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\begin{document}
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\title[Rigidity of Conformal Anosov systems]{Further rigidity properties of
conformal Anosov systems}
\author{R. de la Llave}
\address{Dept. of Mathematics, Univ. of Texas, Austin TX 78712}
\email[R. de la Llave]{llave@math.utexas.edu}
\keywords{Rigidity, conformal structures, Anosov systems,
cohomology equations }
\subjclass{}
\thanks{}
\begin{abstract}
We consider systems that have some hyperbolicity behavior
and which preserve conformal structures on the stable
and unstable bundles. We show that two such systems that are topologically
conjugate are smoothly conjugate. This is somewhat more general
than a conjecture of the author in 2002. Related results
have also been obtained by B. Kalinin and V. Sadovskaia.
\end{abstract}
\maketitle
\section{Introduction}
The goal of this paper is to show rigidity properties of
systems with some hyperbolicity behavior
which preserve conformal structures along stable and unstable
foliations.
The main result is valid only for Anosov diffeomorphisms
and flows, but much of
the other results are valid under less restrictive assumptions
on the hyperbolicity of the system. The only obstruction
to obtaining the full result for partially hyperbolic
systems is a technical regularity result, which we
believe is true and within reach.
We discuss these generalizations in
Remark \ref{rem:further}.
The main result of this paper is the following
\begin{theorem}\label{main}
Let $f$, $g$ be $C^r$ transitive Anosov diffeomorphisms
(resp. $X_t$, $Y_t$ be transitive Anosov flows) of
a compact manifold $M$ of dimension $d$.
$r \in \nat + [0,1)$, $r > 1$.
Assume that
\begin{itemize}
\item[i)] $\dim( E^s), \dim(E^u) \ge 2$.
\item[ii)] There exists a homeomorphism
$h$ such that
\begin{equation} \label{conjugacy}
f \circ h = h \circ g.
\end{equation}
resp.
\begin{equation} \label{conjugacy2}
X_t \circ h = h \circ Y_t.
\end{equation}
\item[iii)] Each of the diffeomorphisms $f,g$
preserve $C^1$ conformal structures on the stable and unstable bundles.
\end{itemize}
Then
\begin{equation}
h \in C^{r'}
\end{equation}
where $r = \begin{cases} r - \epsilon & \text{if}\quad r \in \nat \\
r &\text{if} \quad r \in \nat + (0,1) \end{cases}$.
\end{theorem}
\begin{remark}
We note that the assumption iii) can be weakened in two directions:
a) In \cite{Llave02}, it is shown that if $M= \torus^d$,
-- or has some triviality properties of the tangent bundle --
the fact that $f$ preserves a smooth conformal structure
on the stable and unstable bundles
holds when $f$ is sufficiently $C^1$-close to a linear
automorphism and
\begin{equation}\label{periodic}
f^n(x) = x \implies Df^n(x)|_{E^s_x} = (\mu^s_x)^n \Id_{E^s_x}, \quad
Df^n(x)|_{E^u_x} = (\mu^u_x)^n \Id_{E^u_x}
\end{equation}
Note that in \eqref{periodic} we do not assume that the $\mu_x$
are independent of $x$.
One of the consequences of Theorem \ref{main} is that
if \eqref{periodic} holds and $f$ is $C^1$-close to
a linear automorphism of the torus, then, we have that
indeed that $\mu^s_x$ are independent of $x$.
b) In \cite{KalininLS02}, it is shown that the preservation of
a continuous conformal structure is implied by the preservation
of an structure that belongs to $L^p$ with $p$ high enough
and that $f$ is $C^1$ close to a measure preserving
diffeomorphism. Similarly, in
\cite{KalininLS02}, one shows that the regularity of
the conformal structure can be bootstrapped till it is
roughly $r -1$.
c) The papers \cite{Sadovskaia02}, \cite{KalininS02} do
not require that the systems are transitive. One of the first
steps of those papers is to show that Anosov diffeomorphisms
that preserve conformal structures are volume preserving.
If one used the arguments from these papers for the first part
of the arguments, the assumption that the systems are transitive
could be eliminated for diffeomorphisms.
We have chosen not to follow this route
to have more arguments documented in the literature.
We point, however that we do not use much transitivity. The main
place is in the application of the theory of SRB measures, which includes
transitivity in its assumptions. It is possible that this use of
transitivity
could be eliminated by considering the arguments in the basins of
attraction of the different ergodic components.
\end{remark}
\begin{remark} \label{onedimensional}
The Theorem \ref{main}
remains valid if Assumption i) is
replaced by
$\dim E^s = 1$,
$\dim E^u \ge 2$,
and, whenever $g^n(x) = x$, hence $f^n\circ h(x) = h(x) $
we have
\[
\Spec( Dg^n(x)|_{E^s,x} ) = \Spec( (D f^n) \circ h(x) |_{E^s_{h(x)}})
\]
We fact that $\dim E^s = 1$ implies that the Anosov system is
transitive. See \cite{Newhouse70}. Therefore, for diffeomorphisms,
the assumption of transitivity becomes redundant if one modifies
Assumption i) as indicated. For flows, the assumption of transitivity is
not implied by the manifolds being one-dimensional, \cite{FranksW80}.
We develop the proof of Theorem \ref{main} under the assumptions
indicated in this remark in Section \ref{sec:onedimensional}
\end{remark}
The main technique of this paper is a sequence
of arguments bootstrapping the regularity of the conjugacies.
Many of the arguments have a similar flavor. We derive formulas
for the increments of the derivatives along the (un)stable directions.
These formulas will hold in a rather weak sense. Nevertheless, by
examining them, we will discover that the derivatives -- or the increments --
hold in a somewhat stronger sense. Eventually, we will reach a formula
so strong that can be differentiated term by term.
Therefore, we establish that the conjugacy is smooth when restricted
to the stable and unstable foliations.
Once we have that the conjugacy is smooth along the stable and unstable
foliations, we can appeal to a regularity result
(Lemma 2.5 of \cite{LlaveMM86}) that states that functions
smooth along the stable and unstable manifold are actually smooth.
As is usual in bootstrap arguments, the more complicated parts
happen at the beginning, when we are allowed to assume less
properties of the function. We have indicated the
milestones that we will be accomplishing as titles of the
subsections.
Results very similar to Theorem \ref{main}
have been obtained by more geometric methods
in the remarkable papers \cite{Sadovskaia02}, \cite{KalininS02}.
The geometric argument in
\cite{Sadovskaia02}, \cite{KalininS02}
has several advantages with the argument presented
here. In particular, in the Anosov case, it shows that the
stable and unstable foliations are smooth
and that they preserve an affine connection.
Hence, the papers \cite{Sadovskaia02}, \cite{KalininS02}
can use the results of
\cite{Kanai93}, \cite{BenoistFL92}, \cite{Yue96}, \cite{BessonCG95}
to get, just from the preservation of the
conformal structures along the stable and unstable manifolds
-- an assumption that looks almost local -- global information
about the manifold.
