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\TITLE
Rigidity of
higher dimensional conformal Anosov systems
\AUTHOR Rafael de la Llave
\FROM Department of Mathematics
University of Texas at Austin
Austin TX 78712-1082
\ENDTITLE
\ABSTRACT
We show that Anosov systems in manifolds with trivial tangent bundles and
with the property that the derivatives of
the return maps at periodic orbits are
multiples of the identity in the stable and unstable bundles
are locally rigid. That is, any other smooth map,
in a $C^1$ neighborhood such that it has the same
Jordan normal form at corresponding periodic orbits is
smoothly conjugate to it.
This generalizes results of \cite{CM}.
We present several arguments for the main results.
In particular, we use
quasi-conformal regularity theory.
We also extend the examples of \cite{L2} to
show that some of the hypothesis we make are indeed
necessary.
\ENDABSTRACT
\SECTION Introduction, notation and statement of main results.
\SUBSECTION Introduction.
The main goal of this paper is to prove some rigidity results
for some Anosov systems in higher dimensions.
In our context, rigidity means that, for some systems,
data at periodic orbits determine the smooth conjugacy class.
The data we will consider turn out to be invariants of $C^1$ --- or even
Lipschitz --- conjugacy so that, as a corollary we obtain that systems
in our class which are conjugated by a Lipschitz homeomorphism
are smoothly conjugate.
We will give precise statements later, but we point out that the
main novelty of the results presented here is that they hold
for Anosov systems in arbitrarily high dimensions without using families.
Also, we introduce the use of quasi-conformal techniques.
In spite of the importance of quasi-conformal techniques
in the geometric rigidity program \cite{Mo1},\cite{Mo2} they seem not
to have played much role in the study or rigidity of Anosov
dynamical systems.
We recall that in \cite{L1}, \cite{LM} and, by
other methods in \cite{Po}, \cite{L2}, it was proved that two 2-dimensional Anosov
diffeomorphisms (two 3-dimensional Anosov flows) that are topologically
conjugate and such that the Lyapunov exponents at corresponding periodic
orbits agree are smoothly conjugate.
Since the Lyapunov exponents at periodic orbits are invariants of
Lipschitz conjugacy, we obtain as a corollary that two Lipschitz conjugate
Anosov diffeomorphisms of a 2-dimensional manifold (or Anosov flows
of a 3-dimensional manifold) are smoothly conjugate.
The fact that low dimensionality plays a role was clarified in \cite{L2}
which constructed examples of analytic families of
analytic diffeomorphisms in $\torus^d$ $d\ge4$
which are $C^k$ conjugate but not $C^{k+1}$.
(To the best of our knowledge, the situation in $\torus^3$ remains unexplored.)
The examples in \cite{L2} have the property that the germs of return maps
around periodic orbits are $C^\infty$ equivalent locally, so that, without
extra hypothesis, it is hopeless to try to obtain rigidity out of local
data around periodic orbits for general Anosov systems.
In this paper, we will also extend the examples
of \cite{L2} to show that even assuming
normal forms which are just one non-trivial
Jordan block, rigidity is false. At the end of the
statement of Theorem 1.1, we have collected some
remarks on what is the role of the different assumptions
in this theory. Roughly, it seems to us that the
reason why one needs (in the present state of the theory)
assumptions such as
one-dimensionality or the conformal assumptions used
here is that the one needs to rearrange the
equations obtained taking derivatives of the
equations into coboundary equations that have
much better regularity and existence theory.
For certain automorphisms of $\torus^d$, it was shown in
\cite{CM} that analytic families of analytic diffeomorphisms
that preserved the data at periodic orbits, had to be analytic
changes of variables of the map. The technique used was a
KAM iteration.
In the present paper we present results that generalize the results
of \cite{CM} in that we do not require families, just proximity assumptions.
Also, the class of systems that we can consider are somewhat wider
in principle (as it is sometimes the case in
rigidity theory, one may end up proving
that classes that are wider in principle,
are indeed equivalent. (See e.g Lemma 1.3.)
We also can deal with regularities other that analytic.
We will present two proofs of the main rigidity result, one following
the strategy in \cite{L1},\cite{LM} (we will refer to it as
the ``scattering method'') and another one following the
strategy of \cite{Po},\cite{L2} (we will refer to is
as ``using SRB measures''). Both will require some
technical improvements which may be of independent interest
for the rigidity program.
In the the proof using SRB measures, we need to
introduce quasi conformal estimates and study regularities
in quasi-conformal classes.
The
quasi-conformal classes
appear naturally in this higher dimensional
problems to supplement
the use of invariant measures. This is because
the use of invariant measures gives control over the
Jacobian of the transformation and
eventually shows it is smooth. For one dimensional
functions, the fact that the Jacobian of
a transformation is smooth is clearly enough to show that the
function is smooth. Of course, this is not true in higher dimensions
and we need to supplement the estimates of the Jacobian
with estimates that show that the derivative is ``diagonal''.
Of course, in a manifold, the notion of a derivative being diagonal does not
make sense, since it maps one tangent space into a completely different one.
One possible way out would be to assume that there is a trivialization, but it
is more geometrically natural to use the notion -- in principle weaker -- of
1-quasiconformality. That is, that there exists a metric such that
the lengths of all vectors in the unstable manifold gets expanded by the
same amount.
\SUBSECTION Notation.
We will use $C^\alpha$
$0<\alpha\le \Lip$, $C^{\Lip}$ means Lipschitz functions and $\Lip <1$,
$\Lip >\alpha \quad \forall \alpha <1$.
We also denote by $\|f\|_{C^\alpha}$ the semi-norm
$\sup_{x\ne y} d(f(x),f(y))/d(x,y)^\alpha$.
When $f$ ranges on a Lie group, we can consider
$\|f\|'_{C^\alpha} = \sup_{x\ne y} d(f(x)f^{-1}(y),\Id)/d(x,y)^\alpha$.
These two seminorms are equivalent when the range of $f$ is restricted
to lie in a bounded neighborhood of the identity.
Note that the use of seminorms is rather natural for cohomology equations
since the solutions are defined up to additive (or multiplicative constants).
We recall that, given an Anosov diffeomorphism of (resp.\ an Anosov flow
$X_t$ on) a manifold $M$ we can find a splitting of the tangent bundle
$$T_x M = E_x^s \oplus E_x^u\ \
(\hbox{resp. } T_x M = \{\dot X(x)\} \oplus E_x^s \oplus E_x^u \ )$$
corresponding to stable and unstable directions. For Anosov flows,
we will also introduce the center-stable, center-unstable bundles.
