of correlations for such a random 
perturbation; eventually, this will enable us to obtain an estimate for the decay of correlations for the flow
itself.

The strategy is quite general and one can probably use a large class of random 
perturbations to carry it out; nevertheless, its implementation turns out to be quite
delicate and it appears to be convenient to choose a very special
perturbation. To define such a perturbation a few preliminaries are needed.

Let $d$ be the metric associated to the Riemannian structure, define
$$
d_\ve(x,\,y)=\max\left\{\sup_{t\in[0,\,n_\ve]}d(\phi_t
x,\,\phi_t y);\,\ve^{-1}d(\phi_{n_\ve}x,\,\phi_{n_\ve}y)\right\},
\tag 2.1
$$
where
$$
n_\ve=\inf\left\{n\in\Bbb N\;\Bigg|\;\sup\Sb x\in{\Mp}\\ v\in
E^s(x)\endSb\frac{\|d_x\phi_nv\|} {\|v\|}\leq\ve^2\right\}.
$$

Let $B_\delta^{\ve}(x)$ be the set of points with distance, from $x$, less than
$\delta $  with respect to the distance $d_\ve$.\footnote{The ball $B^1_\delta(x)$
($\ve=1$) is just the usual dynamical ball of size $\delta$ over the trajectory
from time zero to time $n_1$; nevertheless, we will be interested in the case where
$\delta$ is small but fixed and $\ve$ is arbitrarily small. In this situation
$B_\delta^{\ve}(x)$ is more or less a tiny neighborhood of a disk of diameter
$2\delta$, centered at $x$, on the strong stable manifold of $x$. Such a
neighborhood has size $\ve$ along the flow direction and size smaller than $\ve^3$
in the strong unstable direction.}

We are now in a position to define the class of averages we are interested in.
$$
(\Bbb A_{\ve,\delta}^\varphi g)(x)=\int_{B^{\ve}_\delta(x)}d\mu(y)
\frac{(1+\delta\varphi(x))(1+\delta\varphi(y))}{\mu(B^{\ve}_\delta(x))^{\frac
12}\mu(B^{\ve}_\delta(y))^{\frac 12}}
 g(y).
$$
Note that $\Bbb A_{\ve,\delta}^\varphi$ is well defined both as a bounded
operator on $L^2(\M,\,\mu)$ and on $C^{(0)}(\M)$, if $\varphi\in C^{(0)}(\M)$.
Moreover, as an operator on $L^2(\M,\,\mu)$ it is self-adjoint. In appendix I we will
see that there exist  functions $\varphi^{\ve}_\delta\in C^{(0)}(\M)$ such that, for
some fixed $c_0>0$, $\|\varphi^\ve_\delta\|_\infty\leq c_0$, for each $\delta$ and
$\ve$ sufficiently  small, and 
$$
\Bbb A_{\ve,\delta}^{\varphi^{\ve}_\delta} 1=1 .
$$

>From now on we will fix some sufficiently small 
$\delta_0>0$ and set, for each $g\in L^2(\M,\,\mu)$, 
$$
\Bbb A_{\ve} g(x)\equiv\Bbb
A_{\ve,\delta_0}^{\varphi^{\ve}_{\delta_0}}g(x)\equiv\int_\Mp
a_{\ve}(x,\,y)g(y)d\mu(y).
$$

Now we can define the wanted random perturbation of the flow: for each 
$f\in L^2(\M,\,\mu)$
$$
T_{t,\varepsilon}f=(\Bbb A_{\ve} f)\circ\phi_t.
$$
This corresponds to first evolving a point for a time $t$, by the flow, and
then spreading it on a neighborhood of radius $\delta_0$ (in the metric 
$d_\ve$) according to the distribution specified by the average 
$\Bbb A_\ve$.
To such a stochastic process is naturally associated a Markov semigroup 
(simply the adjoint):
$$
\Bbb P_{t,\,\varepsilon}g=\Bbb A_{\ve} (g\circ \phi_{-t}).
$$

By construction, $T_{t,\varepsilon}1=1$ and $\Bbb
P_{t,\varepsilon}1=1$.\footnote{In fact, the reason of the special choice of the
averaging is motivated only by the convenience of having the same invariant measure
for the deterministic flow and its random perturbation.}


The key idea--introduced in a different language by Chernov--is to investigate 
the operator
$\Bbb C_{t,\,\ve}= (\Bbb P_{t,\,\varepsilon}^*)^2 \Bbb P_{t,\,\varepsilon}^2$. 
Such an operator is easily seen to be
$$
\Bbb C_{t,\,\ve}g(x)=\int_{\Mp}C_{t,\,\varepsilon}(x,\,y)g(y)d\mu(y)
$$
where
$$
C_{t,\varepsilon}(x,\,y)=\int_{\Mp^3}d\mu(z_1)d\mu(z_2)d\mu(z_3)
a_\ve(\phi_t x,z_1)a_\ve(\phi_t z_1,z_2)
a_\ve(z_3,z_2)a_\ve(\phi_t y,\phi_{-t} z_3).
$$

The relevance of the previous operator is due to the next estimate.

\proclaim{Theorem 2.1} There exists $\gamma,\,c_1,\,\ve_0\in\Bbb R^+$ such that
for each $\varepsilon<\ve_0$ and for each $x,\,y\in\M$, setting $t_\ve\equiv
c_1\log\ve^{-1}$, 
$$
C_{t_\ve,\,\varepsilon}(x,\,y) \geq\gamma\ve^{\frac 1\alpha-1}>0;
$$
where $\alpha$ is the H\"older continuity of the foliations.
\endproclaim

The proof of theorem 2.1 is the content of section  3.

>From now on let us set $T_\varepsilon\equiv
T_{t_\ve,\,\varepsilon}$, 
$\Bbb P_\ve\equiv\Bbb P_{t_\ve,\,\ve}$.
Define the projector
$$
\Pi g=\int_{\Mp}g.
$$

Notice that $\Pi^*=\Pi$.


A very important, but standard, consequence of theorem 2.1 is the following. 

\proclaim{Lemma 2.2}The spectral radius of $\Bbb C_\varepsilon-\Pi$, as an
operator on $L^2(\M,\,\mu)$, is less than $1-\gamma\ve^{\frac 1\alpha-1}$.
\endproclaim
\demo{Proof}
The first, completely standard, consequence of theorem 2.1 is the
estimate\footnote{Let us briefly recall the argument: consider 
$f\in L^1(\Mp,\,\mu)$ with $\int_{\Mp}f=0$. Remember that $\Bbb C_\ve 1=
\Bbb C_\ve^*1=1$ and let $\Mp^+_\ve=\{x\in\Mp\;|\;\Bbb C_\ve f\geq 0\}$;
$\Mp_+=\{x\in\Mp\;|\; f\geq 0\}$, $\gamma_\ve\equiv \gamma\ve^{\frac 1\alpha-1}$, 
then
$$
\aligned
\|\Bbb C_\ve f\|_1&=2\int_{\Mp^+_\ve}d\mu(x)\int_{\Mp}d\mu(y) C_\ve(x,\,y)
f(y)=2\int_{\Mp}d\mu(y) f(y)\int_{\Mp^+_\ve}d\mu(x)[ C_\ve(x,\,y)-\gamma_\ve]\\
&\leq 2\int_{\Mp_+}d\mu(y)f(y)\int_{\Mp}d\mu(x) [C_\ve(x,\,y)-\gamma_\ve]
=(1-\gamma_\ve)2\int_{\Mp_+}d\mu(y)f(y)\\
&=(1-\gamma_\ve)\|f\|_1.
\endaligned
$$
Accordingly, for each $f\in L^1(\Mp,\,\mu)$, holds
$$
\|(\Bbb C_\ve-\Pi)^nf\|_1=\|\Bbb C_\ve^n(\Idfoot-\Pi)f\|_1\leq
(1-\gamma_\ve)^n\|f\|_1 .
$$
}
$$
\|(\Bbb C_\ve-\Pi)^n\|_1\leq (1-\gamma\ve^{\frac 1\alpha -1})^n.
$$

The operator $\Bbb C_\varepsilon-\Pi$  is self-adjoint, hence
$$
\aligned
\|(\Bbb C_\varepsilon-\Pi)^n\|_2&=
\sup_{g\in \Cal C^{(0)}}    
\frac{\langle (\Bbb C_\varepsilon-\Pi)^ng,\,g\rangle}
{\langle g,\,g\rangle}=
\sup_{g\in \Cal C^{(0)}}    
\frac{\langle (\Bbb C_\varepsilon-\Pi)^{n-1}g,\,
\left(\Bbb C_\varepsilon-\Pi\right) g\rangle}
{\langle g,\,g\rangle} \\    
\leq&\sup_{g\in \Cal C^{(0)}}  
\frac{\|(\Bbb C_\varepsilon-\Pi)^{n-1}g\|_1
\|(\Bbb C_\varepsilon-\Pi)g\|_\infty}  
{\langle g,\,g\rangle} \\    
\leq&\sup_{g\in \Cal C^{(0)}} 
\frac{K_\ve(1-\gamma\ve^{\frac 1\alpha-1})^{n-1}\|g\|_1^2}{\langle g,\,g\rangle} 
\leq \frac{K_\ve}{1-\gamma\ve^{\frac 1\alpha-1}}(1-\gamma\ve^{\frac 1\alpha-1})^n,
\endaligned
$$
where $K_\ve=\|C_{t_\ve,\,\ve}\|_\infty+1$.
\enddemo

\proclaim{Lemma 2.3}  The $L^2(\M,\,\mu)$ norm of $\Bbb
P_\varepsilon^2-\Pi$ is less than $(1-\gamma\ve^{\frac 1\alpha-1})^{\frac 12}$.
\endproclaim
\demo{Proof}
For each $f\in L^2(\M,\,\mu)$, we have
$$
\|(\Bbb P^2_\varepsilon-\Pi)f\|^2_2=\langle f,\,(\Bbb C_\varepsilon-
\Pi)f\rangle\leq \|\Bbb C_\varepsilon-\Pi\|_2\|f\|^2_2,
$$
and the Lemma is proven since, for self-adjoint operators, the norm equals the
spectral radius.
\enddemo

Lemma 2.3 implies a precise control on the rate of correlation decay for the random
flow. Certainly the alert reader has noticed that the estimate holds for all $L^2$
functions, while the correlations for the flow will have a fast decay only for
``smooth" observables. Here lies the strength and the weakness of the present
method. Strength because it uses rough, but easy to obtain, estimates. Weakness because
such estimates are likely to be non optimal.

\vskip1cm
\subhead \S3 Estimating the Kernel
\endsubhead

Here we prove Theorem 2.1. Actually, it would be easier to follow the argument by
simply drawing few pictures. Unfortunately, such pictures are clearer to the
one who draws them than to the one who merely looks. Hence, here I provide the
algebra+analysis and I strongly invite the reader to draw her/his own pictures.

Let us consider the set 
$$
\aligned
\Omega_{xy}=\{(z_1,\,z_2,\,z_3)\in\M^3\;|&\; d_\ve( x,\,\phi_{-t}z_1)\leq\delta_0;\;
d_\ve(z_1,\,z_2)\leq\delta_0;\;\\
&d_\ve(z_3,\,z_2)\leq\delta_0;\;d_\ve(y,\,\phi_{-t}z_3)\leq\delta_0\},
\endaligned
$$
where, through this section, $t=c_1\log\ve^{-1}$ ($c_1$ will be chosen later, just
after lemma 3.1).

There exists $c_2>0$ such that\footnote{See Lemma I.1  in appendix I.}

$$
C_{t,\,\ve}(\phi_{-t} x,\,\phi_{-t} y)\geq
\int_{\Omega_{xy}}d\mu_3(z_1,\,z_2,\,z_3)\frac {c_2}{\mu(B^\ve_{\delta_0}(x))
\mu(B^\ve_{\delta_0}(y))\mu(B^\ve_{\delta_0}(z_1))^2};
$$

where $\mu_i$ is the product measure on $\M^i$.