Besides, being a different method, the present
method has the advantage that it
applies for flows and, we hope, it can apply to
partially hyperbolic systems provided that a regularity result
which we formally conjecture is true.
\section{Proof of Theorem~ \ref{main}}
The proof presented here consists of a sequence
of arguments that bootstrap the regularity of the conjugacy
$h$ using
that it satisfies that equation \eqref{conjugacy}
(or \eqref{conjugacy2}.)
As in \cite{LlaveMM86}, we will bootstrap the
regularity along the stable and unstable manifolds
separately and then, we will appeal to regularity results
to conclude that a function which is regular restricted
to the stable and unstable manifolds is regular in the usual sense.
We emphasize that the fact that we can bootstrap the regularity
along the stable and unstable foliations is independent of
the fact that they are transversal. Hence, this part of
the argument can be carried for partially hyperbolic
systems or for Anosov flows. (We use at certain stages the
fact that the system admits a SRB measure).
In the case of Anosov flows, however, since the conjugacies
have to be smooth along the direction of the flow too, we
are in a situation where we can apply the regularity results
of \cite{LlaveMM86} and obtain the desired conclusions.
We conjecture that the required regularity result
-- an analogue of Lemma 2.5 of \cite{LlaveMM86} --
is true for partially hyperbolic systems
satisfying the accessibility property. If such conjecture
was true, then Theorem \ref{main} will go
through for partially hyperbolic systems that
have accessibility and SRB measures.
We think that it would be extremely interesting to integrate
the arguments presented here with the arguments of
\cite{Sadovskaia02}, \cite{KalininS02}.
In the following, we will present the proof with the notation of
Anosov diffeomorphisms, but we call attention to the fact that
the bootstrap of regularity only uses that the manifolds are
stable and unstable. Hence it applies just the same for
stable manifolds of a flow or a partially hyperbolic system.
We will denote the stable manifold of the point $x \in M$
under the map $f$ as $W^{s,(f)}_x$ but we
will suppress the $(f)$ if it is understood from
the context and will not lead to confusion.
\subsection{$h$ is conformal along the stable and unstable leaves}
The first step of the proof of Theorem~\ref{main} is:
\begin{lemma}\label{isconformal}
$h: W_x^{s, (g)} \rightarrow W_{h(x)}^{s, (f)} $
is quasi-conformal when
$W_x^{s, (g)}, W_{h(x)}^{s, (f)} $
are endowed with the conformal structures preserved by $g,f$
respectively.
\end{lemma}
Of course, there is an analogous result for the unstable manifolds,
which can be deduced from Lemma~\ref{isconformal} by considering
the inverse mappings (a direct proof only requires trivial modifications).
Lemma~\ref{isconformal}
was already established in \cite{Llave02} Section 3.2.
We recall that main idea of the argument was to show
that we can construct a map $k$ such that
$k: W_x^{s, (g)} \rightarrow W_{h(x)}^{s, (f)} $,
$k |_{W_x^{s (g)}}$ has uniformly bounded first derivatives
and
$d_s( h(x), k(x) )$ is bounded, were we denote by $d_s $
the distance measured along $W_x^{s, (g)}$.
Such map -- note that we do not require that $k$ is a local
homeomorphism -- is constructed by convolving with suitable
kernels after taking local coordinates.
By an argument quite similar to that of
\cite{Shub69}, we have:
\begin{equation}\label{scattering}
h = \lim_{n \to \infty} f^n \circ k \circ g^{-n}
\end{equation}
and the limit is reached uniformly.
Using the conformality of $f,g$ and the fact that $k$ is uniformly
quasi-conformal in all the leaves, we obtain that the distortion of
$f^n \circ k \circ g^{-n}|_{W_x^{s (g)}}$ is uniformly bounded.
Since the limit $h$ is a homeomorphism, standard results in
the theory of quasi-conformal maps
show that $h$ is quasi-conformal.
(See \cite{Vaisala71} Theorem 37.2, Corollary 37.3 p. 125 ff.)
We emphasize that the argument to establish
Lemma \ref{isconformal} does not use at all the fact that the
eigenvalues at corresponding periodic points are the same.
We only use that there is a conformal structure present.
\qed
\subsection{ The stable Jacobian of the conjugacy is bounded }\label{isbounded}
We will introduce the notation $J^{s,(g)}[h]$ to denote the
Jacobian of $h$ restricted to the stable leaves. Similar
notations will be used for the Jacobians of other maps,
for the restrictions to the unstable foliations. If the map whose
foliation is understood, we will suppress the superindex in parenthesis
indicating it.
So far, we have shown that $J^{s, (g)}[h]$ is well defined, but we have not
yet established any control on its size.
It is not difficult to
show that $J^{s, (g)}[h]$ is uniformly integrable on balls of size one
of the stable leaves. It suffices to observe
that
\begin{equation}\label{uniformint}
\int_{B_r \cap W^{s,(g)}_x} J^{s, (g)}[h](x) \,dx =
| h(B_r \cap W^{s,(g)}|
\end{equation}
where by $| h(B_r \cap W^{s,(g)}| $ we mean the Riemannian
volume with respect to the metric in $W^{s,(f)}_{h(x)}$.
For subsequent arguments, it will be more useful to
obtain control of $\log J^s[h] (x)$.
In order to obtain control on $\log J^s[h] (x)$, we will use an
argument similar to that in \cite{LivsicS72}. This
argument uses the theory of SRB measures. We refer to
\cite{Sinai72}, \cite{Bowen75}, \cite{Ruelle76}, \cite{BowenR75}
for the classical theory for Anosov or Axiom A systems,
and to \cite{PesinS82} for results for partially hyperbolic
systems.
We point out that in this section, the fact that the systems are
conformal will not play any role. The arguments apply
just as well to any Anosov system such that the conjugacy is
absolutely continuous along stable manifolds.
We denote by $\mu^{+, (g)}$, $\mu^{-,(g)}$, the
future and past SRB measures for the map $g$.
If the map $g$ is understood, we will suppress it from the notation.
One of the characterizations of $\mu^{-,(g)}$ is that
it has a density along the stable directions.
By the fact that $h$ is absolutely continuous
on the stable manifolds and uniformly integrable
on balls of size one, we obtain that
map $h_* \mu^{-,(g)}$ is a measure which is invariant under $f$
and that it has densities along the stable manifolds of $f$.
Hence, we obtain that
\begin{equation}\label{transported}
\mu^{-, (g)} = h_* \mu^{-, (f)}.
\end{equation}
We also recall that another characterization of SRB measure
$\mu^{-,(g)}$ is that they are Gibbs measures with respect to
the potential $\log J^s[g]$. Similarly for $\mu^{-,(f)}$.