$E^{cs}_x = E^s_x \oplus \{\dot X\}$,
$E^{cu}_x = E^u_x \oplus \{\dot X\}$.
We also recall that, associated to the stable, unstable
bundles (and center-stable, center-unstable bundles
in the case of flows), we can associate the stable, unstable, center-stable, center-unstable foliations.
We will denote
by $W^s_x$, $W^u_x$, $W^{cs}_x$, $W^{cu}_x$.
the leaf of one of the foliations passing through the point $x$
When we want to use just an small
disk of the leaf around the point, we will use the notation
$W^{s,\loc}_x$ and similarly for the others.
Since these foliations depend on the map or flow that we are considering,
if there is possibility of confusion,
we will make the dependence on the map
explicit using
$W^{s,(f)}_x$ and the like.
In the course of the proofs we will need properties of these foliations
but we will refer to the literature for the most standard ones.
We recall that when the map is
$C^r$ , $r = 1, 2, \ldots, \infty, \omega$,
these foliations have $C^r$ leaves and their $r$-jets are H\"older
on the manifold.
(In the case that $r = \omega$, the topology of $r$-jets
is induced by a norm that implies the analyticity. )
It then, makes sense to consider the restriction of an
smooth functions to these leaves and the derivatives along these leaves.
Following \cite{LMM}, we will say that a map is in $C^r_s$ if the restriction to
leaves of the stable foliation is $C^r$ and the $r$-jet is continuous on the manifold.
Similarly, we will denote by $D_s$ the differential of the restrictions.
That is, if $\phi: M \mapsto N$,
$D_s \phi (x)$ will be a linear map from $E^s_x$
to $T_{\phi(x)}N$.
In the important case that $M=N$ and
that $\phi = f$,
since $E^s$ is invariant, we can consider
$D_s f$ as a map from $E^s_x$ to $E^s_{f(x)}$.
Analogous definitions are made for the other invariant bundles.
Of course, these concepts
depend on the foliation and, hence on the map. When there is danger of confusion,
we will denote them as $C^r_{s,(f)}$ and $D_{s,(f)}$.
Since we will also be using some parts of the theory of quasi-conformal maps,
we recall that, given a metric, we can define the ``distortion'' of
a derivative by:
$$
K(Df(x) ) =
{ \max_{v \in T_xM, ||v|| =1} || Df(x) v|| \over
\min_{v \in T_xM, ||v|| =1} || Df(x) v||}
\EQ(distortiondef)
$$
We emphasize that the distortion of a map
depends on the metrics on the domain
and the range that we consider.
Notably, if we have two metrics on
a manifold, the identity map, may
have distortion bigger than one
if we consider it
as a map between the manifold endowed with a metric
to the manifold endowed with the other.
This dependence on the target and domain metrics
will not be included in the notation, following customary
practice, but we will try to make clear
what are the metrics in the domain and the
range since it will be necessary to take advantage of
of the freedom afforded by the choice of metrics with special
properties.
For Anosov systems that leave invariant subbundles,
we can also consider distortions along the subbundles.
That is:
$$
K_s(Df(x) ) =
{ \max_{v \in E^s_x, ||v|| =1} || Df(x) v|| \over
\min_{v \in E^s_x, ||v|| =1} || Df(x) v||}
\EQ(distortiondef2)
$$
We will refer to $K_s$ as the stable distortion.
Analogous definitions will hold for all the other invariant subbundles.
Notice that in that case, we only need to have the metric
defined on the subbundle.
If a map is differentiable almost everywhere, we will call the
distortion of the map, the supremum of the distortions of the derivatives
and say that the map is $K$-quasi-conformal if it is differentiable almost
everywhere and the distortion is bounded by $K$.
Similarly, given any metric on the stable bundle, which is
smooth restricted to the stable leafs, (recall that the
stable bundle is the tangent bundle of the stable leafs)
we
will speak about stable quasi-conformal maps when almost everywhere
-- with respect to this metric --
on any leaf of the stable foliation, the map restricted to the stable leaf is
differentiable, the derivative sends elements of the stable bundle into
elements of the stable bundle and the stable distortion is bounded.
Analogous definitions hold for the other bundles.
\SUBSECTION Statement of results.
Our main rigidity is the following:
\CLAIM Theorem(main)
Let $M$ be a compact manifold with a trivial
tangent bundle. Let
$f$ be a $C^r$ $(r=2,3,\ldots,\infty,\omega)$ transitive Anosov
diffeomorphism of (resp.\ $X_t$ a transitive Anosov flow on) $M$
such that:
\item{\rm i)} If $f^n(x) = x$, then
$Df^n(x): E^s_x \longrightarrow E^s_x$ and
$Df^n(x): E^u_x \longrightarrow E^u_x$ are multiples of the identity.
(resp. If $X_T(x) = x$, then
$DX_T(x): E^s_x \longrightarrow E^s_x$ and
$DX_T(x): E^u_x \longrightarrow E^u_x$ are multiples of the identity.)
\item{\rm ii)} Assume moreover that
we can find continuous trivialization
$\Psi^s_x : E^s_x \to \real^s$,
$\Psi^s_x : E^s_x \to \real^s$
such that
$$
\eqalign{
&\Psi^s_{f(x)} D_s f(x) (\Psi_x^s)^{-1} = \gamma_s(x) \Id_s \cr
&\Psi^u_{f(x)} D_u f(x) (\Psi_x^u)^{-1} = \gamma_u(x) \Id_u \cr
}
\EQ(trivialization)
$$
where $\gamma_s$, $\gamma_u$ are real valued functions.
\vskip 1 em
Assume that $g$ is a $C^r$ diffeomorphism of (resp.\ $Y_t$ an Anosov flow on)
$M$ and $h:M\to M$ is a homeomorphism in such a way that
\vskip1pt
\item{\rm a)} $h\circ g = f\circ h$ (resp.\ $h\circ X_t = Y_t \circ h$)
\item{\rm b)} If $f^n(x) = x$, then $Df^n(x)$ and $(Dg^n)\circ h(x)$ have
the same Jordan normal form (resp.\ if $X_T(x)=x$ then $DX_T(x)$,
$(DY_T)$ ($h(x))$ have the same Jordan normal form)
\item{\rm c)} $g$ is sufficiently close to $f$ in $C^1$.