Next, let 
$$
\widetilde\Omega_{xy}=\left\{(z_1,\,z_3)\in\M^2\;|\;d_\ve(x,\,\phi_{-t}z_1)
\leq\delta_0;\;   d_\ve(z_1,\,z_3)\leq\frac{\delta_0}2;\;d_\ve(y,\,\phi_{-t}z_3)
\leq\delta_0\right\}.
$$
Clearly,
$$
\aligned
C_{t,\,\ve}(\phi_{-t}x,\,\phi_{-t}y)
&\geq c_2\int_{\widetilde\Omega_{xy}}d\mu_2(z_1,\,z_3)
\frac{\mu\left(\{z_2\in\M\;|\;d_\ve(z_2,\,z_1)\leq\frac{\delta_0}2\}\right)}
{\mu(B^\ve_{\delta_0}(x))\mu(B^\ve_{\delta_0}(y))\mu(B^\ve_{\delta_0}(z_1))^2}\\
&\geq c_3\int_{\widetilde\Omega_{xy}}d\mu_2(z_1,\,z_3)
\frac1{\mu(B^\ve_{\delta_0}(x))\mu(B^\ve_{\delta_0}(y))
\mu(B^\ve_{\delta_0}(z_1))},
\endaligned
$$
where, again, we have used lemma I.1 of appendix I.

Let $U$ be a neighborhood containing a C-frame $(W_1,\,W_2,\,W_3)$ of size
$\frac{\delta_0}{8}$.\footnote{See appendix III for the definition and 
properties of C-frames.} Let
$W^{u}_\ve(x)$ be the piece of strong unstable manifold of $x$ contained in
$B^\ve_{\delta_0}(x)$. 
If $(z_1,\,z_3)\in\widetilde\Omega_{xy}$ then $z_1$ must be in a small
neighborhood of $\phi_{t}W^{u}_\ve(x)$ and  $z_3$ must be in
a small neighborhood of $\phi_{t}W^{u}_\ve(y)$.

\proclaim{Lemma 3.1} There exists $L(\delta_0)>0$ such that each piece of strong
unstable manifold with diameter larger than $L$ intersects properly\footnote{\rm See
appendix III for the definition of ``proper" intersection.} the given C-frame.
\endproclaim
\demo{Proof}
Here we use the fact that smooth hyperbolic contact flows are mixing \cite{KB}.
Let $(W_1,\,W_2,\,W_3)$, $W_1\cap W_3=\{\bar x\}$, $W_2\cap W_3=\{\bar y\}$ be the
considered C-frame. We will discuss a proper intersection near $\bar x$, the same
arguments hold near $\bar y$. Consider a covering
$\Cal P$ of
$\M$ made of elements approximately rectangular  (that is with faces almost parallel
to the weakly stable and unstable  directions) such that there exists an element
$P_0$ which is contained in a ball of radius $\delta_0^{3/\alpha}/100$ and contains
a ball of radius $\delta_0^{4/\alpha}$ around
$\bar x$. Clearly, if an  unstable manifold intersects $P_0$ and is sufficiently large
then it intersects properly the C-frame. Consider a piece of strong unstable 
manifold $W$. Because of the mixing property there exists $T>0$ such that
$\phi_TP\cap P'\neq\emptyset$ for all $P,\,P'\in\Cal P$. Consider $\phi_{-
T}W$ and assume it crosses completely one element $P$ of the covering\footnote{By this
we mean that $\partial P\cap\phi_{-T}W$ is contained in the faces of $P$ almost
parallel to the weak stable directions.} (this  will always be the case if
$\phi_{-T}W$, and hence $W$, has sufficiently  large diameter, provided we have
chosen the covering with sufficient  overlapping, see \cite{LW} for similar
constructions), then $W\cap P_0\neq 
\emptyset$.
\enddemo

It is now clear that the constant $c_1$ must be chosen such that the minimal
expansion along a piece of trajectory of length $t-n_\ve$ is at least
$\ve^{-1}\delta_0^{-1}L$, whereby $\phi_tW^u_\ve(x)$ and
$\phi_tW^u_\ve(y)$ have diameter larger than $L$ and therefore intersect properly the
C-frame at least once, because of lemma 3.1.

We can then write
$\phi_{t}W^{u}_\ve(x)=\sum_{i=1}^{k_x} W_i(x)$ and
$\phi_{t}W^{u}_\ve(y)=\sum_{i=1}^{k_y} W_i(y)$ where $W_i(x)\cap
C_1\neq\emptyset$ and $W_i(y)\cap C_2\neq\emptyset$, $W_i(x)$ and $W_i(y)$ intersect 
properly the C-frame, and the diameter of
$W_i(x)$ and $W_i(y)$ is contained in the interval $[L,\,c^*L]$ (where $c^*$ depends
only on $(\M,\,\phi_t)$ and $L$).\footnote{Remember that
$C_1=B_{(\delta_0/8)^{3/\alpha}}\cap W^{0s}(\bar x)$ and
$C_2=B_{(\delta_0/8)^{3/\alpha}}\cap W^{0s}(\bar y)$.}

Given any two $W_i(x)$, $W_j(y)$ we know (cfr. theorem III.5) that there exists
$W^s_{ij}$  such that  $W^s_{ij}\cap W_i(x)\neq\emptyset$, 
$W^s_{ij}\cap W_j(y)\neq\emptyset$.  In fact, both intersections consist
of  only one point; let
$\{\xi_{ij}(x)\}\equiv W^s_{ij}\cap W_i(x)$ and 
$\{\xi_{ij}(y)\}\equiv W^s_{ij}\cap W_j(y)$. Moreover, the distance (on $W^s_{ij}$), 
between $\xi_{ij}(x)$ and $\xi_{ij}(y)$ is shorter, in the Riemannian metrics, than
$\frac{\delta_0}8$ (that is, $(W_i(x),\,W_j(y),\,W^s_{ij})$ form a C-frame of size
less than $\frac{\delta_0}2$).

In addition, we can partition $\phi_tB^\ve_{\delta_0}(x)$ by sets 
$\{B_i(x)\}_{i=1}^{k_x}$ and $\phi_tB^\ve_{\delta_0}(y)$ by sets
$\{B_i(y)\}_{i=1}^{k_y}$ such that $B_i(x)$ contains $W_i(x)$, and the analogous
requirement holds for $B_j(y)$.\footnote{Think to, and draw, $B_i(x)$, $B_j(y)$ as
long, distorted, cylinders around $W_i(x)$ and $W_j(y)$, respectively.}

>From appendix III (lemma III.3) we know that, for each $\xi_{ij}(x)$, there exists a
set $\Sigma_{ij}\subset W^u(\xi_{ij}(x))$ such that, for each $z\in\Sigma_{ij}$,
$W^s(z)$ intersects $W^u(\xi_{ij}(y))$ and, calling $\Sigma^\ve_{ij}$ its 
$c_4\delta_0^{\frac 1\alpha}\ve^{\frac 1\alpha}$-neighborhood (for some $c_4$
sufficiently small) in
$W^u(\xi_{ij}(x))$, $\Sigma^\ve_{ij}$ has measure\footnote{Here we mean the measure
$\mu$ restricted to $W^u(\xi_{ij}(x))$.} larger than $c_5\delta_0^{\frac
1\alpha}\ve^{\frac 1\alpha}$. Thus, setting $\widetilde\Sigma_{ij}^{\ve}\equiv
\bigcup\limits_{z\in\Sigma^\ve_{ij}}W^{0s}_{\frac \ve 2}(z)$, it follows
that, if $z_1\in\widetilde\Sigma_{ij}^\ve$, then $\emptyset\neq W^s(z_1)\cap
W^{0u}(\xi_{ij}(y))\equiv\{\tilde z\}$; $d_\ve(z_1,\,\tilde z)\leq \frac {\delta_0}4$
(hence, $B^\ve_{\frac {\delta_0}4}(\tilde z)\subset B^\ve_{\frac{\delta_0}2}(z_1)$)
and there exists $\tau<\frac{\ve\delta_0} 8$ such that $\phi_\tau\tilde z\in
W^{0s}(\xi_{ij}(y))$ (due to the H\"older continuity of the stable foliation). Hence
$$
\widetilde{\Omega}_{xy}\supset\left\{ (z_1,\,z_3)\in\M^2\;|\;
z_1\in\phi_tB^\ve_{\delta_0}(x);\;z_3\in\phi_tB^\ve_{\delta_0}(y);\;
z_1\in\widetilde\Sigma_{ij}^{\ve};\; z_3\in
B^\ve_{\frac{\delta_0}4}(\tilde z)
\right\},
$$
and we can write 
$$
C_{t,\,\ve}(\phi_{-t}x,\,\phi_{-t}y)
\geq c_3\sum_{ij}\int_{B_i(x)\cap {\widetilde\Sigma}^\ve_{ij}}d\mu(z_1)
\frac{\mu(B_j(y)\cap B^\ve_{\frac{\delta_0}4}(\tilde z))}
{\mu(B^\ve_{\delta_0}(x))\mu(B^\ve_{\delta_0}(y))\mu(B^\ve_{\delta_0}(z_1))}.
$$

To conclude other two estimates are needed.

\proclaim{Lemma 3.2} For each $z_1\in\widetilde\Sigma^{\ve}_{ij}\cap B_i(x)$, holds
$$
\mu(B^\ve_{\frac{\delta_0}4}(\tilde z)\cap B_j(y))\geq
c_5\delta_0^{-2d}\ve^{-1}\mu(B^\ve_{\delta_0}(z_1))\mu(B_j(y)).
$$
In addition,
$$
\mu(\widetilde\Sigma^{\ve}_{ij}\cap B_i(x))
\geq c_6\ve^{\frac 1\alpha}\mu(B_i(x)).
$$
\endproclaim

Dear reader, if you have drawn a picture now you will ``see" lemma 3.2 just by
looking at your picture; at any rate, the non-believer can find the proof in appendix
IV.

Accordingly,
$$
\aligned
C_{t,\,\ve}(\phi_{-t}x,\,\phi_{-t}y)\geq&\sum_{ij}
c_{16}\int_{\widetilde\Sigma^{\ve}_{ij}\cap
B_i(x)}d\mu(z_1)\frac{\mu(B_j(y))}{\delta_0^{2d}\ve\mu(B^\ve_{\delta_0}(x))
\mu(B^\ve_{\delta_0}(y))}\\
\geq&c_{17}\delta_0^{-2d}\ve^{\frac 1\alpha -1}\sum_{ij}\frac{\mu(B_i(x))
\mu(B_j(y))}{\mu(B^\ve_{\delta_0}(x))\mu(B^\ve_{\delta_0}(y))}\\
=&c_{17}\delta_0^{-2d}\ve^{\frac 1\alpha-1}\equiv\gamma\ve^{\frac 1\alpha -1}.
\endaligned
$$

\vskip1cm
\subhead \S 4 Decay of correlations
\endsubhead

In this section we will see that the results collected up to now yield a deep
knowledge on the behavior of the correlations of the flow $\phi_t$.

Let us define the following H\"older norms:
$$
\aligned
\|f\|_{u,\beta}&=\sup\Sb x\in\Mp;\;y\in W^u(x)\\d_\ve(x,\,y)\leq \delta_0\endSb
\frac{|f(x)-f(y)|}{d_\ve(x,\,y)^\beta}\\
\|f\|_{s,\beta}&=\sup\Sb x\in\Mp;\;y\in W^s(x)\\d(x,\,y)\leq \delta_0\endSb
\frac{|f(x)-f(y)|}{d(x,\,y)^\beta}\\
\|f\|_{0,\beta}&=\sup\Sb x\in\Mp\\ \tau\in[-\delta_0,\,\delta_0]\endSb
\frac{|f(x)-f(\phi_\tau x)|}{\tau^\beta}\\
\|f\|_{0s,\beta}&=\|f\|_{s,\beta}+\|f\|_{0,\beta}
\endaligned
$$


The basic approximation, without which all the present approach would be useless, is
contained in the following Proposition.