The measure $h_* \mu^{-,(f)}$ is invariant under $g$ and
and Gibbs with respect to the potential $\log J^s[f]\circ h$.
The fact that we have \eqref{transported}
implies, by the theory in \cite{Sinai72} Theorem 4
-- see also \cite{Bowen75} Proposition 4.5 -- that, for
a H\"older function $\Psi$ and a constant $K$, we have:
\begin{equation} \label{cohomologous}
\log J^s[f]\circ h - \log J^s[g] = \Psi\circ g - \Psi + K
\end{equation}
Moreover, $K$ is the difference between the topological
pressures of the functions $\log J^s[f]\circ u $ and
$ \log J^s[g]$.
It is not hard to show, using the thermodynamic formalism that
$K = 0$ using the fact that $f,g$ are conjugate and, therefore
have the same entropy. Nevertheless, we will present an elementary
proof below.
We will show that, if $K=0$, then,
$\log J^s[h]$ is bounded.
The same argument will reach a contradiction
with \eqref{uniformint} under the assumption that $K \ne 0$.
We start by showing that if $K$ in \eqref{cohomologous} is $0$,
then $\log J^s[h]$ is bounded.
The argument will be very similar to the one
we applied to use that the function $h$ is
quasi-conformal. In the case that the manifolds are one-dimensional,
it was used in \cite{Llave87}.
We first note that to show that $\log J^s[h]$ is bounded
it suffices to show that,
with the same function $k$ as in \eqref{scattering},
we have
\begin{equation}\label{jacobianbounds}
|\log J^{s,(g)} [f^n\circ k \circ g^{-n}](x) | \le C
\end{equation}
where $C$ is independent of $n,x$.
Indeed,
Given a ball $B_r$ in a stable manifold
of $g$, and $\sigma > 0$, by the uniform convergence
of $h_n \equiv f^n \circ k \circ g^{-n}$,
we can find $n$ such that
$ B_r \subset h^{-1} \circ h_n ( B_{r +\sigma})$,
(where $B_{r +\sigma}$ is, a ball with the same center and
radius $r + \sigma$).
Then, $h(B_r) \subset h_n( B_{r +\sigma})$.
By the assumption that $J^s[h_n]$ is uniformly bounded, we
can bound $|h(B_r) | \le C |B_{r+\sigma}|$,
where, again, $|S| $ denotes the Riemannian volume on $W^{s,(f)} $.
Taking $\sigma = 1/10 r$, we can bound
$|B_{r+\sigma}| \le C |B_r| $.
Hence, under \eqref{jacobianbounds} we have
shown that
$|h(B_r) | \le C |B_{r}|$.
To establish \eqref{jacobianbounds} we expand the logarithm in the
LHS
\begin{equation} \label{jacobianscatter}
\begin{split}
\log J^{s,(g)}[ f^n \circ k \circ g^{-n}](x) = &
\sum_{i=1}^n \log J^s[f]\circ f^i \circ k \circ g^{-n}(x)
\\
& + \log J^s[k] \circ g^{-n} (x)
\\
& + \sum_{i =1}^n \log J^s[g^{-1}]\circ g^{-i + n} \circ g^{-n}(x)
\end{split}
\end{equation}
We want to show that the LHS of \eqref{jacobianscatter} is
bounded uniformly in $n,x$.
Since $\log J^s[k] $ is bounded and for every
function $\Gamma$ we have
$\sup \Gamma \circ g^{-n} (x) = \sup \Gamma(x)$,
to obtain that the LHS of \eqref{jacobianscatter} is bounded
it suffices to show -- relabeling the indexes in the last sum -- that:
\begin{equation} \label{jacobianscatter2}
\sum_{i=1}^n \log J^s[f]\circ f^i \circ k
+ \sum_{i =1}^n \log J^s[g^{-1}]\circ g^{i}
\end{equation}
is bounded.
The boundedness of
\eqref{jacobianscatter}
follows from the identity -- derived by applying \eqref{cohomologous}
repeatedly --
\begin{equation}\label{auxiliary}
\sum_{i=1}^n \log J^s[f]\circ f^i \circ h
+ \sum_{i =1}^n \log J^s[g^{-1}]\circ g^{i} = \Psi\circ g^i - \Psi
\end{equation}
which is bounded.
We also observe that $d(f^i\circ h(x), f^i\circ k(x)) \le C \lambda^{i}$
-- where $\lambda < 1, C$ are independent of $x$ -- and that
$J^s[f]$ is H\"older so that we can bound
\[
| \log J^s[f]\circ f^i \circ h
- \log J^s[f]\circ f^i \circ | \le C \lambda^{i \alpha}
\]
Hence, the difference between \eqref{jacobianscatter}
and \eqref{auxiliary} can be bounded by a geometric series.
To reach a contradiction with the fact that $K \ne 0$,
we note that the argument presented above remains the same
except that \eqref{auxiliary} is replaced by
\begin{equation}\label{auxiliary2}
\sum_{i=1}^n \log J^s[f]\circ f^i \circ h
+ \sum_{i =1}^n \log J^s[g^{-1}]\circ g^{i} = \Psi\circ g^i - \Psi +n K
\end{equation}.
Therefore, we obtain, even in the case that $K \ne 0$ that
$J^s[ f^n \circ k \circ g^{-n}](x) e^{-nK}$ remains bounded.
If $K > 0 $, this is a contradiction with the fact that
given a ball $B \subset W^{s, (g)}_x$,
$f^n \circ k \circ g^{-n} B $ converges uniformly to
$h(B)$, which is a bounded set. In particular, the measure of
$f^n \circ k \circ g^{-n}(B) $ should remain bounded.
We can reach a similar contradiction with $K < 0$ by reversing the
roles of $f$, $g$ and those of $h$, $h^{-1}$ in the argument above.
\begin{remark}
Once that we have that $\log J^s[h]$ is bounded, we note that
$\log J^s[h]$ satisfies the cohomology equations \eqref{cohomologous}.
It is well known \cite{PollicottY99}, \cite{Llave01},
that the bounded measurable solutions of
\eqref{cohomologous} for systems that
have ergodic SRB measures are unique up to an additive constant
and, furthermore are H\"older.
This shows that the $\log J^s[h] = \Psi + C $. Hence,
$\log J^s[h] $ is H\"older.
Of course, in the one-dimensional case, where the
Jacobian can be identified with the derivative, this shows that $h$ is
$C^{1+\alpha}$.
Later we will do better exploiting the quasi-conformal structure.
\end{remark}
\begin{remark}
Another corollary of the arguments of this section is that if
the Jacobians along stable directions match and the system
$g$ is Anosov transitive, the conjugacy is absolutely continuous
on stable leaves.
To establish this result, we only need to point out that we can use
Livsic Theorem to show that there exists a $\Psi$ satisfying
\eqref{cohomologous} with $K = 0$. From them on, we can use the same
argument as that presented here.