(resp. $X$ is sufficiently $C^1$ close to $X$.)
\vskip1pt
Then, $h$ is $C^{r - \epsilon}$ for all $\epsilon > 0$.
In other words, the diffeomorphisms satisfying condition $i)$ and $ii)$ in
\clm(main) are ``locally rigid''. (Locally refers to the fact that
we need to assume that the diffeomorphism $g$ is in a neighborhood.)
This is, of course, more restrictive than proving the result
for all diffeomorphisms but is less restrictive
than assuming that there is a family which satisfies the same
assumptions for all its elements.
Note that the condition $i)$ can be
verified by knowing the conjugacy class of the
derivatives at periodic points and that condition
$b)$ is a necessary condition for Lipschitz conjugacy.
Examples of diffeomorphisms and flows satisfying condition $i)$ and $ii)$ of
\clm(main) can be readily constructed by taking linear isomorphisms
of tori or of nilmanifolds with the appropriate spectrum.
There is an small redundancy between conditions $i)$ and $ii)$.
We will show that if there is a diffeomorphism $f$
satisfying conditions $i)$ and $ii)$, then, in a $C^1$ neighborhood,
all the diffeomorphisms satisfying condition $i)$ also satisfy condition $ii)$.
(See Proposition 2.6.) In particular, given a
linear automorphism satisfying $i)$, we can find
a neighborhood so that all the diffeomorphisms satisfying $i)$
also satisfy $ii)$.
We also note that it follows from assumption $ii)$ that the
trivializations are somewhat smoother. Indeed, we will
show that it is possible to choose
$ \gamma_s \Psi^s , \Psi^u \in C^{r-1}$. (See Proposition 2.5. )
\REMARK
We are not sure that assumption $ii)$ is necessary.
The reason why it enters in the proof is that the derivative of
the conjugacy equation ( $a) $ in \clm(main) ) is
$$
Df\circ h Dh = Dh \circ g Dh
\EQ(conjder)
$$
Given that the conjugacy itself can be constructed, we can consider
\equ(conjder) as an equation for $Dh$. Unfortunately the only
equations for which have a good theory (the smooth Livisic theory)
is for equations of
the form
$$
Dh\circ g Dh^{-1} = \eta(x)
\EQ(coboundary)
$$
Note that \equ(coboundary) does not have a geometric meaning.
The maps $Dh$ and $Dh \circ $
are supposed to be maps from some tangent spaces so that the product in
\equ(coboundary) does not make geometric sense. Nevertheless, with
a trivialization, we can indeed make sense of equations
such as coboundary and apply the rich smooth Livsic theory.
Such rearrangements already were implicit in the one dimensional
results.
It seems possible to us that, using more systematically the
existence of a conformal metric, one could
perhaps eliminate this assumption.
\REMARK
We note, however that condition $ii)$ of \clm(main) is satisfied for
nilmanifolds and infranilmanifolds. These are the only manifolds
for which examples of Anosov systems are known and, indeed, it is
an outstanding conjecture that these are the only manifolds
for which Anosov systems exist and that they are all
transitive. For flows, the situation is much less clear,
since it is known that there are non-algebraic Anosov flows.
(See \cite{HT})
\REMARK
The reason why we need to assume $c)$ is related to the fact that
we only have existence results for
Livsic equation \equ(coboundary) in non-commutative groups
when the range is sufficiently close to the identity.
The result without such assumptions was stated in
\cite{Li2}. It is nevertheless well known and accepted
that the proof presented has serious gaps.
On the other hand, the smooth Livsic theorem
as we will state it does not require smallness assumptions.
\REMARK
Note that we do not need to assume that the Lyapunov exponents at periodic
orbits are the same for all the periodic orbits as it is the case for the maps
that are produced using automorphisms of tori or of nilmanifolds.
We only need that the return maps at periodic orbits are
diagonal on the stable and unstable manifolds.
It could well happen that the spectrum at different periodic
points was different.
Nevertheless, we will show (See Lemma 1.3 ) that when the stable
and unstable manifolds are two dimensional, the fact that the return
maps are diagonal implies that the Lyapunov exponents are constant.
We also note that, as a part of
the proof, we also obtain are result that may be
of independent interest.
\CLAIM Theorem (conformal)
Let $f$ be a $C^r$ transitive Anosov diffeomorphism
on a compact manifold $M$ with a trivial tangent bundle.
(resp. $X_t$ a transitive Anosov flow).
Assume that:
\item{$i)$} $f$ (resp. $X_t$) satisfies assumption $i)$ $ii)$ of \clm(main).
\vskip1pt
Then
there exists a $C^{r-1}_u$ metric $g^u$ on the bundle $E^u$ such that,
in that metric,
$K_u (Df(x) ) = 1 \forall x \in M$
(resp. $K_u( DX_t(x) = 1 \forall x \in M, t \in \real$.)
And analogous results for the stable bundle and, in the case of
flows, for the center-unstable, center-stable.
\REMARK In the above result if
one makes assumption $i)$, $ii)$ for
one of the invariant bundles, it is possible
to recover the conclusions for that bundle.
A consequence of
\clm(conformal) is
the following:
\CLAIM Lemma (Liouville)
Assume that a transitive Anosov system
satisfies the conclusions of \clm(conformal)
(In particular, i) and ii) of \clm(main) )
and that the unstable manifold is two-dimensional.
Then, the unstable Lyapunov exponents are constant.
\PROOF
Consider the stable manifold of a fixed point.
The metric $g^u$ on which the map is confomal
restricted to the unstable manifold is
differentiable and bounded with respect to the
natural Riemannian metric.
Recall that the stable manifold is topologically $\real^2$.
By the Riemann measurable mapping theorem, we can find a
bounded and differentiable change of
variables that reduces the metric $g^u$ to standard and
$f$ to conformal.
Now, given the fact that the map is bounded, we discover that
restricted to the unstable manifold in these coordinates
$|f(x)| \le K |x|$ and is conformal. By Liouville's theorem,
we obtain that $f$i reduces to multiplying by a constant in these
coordinates.
Now, we recall that, by the transitivity of the
Anosov diffeomorphism, the unstable manifold is dense.
It follows that we can approximate arbitrary well the exponents in
a periodic orbit by the exponents of a finite orbit in the unstable
manifold. But those are constant.