\proclaim{Proposition 4.1} For each $f,\,g\in C^{(1)}(\M)$ holds:
$$
\aligned
\bigg| \int_\Mp f g\circ\phi_{-2k n_\ve}-\int_\Mp &f(\Bbb P_\ve)^{2k}g\bigg|
\leq kc_{18}\big\{\|g\|_1\|f\|_{s,1}\\
&+\|g\|_1\|f\|_\infty+\|f\|_{01}\|g\|_\infty+\|g\|_{u,\beta}\|f\|_1\big\}.
\endaligned
$$
\endproclaim
\demo{Proof}
Define $C^u_\beta(\M)=\{f\in
C^{(0)}(\M)\;|\;\|f\|_\infty+\|f\|_{u,\beta}<\infty\}$ and $C^{0s}(\M)=\{f\in
C^{(0)}(\M)\;|\;\|f\|_\infty+\|f\|_{0s,1}<\infty\}$. Clearly,
$\|f\circ\phi_t\|_{s,\beta}\leq \lambda^{\beta t}\|f\|_{s,\beta}$; $\|f\circ
\phi_t\|_{0,\beta}=\|f\|_{0,\beta}$, hence $\phi_t:C^{0s}(\M)\to C^{0s}(\M)$.
Moreover, $\P_\ve: C^{u}_\beta(\M)\to C^u_\beta(\M)$ due to the following lemma.

\proclaim{Lemma 4.2} For each $\rho<1$, $\beta<\alpha$ there exists $c_{19}>0$ such
that, for
$\delta_0$ sufficiently small, holds
$$
\|\P_\ve f\|_{u,\,\beta}\leq
c_{19}\left(\ve^\rho\|f\|_\infty+\ve^{2\beta}\|f\|_{u,\beta}\right).
$$
\endproclaim
\demo{Proof}
>From appendix I (at the very end) we know that, choosing $\delta_0$ sufficiently
small,
$$
\|\varphi\|_{u,\,\beta}\leq c_{20}\ve^\rho.
$$
Using lemma I.3 we can compute
$$
\aligned
\|\P_\ve f\|_{u,\beta}&=\|\Bbb A_\ve f\circ\phi_{-t_\ve}\|_{u,\beta}\\
&\leq c_{21}\ve^\rho\|f\|_\infty+(1+c_{21}\delta)
\|(1+\delta\varphi)f\circ\phi_{-t_\ve}\|_{u,\beta}\\
&\leq c_{22}\ve^\rho\|f\|_\infty+(1+c_{22}\delta)
\|f\circ\phi_{-t_\ve}\|_{u,\beta}\\
&\leq c_{19}\ve^\rho\|f\|_\infty+c_{19}\ve^{2\beta}
\|f\|_{u,\beta}
\endaligned
$$
\enddemo

\proclaim{Lemma 4.3}For each $f_1\in C^{0s}(\M)$ and $f_2\in C^u(\M)$ hold
$$
\aligned
\left|\int_\Mp f_1f_2-\int_\Mp f_1\Bbb A_\ve f_2
\right|&\leq c_{24}\ve^\rho\|f_1\|_\infty\|f_2\|_1\\
&+c_{24}(\|f_1\|_{s,1}\|f_1\|_1+\ve\|f_1\|_{0,1}\|f_2\|_1
+\|f_2\|_{u,\beta}\|f_1\|_\infty).
\endaligned
$$
\endproclaim
\demo{Proof}

The result follows by direct computation:\footnote{Given a
couple of points $x,\, y\in\Mp$, sufficiently close, it exists only one point that
belongs to $W^{0s}(x)\cap W^u(y)$, let us designate it by $[x,\,y]$. If we consider
the set $\Omega_{\delta_0}^*=\{(x,\,y)\in\Mp^2\;|\; 
d_\ve(x,\,y)\leq {\delta_0}\}$, then, for
${\delta_0}$ sufficiently small, we can define the function
$\Psi:\Omega_{\delta_0}^*\to\Mp^2$ by
$$
\Psi(x,\,y)=([x,\,y],\,[y,\,x])\equiv (\xi,\,\eta).
$$
In appendix v it is shown that the Jacobian of $\Psi$ differs from $1$ by $\Cal
O(\ve^{3\alpha})$; while
$$
\mu_2(\Psi(\Omega_{\delta_0}^*)\Delta\Omega_{\delta_0}^*)\leq \hbox{const}
\ve\mu_2(\Omega_{\delta_0}^*).
$$}
$$
\aligned
\int_\Mp f_1&\Bbb A_\ve f=\int_{\Omega_{\delta_0}^*}d\mu_2(x,\,y) 
\frac{f_1(x)(1+{\delta_0}\varphi(x))(1+{\delta_0}\varphi(y))f_2(y)}
{\mu(B^\ve_{\delta_0}(x))^{\frac 12}\mu(B^\ve_{\delta_0}(y))^{\frac 12}}\\
&=\int_{\Psi(\Omega^*_{\delta_0})} d\mu_2(\xi,\,\eta)\frac{f_1([\xi,\,\eta])
(1+{\delta_0}\varphi([\xi,\,\eta]))
(1+{\delta_0}\varphi([\eta,\,\xi]))f_2([\eta,\,\xi])}
{\mu(B^\ve_{\delta_0}([\xi,\,\eta]))^{\frac 12}
\mu(B^\ve_{\delta_0}([\eta,\,\xi]))^{\frac 12}}\\
&\quad+\Cal O(\ve^{3\alpha}\|f_1\|_\infty \|f_2\|_1)\\
&=\int_{\Omega^*_{\delta_0}} d\mu_2(\xi,\,\eta)\frac{f_1(\xi)(1+{\delta_0}
\varphi(\eta))(1+{\delta_0}\varphi(\xi))f_2(\xi)}
{\mu(B^\ve_{\delta_0}(\xi))^{\frac 12}\mu(B^\ve_{\delta_0}(\eta))^{\frac 12}}\\
&\quad+\Cal O(\ve^{\rho}\|f_1\|_\infty \|f_2\|_1)+\Cal O(\|f_1\|_{s,1}\|f_2\|_1
+\ve\|f_1\|_{0,1}\|f_2\|_1+\|f_2\|_{u,\beta}\|f_1\|_1).
\endaligned
$$
\enddemo

Then,
$$
\aligned
\bigg|\int_{\Mp}fg\circ\phi_{-2kn_\ve}-&
\int_{\Mp}f\Bbb P^{2k}_\ve g\bigg|
\leq\sum_{i=0}^{k-1}
\left|\int_{\Mp}f\circ\phi_{2(k-i)n_\ve}\Bbb P_\ve^{2i} g\right.\\
&\left.-\int_{\Mp}f\circ\phi_{2(k-i-1)n_\ve}\Bbb P^{2(i+1)}_\ve g\right| .
\endaligned
$$

Using lemma 4.3 we get
$$
\aligned
\left|\int_{\Mp}fg\circ\phi_{-2kn_\ve}\right.&\left.-
\int_{\Mp}f\Bbb P^{2k}_\ve g\right|
\leq kc_{18}\big\{\|f\|_{s,1}\|g\|_1\\
&+\ve^\rho\left[(\|f\|_\infty\|g\|_1+\|f\|_{0,1}\|g\|_1
+\|g\|_{u,\beta}\|f\|_1)\right]\big\}. 
\endaligned
$$
\enddemo

In conclusion, for each $f,\,g\in C^{(1)}(\M)$, $\int_{\Mp}g=0$, we have
$$
\aligned
\left|\int_{\Mp}f g\circ\phi_{-t}\right|
&\leq\left|\int_{\Mp}f\circ\phi_{n_\ve}g\circ\phi_{-t+n_\ve}-
\int_{\Mp}f\circ\phi_{n_\ve}\Bbb P^{[\frac t{n_\ve}]-1}
g\circ\phi_{-t+[\frac t{n_\ve}]n_\ve+n_\ve}\right|\\
&+\left|\int_{\Mp}f\circ\phi_{n_\ve}\Bbb P^{[\frac t{n_\ve}]-1}
g\circ\phi_{-t+[\frac t{n_\ve}]n_\ve+n_\ve}\right|\\
&\leq \frac t{n_\ve}\ve^\rho
c_{25}(\|g\|_{u,\beta}\|f\|_\infty+\|f\|_{0s}\|g\|_\infty)+\|f\|_\infty\|g\|_\infty
e^{-\gamma\ve^{\frac 1\alpha-1}\frac t{2n_\ve}}.
\endaligned
$$
If $\alpha=1$, then $\gamma_\ve=\gamma$ does not depend on $\ve$ and
it suffices to choose
$\ve=e^{-\sqrt{\gamma t}}$ to conclude  the argument obtaining
$$
\left|\int_{\Mp}f g\circ\phi_{-t}\right|\leq
c_{20}\sqrt{t}e^{-\sqrt{\gamma
t}}\left(\|g\|_{u,\beta}\|f\|_\infty+\|f\|_{0s,1}\|g\|_\infty
+\|f\|_\infty\|g\|_\infty\right), 
$$
which proves theorem A (the result is easily extended to H\"older 
observables via an approximation argument).

If $\alpha<1$ then the choice $\ve=bt^{-\frac\alpha{1-\alpha}}
(\log t)^{\frac {2\alpha}{1-\alpha}}$, for some $b$ sufficiently large, yields
$$
\left|\int_{\Mp}f g\circ\phi_{-t}\right|\leq
c_{20}(1+|t|)^{-\frac{(\rho+1)\alpha-1}{1-\alpha}}\left(\|g\|_{u,\beta}\|f\|_\infty
+\|f\|_{0s,1}\|g\|_\infty+\|f\|_\infty\|g\|_\infty\right) , 
$$
which proves theorem B.

\vskip1cm
\subhead Appendix I (Averages)
\endsubhead
\vskip.7cm

In this appendix we prove that there exists a special average satisfying the
requirement stated in section 2 (that is $\Bbb A_\ve^\varphi1=1$).

In the following we will use $c_i$  to designate any constant that depends only on 
$(\M,\,\phi_1,\,\mu)$.\footnote{The $c_i$ in this appendix have no relation with the
constants in the main text or in other appendices bearing the same name.}

First we need some informations on the measure of the balls in the $d_\ve$ metrics.

\proclaim{Lemma I.1} There exists $\delta_0>0$, $c_0>1$ and $\theta\in(0,\,1)$, such
that for each $\delta\leq\delta_0$; $\ve$ sufficiently small; $x,\,y\in \M$  
$$
\e{-c_0d_\ve(x,\,y)}\leq\frac{\mu(B^\ve_\delta(x))}{\mu(B^\ve_\delta(y))}
\leq\e{c_0d_\ve(x,\,y)}
$$
and, if $y\in W^{u}(x)$,
$$
\e{-c_0\ve\delta^{-1}
d_\ve(x,\,y)}\leq\frac{\mu(B^\ve_\delta(x))}{\mu(B^\ve_\delta(y))}
\leq\e{c_0 \ve \delta^{-1}d_\ve(x,\,y)};
$$
in addition,\footnote{{\rm Given two sets $A$ and $B$ we will use $A\Delta B$ to
designate the symmetric difference, i.e. $A\Delta B=(A\cup B)\backslash(A\cup B)$.}}
$$
\frac{\mu(B^\ve_\delta(x)\Delta B^\ve_\delta(y))}{\mu(B^\ve_\delta(x))}
\leq c_0\delta^{-1}d_\ve(x,\,y),
$$
and
$$
\frac{\mu(B^\ve_{c_0^{-1}\delta}(x))}{\mu(B^\ve_\delta(x))}\geq \theta.
$$
\endproclaim

The proof of the above Lemma is the content of appendix II.