\end{remark}
\subsection{The derivative of the conjugacy is bounded } \label{sec:Dhbounded}
One important reason for us is why the estimates of the Jacobians are
so useful for us is that, for conformal mappings, estimates on the
Jacobian imply estimates on the derivatives.
Let $M$, $\tilde{M}$ be two manifolds, $E$, $\tilde{E}$ be subbundles
of $TM$, $T\tilde{M}$ respectively.
Let $\sigma$, $\tilde{\sigma}$ be Riemannian metrics on $E$,
$\tilde{E}$ continuous over the corresponding manifolds.
Given a mapping $F:M\to \tilde{M}$ such that $DF(x):E_x \to
\tilde{E}_{F(x)}$ we define
\[
K_{E, \tilde{E}, \sigma, \tilde{\sigma}}(x) = \max_{\substack{v \in
E_x \\
\norm{v}_\sigma =1}}
\norm{DF(x)v}_{\tilde{\sigma}}\Big/\min_{\substack{v\in E_x \\
\norm{v}_\sigma=1}} \norm{DF(x)v}_{\tilde{\sigma}}
\]
We say that $F$ is $A$-quasi-conformal when $K_{E, \tilde{E}, \sigma,
\tilde{\sigma}}(x) \leq A$.
\begin{proposition} \label{Jacobianbounds}
Let $F:M \to \tilde{M}$ be $E$, $\tilde{E}$, $\sigma$,
$\tilde{\sigma}$, $A$-quasi-conformal then, for a constant $C$ which
depends only on $\sigma$, $\tilde{\sigma}$ and the background metric
we have
\[
\norm{DF(x)} \leq C A^{\frac{\nu-1}{\nu}}J_F^{1/\nu}
\]
\end{proposition}
The proof of the proposition is very easy.
Using the singular value decomposition of linear algebra, we know that
there are matrices $U$, $V$ such that $U$ is $\sigma_{F(x)}'$ unitary,
$V$ is $\sigma_x$ unitary and $UDF(x)V \equiv D$ is diagonal with
pointwise entries $d_1\leq d_2 \cdots \leq d_\nu$.
Then we have:
\begin{equation*}
\begin{split}
\norm{DF(x)}_{\sigma,\sigma'} & = d_\nu \\
K\big(DF(x)\big) & = \frac{d_\nu}{d_1} \\
J_{\sigma, \sigma'}\big(DF(x)\big) & = d_1 \cdots d_\nu
\end{split}
\end{equation*}
Because of the continuity of the metric, the norm and the Jacobian
with respect to the background metric are equivalent to those with
respect to $\sigma$, $\tilde{\sigma}$.
Therefore, we have
\[
A^{-(\nu-1)} \norm{DF(x)}^\nu_{\sigma, \tilde{\sigma}} \leq J_{\sigma,
\tilde{\sigma}}\big(DF(x)\big) \leq \norm{DF(x)}^\nu_{\sigma, \tilde{\sigma}}
\]
\qed
As an immediate consequence of Proposition \ref{Jacobianbounds} we
obtain
\begin{equation} \label{bounded}
\norm{D_{s,(g)} h} \in L^\infty\ .
\end{equation}
where by $D_{s,(g)}$ we mean, following \cite{LlaveMM86},
the derivative along the leaves of the stable foliation for
$g$.
When there is no possibility of confusion we will omit the map
$g$ from the notation.
Of course, a similar argument shows that $D_{u,(g)} h \in L^\infty$.
The result \eqref{bounded}
implies that $h$ is Lipschitz restricted to the stable and
unstable manifold.
We also note that a similar argument with $f$, $g$ exchanged shows
that $D_s(h^{-1})$ is uniformly bounded.
Hence $\norm{(D_s h)^{-1}}$ is also uniformly bounded.
\begin{remark}
For Anosov systems, because of the local
product structure, we have shown that $h$ is Lipschitz.
In the case that the system is partially hyperbolic, we
would have established some H\"older exponents which are
depending only on the accessibility properties. In particular,
they are independent of $h$.
This will not be a difficulty for the subsequent argument since
we will continue bootstrapping the regularity along stable and
unstable manifolds separately.
\end{remark}
\subsection{Further regularity along stable and unstable directions} \label{sec:further}
The next part of the argument will be based on the observation that,
taking derivatives along the stable directions of $g$ in
\eqref{conjugacy} we have:
\begin{equation} \label{coboundary}
D_sh(x) = Df^{-1}\circ h\circ g(x) D_s h \circ g(x) D_sg(x)\ .
\end{equation}
(Note that we have omitted the indexes that indicate
which maps give rise to the stable foliations that
originate the operators $D_s$. All of them in this case are
with respect to the foliations invariant under $g$. )
Equation \eqref{coboundary}
can be considered as saying that $D_sh$ is a coboundary between
$D_sf^{-1}\circ h\circ g(x)$ and $D_sg(x)$.
By the results in the previous section, we already have
that $D_s h$ is an $L^\infty$ coboundary.
There are already several papers in the literature which show that
coboundaries which are measurable are automatically smooth (see,
e.g. \cite{NiticaT98} for the case of continuous functions).
In this paper we will adapt the method of \cite{Llave02-boot}.
We denote by $\LL(X, Y)$ the space of Linear operators from $X$ to $Y$
endowed with the supremum norm.
We note that $D_s h(x) \in \LL(E_x^{s, (g)}, E_{h(x)}^{s, (f)})$.
Similarly
\begin{equation*}
\begin{split}
Df^{-1} & \in \LL(E_x^{s, (f)}, E_{f^{-1}(x)}^{s, (f)}) \\
Dg(x) & \in \LL(E_x^{s, (f)}, E_{f^{-1}(x)}^{s,f})
\end{split}
\end{equation*}
We introduce the operator
$\eta(x)$
\[
\eta(x): \LL(E_{g(x)}^{s,(g)},
E_{h\circ g(x)}^{s,(f)}) \to \LL(E_x^{s,(g)}, E_{h(x)}^{s,f}
\]
by
\[
\eta(x) \Gamma_{g(x)} = D_sf\circ h\circ g(x) \Gamma_{g(x)} D_sg(x)
\]
We also introduce the notation
\[
\Psi(x) = D_sh(x)
\]
Hence, the equation \eqref{coboundary} can be written more compactly
as:
\begin{equation} \label{coboundary2}
\Psi(x) = \eta(x) \Psi\circ g(x)\ .
\end{equation}
Iterating \eqref{coboundary} we have
\begin{equation} \label{coboundaryn}
\Psi(x) = \eta(x) \cdots \eta\circ g^n(x) \Psi \circ g^{n+1}(x)
\end{equation}
The subsequent analysis will use heavily expressions such as
\eqref{coboundaryn} to bootstrap the regularity of $\Psi$.