\QED
Note that a corollary of \clm(conformal) is
that if $f$ is an Anosov system satisfying
assumption $i)$ of \clm(main), then,
for any metric $K_u(f^n) \le C$,
$K_s(f^n) \le C$ with $C$
independent of $n$.
We will also prove the following result:
\CLAIM Theorem (almostLip)
Let $f$ be a transitive Anosov diffeomorphism
satisfiying $i)$, $ii)$ of \clm(main).
Then, given $\epsilon > 0$
we can find $\delta > 0 $ such that
if $d_{C^1}(f,g) \le \delta$,
the homeomorphism $h$
satisfying
$$ f\circ h = h \circ g ; \qquad d_{C^0}(h,\Id) \le K \delta, $$
given by
structural stability satisfies
$$ h, h^{-1} \in C^{1 -\epsilon}.$$
This result is very similar to the
result of \cite{PV}, which used very heavily that
the systems had one-dimensional stable and unstable leaves.
We postpone the proof of this theorem till
a subsequent section.
Another fact that reinforces the idea
that systems satisfying $i)$ and $ii)$
of \clm(main) are in some respects
one dimensional is the following observation.
\CLAIM Lemma (smoothfoliation)
Let $f$ be a transitive Anosov flow
satisfying $i)$ $ii)$. Then, the Anosov
splitting is $C^{1+\alpha}$ for
every $\alpha > 0$.
\PROOF
The standard proof of the
regularity of the splitting is
to use the invariant section theorem.
We see that the
action on the $C^r$ sections of
the Grasmannian of maps close to the
stable bundle has the norm
$$
\sup_x | Df(x)|^r |D_s f(x)| |(D_u f)^{-1}(x) |
\EQ(bundlesmooth)
$$
Hence, provided that $r$ is such that
\equ(bundlesmooth) is smaller than $1$, we obtain that the
bundle is $C^r$. (In the original paper
\cite{HP}, \equ(bundlesmooth) is substituted by the
expression that is obtained taking the sup in all
the factors. In \cite{HK}, one can find the remark
that it is \equ(bundlesmooth) what is needed.
Morever, using the fact that the invariant splitting
for $f$ is the same as that for $f^n$ one can use
$f^n$ in place of $f$ in \equ(bundlesmooth). )
Of course, we can obtain the same result in
any smooth metric.
We have shown in \clm(conformal) that we can
obtain a metric $g_u \in C^{r-1}_u $
in the stable bundle for which
$|Df(x)| | Df^{-1}(f(x))| = 1$.
We also have a metric on the
stable bundle so that
$|D_sf(x)| \le \lambda < 1$.
We can consider a metric which is the
direct sum of the two for which
obviously,
\equ(bundlesmooth) is satisfied for $r > 1$.
This metric is not smooth, but if we replace it by
an smooth approximation, the bounds do not change much and
the argument shows that indeed there is a $C^{1+\alpha}$ splitting.
\QED
\SECTION Preliminaries for the proofs of \clm(main)
\SUBSECTION Overview of the arguments
The proofs we will present have several recognizable milestones
\item{1)} Replacing $h$ by a smoother function.
\item{2)} Proving that $h$ is absolutely continuous
\item{3)} Bootstrapping the smoothness
For each of the steps 2) and 3) we will present two independent proofs.
For 2) we will show how to obtain the result using the scattering method
of \cite{L1}, \cite{LM} and how
--- when dimension of the stable or unstable manifolds is bigger than 1 ---
using quasi-conformal estimates.
For 3) we will present a bootstrap argument similar to that in \cite{L1}, \cite{LM}
and another one based on the use of SRB measures.
The latter one has the advantage that when the stable or unstable
manifolds are 1-dimensional, it makes unnecessary step~2.
Also, it is the only bootstrap we are aware of that can establish
analytic regularity.
We hope that presenting alternative arguments may be useful to the
readers interested in extending the theory.
For the experts, we point out that
the step of
local smoothing has been significantly
stremlined because we have
removed the restriction that the smoothed map is
a homeomorphism.
(The need to get a
local homeomorphism
was the main reason why the result in \cite{L1} was only local.)
This is accopplished by observing that the construction
of \cite{Sh} produces a homeomorphism
even if the starting point of the iterative process
is not a homemorphism.
(See Section 2.2)
As usual in many rigidity results
for Anosov systems, an important role
will be played by theorems that assert existence and
smoothness of solutions of cohomology equations.
Here, we will have to consider cohomology equations over
non-commutative groups and we review briefly the theory
we need.
To simplify the notation we will do the detailed work only with
diffeomorphisms
and deduce the results for flows by taking time one maps of the later.
This will require that our regularity results are worked both for
Anosov diffeomorphisms and for time maps of Anosov flows.
If we take $f=X_1$, $g= Y_1$, the conjugacy between $X$ and $Y$ is a
conjugacy between $f,g$.
In the proof we will show that provided that $f,g$ have stable and unstable
manifolds, the conjugacies are smooth when restricted to the stable
and unstable manifolds.
If $f$ and $g$ are Anosov diffeomorphisms, by Lemma 2.3 of \cite{LMM}
or \cite{Jo}, this implies that they are smooth.
If $f$ and $g$ are time one maps of Anosov flows, we only have to remark
that smoothness of $h$ along the direction of the flow is obvious and that
the regularity lemmas of \cite{LMM}, \cite{Jo} can be made to work in this
situation (Lemma~2.3 of \cite{LMM} does not require any change.
The result in \cite{Jo} only works when we have two transverse foliations,
but we can first work on the leaves of $W_x^{cs}$ and apply \cite{Jo} to
the orbits and the stable foliation and then apply the result to the
transverse foliations of $W_x^{cs}$ and $W_x^u$.)
\SUBSECTION M.~Shub's theory of conjugacy of expanding maps and local smoothing.
We recall that in \cite{Sh} it was proved that if $f,g$ are expanding maps
of a manifold in the same homotopy class
$$h= \lim_{n\to\infty} f^n g^{-n}$$
exists and is a homeomorphism that satisfies
$$h\circ g = f\circ h$$
Here we want to state another theorem that is proved by the same methods
as the result in \cite{Sh}.
We will be considering maps that have a contracting foliation.