Let us  get to the point: a little algebra shows that the equation $\Bbb
A_\ve^\varphi1=1$ is equivalent to
$$
\varphi=\frac 12(\Id-\Bbb B^\ve_\delta)\varphi-\frac 12
h_\delta^\ve -\frac\delta 2H_\delta^\ve(\varphi)\equiv \Bbb L^\ve_\delta\varphi
\tag I.1
$$
where
$$
\aligned
&\Bbb B_\delta^\ve f(x)
  =\int_{B^\ve_\delta(x)}
  \frac{f(y)}{\mu(B^\ve_\delta(x))^{\frac 12}\mu(B^\ve_\delta(y))^{\frac12}}
d\mu(y);\\
&h_\delta^\ve(x)=\delta^{-1}[\Bbb B^\ve_\delta 1-1] ;\\
&H_\delta^\ve(f)=f\Bbb B^\ve_\delta f+f h_\delta^\ve .
\endaligned
$$
Note that\footnote{The following estimates rest on Lemma I.1.}
$$
\aligned
&\|h^\ve_{\delta}\|_\infty\leq c_0;\\
&\|H_\delta^\ve(f)\|_\infty\leq c_0(\|f\|_\infty^2+\|f\|_\infty) .
\endaligned
$$

Let us define, for each $f\in C^{(0)}(\M)$, the norm
$$
\|f\|_*=\sup\Sb{x,\,y\in\Mp}\\
d_\ve(x,\,y)\leq\delta_0\endSb\frac{|f(x)-f(y)|}{d_\ve(x,\,y)}.
$$

>From now on I will suppress all the
subscripts and superscripts involving $\ve$ and $\delta$ provided it can be done
without creating confusion.

The most relevant fact about the operator $\Bbb B$ is contained in the following.

\proclaim{Lemma I.2} There exists $\theta_*<1$, independent on $\ve$ and
$\delta\leq \delta_0$,  such that, viewing $\Bbb B$ as an operator on $\Cal
C^{(0)}(\M)$,
$$
\|(\Id-\Bbb B)^2\|_\infty\leq 4\theta_*.
$$
\endproclaim
\demo{Proof}
For each $f\in C^{(0)}(\M)$ and $x\in\M$ holds\footnote{Notice that lemma I.1 implies 
$\|\Bbb B f\|_*\leq c_0\delta^{-1}\|f\|_\infty$.}
$$
\aligned
|(\Id-\Bbb B)&\Bbb B f(x)|\leq\frac 1{\mu(B^\ve_\delta(x))}\int_{B^\ve_\delta(x)}
d\mu(y)|\Bbb B f(x)-\Bbb B f(y)|+ c_1\delta\|f\|_\infty\\
&\leq \frac 1{\mu(B^\ve_\delta(x))}\int_{B^\ve_\delta(x)}d\mu(y)\min\{2\|\Bbb B
f\|_\infty;\,
\|\Bbb B f\|_*d_\ve(x,\,y)\}+ c_1\delta\|f\|_\infty\\
&\leq \frac {1+c_1\delta}{\mu(B^\ve_\delta(x))}\int_{B^\ve_\delta(x)}d\mu(y)
\min\{2\|f\|_\infty;\,c_0\delta^{-1}\|f\|_\infty
d_\ve(x,\,y)\}+c_1\delta\|f\|_\infty\\ 
&\leq\frac{(1+c_1\delta)\|f\|_\infty}{\mu(B^\ve_\delta(x))}
\left\{\mu(B^\ve_{c_0^{-1}\delta}(x))+2\mu(B^\ve_\delta(x)\backslash
B^\ve_{c_0^{-1}\delta}(x))\right\}+c_1\delta\|f\|_\infty\\ 
&\leq (2-\theta)(1+c_1\delta)\|f\|_\infty<2\|f\|_\infty.
\endaligned
$$
Then, by choosing $\delta_0$ sufficiently small,
$$
\|(\Id-\Bbb B)^2\|_\infty\leq \|\Id-\Bbb B\|_\infty+\|(\Id-\Bbb B)\Bbb B
\|_\infty\leq 2+(2-\theta)(1+c_1\delta)\leq 4\theta_* .
$$
\enddemo

By using the above estimates follows immediately that the operator
$\Bbb L^2$ is a contraction on $K_a=\{f\in C^{(0)}(\M)\;|\;\|f\|_\infty\leq a\}$ for
$\delta_0$ sufficiently small and $a$ appropriately chosen. Therefore, equation (I.1) has a
unique solution in $K_a$. Since (I.1) has a solution,
we can rewrite it in the equivalent form
$$
\varphi=-h_\delta^\ve-\Bbb B_\delta^\ve\varphi-\delta H_\delta^\ve(\varphi)
\tag I.2
$$

For it holds\footnote{The proof is simple and relies on Lemma I.1. Here is the idea:
$$
\aligned
|\Bbb B^\ve_\delta1(x)-\Bbb B^\ve_\delta1(y)|&=\frac {\left|\int_{B^\ve_\delta(x)}
\frac {d\mu(z)}{\mu(B^\ve_\delta(z))^{\frac 12}}-\int_{B^\ve_\delta(y)}\frac
{d\mu(z)}{\mu(B^\ve_\delta(z))^{\frac 12}}\right|}{\mu(B^\ve_\delta(x))^{\frac 12}}
+\left|\frac{\mu(B^\ve_\delta(y))^{\frac 12}}{\mu(B^\ve_\delta(x))^{\frac 12}}
-1\right|\Bbb B1(y)\\ 
&\leq\frac{\Bigg|\int_{B^\ve_\delta(x)}d\mu(z)
\left[1-\frac{\mu(B^\ve_\delta(x))^{\frac 12}}
{\mu(B^\ve_\delta(z))^{\frac 12}}\right]
-\int_{B^\ve_\delta(y)}d\mu(z)\left[1-\frac{\mu(B^\ve_\delta(x))^{\frac 12}}
{\mu(B^\ve_\delta(z))^{\frac 12}}\right]\Bigg|}{\mu(B^\ve_\delta(x))^{\frac 12}}\\ 
&\ \ \ +\frac{|\mu(B^\ve_\delta(x))-\mu(B^\ve_\delta(y))|}{\mu(B^\ve_\delta(x))}
+e^{c_0\delta}c_0\delta d_\ve(x,\,y)\\  
&\leq \frac{c_0\delta\mu(B^\ve_\delta(x)\Delta B^\ve_\delta(y))}
{\mu(B^\ve_\delta(x))}+2e^{c_0\delta}c_0\delta d_\ve(x,\,y)\\ 
&\leq(c_0^2+2e^{c_0\delta}c_0\delta) d_\ve(x,\,y)\leq c_2 d_\ve(x,\,y).
\endaligned
$$
}
$$
\aligned
&\|h^\ve_{\delta}\|_*\leq c_0\delta^{-1};\\
&\|H_\delta^\ve(f)\|_*\leq c_0\delta^{-1}(\|f\|_\infty^2+\|f\|_\infty)+c_0\|f\|_*(
\|f\|_\infty+c_0) ;
\endaligned
$$
equation (I.2) immediately implies that $\varphi\in\Cal C^{(1)}(\M)$ and
$$
\|\varphi\|_*\leq b\delta^{-1}
$$
for some $b$ independent on $\delta$ and $\ve$.


The only property of the function $\varphi$ that is used in the paper and 
it is not a consequence of what we have done so far is the estimate on the unstable
derivative used in section four. To gain this further inside more work is needed.


Given a function $f$ let us define a H\"older norm in the
unstable direction by
$$
\|f\|_{u,\,\beta}\equiv\sup_{x\in\Mp}\sup\Sb y\in W^u(x)\\d_\ve(x,\,y)\leq 
\delta_0\endSb\frac{|f(x)-f(y)|}{d_\ve(x,\,y)^\beta}.
$$

\proclaim{Lemma I.3} For each $f\in\Cal C^{(1)}(\M)$, $\beta\leq \alpha$,
$$
\|\Bbb B f\|_{u,\,\beta}\leq
(1+c_3\delta)\|f\|_{u,\,\beta}+c_3\ve\delta^{-\beta}\|f\|_\infty.
$$
\endproclaim

The proof of Lemma I.3 is contained in appendix II. As an immediate consequence  
$$
\aligned
\|h\|_{u,\,\alpha}&\leq c_4\ve\delta^{-\alpha}\\
\|\Bbb L f\|_{u,\,\alpha}&\leq
(1+c_5(\|f\|_\infty+1)\delta)\|f\|_{u,\,\alpha}
+c_5(1+\|f\|_\infty)\delta\ve\delta^{-\alpha} .
\endaligned
$$

Iterating the second of the above inequalities yields, remembering that 
$\|f\|_\infty\leq a$ implies $\|\Bbb L^i f\|_\infty\leq a$,
$$
\|{\Bbb L}^n f\|_{u,\,\alpha}\leq e^{nc_6(1+a)\delta}
\left(\| f\|_{u,\,\alpha}+c_7\ve\delta^{-1-\alpha}\right).
$$
We have already seen that $\Bbb L^2$ is a contraction, hence by defining 
$\tilde\varphi_n=\Bbb L^n (0)$ we have a sequence converging exponentially fast, in
the sup norm, to $\varphi$. Whence, for each
$x\in\M$ and $y\in W^{u}(x)$, setting $n_{xy}=c_8\ln[\ve
d_\ve(x,\,y)^{\alpha-\epsilon}]^{-1}$ (where $\epsilon$ will be chosen later),
$$
\frac{|\varphi(x)-\varphi(y)|}{d_\ve(x,\,y)^{\alpha-\epsilon}}\leq
\frac{|\tilde\varphi_{n_{xy}}(x)-\tilde\varphi_{n_{xy}}(y)|}{d_\ve(x,\,y)^{\alpha-\epsilon}}
+c_9\ve
$$
where $c_8$ has been chosen as to obtain the last term in the above expression.

Accordingly, for each $x,\, y\in\M$, $d_\ve(x,\,y)\leq\delta_0$,
$$
\aligned
\frac{|\varphi(x)-\varphi(y)|}{d_\ve(x,\,y)^{\alpha-\epsilon}}&\leq
d_\ve(x,\,y)^\epsilon\|\Bbb L^{n_{xy}}(0)\|_{u,\,\alpha}+ c_9\ve\\
&\leq d_\ve(x,\,y)^\epsilon e^{c_{10}\delta\ln
\ve^{-1}d_\ve(x,\,y)^{-1}}c_7\ve\delta^{-1-\alpha} +c_9\ve\\
&\leq
d_\ve(x,\,y)^{\epsilon-c_{10}\delta}\ve^{1-c_{10}\delta}\delta^{-1-\alpha}
+c_9\ve\leq c_{11}\ve^{1-2\epsilon} ,
\endaligned
$$
where we have chosen $\epsilon=c_{10}\delta$ and $\ve$ sufficiently
small($\ve<\delta^{\frac{1+\alpha}{c_{10}\delta}}$). That is
$$
\|\varphi\|_{u,\,\alpha-c_{10}\delta}\leq c\ve^{1-c_{12}\delta}.
$$


\vskip1cm
\subhead Appendix II (Balls)
\endsubhead

This appendix is dedicated to the task of proving Lemma I.1 and Lemma I.3.
The reader be advised that throughout this appendix ``$c$" will stand for a
generic constant (not always the same) depending on the manifold $\M$,
on $\phi_1$ but $\underline{\hbox{not}}$ on $\varepsilon$ or $\delta$.