The analysis will be somewhat similar to that of \cite{Llave02-boot}.
For the moment, we will just estimate
$\eta(x) \cdots \eta\circ g^n(x)$.
The following bounds will be crucial for subsequent analysis:
\begin{lemma} \label{uniform}
With the notation above, we have
\[
\norm{\eta(x) \cdots \eta \circ g^n(x)} \leq C
\]
where $C$ depends only on the properties of the metrics.
In particular, it is independent of $n$.
\end{lemma}
\begin{proof}
Clearly we have
\begin{equation} \label{easybound}
\norm{\eta(x) \cdots \eta \circ g^n(x)} \leq \norm{Df^{-n}\circ h\circ
g(x)} \norm{Dg^n(x)}
\end{equation}
By \eqref{easybound} and Proposition \ref{Jacobianbounds}, to prove
Lemma \ref{uniform}, it suffices to bound uniformly in $n$, $x$.
\[
E(n,x) = (J^s[f^{-n}])\circ h \circ g^n(x) J^s[g^n](x)
\]
We note that because of \eqref{conjugacy} we have
\begin{equation}
E(n,x) = J^s[h](x)\big/ J^s[h]\circ g^n(x)
\end{equation}
and the R.H.S. is bounded uniformly in $n$, $x$
because of the fact that the stable Jacobian is H\"older, and
bounded above and below away from zero.
\end{proof}
\begin{remark}
Even if we will not use it here, we note that the
fact that the maps are conformal,
show that the rates of growth in along all the directions in
an orbit are the same.
The proof of stable foliations in
\cite{HurderK90} allow us
to conclude that the stable and unstable foliations are $C^1$.
Actually, following \cite{HirschP68}, once we know that the
foliations are $C^1$, we can use the fact that the derivatives of the
jet satisfy an equation that can be analyzed with the fiber
contraction theorem, to obtain that the foliations are $C^{1+\gamma}$.
Even if these considerations will not play any role in our
discussions, the regularity of the foliations is a very important part
of the argument in \cite{Sadovskaia02} \cite{KalininS02} and it allows
them to reconstruct global properties of the manifold.
\end{remark}
\subsection{$D_sh$ is H\"{o}lder}
The following will be the main result of this section.
\begin{lemma} \label{dsholder}
Assume that $D_sg$, $D_sf\circ h$ are $C^{\alpha, (s)}$, $\alpha>0$
(resp. $C^{\alpha, (u)}$, $\alpha >0$).
Then
\[
D_sh \in C^{\alpha, (s)} \quad (\text{resp. } C^{\alpha, (u)}).
\]
\end{lemma}
Note that we have assumed $f\in C^r$ with $r>1$.
We have also shown that $h$ is Lipschitz restricted to the and unstable stable
foliation
Hence, using that the map is Anosov, we can take
\[
\alpha = \min(r-1, \Lip).
\]
(In the case that the map is partially hyperbolic, we will need to
rearrange the argument slightly. )
In the case that $\alpha = \Lip$ --- which is the case relevant for
higher regularity --- we obtain that $D_sh$ is differentiable almost
everywhere along the stable and unstable leaves.
We also call attention that in the course of the proof
of Lemma~\ref{dsholder}, we will
introduce a remarkable formula \eqref{incrementsum} giving the
``increments'' of $D_sh$ along the stable manifolds.
This formula will be the basis for subsequent bootstraps.
\medskip
We will need to introduce some notation which is very similar to that
of \cite{Llave02-boot}, \cite{KalininLS02}.
Given a vector field $u$ in $M$ we denote by $\delta_u(x) =
\exp_x(u(x))$ where $\exp$ denotes the Riemannian geometry
exponential.
In subsequent arguments we will need that the vector fields $u$ are
such that $\delta_u$ preserves the (un)stable foliation of $g$ and
that they are absolutely continuous.
The reason to require that $\delta$ is absolutely continuous is that
we are going to consider $\Psi \circ\delta$ where $\Psi \in L^\infty$
and evaluate identities that hold for general points $x$ at
$\delta(x)$.
We note that maps satisfying the previous requirements are plentiful
because of the absolute continuity of the foliation.
That is
\begin{proposition}\label{abundant}
If $d(x,y) \leq \ep_0$, $y \in W_x^{s,(g)}$ then there exists $\delta$
as above, such that
\[
\delta(x) =y\ , \quad \norm{\delta -\Id}_{L^\infty} \leq Cd(x,y)\ , \quad
\norm{J [\delta] }_{L\infty} \leq C\ .
\]
\end{proposition}
The proof of the proposition is obvious if we consider a coordinate
patch such as those introduced in Theorem 9 in
\cite{Anosov69}. There, it is shown how it is possible
to make changes of variables that reduce the stable foliations
to a product foliation in $\real^d$. This changes of variables are, moreover
absolutely continuous. Hence, the problem considered in
Proposition \ref{abundant} can be refered just to
the a trivial foliation.
We will also need to compare tangent spaces at nearby points.
We just recall the device of \emph{``connectors''} introduced in
\cite{HirschPPS69}.
Let $x$, $y$ be points which are close enough that we can find a
unique shortest geodesic.
Define by $S_x^y:T_xM \to T_yM$ to be the map obtained by transporting
the vectors from $T_xM$ to $T_yM$ along the shortest geodesic.
Note that since we are assuming that the background metric is
analytic, we obtain that $S_x^y$ depends analytically on $x$, $y$
provided that $d(x,y) \leq \ep_0$ for some $\ep_0>0$.
Note that, if both sides can be defined, we have
\[
S_x^z = S_y^z \circ S_x^y
\]
In particular
\begin{equation} \label{identity}
\Id = S_y^x \circ S_x^y\ .
\end{equation}
Using the notations of $\Psi = D_sh$ and $\eta$ introduced in section
\ref{sec:Dhbounded} we will show that
\[
\norm{\Psi(x) - S_{\delta_v(x)}^x \Psi
\circ \delta_v(x)} \leq C\norm{v}_{C^0}^\alpha
\]
provided that $\delta_v$ preserves the stable foliation.
Later we will provide with a different argument to show a similar
result when $\delta_v$ preserves the unstable foliation.
Assuming that $\delta$ preserves the stable foliation of $g$ and that
$\delta(x)-x$ is small enough so that we can define the transport, then we
see that $g^n\circ \delta(x)$, $g^n(x)$ are also close and we define the
$S_{g^n\circ \delta(x)}^{g^n(x)}$.