That is, we assume that and we can find $C>0$, $0<\lambda <1$, $\lambda<\mu$
$$
\eqalign{
&T_x M = E_x^s \oplus E_x^{ns} \cr
& v\in E_x^s \Leftrightarrow |Df^n (x) v| \le C\lambda^n |v|\qquad
n\ge 0 \cr
& v\in E_x^{ns} \Leftrightarrow |Df^n (x) v|\le C\mu^n |v|\qquad
n\le 0
}
\EQ(hyperbolic)
$$
We recall (see \cite{HP}, \cite{Fe1}, \cite{Fe2})
that we can associate a strong stable
foliation whose leaves are characterized by
$$
y\in W_x^s \Leftrightarrow d(f^nx,f^ny) \le C\lambda^n d(x,y)\qquad n\ge0
\EQ(manifoldrates)
$$
This foliation has $C^r$ leaves, and the $C^r$ jets of the leaves are
H\"older functions on the manifold.
Also, $T_xW_x^s = E_x^s$.
We also remark that we can introduce an adapted metric so that all the
$C$'s in the previous estimates \equ(hyperbolic) \equ(manifoldrates)
become~$1$.
We will do that since it simplifies the estimates without any loss of
generality.
Note that because the leaves are smooth, it makes sense to restrict the
ambient Riemannian metrics to them.
If $x$ and $y$ are in the same leaf, we define $d_s(x,y)$ to be the distance
along $W_x^s$ considered as a Riemannian manifold on its own.
We emphasize that, even if $d_s (x,y)\le d(x,y)$, $d_s(x,y)$ can be much
bigger than $d(x,y)$ since $W_x^s$ can wrap around the manifold and come
back close to itself.
For the applications we have in mind (Anosov systems and time-1 maps of
Anosov flows) we can take $\mu$ to be equal to 1.
In that case, we point out that the stable foliation is invariant under
homeomorphisms.
That is, if $f,g$ are as above, and $h\circ f= g\circ h$, with $h$
homeomorphism then,
$$
\eqalign{
y\in W_x^{s,f} &\Leftrightarrow d(f^n(x),f^n(y)) \to 0 \Leftrightarrow\cr
\Leftrightarrow d(h\circ f^n (x),h \circ f^n(y)) \to 0 \cr
&\Leftrightarrow d(g^n \circ h(x),g^n \circ h(y))\to 0 \Leftrightarrow
h(y)\in W_{h(x)}^{s,g}\cr}
\EQ(argument)
$$
Note, however, that we have formulated the following result
in such a way that this
is not needed.
\CLAIM Theorem(shub)
Let $f,g$ be diffeomorphic with stable foliations.
Let $h$ be a homeomorphism of the manifold such that:
\item{$i)$}$h\circ f= g\circ h$ ( and, therefore, by \equ(argument) $h(W_x^{s,f}) = h(W_x^{s,g})$ )
\vskip 1 pt
Let $k$ be a map --- not necessarily invertible --- such that
\item{$ii)$}$k(W_x^{s,f}) \subset W_{h(x)}^{s,g}$
\item{$iii)$} $\sup_x d_s (k(x),h(x)) < \infty. $
Then
$$h(x) = \lim g^n \circ h\circ f^{-n} (x)
\EQ(scattering)$$
and the limit is reached uniformly in the distance $d_s$.
\PROOF
We just observe that if we consider the space of maps $k$ satisfying ii)
in \clm(shub) topologized with
$$d(k_1,k_2) = \sup_x d_s (k_1(x), k_2 (x))$$
it is a complete space and that the map $k\to \C(k) =f\circ k\circ g^{-1}$ is
a contraction because
$$\eqalign{ d_s(f\circ k_1\circ g_{(x)}^{-1}, f\circ k_2\circ g^{-1}(x))
& \le \lambda d_s (k_1\circ g^{-1}(x), k_2\circ g^{-1}(x)) \cr
& \le \lambda \sup_y d_s (k_1(y), k_2 (y))\cr}$$
Since that last term is independent of $x$, when we take $\sup_x$ in the
left hand side, we prove that $\C$ is a contraction and hence, $\C^n(k)$
converges to the unique fixed point of $\C$.
\QED
The space of maps satisfying ii) is not empty because we assume that
there exists an $h$ that satisfies that.
\REMARK
Even if we will not need it, we point out
the map $k$ produced in \clm(shub) is a homeomorphism if it
is homotopic to the identity.
This is because, if we reverse the role of $f$ and $g$ we can obtain
another map $\tilde k$ that satisfies $\tilde k \circ g = f \circ \tilde k$.
Then we have, $ k\circ \tilde k \circ g = g \circ k \circ \tilde k$,
$\tilde k \circ k \circ f = f \circ \tilde k \circ k$. Since
$f$ and $g$ are expansive, we obtain the $k \circ \tilde k = \Id$,
$\tilde k \circ k = \Id$.
We will also need the following result that tells us that we can
start the scattering process in \clm(shub) from a slightly smooth
map.
\CLAIM Lemma(smoothing)
Let $f,g$ be either a $C^r$ Anosov $(r=1,\ldots,\infty,\omega)$ Anosov
diffeomorphism or a time one map of an Anosov flow.
Let $h$ be a homeomorphism satisfying:
\vskip1pt
\item{\rm i)} $h\circ f= g\circ h$
\vskip1pt
\noindent (Hence $h(W_x^{s,f}) = W_{h(x)}^{s,g}$.)
Then, we can find a map $k$ that:
\vskip1pt
\item{\rm a)} $k(W_x^{s,f}) = W_{h(x)}^{s,g}$
\item{\rm b)} $\sup_x d_s (k(x),h(x)) <\infty$
\item{\rm c)} $k\in C_s^{\infty,f}$
\item{\rm d)} The $r$-jets of $k$ are $C^\alpha$ on the manifold.
\item{\rm e)} The map $k$ can be chosen to
be as $C^0$ close to $h$ as we desire.
\PROOF
The idea of the proof is the observation that, given a map $h$ that sends
the stable foliation of $f$ into that of $g$, we can produce --- using local
coordinates --- another map which is smooth in all the places where $h$ was
and, moreover, in a neighborhood, is as smooth as in the claim of
the result.
By repeating the process a finite number of times, we end up with a map
which is smooth everywhere.
Recall that, by the standard theory of invariant (un)stable foliations,
(see e.g., \cite{A, \S16}), in any neighborhood of a point, we can find
a coordinate system that sends the foliation into the standard
foliation obtained taking Cartesian products of open sets in
Euclidean spaces.