To address the problem it is convenient to start by introducing appropriate 
foliations.\footnote{The following computations may look a bit cumbersome; yet, on
the one hand, I do not know of a simpler approach; on the other hand, the idea to
compute volumes by foliating them is a  very efficient and very old one \cite{Arc}.} 


Given a point $x\in\M$ and a ball $B_{\delta_*}(x)$ (the ball, centered at $x$, of
radius $\delta_*$ in the metric $d$) we will call a smooth foliation $\F=\{\F\}$ of a
neighborhood of
$x$ ``unstable-like" if it consists of $(d+1)$-dimensional  manifolds uniformly
transversal to $W^s(x)$ and ``stable-like" if it consists of $d$-dimensional 
manifolds uniformly transversal to $W^{0u}(x)$. In addition, we will call an
unstable-like foliation $\F$  ``adapted to $B_{\delta_*}(x)$" if the neighborhood
foliated by the leaves of $\F$ contains the neighborhood
$B^\ve_{4\delta_*}(x)$ and if, for each
$\gamma\in \F$, $\gamma\cap\partial B_{\delta_*}(x)\neq\emptyset$ implies
$\gamma\subset\partial B_{\delta_*}(x)$;\footnote{Note that this implies that the 
curvature of the manifolds in $\F$ cannot be less than $\delta^{-1}_*$, yet it can be
arranged so that it is less than
$c\delta^{-1}_*$, also it can be arranged for the conditional measure on the fibers
to have smooth densities (both along the same fiber and from fiber to fiber), with
respect to the restriction of
$\mu$ to the fibers.} moreover, if $\gamma\cap W^s(x)\neq\emptyset$,
then $\partial(B^\ve_{4\delta_*}(x)\cap \gamma)\subset
\partial B^\ve_{4\delta_*}(x)$ and $\gamma\subset B^\ve_{8\delta_*}(x)$. Analogously,
we will call a stable-like foliation $\F$  ``adapted to
$B_{\delta_*}(x)$" if the neighborhood foliated by the leaves of $\F$ contains 
the neighborhood $\phi_{n_\ve}B^\ve_{4\delta_*}(\phi_{-n_\ve}x)$ and if for each
$\gamma\in \F$ $\gamma\cap\partial
\phi_{n_\ve}B_\delta(\phi_{-n_\ve}x)\neq\emptyset$ implies
$\gamma\subset\partial \phi_{n_\ve}B_\delta(\phi_{-n_\ve}x)$; moreover, if $\gamma\cap
W^{0u}(x)\neq\emptyset$, 
then $\partial(\phi_{n_\ve}B^\ve_{4\delta_*}(\phi_{-n_\ve}x)\cap \gamma)
\subset\partial\phi_{n_\ve}B^\ve_{4\delta_*}(\phi_{-n_\ve}x)$ and
$\gamma\subset \phi_{n_\ve}B^\ve_{8\delta_*}(\phi_{-n_\ve}x)$.

\demo{Proof of Lemma I.1}

Clearly it suffices to discuss the situation $d_\ve(x,\,y)\ll\delta$. We will consider
three cases: i) $y\in W^{u}(x)$ ii) $y=\phi_\tau
x$ iii) $y\in W^{0s}(x)$. The first inequality of Lemma
I.1 follows from a trivial combination of similar inequalities for the cases (i),
(ii) and (iii). Here we will discuss explicitly only (i), since it yields the second
inequality of Lemma I.1 (stronger than what it is needed to prove the first
inequality) and the proofs of (ii), (iii) follow along the same lines with only
minor, and obvious, changes.

The idea is to use a stable-like foliation $\F$ adapted to
$B_{\delta\ve}(\phi_{n_\ve}x)$. Given such a foliation we can obtain an associated
stable-like foliation adapted to $B_{\delta\ve}(\phi_{n_\ve}y)$ by introducing a
diffeomorphism 
$\psi:\M\to \M$ such that
$\psi(B_{\delta\ve}(\phi_{n_\ve}x))=B_{\delta\ve}(\phi_{n_\ve}y)$.  A moment
reflection shows that it can be arranged so that
$\psi$, restricted to the set $\phi_{n_\ve}B^\ve_{16\delta}(x)$, is
measure preserving, and
$$
\|D\psi-\Id\|_\infty\leq c d(\phi_{n_\ve}x,\,\phi_{n_\ve}y)\leq c\ve d_\ve(x,\,y).
$$
The foliation $\F'=\psi\F$ is the wanted foliation of
$B_{\delta\ve}(\phi_{n_\ve}y)$.

The next step is to induce foliations in a neighborhood of $x$. Let us define
$\F_*=\phi_{-n_\ve}\F$; $\F'_*=\phi_{-n_\ve}\F'$;
$\widetilde\F_*=\{\gamma_*\cap B_\delta(x)\}_{\gamma_*\in\F_*}$;
$\widetilde\F'_*=\{\gamma_*'\cap B_\delta(y)\}_{\gamma'_*\in\F_*}$ and
$\widetilde\F=\phi_{n_\ve}\widetilde\F_*$,
$\widetilde\F'=\phi_{n_\ve}\widetilde\F'_*$.\footnote{It is essential to remark that
$\gamma_*\cap B^\ve_\delta(x)$, where $\gamma_*\in\F_*$, is the 
connected component of $\gamma_*\cap B_\delta(x)$ intersecting
$\phi_{-n_\ve}W^{0u}_{\ve\delta}(\phi_{n_\ve}x)$ and the same holds for
$\gamma_*'\in\F_*'$.} By construction
$\widetilde \F_*$ is a foliation of $B^\ve_\delta(x)$ and $\widetilde
\F'_*$ is a foliation of $B^\ve_\delta(y)$. Moreover $\widetilde
\psi=\phi_{-n_\ve}\circ\psi\circ\phi_{n_\ve}$ establishes a correspondence among
the leaves of $\F_*$ and $\F'_*$. Unfortunately $\widetilde \psi$ is not suitable
to establish a correspondence point by point since, due to the possibility of
different expansion rates (in higher dimensions), it is not possible to bound
$d(z,\,\widetilde\psi (z))$ effectively in terms of $d_\ve(x,\,y)$. It is therefore
more convenient to establish, between corresponding leaves, a pointwise
correspondence by using an unstable-like foliation.\footnote{To use the weak-unstable
foliation itself it is not advisable since such a foliation, in more than three
dimensions, may be only H\"older continuous. In fact, we will be forced to use it in
the proof of Lemma I.3, due to that we will be able to obtain only H\"older
estimates.} To be concrete one can introduce a coordinate system in a neighborhood of
$x$ (e.g. the one induced by the exponential map) and consider a foliation $\F^u$
made of planes (with respect to the Euclidean structure of the chart) parallel to the
weak-unstable direction at $x$.  One can then define
$\theta_{\gamma_*}:\gamma_*\to\gamma_*'\equiv\widetilde\psi(\gamma_*)$ by
$\theta_{\gamma_*}(z)\equiv\gamma^u(z)\cap\gamma_*'$ (where $\gamma^u(z)\in\F^u$ is
the leaf containing $z$). 

The first important property of the above construction is that for each
$z\in\tilde\gamma_*\in\widetilde\F_*$ 
$$
d(z,\,\theta_{\gamma_*}(z))\leq c\ve d_\ve(x,\,y).
\tag II.1
$$
To see this define $z_1=\phi_{n_\ve}(z)$ and $w_1=W^{0u}(z_1)\cap\gamma'$
($\gamma'=\phi_{n_\ve}(\gamma'_*)$). Since the foliations $\F$ and $\F'$ are smooth
and uniformly transversal to the unstable direction, it follows, by construction, that
$d(z_1,\,w_1)\leq c\ve d_\ve(x,\,y)$. Accordingly, setting $w=\phi_{-n_\ve}(w_1)$, 
$d(z,\,w)\leq c\ve d_\ve(x,\,y)$. Since the stable and unstable foliation are, at
worst, H\"older we have  $d(w,\,\theta_{\gamma_*}(z))\leq c d(z,\,w)$ as well, which
implies (II.1).

Let us define $\hat B(x)=\{z\in\M\;|\;\gamma_*(z)\cap
B^\ve_\delta(x)\neq\emptyset\}$ and $\hat B(y)=\{z\in\M\;|\;\gamma'_*(z)\cap
B^\ve_\delta(y)\neq\emptyset\}$.\footnote{Note that, by construction, both $\hat
B(x)$ and $\hat B(y)$ consist of a ``pile" of fibers of diameter larger than
$4\delta$.} By construction $\hat B(x)$ is $\F_*$ measurable and
$\hat B(y)$ is $\F'_*$ measurable (i.e., they are measurable with respect to the
$\sigma$-algebra associated to the partition of $\M$ induced by $\F_*$ and $\F_*'$,
respectively); moreover,
$B^\ve_\delta(x)=\hat B(x)\cap B_\delta(x)$, $B^\ve_\delta(y)=\hat B(y)\cap
B_\delta(y)$ and $\widetilde\psi \hat B(x)=\hat B(y)$.

The second key fact is \footnote{The point here is that the tangent spaces
${\Cal T}_z\gamma_*$ and ${\Cal T}_{z'}\gamma'_*$  ($z'=\theta_{\gamma_*}(z)$) form
an ``angle" (e.g. in the above mentioned chart) smaller than $c\ve^2 d_\ve(x,\,y)$.
The proof of this fact is left to the reader but can be obtained straightforwardly by
the same estimates that allow to prove the H\"older continuity of the stable
distribution (see \cite{KH}). Once such an estimate on the tangent spaces is
obtained the result follows by direct computation.}
$$
\sup_{\xi\in \hat B(y)}|1-
\frac{d(\theta_{\gamma_*})_*\nu_{\gamma_*}}{d\nu_{\gamma'_*}}(\xi)|\leq c
\ve d_\ve(x,\,y),
\tag II.2
$$
where $\nu_\gamma$ is the measure on $\gamma$ obtained by restricting the
Riemannian metric to $\gamma$ and, in general, for a map $\theta$ and a measure
$\nu$ we use the notation $\theta_*\nu(f)\equiv \nu(f\circ\theta)$.

We are now in position to compute.\footnote{Notice that we use indifferently the same
symbol for a partition and for the associated $\sigma$-algebra; hence
$\mu(\cdot\;|\;\F)$ means the conditional measure with respect to the
$\sigma$-algebra associated to the partition of $\Mp$ induced by the 
foliation $\F$ (that is $\F\cup\{\cup_{\gamma\in\F}\gamma\}^c$). 
Also, here and in the following we
will use $\chi_A$ to indicate the characteristic function of the set $A$.}
$$
\aligned
\mu(B^\ve_\delta(y))=&\mu(\mu(\chi_{B^\ve_\delta(y)}\;|\;\F'_*))=
\mu(\chi_{\hat B(y)}\mu(\chi_{B_\delta(y)}\;|\;\F'_*))\\
=&\mu(\chi_{\hat B(x)}\mu(\chi_{B_\delta(y)}\;|\;\F'_*)\circ\tilde\psi).
\endaligned
\tag II.3
$$
If we adopt the convention of calling $\gamma(z)$ the fiber of a generic smooth
foliation $\F$ containing the point $z$ ($\gamma(z)=\emptyset$ if $z$ is not covered
by $\F$), then it is well known that for each integrable function $f$
$$
\mu(f\;|\;\F)(z)=\int_{\gamma(z)}f(\xi)J_{\gamma(z)}(\xi)
d\nu_{\gamma(z)}(\xi) .
\tag II.4
$$