Evaluating \eqref{coboundaryn} at $\delta(x)$ and inserting factors
$S_{g^i(x)}^{g^i\circ \delta(x)} S^{g^i(x)}_{g^i\circ \delta(x)}$, which are
just the identity, we obtain:
\begin{equation}\label{monster}
\begin{split}
S_{\delta(x)}^x \Psi\circ \delta(x) = &
S_{\delta(x)}^x \eta \circ \delta(x)
S_{g \circ \delta(x)}^{g(x)}
S^{g \circ \delta(x)}_{g(x)}
\eta \circ g \circ \delta(x)
S_{g^2 \circ \delta(x)}^{g^2(x)}
S^{g^2 \circ \delta(x)}_{g^2(x)} \\
& \cdots \\
& \cdot S_{g^n \circ \delta(x)}^{g^n(x)}
S^{g^n \circ \delta(x)}_{g^n(x)}
\eta \circ g^n \circ \delta (x) \\
& \cdot S_{g^{n+1} \circ \delta(x)}^{g^{n+1}(x)}
S^{g^{n+1} \circ \delta(x)}_{g^{n+1}(x)}
\Psi \circ g^{n+1} \circ \delta(x)
\end{split}
\end{equation}
Introducing the notation
\begin{equation} \label{etatilde}
\begin{split}
\tilde{\eta}\circ \delta(x) & = S^x_{\delta(x)}\eta\circ \delta(x)
S_{g(x)}^{g\circ\delta(x)} \\
\tilde{\Psi} \circ \delta(x) & = S^x_{\delta(x)}\Psi\circ \delta(x)
\end{split}
\end{equation}
We see that \eqref{monster} can be written
\begin{equation}\label{translated}
\tilde{\Psi}\circ \delta(x) = \tilde{\eta}\circ \delta(x) \tilde{\eta}\circ
g \circ \delta(x) \cdots \tilde{\eta}\circ g^n\circ\delta(x)
\tilde{\Psi}\circ g^{n+1}\circ \delta(x)
\end{equation}
\begin{remark}
There is a nice geometric intuition behind the calculation leading to
\eqref{monster}.
Namely, we are trying to analyze the objects along the orbit
$g^i\circ \delta(x)$
by referring them to objects in the orbit $g^i(x)$.
This is accomplished, of course by conjugating with the
identifications $S$.
This will allow us to
compare operators along two neighboring orbits
by referring
them to common points.
\end{remark}
Adding and subtracting in \eqref{coboundaryn} and \eqref{monster}, we
obtain:
\begin{equation} \label{monster2}
\begin{split}
\Psi(x) - \tPsi \circ \delta(x) = &
\big[ \eta(x) - \teta \circ \delta (x) \big]
\teta \circ g \circ \delta(x) \cdots
\teta\circ g^n \circ \delta(x)
\tPsi\circ g^{n+1} \circ \delta(x) \\
& + \quad \cdots \quad + \\
& \eta(x) \cdots \eta \circ g^{i-1} (x)
\big[ \eta\circ g^i(x) - \teta \circ g^i \circ \delta(x) \big]
\teta \circ g^{n} \circ \delta(x)
\tPsi \circ g^{n+1} \delta(x) \\
& + \quad \cdots \quad + \\
& + \eta(x) \cdots \eta \circ g^{n-1} (x)
\big[ \eta\circ g^n(x) - \teta \circ g^n \circ \delta(x) \big]
\tPsi \circ g^{n+1} \delta(x) \\
& + \eta(x) \cdots \eta \circ g^{n} (x)
\big[ \Psi \circ g^{n+1} (x) - \tPsi \circ g^{n+1} \circ \delta(x) \big]
\end{split}
\end{equation}
We denote by $T_i(x)$ the general term of \eqref{monster2}, namely:
\[
T_i(x) = \eta(x) \cdots \eta\circ g^{i-1}(x)[\eta\circ g^i(x) -
\tilde{\eta}\circ g^i \circ \delta(x)]\Psi\circ g^{i+1}\circ \delta(x)
\]
We want to show that
\begin{equation} \label{incrementsum}
\Psi(x) - \tilde{\Psi}\circ \delta(x) = \sum_{i=1}^\infty T_i(x)
\end{equation}
almost everywhere $x$.
First of all we want to show that $\sum_{i=1}^\infty T_i(x)$ converges
almost everywhere as well as in $L^\infty$ sense. This follows from
\begin{equation} \label{exponentialbounds}
\norm{T_i}_{L^\infty} \leq \lambda_c^{i\alpha} C \norm{\delta-\Id}_{C^0}^\alpha
\end{equation}
To prove \eqref{exponentialbounds} we use that, by Lemma \ref{uniform}
the factors $\eta(x) \cdots \eta\circ g^{i-1}(x)$ are uniformly
bounded independently of $i$, $x$.
Also $\tilde{\Psi} \circ g^{n+1} \circ \delta(x)$ is uniformly bounded
in $L^\infty$.
Because $\eta$ is $C^\alpha$ and $\delta$ is adapted to the stable
foliation we have
\[
\norm{\eta\circ g^i - \tilde{\eta}\circ g^i \circ \delta}_{L^\infty}
\leq \lambda^{i\alpha}C\norm{\delta-\Id}^\alpha_{L^\infty}
\]
This proves \eqref{exponentialbounds}.
To estimate the last term in \eqref{monster2},
we follow an argument in \cite{NicolP99}.
We note that, by Lusin theorem, $\Psi$ is continuous in as set of
measure at least $1-\ep$.
We obtain that, because of the ergodicity of $g$,
$\Psi\circ g^{n+1}(x) - \tilde{\Psi}\circ g^{n+1}\circ \delta(x)$
converges to zero along a subsequence.
Hence, we obtain that \eqref{incrementsum} holds almost everywhere.
This shows that $D_sh \in C^{\alpha, s}$.
We note that if $\Psi$ satisfies \eqref{coboundary} then we also have
\[
\Psi(x) = \eta^{-1}\circ g^{-1}(x) \Psi\circ g^{-1}(x)\ .
\]
Hence, applying the result we have for the regularity along the stable
directions, we obtain the claim for the unstable directions.
\qed
One consequence that we will use later is that the formula
\eqref{incrementsum} holds not only almost everywhere but everywhere.
Hence, we do not need to use the device of studying increments by
comparing with absolutely continuous vector fields.
We have when $y \in W_x^{s(g)}$
\begin{equation} \label{incrementsum2}
\begin{split}
\Delta(y) & \equiv \Psi(x) - S_y^x\Psi(y) \\
& = \sum_{i-1}^\infty \eta(x) \cdots \eta\circ g^i(x) [\eta\circ
g^i(x) - \tilde{\eta}\circ g^i(y)] \Psi\circ g^{i+1}(y)
\end{split}
\end{equation}
\subsection{$D_s h$ is differentiable}
\begin{lemma} \label{differentiable}
If $D_sf^{-1}\circ h$, $Dg \in C^{1+\alpha, (s)}$, $\alpha>0$
(resp. $\in C^{1+\alpha, (u)}$, $\alpha>0$) then $D_sh \in
C^{1+\alpha, (s)}$. (resp. $C^{1+\alpha, (u)}$.)