More precisely, denoting by $B_\rho^s, B_\rho^u$ the unit balls in
$\real^s,\real^u$, we can find maps
$$\Lambda_x^f : B_\rho^s\times B_\rho^u \to M $$
in such a way that
\item{a)} $\Lambda_x^f (0,0) = x$
\item{b)} $\Lambda_x^f (B_\rho^s \times \{y\}) =
W_{\Lambda_x^f(0,y)}^{s,f,\hbox{loc}}$
\item{c)} The mapping $\Lambda_x^f (\cdot,y)$ is a $C^\infty$ local
diffeomorphism and the $C^\infty$ jets of these maps depend continuously
on $y$.
\item{d)} A finite number of these coordinate patches covers the manifold.
Of course, analogous maps can be constructed for $y$.
If $h$ is a map that sends the stable foliation of $f$ into that of $g$,
we have:
$$\Lambda_{h(x)}^{g^{-1}} \circ h\circ \Lambda_x^f (s,u)
= (\alpha (s,u),\beta(u))$$
with $\alpha,\beta$ continuous mappings on an open neighborhood of the origin.
We can now approximate the function $\alpha(s,u)$ by another function
$\tilde\alpha$ such that
\item{a)} $\tilde\alpha (s,u)$ agrees with $\alpha (s,u)$ if
$|s| >\tilde\rho$, $|u| >\tilde\rho$
\item{b)} $\tilde\alpha (\cdot,u)$ is $C^\infty$ and the $C^\infty$ jet
depends continuously on $u$.
\item{c)} The modulus of continuity of $\tilde\alpha (\cdot,u)$ is not
bigger than that of $\alpha(\cdot,u)$.
\noindent
(A convenient way of doing this is setting
$$
\tilde\alpha (s,u) = \int K(s,s';u) \alpha (s',u)\,ds
$$
where
$K(s,s';u) = \delta (s-s')$ if $u$ or $s$ are large.)
The map $\tilde h = \Lambda_{h(x)}^g (\tilde\alpha,\beta)\circ
\Lambda_x^{f^{-1}}$ agrees with $h$ outside of a neighborhood --- so that it
can be extended by $h$.
Notice also that $\tilde h$ is certainly $C_s^{\infty,f}$ in the places where
$h$ is and also in a neighborhood of $x$.
By repeating the process a finite number of times we end up with a map that
satisfies the conclusions of the theorem.
\QED
\REMARK
We also note that the proofs we present could work with considerably
less smoothness of the map $k$.
For the first proof using the scattering method it would suffice that $k$
is in $C_s^{1,f}$ and for the proof using quasi-conformal estimates it
would suffice that it is quasi-conformal.
\SUBSECTION \vbox{\hbox{Smooth Livsic theory for non-commutative maps.}
\hbox{Existence and properties of trivializations.}
\hbox{ Proof of \clm(conformal)}
}
A very important role in the argument will be played
by the Livsic theorems that discuss the existence
and smoothness of solutions to cohomology equations
over transitive Anosov systems.
In this sections, we will review them briefly and
show how they can be used to prove existence
and regularity of trivializations and also
to prove \clm(conformal).
We will find very important the following results.
\CLAIM Theorem(Livsic)
Let $f$ be a transitive Anosov diffeomorphism of (resp.\ $X_t$ a
transitive Anosov flow on) a compact manifold $M$.
Let $G$ be a Lie group, $\G$ its algebra.
Let $\eta :M\to G$ (resp.\ $\eta :M\to \G$) be a $C^\alpha$ function
$0<\alpha \le \Lip$.
\vskip1pt
Assume that:
{\rm i)}
Whenever $f^n(x) =x$, then $\eta (f^{n-1}(x)) \cdots \eta (x)=\Id$
(resp.\ whenever $X_T(x)=x$ then $\Gamma(t)$ the solution of
${d\over dt}\Gamma(t) =\eta (X_t(x))\Gamma(t)$;
$\Gamma(0)=\Id$ verifies $\Gamma(T) = \Id$)
\vskip1pt
{\rm ii)}
If $G$ is neither compact, abelian nor nilpotent assume $\eta (M)$ lies
in a sufficiently small neighborhood of the identity in $G$.
\vskip1pt
Then,
\vskip1pt
{\rm a)}
There exist a $\varphi :M\to G$ such that
$$\varphi (f(x))\ \eta (x) = \varphi (x)
\EQ(cohomology)$$
(resp.
$$-\varphi^{-1}(x)\ \eta (x) = \dot\varphi (x)
\EQ(cohomology2)$$
where $\dot\varphi (x) = {d\over dt} \varphi (X_t(x))|_{t=0}$)
\vskip1pt
{\rm b)}
$\varphi$ is $C^\alpha$ and $\|\varphi \|_{C^\alpha} \le K\|\eta\|_{C^\alpha}$
\REMARK
The statement of \clm(Livsic) for diffeomorphisms can be found in \cite{Li2}.
In that paper, one can also find a statement of the theorem without
assumption ii).
Unfortunately, it seems well known among the experts that the
argument presented there for this part is false
and indeed, proving \clm(Livsic)
without assumption $ii)$ is
one of the outstanding problems. The argument for the result
as stated can be found in \cite{Li2}.
More modern expositions can be found in \cite{NT1} and
\cite{L3}.
\REMARK
Note that \equ(cohomology2) is an infinitesimal version of \equ(cohomology).
If we write, $f(x) = 1+\varep \dot X$ and
$\eta (x) = \Id + \varep \hat\eta (x)$ --- with $\hat\eta$ an element of
the Lie algebra! --- then, substituting
$$\varphi (x+\varep \dot X)^{-1} = \varphi^{-1} (x) - \varep \varphi^{-1}
(x) \dot\varphi (x) \varphi^{-1}(x)$$
into \equ(cohomology) we derive \equ(cohomology2).
This heuristic way of thinking will show how we can go from one to the
other by ``integrating'' or ``taking derivatives.''
\REMARK
Note that condition i) is clearly a necessary condition for the
existence of solutions to \equ(cohomology).
In effect, if such solution existed
$$\eqalign{
&\eta (f^{n-1}(x))\cdots \eta (x) =\cr
&(\varphi (f^n(x)))^{-1} \varphi (f^{n-1}(x)) \
(\varphi (f^{n-1}(x)))^{-1} \varphi (f^{n-2}(x))\cdots
(\varphi (f(x)))^{-1} \varphi (x) =\cr
&= (\varphi (f^n x))^{-1} \varphi (x) = \Id\cr}$$
A similar argument works for the case of flows.