A direct computation yields\footnote{Just remember that for each foliation
$\F$, $\gamma\in \F$,  each function $f$ and diffeomorphism $G: \Mp\to\Mp$, letting
$\F'=G\F$, $\gamma'=G\gamma$, (see \cite{KS})
$$
\aligned
&\int_{\gamma}fd\nu_{\gamma}=\int_{\gamma'} f\circ G^{-1}
|\det(D G^{-1}|_{\gamma'})|d\nu_{\gamma'};\\
&\mu(f|\F)\circ G^{-1}=\frac{\mu(f\circ G^{-1}|\det D G^{-1}||\F')} 
{\mu(|\det D G^{-1}||\F')},
\endaligned
$$
and use the obvious change of variable in the representation (II.4).}
$$
J_{\gamma(z)}(\xi)=J_{\gamma_*(\phi_{-n_\ve}z)}
(\phi_{-n_\ve}\xi)\left|\det\left(d_{(\phi_{-n_\ve}\xi)}
\phi_{n_\ve}\big|_{\gamma_*(\phi_{-n_\ve}z)}\right)\right|^{-1}.
\tag II.5
$$
In addition, by construction of $\F$, there exists a smooth function
$j:\M\to\Bbb R$ such that
$$
J_{\gamma}(\xi)=\frac{j(\xi)}{Z_{\gamma}};\quad
Z_\gamma=\int_\gamma j(\xi)d\nu_\gamma(\xi).
$$
Then, for the foliation $\F'$, holds
$$
J_{\gamma'}(\xi)=\frac{j'(\xi)}{Z_{\gamma}};\quad
j'(\xi)=j(\psi^{-1}(\xi))|\det(D_{\psi^{-1}\xi}\psi|_{\gamma})|^{-1}.
$$
Let $\widetilde\gamma_*\in\widetilde\F_*$ and
$\widetilde\gamma'_*=\widetilde\psi\widetilde\gamma_*\in\widetilde\F'_*$, then
$$
\left|1-\frac{\nu_{\gamma_*}(\widetilde\gamma_*)}
{\nu_{\gamma'_*}(\widetilde\gamma'_*)}\right|\leq c\ve d_\ve(x,\,y).
$$
To see this, on the one hand, calling
$\bar\gamma=\theta_{\gamma_*}(\tilde\gamma_*)\subset\gamma'_*$,
(II.2) implies
$$
\left|1-\frac{\nu_{\gamma_*}(\widetilde\gamma_*)}
{\nu_{\gamma'_*}(\bar\gamma)}\right|\leq c\ve d_\ve(x,\,y).
\tag II.6
$$
On the other hand, geometric considerations yield immediately
$$
\frac{\nu_{\gamma'_*}(\widetilde\gamma'_*\Delta\bar\gamma)}
{\nu_{\gamma'_*}(\widetilde\gamma'_*)}\leq c\ve d_\ve(x,\,y).
\tag II.7
$$
The usual distortion estimates \cite{Ma} imply
$$
\left|1-\frac{\nu_{\gamma}(\widetilde\gamma)}
{\nu_{\gamma'}(\widetilde\gamma')}\right|\leq c\ve d_\ve(x,\,y).
\tag II.8
$$
In addition, for each $\xi\in\widetilde\gamma$ and $\xi'\in\widetilde\gamma'$,
$d(\xi,\,\xi')\leq\ve d_\ve(x,\,y)$,
$$
\left|1-\frac{J_{\gamma}(\xi)}{J_{\gamma'}(\xi')}\right|\leq c\ve
d_\ve(x,\,y).
\tag II.9
$$
Using again distortion estimates, the definitions of $J,\,J'$ and (II.5), we have
$$
\left|1-\frac{J_{\gamma_*}(\xi_*)}{J_{\gamma'_*}(\xi'_*)}\right|\leq c\ve
d_\ve(x,\,y),
$$
for each $\xi_*\in\widetilde\gamma_*$ and $\xi'_*\in\widetilde\gamma'_*$,
$d(\phi_t\xi_*,\,\phi_t\xi'_*)\leq c\ve d_\ve(x,\,y)$ for each $t\in[0,\,n_\ve]$.
Further, it is easy to show that both $J_{\gamma_*}$ and $J_{\gamma'_*}$
are uniformly bounded by $c\nu_{\gamma_*(x)}(\tilde\gamma_*(x))^{-1}$.

All the above facts together allow to continue the computation started in (II.3)
$$
\aligned
\mu(\chi_{B_\delta(y)}|\F'_*)(\tilde\psi z)&=\int_{\gamma'_*(\tilde\psi z)}
\chi_{B_\delta(y)}(\xi)J_{\gamma'_*(\tilde\psi z)}(\xi)d\nu _{\gamma'_*(\tilde\psi
z)}(\xi)\\
&=\int_{\tilde\gamma'_*(\tilde\psi z)}J_{\gamma'_*(\tilde\psi z)}(\xi)
d\nu_{\gamma'_*(\tilde\psi z)}(\xi)\\
&=(1+\Cal O(\ve d_\ve(x,\,y)))\int_{\bar\gamma(\tilde\psi z)}J_{\gamma'_*(\tilde\psi
z)}(\xi) d\nu_{\gamma'_*(\tilde\psi z)}(\xi)\\
&=(1+\Cal O(\ve d_\ve(x,\,y)))\int_{\tilde\gamma_*(z)}J_{\gamma'_*(\tilde\psi
z)}(\theta\xi) d\nu_{\gamma_*(z)}(\xi)\\
&=(1+\Cal O(\ve d_\ve(x,\,y)))\int_{\tilde\gamma_*(z)}J_{\gamma_*(z)}(\xi)
d\nu_{\gamma_*(z)}(\xi)\\
&=(1+\Cal O(\ve d_\ve(x,\,y)))\mu(\chi_{B_\delta(x)}|\F_*)(z).
\endaligned
$$

Whence,
$$
\mu(B^\ve_\delta (y))=(1+\Cal O(\ve d_\ve(x,\,y)))\mu(B^\ve_\delta(x)).
\tag II.10
$$

This concludes the proof of case (i). Case (ii) and (iii) can be dealt in the same way,
only in (iii) one must start with an unstable-like foliation $\F$ adapted to
$B_\delta(x)$ and perform the computation in $B_\delta(\phi_{n\ve}x)$.

The first and second inequality of the Lemma are then proven by interchanging the 
role of $x$ and $y$ and thanks to the compactness of $\M$.

The third inequality is much simpler; just notice that\footnote{In fact, if $z\in 
B^\ve_\delta(x)\Delta
B^\ve_\delta(y)$, then there are two possibilities: either $d_\ve(x,\,z)\leq\delta$
and $d_\ve(y,\,z)\geq\delta$ (which implies $\delta\leq
d_\ve(y,\,z)+d_\ve(x,\,y)\leq d_\ve(x,\,y)+\delta$) or $d_\ve(x,\,z)\geq\delta$ and
$d_\ve(y,\,z)\leq\delta$ (which implies $d_\ve(x,\,z)\leq
d_\ve(x,\,y)+d_\ve(y,\,z)\leq d_\ve(x,\,y)+\delta$).}
$$
B^\ve_\delta(x)\Delta
B^\ve_\delta(y)\subset\{z\in\M\;|\;\delta-d_\ve(x,\,y)\leq d_\ve(x,\,z)\leq
\delta+d_\ve(x,\,y)\}.
\tag II.11
$$

That is to say that the set we are interested in is contained in a small neighborhood
of $\partial B^\ve_\delta(x)$ which measure is easily estimated, using the same
arguments employed in deriving (II.6)--(II.9), by a constant times
$\delta^{-1}\mu(B^\ve_\delta(x))d_\ve(x,\,y)$. The fourth inequality is more of the
same.
\enddemo

\demo{Proof of Lemma I.3}

The already obtained estimates on the measure of balls allow us to write for each
$x\in\ M$, $y\in W^{u}(x)$ and $d_\ve(x,\,y)\leq \delta_0$,
$$
\|\Bbb B f(x)-\Bbb B f(y)\|\leq c\ve\delta^{-1} d_\ve(x,\,y)\|f\|_\infty
+\frac{\left|\int_{B^\ve_\delta(x)}\frac
{f(z)}{\mu(B^\ve_\delta(z))^{\frac 12}}-\int_{B^\ve_\delta(y)}\frac
{f(z)}{\mu(B^\ve_\delta(z))^{\frac 12}}\right|}{\mu(B^\ve_\delta(x))^{\frac 12}}.
$$

To proceed we will use again the previously introduced foliations, only now we will
establish the pointwise correspondence among fibers by using the weak-unstable
foliation. Namely, let $\theta^u_{\gamma_*}:\gamma_*\to\gamma_*'$ be defined by
$\theta^u_{\gamma_*}(z)\equiv W^{u}(z)\cap\gamma'_*$. The considerations previously
carried out in the proof of Lemma I.1 can be applied again only remembering that the
function $\theta^u_{\gamma_*}$ is now only H\"older continuous, together with its
Jacobian. 

Thus\footnote{As already remarked the computation carried out in (II.10), and all the
previous relevant inequalities, can be extended to the present context. The only
substantial difference is a consequence of the lack of regularity of the unstable
foliation that allows only the weaker estimate
\cite{Ma}
$$
|1-\frac{d\theta^u_*\nu_{\gamma_*}}{d\nu_{\gamma'_*}}|\leq c
d_\ve(x,\,y)^\alpha\ve^{2\alpha} .
$$
This means that, for each $g\in L^\infty(\Mp)$,
$$
\aligned
\mu(\chi_{B^\ve_\delta(y)}g)&=\mu(\chi_{\hat B(y)}\mu(\chi_{B_\delta(y)}g|\F'_*))
=\mu(\chi_{\hat B(x)}\circ\tilde\psi^{-1}\mu(\chi_{B_\delta(y)}g|\F'_*))\\
&=\mu(\chi_{\hat B(x)}\mu(\chi_{B_\delta(y)}g|\F_*)\circ\tilde\psi)
=(1+\Cal O(\ve^{2\alpha}d_\ve(x,\,y)^\alpha))
\mu(\chi_{\hat B(x)}\mu(\chi_{B_\delta(x)}g\circ\theta^u|\F_*))\\
&=(1+\Cal O(\ve d_\ve(x,\,y)^\alpha))
\mu(\chi_{B^\ve_\delta(x)}g\circ\theta^u),
\endaligned
$$
where, in the last equality, we have assumed $\alpha\geq \frac 12$, which is the
only case in which theorem B yields an interesting result.}
$$
\aligned
\|\Bbb B f(x)-\Bbb B f(y)\|&\leq
c\ve\delta^{-\alpha}d_\ve(x,\,y)^\alpha\|f\|_\infty+\frac
1{\mu(B^\ve_\delta(x))}\bigg|\int_{B^\ve_\delta(x)}\frac {\mu(B^\ve_\delta(x))^{\frac
12}f(z)}{\mu(B^\ve_\delta(z))^{\frac 12}}\\
&\ \ \ -\int_{B^\ve_\delta(x)}\frac {\mu(B^\ve_\delta(x))^{\frac 12}f(\theta^u
z)}{\mu(B^\ve_\delta(\theta^u z))^{\frac 12}}\bigg|\\ 
&\leq c\ve\delta^{-\alpha}d_\ve(x,\,y)^\alpha\|f\|_\infty
+(1+c\delta)\|f\|_{u,\,\alpha}d_\ve(x,\,y)^\alpha,
\endaligned
$$
since
$d_\ve(z,\,\theta^u(z))=\ve^{-1}d(\phi_{n_\ve}(z),\,\phi_{n_\ve}(\theta^u(z))\leq
(1+c\ve)d_\ve(x,\,y)$ (and, at the same time, $d_\ve(z,\,\theta^u(z))\geq
(1-c\ve)d_\ve(x,\,y)$). We finally obtain the wanted inequality:
$$
\|\Bbb B^\ve_\delta f\|_{u,\,\beta}\leq
(1+c\delta)\|f\|_{u,\,\beta}+c\ve\delta^{-\beta}
\|f\|_\infty.
$$
\enddemo

\vskip1cm
\subhead Appendix III (C-frames)
\endsubhead

This appendix is dedicated to the definition and the study of the properties of
C-frames.

The notion of C-frames has been introduced by N. Chernov \cite{Ch} (he called them
H-frames) in the two dimensional case; here we generalize the construction to
arbitrary dimensions. Also, we do not use the notion of ``uniform
non-integrability." Chernov expresses such a condition in terms of the function
$\tau$ defined in the proof of Lemma III.3. His condition reads
$$
c^{-1}\|z\|\|v\|\leq \tau(z,\,v)\leq c\|z\|\|v\|.
$$
For contact flows on three dimensional manifolds (or, more generally, for contact
flows with Lipschitz foliations) it is possible to see that such a  condition is
always satisfied. In the higher dimensional situation the story is different. In fact
it is known that the regularity of $\tau$ is related to the regularity of the
foliation \cite{Ham}, hence Chernov condition cannot hold if the foliation is only
H\"older.