\end{lemma}
The proof of Lemma \ref{differentiable} is very simple.
Given \eqref{incrementsum2} it is natural to guess what the formula
for the derivative should be (see \eqref{guess}). Then we will prove that
this formula is indeed a derivative by estimating the remainder.
We call attention to the fact that the formula \eqref{guess} will be
the basis for subsequent bootstraps.
We guess
\begin{equation}\label{guess}
\begin{split}
G(x) & =
\sum_{i=1}^\infty \eta(x) \cdots \eta\circ g^i(x)
(D_s\tilde{\eta} \circ g^i)(x)\Psi\circ g^{i+1}(x) \\
& =
\sum_{i=1}^\infty \Psi(x) \Psi^{-1} \circ g^{n-1}(x)
(D_s\tilde{\eta} \circ g^i)(x)\Psi\circ g^{i+1}(x) \\
\end{split}
\end{equation}
First of all, we observe that the formula in \eqref{guess} converges
uniformly.
The reason is that, because of the chain rule we have for a vector
field $X$ tangent to the stable direction:
\[
D_s (\tilde{\eta}\circ g^i)(x)\,
X (\tilde{\eta}\circ g^i)(x) = (D_s\eta)\circ g^i Dg^i(x) X
\]
Because the vector field $X$ is tangent to the stable direction
we see that $ \norm{SDg^i(x) X} \le \lambda^i \norm{X}$.
Hence, $\norm{ D_s \tilde \eta \circ g^i(x)} \le C \lambda^i$
and, therefore,
the general term of \eqref{guess} can
be bounded by a decreasing exponential.
Once we know that the formula \eqref{guess} makes sense, to
establish that it is a true derivative, we just need
to estimate the error
in the linear approximations.
This will be easy because the formula for the increments is given in
\eqref{incrementsum}. When we substract the
linear approximation given by \eqref{guess} from
the increment formula \eqref{incrementsum}, we obtain:
\begin{equation} \label{errorexpression}
\begin{split}
\Psi(x) &- S_y^x\Psi(y) - G(x)[y-x] \\
= &
\sum_i \eta(x) \cdots \eta\circ g^i(x) [\eta g^i(x) - \tilde{\eta}\circ
g^i(y) - D_y \tilde{\eta}\circ g^i(y)] \Psi\circ g^{i+1}(x) \\
& + \sum_i \eta(x) \cdots \eta\circ g^i(x)[\eta \circ g^i(x) -
\tilde{\eta}\circ g^i(y)] [\Psi\circ g^{i+1}(x) -
\tilde{\Psi}\circ g^{i+1}(y)]
\end{split}
\end{equation}
The desired result will be established when we estimate
the \eqref{errorexpression} and show that it can be bounded
by something that is $o(d_s(x,y))$. We will start by
bounding the general term of each of the sums in
\eqref{errorexpression} and then add up all the estimates
obtained for each of the terms.
We again note that the factor $\eta(x)\cdots \eta\circ g^i(x)$ are
bounded uniformly.
So is $\Psi\circ g^{i+1}(x)$.
Because $\eta \in C^{1+ \alpha, (s)}$, we have that
\[
\abs{[\eta\circ g^i(x) - \tilde{\eta}\circ g^i(y) - [D_s\tilde{\eta}
g^i](x)[x-y]]} \leq \lambda^{(1+\alpha)i}d_s(x-y)^{1+\alpha}
\]
This shows that the first sum in \eqref{errorexpression} can be
estimated by $Cd(x,y)^{1+\alpha}$.
Now we estimate the second sum in \eqref{errorexpression}.
Because $\Psi$ is $C^{\Lip, (s)}$ we have:
\[
\abs{\Psi\circ g^i(x) - \tilde{\Psi}\circ g^i(y)} \leq C \lambda^i
d_s(x,y)\ .
\]
Since $\eta$ is Lipschitz we can bound:
\[
\norm{\eta\circ g^i(x) - \tilde{\eta}\circ g^i(y)} \leq C \lambda^i d_s(x,y)
\]
Hence the second sum can be bounded by $ C d_s(x,y)^2$
This finished the proof that \eqref{guess} is indeed the derivative
along the stable direction.
This finished the proof of Lemma \ref{differentiable}.
\subsection{Higher derivatives along the stable directions}
Note that we have also established \eqref{guess} a formula for the
derivative along the stable direction.
Establishing the existence of higher order derivatives is now very
easy because we can take derivatives of \eqref{guess} term by term.
The proof is from now on, quite similar to the proofs in
\cite{LlaveMM86} and \cite{BanyagaLW96}.
We just need to show that the expression for the derivative
given in \eqref{guess} can be differentiated term by
term. These formal derivatives will converge uniformly.
Hence, it is standard to conclude that these sums are
indeed the true derivatives.
The estimates needed are a particular case of
Lemma 4 of \cite{BanyagaLW96} -- which are very similar
to those of \cite{LlaveMM86} p. 587.
Following \cite{LlaveMM86}, for $k \in \nat$,
we say that a function $\varphi$ is
in $C^{k, (g)}_s$ when the restriction to the stable leafs of $g$ are
uniformly $C^k$ and the $k$ jets of these restrictions
are continuous over the manifold.
We say that $\varphi \in C^{k + \alpha, (g)}_s$
when $\varphi \in C^{k,(g)}_s $ and the
$k$ jets are uniformly H\"older when restricted to
the stable leaves of the stable leaves.
The space
$C^{k,(g)}_s $ is a Banach space when endowed with the norm
of the sup of the $k$-jets.
Similarly, $C^{k+\alpha,(g)}_s $ is a Banach space when endowed with
the norm obtained taking the maximum of the $C^{k,(g)}$ norm
and the seminorm $H^\alpha_{s,(g)}$ for the $k$-jet.
Clearly, we have
\[
\varphi \in C^{k,(g)}_s \iff
D_{s,(g)} \varphi \in C^{k -1,(g)}_s
\]
and the same with fractional exponents.
\begin{lemma}\label{induction}
Assume that $\Psi, D_s\eta \in C^{t,(g)}_s$
$t \in \nat + [0,\Lip]$ and that $g \in C^r$,
$r \ge t$, $t > 0$.
Then, the sum in \eqref{guess} converges uniformly
in $C^{t, (g) }_s$.
\end{lemma}
\begin{proof}
This lemma is very similar to the estimates in
\cite{LlaveMM86} p. 587 and \cite{BanyagaLW96} Lemma 4.