The regularity of the solutions is
\equ(cohomology), \equ(cohomology2) is
addressed by the following result.
\CLAIM Theorem(smoothlivsic)
Let $f$ be a $C^{r+1}$ $r= 1,2,\ldots,\infty,\omega$
Anosov diffeomorphism of (resp.\ $X_t$ a $C^{r+1}$ Anosov flow on)
a compact manifold $M$.
Let $\eta(x)$ be a $C^r_s$ function from $M$ to a Lie group $G$
(resp.\ to a Lie algebra $\G$).
Let $\phi$ be a continous function from $M$ to $G$
such that it satisfies \equ(cohomology) (resp. \equ(cohomology2))
\vskip1pt
Then $\phi$ is $C^{r}_s$.
Analogous ressults hold for regularity along the unstable
foliation.
In particular, if $\eta \in C^r \subset C^r_s \cap C^r_u$,
then, $\phi \in C^r_s \cap C^r_u \subset C^{r-1 + \Lip}$.
The inclusion $ C^r_s \cap C^r_u \subset \Lambda^r \subset C^{r-\epsilon}$
was established in \cite{LMM} for $r \le \infty$.
Another proof can be found in \cite{Jo}.
The paper \cite{HK} contains a proof for
the $r = \infty$ case. Even if the method of
proof there leads to a higher loss that
$\epsilon$ (with this method
one can only conclude that
$ C^r_s \cap C^r_u \subset C^{r-n-2}$),
the method can be extended (See \cite{L4}) to
conclude the result for $r = \omega$.
The proof of \clm(smoothlivsic) can be
found in \cite{NT1} and in \cite{L3}.
As a simple corollary of
\clm(smoothlivsic) is the following:
\CLAIM Proposition (smoothtrivial)
Let $f$ be as in \clm(main). Then,
it is possible to choose
$\gamma_s, \Psi^s \in C^{r-1}_s$.
Analogous results hold for
the trivialization of the unstable subbundle.
Actually we will show that the only thing that we need to
do to obtain an
smooth $\Psi^s$ is to multiply $\Psi^s$ by a real valued function.
\PROOF
We can assume that there is a $C^{r -1}$ metric on the manifold.
For $r \le \infty$ this is an easy argument based on particions
of unity. For $r = \omega$ it is a deep theorem
in \cite{Gr}.
We note that we can multiply
$\Psi^s$ by the factor
$(\det \Psi_s)^{1/d_s}$, where $d_s$ is the dimension
of the stable subbundle without altering that it is
a trivialization -- note that the determinant
cannot vanish because of the assumption that it
is a trivialization which is invertible-- . Hence, we can assume without loss of
generality that $\det \Psi^s = 1$. Of course,
the $\gamma_s$ gets modified accordingly.
When these normalizations are enforced,
we obtain that
$$
\gamma_s = \det D_s f
$$
which, of course belongs to $C^{r-1}_s$.
We can then write
\equ(trivialization)
as
$$
\Psi_{f(x)} \Psi_x^{-1} = (1/\gamma_s) D_s f(x)
$$
This is an equation that falls under the scope of
\clm(smoothlivsic).
\QED
\CLAIM Proposition (stabletrivial)
Assume that $f$ is an Anosov diffeomorphism in
a manifold satisfying $ii)$ of \clm(main).
Then, there exist a
$C^1$ neighborhood $V$ of $f$ such
that if $g \in V$ satisfies $i)$, it also satisfies $ii)$.
\PROOF
We denote by $\Psi^{s,(g)}, \Psi^{s,(f)}$ the trivializations
corresponding to the maps $f$, $g$ and, similarly $\gamma_s^{(f)}, \gamma_s^{(g)}$.
As in \clm(smoothtrivial), we can assume that there is an
analytic metric in the mainfold and
that $\gamma_s^{(f)} = (\det D_s f)^{1/d_s}$,
$\gamma_s^{(g)} = (\det D_s g)^{1/d_s}$.
We note that because
$
\Psi^{s,(f)} \circ f D_s f (\Psi^{s,(f)})^{-1} = \gamma_s^{(f)} \Id
$ and $f,g$ are $C^1$ close, we have
$$
\Psi^{s,(f)} \circ g D_s g (\Psi^{s,(f)})^{-1} = \gamma_s^{(f)} \Id + \Delta
$$
where $\Delta$ is small.
Note also that
$$
(1/\gamma_s^{(g)} )\Psi^{s,(f)} \circ g D_s g (\Psi^{s,(f)})^{-1} =
\gamma_s^{(f)}/\gamma_s^{(g)} \Id +
(1/\gamma_s^{(g)})\Delta
$$
satisfies the hypothesis of \clm(Livsic) when $g$ satisfies hypothesis $i)$ of
\clm(main).
Then, we conclude that we can find $\Gamma$ such that
$$
\Gamma \circ g ( \gamma_s^{(f)}/\gamma_s^{(g)} \Id +
(1/\gamma_s^{(g)}\Delta ) (\Gamma \circ g )^{-1} = \Id
$$
Hence, we can take
$\Psi_s^{(g)} = \Gamma \Psi_s^{(f)} $.
\QED
Once we have the trivilizations, it is elementary
to give a proof of \clm(conformal) in our case.
The metric is just the pull back of the
standard metric of $\real^n$ given the trivializations.
We point out that it is possible
to give a much more geometric proof that does not require
that the bundle of the manifold is trivial.
The idea is that we fix the metric at one point
with a dense orbit. We can propagate the metric
along the orbit by just doing a push forward
and multiplying by a conveniently chosen factor.
The same arguments used in the proof of
the non-commutative Livsic theorem yield
the result. Full details of this construction appear
in \cite{L3}. Nevertheless, for our purposes, the
proof using the trivialization is enough.
\SECTION Almost everywhere differentiability of the conjugacy.
The main goal of this section is
to present two proofs of the following result.
\CLAIM Theorem(abscont)
With the notations above, and in the conditions of \clm(main) and
\clm(shub), the map
$$h= \lim_{n\to\infty} g^n \circ k\circ f^{-n}$$
is absolutely continuous
when restricted to any $W_x^{s,f}$.
In a first subsection, using the scattering method, we will
prove it is
Lipschitz and in a second subsection, using quasi-conformal
maps, we will show that the limit is quasiconformal.