\proclaim{Definition III.1} By C-frame we mean three $d$-dimensional manifolds
$W_i$ such that 
$$
W_1\cap W_3=\{x\};\;W_1\cap W_3=\{y\};
$$ 
where $x$ and $y$ are two different points in $\M$. In addition, $W_1\subset
W^u(x)$ and  $W_2\subset W^u(y)$ while $W_3\subset W^s(x)\cap W^s(y)$.
\endproclaim

We say that a C-frame is of size $\delta$ if the diameter of $W_3$ is between $\frac
\delta 2$ and $\delta$, $\frac \delta 6\leq d(x,\,y)\leq \frac \delta 3$, and
$W_1\subset B_{2\delta(x)}$, $\partial(W_1\cap B_\delta(x))=W_1\cap\partial
B_\delta(x)$; $W_2\subset B_{2\delta(x)}$,
$\partial(W_2\cap B_\delta(y))=W_2\cap\partial B_\delta(y)$.

Let us call $C_1\equiv B_{\delta^{3/\alpha}}(x)\cap W^{0s}(x)$ and  
$C_2\equiv B_{\delta^{3/\alpha}}(y)\cap W^{0s}(y)$.\footnote{Here ``$\alpha$" is the 
H\"older regularity of the distributions $E^u(x)$, $E^s(x)$ (see section one for more
details).}

\proclaim{Definition III.2} We say that a connected piece of unstable manifold $W_*$
intersects properly a C-frame if
$$
(W_*\cap C_1)\cup(W_*\cap C_2)\neq\emptyset,
$$
and $\partial(W_*\cap B_{2\delta}(x))=W_*\cap\partial B_{2\delta}(x)$.
\endproclaim

Let us investigate a bit more the structure of the C-frames.
Consider a sufficiently small neighborhood $U$. For $\delta_0$ small enough, construct
a C-frame of size $\delta_0$ in $U$.

\proclaim{Lemma III.3} Let $\Sigma$ be the set of points $z\in W_1$ such that
$\{W_1,\, W^s(z),\, W_2\}$ form a C-frame. Then there exists $\ve_0,\, c'>0$
such that for each $\ve\leq\ve_0$ the $\ve$-neighborhood of $\Sigma$ in $W_1$ has
measure larger than $\ve c'$.\footnote{{\rm Of course, we mean the measure $\mu$
restricted to $W_1$. In the case $d=1$, Chernov's case, $\Sigma$ would consist of
only one point and the Lemma would be trivial.}}
\endproclaim
\demo{Proof}
In the following we introduce a Riemannian metric in which $E^s(x)$ and $E^u(x)$
are orthogonal and $\e{\cdot}$ sends a neighborhood of $0$, in $E^u(x)$, in a
neighborhood of $x$, in $W^u(x)$ and a neighborhood of $0$, in $E^s(x)$, in a
neighborhood of $x$, in $W^s(x)$. From now on $\e{\cdot}$ will always be referred to
such a metric.
 
Consider $z\in E^s(x)$ such that $\e{z}=y$.
Let $v\in E^u(x)$ and $\tau(z,\,v)$ be the distance, along the flow direction, 
between $W^u(y)$ and $W^s(\e{v})$. By construction $\tau(z,\,0)=0$. Moreover, by
using the contact structure one can obtain the formula \cite{KB}
$$
\tau(z,\,v)=d\omega(z,\,v)-f(z,v)
$$
where $f$ is a continuous function such that $|f(z,v)|= o(\|z\|^2+\|v\|^2)$.
Since in the following $z$ is fixed we will use $f(v)$ to designate $f(z,v)$.
We want to study $\Sigma_*=\{v\in E^u(x)\;|\;\tau(z,\,v)=0\}$. The set
$\Sigma_*$ is then uniquely determined by the equation
$$
d\omega(z,\,v)=f(v) .\tag III.1
$$

It is well known that there exists a $d-1$ dimensional linear space 
$\Bbb V\subset E^u(x)$ such that $d\omega(z,\,w)=0$ for all $w\in\Bbb V$. Let $\bar
w\in  E^u(x)$ be perpendicular to $\Bbb V$ and such that
$d\omega(z,\,\bar w)=\delta_0^2$. Setting $c_1=\frac{d\omega(z,\,\bar
w)}{\|z\|\|\bar w\|}$, we can choose $\delta_0$ so small that $|f(v)|\leq \frac
{c_1}{100}(\|z\|^2+\|v\|^2)$.

For each $\xi\in   E^u(x)$ we will use the decomposition
$\xi=a\bar w+\eta$ with $\eta\in\Bbb V$. Then (III.1) reads
$$
\delta_0^2 a=f(a\bar w+\eta).\tag III.2
$$

We will show that (III.2) has at least one solution for each $\eta\in
B_{\frac{\delta_0} 6}(0)$. To see this define $g_\eta(a)=\delta_0^{-2}f(a\bar
w+\eta)-a$.

\proclaim{Sub-Lemma III.4} For each 
$\eta\in\Bbb V$ with $\|\eta\|\leq \|z\|$, there exists $\bar a\in\Bbb R$, 
$|\bar a| \leq\frac{c_1}{36}$, such that
$$
g_\eta(\bar a)=0.
$$
\endproclaim
\demo{Proof}
There are three possibilities to consider: $g_\eta(0)=0$, $g_\eta(0)>0$,
$g_\eta(0)<0$. If the first possibility occurs, then the Lemma is trivially true.
The other two cases can be treated in complete analogy, we will consider explicitly 
$g_\eta(0)>0$.
By construction $\|z\|\in[\frac{\delta_0}6,\,\frac{\delta_0}3]$, so for each $|a|
\leq\frac{c_1}{36}$
$$
|a|\|\bar w\|\leq \frac{\delta_0}6\leq \|z\|.
$$
Hence $\e{a\bar w+\eta}\in W_1$ and
$$
\aligned
|f(a\bar w+\eta)|&\leq \frac{c_1}{100}\left[ (|a|\|\bar w\|+\|\eta\|-\|z\|)^2+
2\|z\|(|a|\|\bar w\|+\|\eta\|)\right]\\
&\leq \frac{c_1}{100}(3\|z\|^2+\frac{2|a|\delta_0^2}{c_1})\leq
(\frac{c_1}{300}+\frac{|a|}{50})\delta_0^2 .
\endaligned
$$
Therefore, $g_\eta(\frac{c_1}{36})<0$.
Since  $g_\eta$ is a continuous function the lemma is proven.
\enddemo

Next, let $\theta(\eta)$ be the smallest solution provided by the above theorem.
Clearly the set
$$
\Sigma=\{\e{\theta(\eta)\bar w+\eta}\;|\;\eta\in\Bbb
V,\,\|\eta\|\leq\frac{\delta_0} 6\}
$$
is made up of points whose stable manifold intersect the unstable manifold of $y$; 
moreover 
$$
d(x,\,\e{\theta(\eta)\bar w+\eta})\leq\|\theta(\eta)\bar w+\eta\|\leq\delta_0 .
$$
We are left with the task of estimating the measure of and $\ve$-neighborhood of
$\Sigma$. By the exponential map we can reduce it to an estimate in $\Cal T_x
W^u(x)$. Namely there exists $c_0,\,c_1$ such that (by $m(\cdot)$ we mean the
Lebesgue measure in $\Bbb R^d$)
$$
\aligned
\mu_u(\Sigma_\ve)&\geq c_0 m(\{v\in  E^u(x)\;|\;\exists\eta\in\Bbb V:
\|\theta(\eta)\bar w+\eta-v\|\leq c_1\ve\})\\
&\geq c_0 m(\{a\bar w+\eta\in  E^u(x)\;|\;v\in\Bbb V,\,\|\eta\|\leq
\delta_0,\,|\theta(\eta)-a|\leq c_1\ve\})\\
&=c_2\delta^{d-1}\ve.
\endaligned
$$
\enddemo

There exits another way to construct a multitude of C-frames 
near a given one. The next theorem is due to Chernov in the two dimensional case and
very little changes in higher dimensions.

\proclaim{Theorem III.5} Consider a C-frame of size $\delta$ and two pieces of 
unstable manifold $\widetilde W_1$ and $\widetilde W_2$ that intersects $C_1$
and $C_2$, respectively. Then there exists a piece of stable manifold 
$\widetilde W_3$ such that $(\widetilde W_1,\,\widetilde W_2,\,\widetilde W_3)$ form
a C-frame of size larger than $\delta/2$ and smaller than $2\delta$.
\endproclaim
\demo{Proof}
Consider $\hat x=W^{0s}(x)\cap \widetilde W_1$ and $\hat y=W^{0s}(x)\cap 
\widetilde W_2$. Call $\Psi$ the projection, along the stable manifold, of
$\widetilde W_1$ into $W^{0u}(\hat y)$. By construction, the distance between
$\Psi(\hat x)$ and $\hat y$ it is $\Cal O(\delta^3)$ (it follows from the H\"older
continuity of the stable foliation). Due to the possible lack of regularity of the
stable foliation $\Psi(\widetilde W_1)$  may not be a regular manifold, yet (in
analogy with lemma III.3) we can construct an H\"older continuous
chart (namely $\Psi\circ\e{}: E^u(\hat x)\to\Psi(\widetilde W_1)$), also $\Psi
(\widetilde W_1)$ is clearly a codimension one set in $W^{0u}(\hat y)$ which separate
it into two connected components. Lemma III.3 implies that
$\Psi(\widetilde W_1)$ moves in the flow direction by at least $\Cal O(\delta^2)$ so
it will necessarily intersect $W^u(\hat y)$ at some point
$z$. Whereby, we have the C-frame $(\widetilde W_1,\,\widetilde W_2,\,\widetilde
W_3)$, and its size is easily checked.
\enddemo

\vskip1cm
\subhead Appendix IV (Product Sets)
\endsubhead

The proof of  lemma 3.2 is the content of this appendix.

First remember that, by construction, $d(\tilde
z,\,W^u(\xi_{ij}(y))\leq\frac\ve 8 \delta_0$.
If we define $z_*=\{\phi_t\tilde z\}_{t\in[-\ve\delta_0,\,\ve\delta_0]}\cap
W^u(\xi_{ij}(y))$, then $B^\ve_{\frac{\delta_0}8}(z_*)\subset
B^\ve_{\frac{\delta_0}4}(\tilde z)$, and
$z_*\in W_j(y)\subset B_j(y)$, $d_\ve(z_*,\,z_1)\leq \frac 3 8\delta_0$. 

We will obtain the first inequality by using the product structure of $B_i(y)$. To be
more precise
$$
\mu(B^\ve_{\frac{\delta_0}4}(\tilde z)\cap
B_j(y))\geq\mu(B^\ve_{\frac{\delta_0}8}(z_*)\cap B_j(y))=\mu(\phi_{n_\ve}
(B_j(y)\cap B_{\frac{\delta_0}8}(z_*))\cap B_{\frac{\delta_0\ve}8}(\phi_{n_\ve}z_*)).
$$
In analogy with the approach used in appendix II we can consider an unstable like
foliation $\F$ adapted to $B_{\delta_0}(y)$ (see appendix II for definitions). Let
$\hat{\Cal F}=\phi_t\Cal F$ and
$\hat{\Cal F}_*=\{p\cap B_{\delta_0/8}(z_*)\}_{p\in\hat{\Cal F}}$. We can then extend
$\hat{\Cal F}_*$ to a smooth foliation $\F_0$ such that
$B_{\frac{\delta_0}8}(z_*)$ is measurable in the associated $\sigma$-algebra; finally,
we can consider the foliation
${\Cal F}_1=\phi_{n_\ve}\F_0$. 

Since $\phi_{n_\ve}B_j(y)$ is foliated by $\F_1$, which is almost
the weak-unstable foliation, it follows that there  exists $c_7$
such that for each $p\in\F_1$,
$$
c_7^{-1}\geq \frac{\mu(B_{\delta_0\ve/8}(\phi_{n_\ve}z_*)\;|\;
\F_1 )(p)}{\mu(B_{\delta_0\ve/8}(\phi_{n_\ve}z_*)\;|\;
\F_1)(p_*)}\geq c_7
$$
where $p_*\in\F_1$ and $\phi_{n_\ve}z_*\in p_*$.