The estimates in the above papers give
\begin{equation}
\begin{split}
& \norm{ D_s^j \Psi^{-1} \circ g^{n-1}}_{L^\infty} \le C_j
\norm { \Psi^{-1}}_{ C^{j}_s } \lambda^{j(n+1)} j^{n+1}
\\
&\norm{D_s\tilde{\eta} \circ g^i }_{L^\infty} \le C_j
\norm { D_s \tilde \eta}_{ C^{j}_s }\lambda^{j i } j^i \\
&\norm{ \Psi\circ g^{i+1}}_{L^\infty} \le C_j
\norm { \Psi }_{ C^{j}_s }\lambda^{j (i+1)} j^{i +1}\\
\end{split}
\end{equation}
where $C_j$ are numbers that depend on $j$ and on $\norm{g}_{C^j}$
but which are independent of $i, n$.
Using Leibniz rules for the derivatives, we obtain that, as
claimed the $C^{r, (g)}_s$ norm of the general term in
\eqref{guess} is bounded by a decreasing exponential.
When we need to consider fractional derivatives, the arguments
we
We denote by $d_s(x,y)$ the distance of
two points in the same leaf of the stable foliation
measured along the leaves. We also denote
\[
H^\alpha_s (\eta) = \sup_{ 0 \le d_s(x,y) \le \gamma }
( \eta(x) - \eta(y))/ d_s(x,y)^\alpha
\]
We will assume without loss of generality that the metric
is adapted. That is, $d_s(f(x), f(y) ) \le \lambda d_s(x,y) $
for some $\lambda < 1$. This immediately
leads to
\begin{equation}\label{Halphacomp}
H^\alpha_s ( \eta \circ f) \le H^\alpha_s(\eta) \lambda^\alpha.
\end{equation}
The bounds for the $H^\alpha_s$ of the high derivatives follow
by estimating the same formulas that we use for the integer derivatives.
\end{proof}
Applying repeatedly, Lemma \ref{induction}, we obtain the desired result.
Note that, if we assume that
$\Psi, \eta \in C^{j, (g)}_s$, then Lemma \ref{induction} shows that
the sum in \eqref{guess} converges in
$C^{j,(g)}_s$, therefore, $G = D_s h $ is in $C^{j,(g)}_s$ and,
therefore $h \in C^{j+1, (g)}_s$ and therefore
$\Psi, \eta \in C^{j+1, (g)}_s$.
The induction stops when we reach the regularity assumed in
$g, f$.
\begin{remark}
Note that the argument presented to obtain the higher derivatives
is very reminiscent of the argument presented
for the bootstrap of regularity of
conjugacies in \cite{Llave92}. In Theorem 6.1 of
\cite{Llave92}, it is established that once
a conjugacy is smoother than an critical value, determined
out of the bunching of the spectrum of expansions and
contraction, then the conjugacy is as smooth as the map.
(See also the last parts of \cite{NiticaT98}. )
For maps preserving a conformal metric, the expansions of all
the vectors along the stable manifold is roughly the same.
Hence, the critical value for Theorem 6.1 in
\cite{Llave92} -- expressed in terms of
bunching properties of the expansion properties --
is $1 +\epsilon$. Hence, once that we have
that $h \in C^{1 + \alpha}$, it is also possible to
just adapt the arguments in \cite{Llave92}.
\end{remark}
\section{Regularity in the usual classes} \label{sec:usual}
We have so far shown that the conjugacy $h \in C^{r, (g)}_s$.
A similar argument shows that $h \in C^{r, (g)}_u$.
For the case of difeomorphisms, we are done by invoking
a regularity lemma that says that functions with the above
properties are $C^{r'}$. In \cite{LlaveMM86} one
can find a prove of this result using elliptic regularity
theory. Other proofs are available in \cite{Journe88}.
(Even other versions less technically optimal appear in
\cite{HurderK90}, \cite{Llave92}, \cite{Llave97}).
For the case of flows, we point out that exactly the same argument applies
for the stable and unstable manifolds of the flow.
It is also trivial to observe that if
\eqref{conjugacy2} holds, then the conjugacy is smooth along the
direction of the flow.
Then, we can apply the regularity results to obtain that
the conjugacy is $C^{r',X}_{cs}$ -- where the subindex $cs$ refers
to the center stable direction. Then, a further application of the regularity
result considering now the $cs$ foliation and the unstable one
finishes the proof of Theorem \ref{main}.
\begin{remark} \label{rem:further}
It is interesting to note that, except for Section \ref{sec:usual},
all the previous sections work for partially hyperbolic maps
that have SRB measures with ergodic properties.
It is reasonable to conjecture that there should be an analogue of
the regularity result for foliations of Anosov systems for
foliations of partially hyperbolic systems that have a
certain accessibility property. Following the
scheme of proof in \cite{LlaveMM86}, it seems that it
would be enough to use some version of the theory of
hypoelliptic equations rather than the elliptic regularity theory used
in \cite{LlaveMM86}.
One would expect that the regularity recovered, would be a fraction of
the regularity of the derivatives obtained.
Once such result is on hand, then it will automatically follow
a version of Theorem \ref{main} for partially hyperbolic systems.
We hope to come to this problem in a the future.
\end{remark}
\subsection{Results for systems with one-dimensional
foliations}\label{sec:onedimensional}
We will present two arguments that are very similar to
the arguments in \cite{Llave87}, \cite{LlaveM88} and
in \cite{Pollicott88}, \cite{Llave92}.
In the first argument, we note that, because of
the condition on the eigenvalues, the fact that the system is
transitive and the result of \cite{Livsic72}, we can find a
function $\Psi$ satisfying \eqref{cohomologous} (with $K= 0$).
Then, we can repeat the argument as in
Section\ref{isbounded}. We obtain that the Jacobian of the derivative
is bounded. Of course, in the one-dimensional case, the Jacobian of
the derivative is the same as the derivative. Hence, showing that the
derivative is bounded is obvious.
An alternative argument is based on the fact that the
SRB measure is also characterized as the weak limit of
measures based on periodic orbits with weights which are proportional
to the stable Jacobians. Given the equality of the stable derivatives
at corresponding periodic points, we obtain that the SRB measure
is transported by the conjugacy.
>From then, one can use that the
densities are transported and use that they are smooth to get that the
derivatives are bounded.
The argument presented before applies also to
the one-dimensional case.
Using the commutativity of
the derivatives one can use simpler arguments.
Several different arguments to bootstrap regularity taking advantage of
the fact that the stable manifold is one-dimensional can be found in
\cite{Llave87}, \cite{Llave92}, \cite{Llave97}. We refer to
\cite{Llave97} for a comparison between these one-dimensional
arguments.
\section{Acknowledgments}
The work of the author has been supported by NSF grants.
I am also grateful to V. Sadoskaia and B. Kalinin for making
very valuable comments on \cite{Llave02}. In particular,
spotting incomplete arguments in support of
Conjecture 1.3 there, which is a particular case of
Theorem \ref{main} of this paper.
I also thank them for very valuable discussions and for
encouragement on the subject of this paper. I thank
A. Gonz\'alez for a careful reading of this paper.
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\end{document}