In addition, by the usual distortion estimates, there exists $c_8$ such that, for
all $\hat p\in\hat\F_*$ and $\hat p_*\in\phi_{-n_\ve}p_*$,
$$
c_8\leq\frac{\mu(B^\ve_{\frac{\delta_0}8}(z_1)\;|\;
\F_0)(\hat p)}{\mu(B^\ve_{\frac{\delta_0}8}(z_1)\;|\;\F_0)(\hat p_*)}
\leq c_8^{-1}.
$$

Thus, 
$$
\mu(B^\ve_{\frac{\delta_0}8}(z_*))=\mu(\chi_{B_{\frac{\delta_0}8}(z_*)}
\mu(B^\ve_{\frac{\delta_0}8}(z_*)|\F_0))\leq c_8
\mu(B^\ve_{\frac{\delta_0}8}(z_*)|\F_0)(p_*)
\mu(B_{\frac{\delta_0}8}(z_*)).
$$

Next, let $\hat B=\bigcup\limits_{p\in\hat\F_*} p\supset
B_j(y)\cap B_{\frac{\delta_0}8}(z_*)$. By construction $\hat B$ is $\F_0$ measurable.
In the same way as before
$$
\mu(B_j(y)\cap B_{\frac{\delta_0}8}(z_*))=\mu(\chi_{\hat B}\mu(B_j(y)|\F_0))\leq
c_9\mu(\hat B)\mu(B_j(y)|\F_0)(p_*)\leq c_{10}\ve\mu(\hat B),
$$
since $B_j(y)\cap p_*$ consists of a strip of width $\ve\delta_0$ (in the flow
direction) in $p_*$ (which is approximately a
$d+1$ dimensional disk of size $\delta_0$) and the conditional measure
$\mu(\cdot|\F_0)$ is uniformly equivalent to the restriction, on the fibers, of the
Riemannian measure (the proof is standard, see appendix II for more details).

The last fact to take into account is that, for each $p\in\F_0$, $p\subset\hat B$,
holds $p\cap B_j(y)\supset p\cap B^\ve_{\frac{\delta_0}8}(z_*)$.

Putting together all the above estimates yields
$$
\aligned
\mu(B_j(y)\cap B^\ve_{\frac{\delta_0}8}(z_*))&\geq
\mu(\chi_{\hat B}\mu(B^\ve_{\frac{\delta_0}8}(z_*)|\F_0))
\geq c_8\mu(\hat B)\mu(B^\ve_{\frac{\delta_0}8}(z_*)|\F_0)(p_*)\\
&\geq c_8^2\mu(\hat B)\mu(B_{\frac{\delta_0}8}(z_*))^{-1}
\mu(B^\ve_{\frac{\delta_0}8}(z_*))\\
&\geq c_{11}\delta_0^{-2d-1}\ve^{-1}\mu(B_j(y)\cap B_{\frac{\delta_0}8}(z_*))
\mu(B^\ve_{\frac{\delta_0}8}(z_*)).
\endaligned
$$

The first inequality of the Lemma follows by using the foliation $\hat{\Cal F}$ and
distortion estimates.\footnote{Just choose $m\in\Bbb N$ such that
$\phi_{-m}B_j(y)$ is contained in a ball of radius $\delta_0$ (this can be achieved
by a fixed $m$, independent of $y$ and $\ve$). It is then easy to compare the measure
of $\phi_{-m}B_j(y)$ with $\phi_{-m}(B_j(y)\cap B_{\frac{\delta_0}8}(z_*))$ by using
the foliation $\phi_{-m}\hat{\Cal F}$. It follows that $\mu(B_j(y)\cap
B_{\frac{\delta_0}8}(z_*))$ is proportional to $\mu(B_j(y))$. Moreover, by Lemma I.1
$\mu(B^\ve_{\frac{\delta_0}8}(z_*))\geq
e^{-c_0\frac{\delta_0}4}\mu(B^\ve_{\frac{\delta_0}8}(z_1))\geq c_{12}
\mu(B^\ve_{\delta_0}(z_1))$.}

The second inequality is obtained in a similar way. Construct the foliations
$\hat\F,\,\hat\F_*,\,\F_0$ as before, but with respect to $x$ instead than to
$y$. Then notice that for each 
$p\in\hat{\Cal F}_*$
$$
\mu(\widetilde\Sigma^\ve_{ij}\;|\;\hat{\Cal F}_*)(p)\geq c_{13}\ve^{\frac 1\alpha +1}
$$
due to the results of appendix III. In addition, $\hat B(x)$ is defined as $\hat
B(y)$,
$$
\mu(\widetilde\Sigma^\ve_{ij}\cap B_i(x))\geq\mu(\chi_{\hat B(x)}
\mu(\widetilde\Sigma^\ve_{ij}\;|\;\F_0))
\geq c_{14}\ve^{\frac 1\alpha+1}\mu(\hat B(x))
\geq c_{15}\ve^{\frac 1\alpha} \mu(B_i(x)).
$$


\vskip1cm
\subhead Appendix V (A change of coordinates)
\endsubhead

In this appendix we study the change of coordinates used in section four to prove 
lemma 4.3.

Given a couple of points $x,\, y\in\M$, sufficiently close, it exists only one point 
that belongs to $W^{0s}(x)\cap W^u(y)$, let us designate it by $[x,\,y]$.
If we consider the set $\Omega_\delta^*=\{(x,\,y)\in\M^2\;|\; d_\ve(x,\,y)\leq
\delta\}$, then, for $\delta$ sufficiently small, we can define the function
$\Psi:\Omega_\delta^*\to\M^2$ by
$$
\Psi(x,\,y)=([x,\,y],\,[y,\,x]).
$$
The above function is used as a change of coordinates in section four.
The properties of $\Psi$ needed in the paper are summarized by the following.

\proclaim{Lemma V.1}There exists a constant $c$ such that\footnote{{\rm By
$\Psi_*(\mu)$  we mean the measure defined by $\Psi_*(\mu)(f)\equiv\mu(f\circ\Psi)$
for each $f\in\Cal C^{(0)}(\Omega^*_\delta)$. Here, again, $\mu_2=\mu\times\mu$.}}
$$
\sup_{\Omega^*_\delta}
|1-\frac{d\Psi_*(\mu_2)}{d\mu_2}(x,\,y)|\leq c\ve^{3\alpha} ;
$$
and, for each $x,\,y\in\M$, $d_\ve(x,\,y)\in[\frac \delta 2,\,2\delta]$, 
$$
e^{-c\ve d_\ve(x,\,y)}\leq
\frac{d_\ve(x,\,y)}{d_\ve([x,\,y],\,[y,\,x])}\leq e^{c\ve d_\ve(x,\,y)}.
$$
\endproclaim
\demo{Proof}
To study the first inequality it is convenient to introduce a new system of 
coordinates. Namely, given $\xi\in\M$ we can define $\Theta_\xi$ from  $\Bbb
R^{2d+1}$ to a neighborhood of $\xi$, by introducing local coordinates $\zeta$ on
$W^{0s}(\xi)$ and $\eta$ on $W^u(\xi)$ and then defining
$\Theta_\xi(\zeta,\,\eta)=W^u(\zeta)\cap W^{0s}(\eta)$. By the general theory of
Anosov systems \cite{Ma}, \cite{HK} follows that $\Theta_\xi$ it is H\"older
continuous and that
$$
\frac{d(\Theta_\xi)_*\nu}{d\mu}=\rho_\xi,
$$
where $\nu$ is Lebesgue measure $\Bbb R^{2d+1}$, and
$\rho_\xi$ is H\"older continuous as well.

For each $x\in\M$ we can use such coordinates. Let
$\Theta_2(\zeta_1,\,\zeta_2)\equiv (\Theta_x(\zeta_1),\,\Theta_x(\zeta_2))$;
$\rho_2(\zeta_1,\,\zeta_2)\equiv\rho_x(\zeta_1)\rho_x(\zeta_2)$; then, for each
$f\in C^{(0)}(\M^2)$, $\text{supp }f\subset\Omega^*_\delta$
$$
\Psi_*\mu_2(f)=\mu_2(f\circ\Psi)=\Theta_{2*}\nu_2(\rho_2^{-1}f\circ\Psi).
$$
Since, for each $\zeta_1,\,\zeta_2\in\Omega^*_\delta$,
$$
\aligned
\frac{\rho_2(\Psi(\zeta_1,\,\zeta_2))}{\rho_2(\zeta_1,\,\zeta_2)}&=\frac
{\rho_x([\zeta_1,\,\zeta_2])\rho_x([\zeta_2,\,\zeta_1])}{\rho_x(\zeta_1)
\rho_x(\zeta_2)}\\
&=1+\Cal O(d(\zeta_1,\,[\zeta_2,\,\zeta_1])^\alpha)+
d(\zeta_2,\,[\zeta_1,\,\zeta_2])^\alpha))=1+\Cal O(\ve^{3\alpha}\delta^\alpha),
\endaligned
$$
where we have used the contraction in the unstable direction to get $3\alpha$.

Therefore,\footnote{Note that $\Theta_2^{-1}\circ\Psi\circ\Theta_2$ is linear and its
Jacobian is $1$.}
$$
\aligned
\Psi_*\mu_2(f)&=\nu_2((\rho_2^{-1}f)\circ\Theta_2
\circ(\Theta_2^{-1}\circ\Psi\circ\Theta_2))+\Cal O(\ve^{3\alpha}\|f\|_\infty)\\
&=\nu_2((\rho_2^{-1}f)\circ\Theta_2)+\Cal O(\ve^{3\alpha}\|f\|_\infty)
=\mu_2(f)+\Cal O(\ve^{3\alpha}\|f\|_\infty).
\endaligned
$$

Let us consider now the second inequality: here it is convenient to divide the 
argument in different cases. 

If $d(x,\,y)\geq \ve^{-1}d(\phi_{n_\ve}x,\,\phi_{n_\ve}y)$, then
$$
\aligned
d(x,\,y)&\leq d(x,\,[y,\,x])+d([y,\,x],\,[x,\,y])+d([x,\,y],\,y)\\
&\leq c\ve^2\delta+d([y,\,x],\,[x,\,y]).
\endaligned
$$
In an analogous way we get
$$
|d(x,\,y)-d([y,\,x],\,[x,\,y])|\leq c\ve^2\delta.
$$

If $d(x,\,y)\leq \ve^{-1}d(\phi_{n_\ve}x,\,\phi_{n_\ve}y)$, then
$$
|d(\phi_{n_\ve}x,\,\phi_{n_\ve}y)-d(\phi_{n_\ve}[y,\,x],\,\phi_{n_\ve}[x,\,y])|
\leq c\ve^2\delta.
$$
A fast look at the different cases yields the wanted result.
\enddemo

Finally we have.
\proclaim{Lemma V.2} 
$$
\mu_2(\Psi(\Omega^*_{\delta_0})\Delta\Omega^*_{\delta_0})\leq c\ve
\mu_2(\Omega^*_{\delta_0}).
$$
\endproclaim
\demo{Proof}
Due to Lemma V.1 
$$
\Psi(\Omega^*_{\delta_0})\Delta\Omega^*_{\delta_0}
\subset\{(\xi,\,\eta)\in\M^2\;|\;d_\ve(\xi,\,\eta)\in[\delta_0(1-c\ve),\,
\delta_0(1+c\ve)]\}.
$$
In addition,
$$
\mu_2(\Omega^*_{\delta_0})=\int_{\Mp}d\mu(x)\mu(B^\ve_{\delta_0}(x))
$$
and
$$
\mu_2(\Psi(\Omega^*_{\delta_0})\Delta\Omega^*_{\delta_0})\leq
\int_{\Mp}d\mu(x)\mu(B^\ve_{\delta_0}(x)\backslash B^\ve_{\delta_0(1-c\ve)}
(x)).
$$
Accordingly the result follows in the same way discussed in Lemma I.1.
\enddemo


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\enddocument