\SUBSECTION Proof of \clm(abscont) using the scattering method
We note that, by \clm(shub) the maps $h_n= g^n\circ k\circ f^n$ are
converging uniformly so that it suffices to show that the stable derivatives
of these maps are bounded independently of $x$ and of $n$.
Since the trivialization maps $\Psi$ and their inverse norms which are bounded
independently of $x$ and $n$, it suffices to bound
$$\|\Psi_{R_n(x)}^{s,g} D_s h_n (x) (\Psi_x^{s,f})^{-1}\| \equiv E_n(x)
\EQ(toestimate)$$
independently of $n$ and $x$.
Taking $D_s$ in the definition of $h_n$, applying the chain rule and
introducing the notation
$$\eqalign{
M(x) & = \Psi_{g(x)}^{s,g} D g(x) (\Psi_x^{s,g})^{-1}\cr
N(x) & = \Psi_x^{s,f} Df^{-1} (x) (\Psi_{f(x)}^{s,f})^{-1}\cr}
\EQ(notation)$$
we see that we can rewrite \equ(toestimate) as:
$$\eqalign{
E_n(x) &= \|M(g^{n-1}\circ k\circ k\circ f^{-n}(x) \cdots
M(k\circ f^{-n}(x)) \Psi_{kf^{-n}(x)}^{s,g} D_s k(f^{-n}(x)) \cr
&\qquad (\Psi_{f^{-n}(x)}^{s,f})^{-1} N(f^{-n+1} (x)) \cdots N(x)||
}
\EQ(toestimate2)$$
Note that the $\Psi$'s that appear in the previous formula and the
$D_sk$ are uniformly bounded, so that it suffices to show that the product
of $M$'s and $N$'s is uniformly bounded.
Note that the $M$ and $N$'s are matrices that are multiples of the
identity so that the proof can be carried out in a manner very similar
to that in \cite{L1}.
We define $m(x)$, $\ell(x)$ by:
$$e^{m(x)} \Id = M(x)\qquad e^{\ell(x)} \Id = N(x)$$
Then, with this notation, to show that \equ(toestimate2) is bounded it
suffices to show that
$$\sup_x \big| [m\circ g^{n-1}\circ k+\cdots+ m\circ k+
\ell\circ f +\cdots +
\ell\circ f^n] \circ f^{-n} (x) \big|
\EQ(toestimate3)$$
is bounded independently of $n$.
Note first that in taking suprema over $x$ we can ignore the last $f^{-n}$.
We also note that since
$d(g^{n-1}\circ h(x), g^{n-1}\circ k(x)) \le K\lambda^n$ for some
$0< \lambda <1$ and since $m$ is a $C^\alpha$ function
$$\eqalign{
&\sup_x |m\circ g^{n-1} \circ k(x) +\cdots + m\circ k(x) - m\circ g^{n-1}
\circ h(x) +\cdots + m\circ h(x)|\cr
&\qquad \le \sup_x |m\circ g^{n-1} \circ k(x) - m\circ g^{n-1}\circ h(x)|\cr
&\qquad\qquad +
\sup_x |m\circ g^{n-2} \circ k(x) - m\circ g^{n-2} \circ h(x)|\cr
&\qquad\qquad + \cdots + \cr
&\qquad\qquad + \sup_x |m\circ k(x) - m \circ h(x)| \cr
&\qquad \le K \lambda^{\alpha n} + K\lambda^{\alpha (n-1)} +\cdots +
K\lambda \le K\cr}$$
so that to show that \equ(toestimate3) is bounded it suffices to show that
$$\eqalign{
&\sup_x |m\circ g^{n-1}\circ h(x) +\cdots + m\circ h(x)
+ \ell\circ f^n +\cdots + \ell\circ f| =\cr
&\qquad = \sup_x |m\circ h\circ f^{n-1} (x) +\cdots + m\circ h(x) +\cr
&\qquad\qquad \ell\circ f^{n-1} (x) +\cdots + \ell(x)
+ [\ell\circ f^n (x) - \ell(x)] |\cr}$$
where we have used that $h$ satisfies the intertwining relation.
Ignoring the terms in $[\cdots]$, which are obviously bounded, we have
that the term to estimate becomes, calling $r(x) = m\circ h(x)+\ell(x)$
$$\sup_x |r\circ f^{n-1} (x) +\cdots + r(x)|
\EQ(toestimate4)$$
We now observe that, if $f^n(x)=x$
$\sum_{i=0}^{n-1} n(f^i(x))$ is the Lyapunov exponent of $f^{-n}(x)$ and
$\sum_{i=0}^{n-1} m\circ h\circ f^i (x)$ is the Lyapunov exponent of $g^n$
at $h(x)$.
By the hypothesis of equality of Lyapunov exponents at periodic orbits,
they are equal.
Hence $r$ satisfies the compatability conditions of the commutative Livsic
theorem. Hence:
$$r(x) = -s(f(x)) + s(x)$$
and substituting in \equ(toestimate4) we obtain a telescoping sum that
reduces the sum to
$$\sup_x |-s\circ f^n(x) + s(x)|$$
This is clearly bounded and the theorem is established.
\QED
\SUBSECTION Proof of \clm(abscont) using quasi-conformal estimates.
We will
show that
$$h= \lim_{n\to \infty} g^n \circ k\circ f^{-n}$$
is quasi-conformal when restricted to any $W_x^{s,f}$.
We estimate the distortions of the
$h_n =g^n\circ k\circ f^{-n}$
when restricted to the stable manifold.
We claim that they are uniformly bounded.
Recall that $K(h_n\mid W_x^f) = \sup_x K(D_sh_n)$.
Since the distortion of $\Psi_x^{s,f},\Psi_x^{s,g}$ and their inverses
are uniformly bounded to show that $K(D_sh_n(x))$ is uniformly bounded
it suffices to show that
$K(\Psi_{h_n(x)}^{s,g} D_sh_n(x) (\Psi_x^{s,f})^{-1})$
is uniformly bounded.
Again, using the chain rule and introducing the notation
\equ(notation) we see that
$$\eqalign{
&K(\Psi_{h_n(x)}^{s,g} D_s h_n (x) (\Psi_x^{s,f})^{-1}) =\cr
&= K(M(g^{n-1}\circ k\circ f^n(x)) \cdots
M(k\circ f^{-n}(x)) (\Psi_{kf^{-n}(x)}^{s,g})^{-1} D_s k(f^{-n} x)\cr
&\qquad ((\Psi_{f^{-n}(x)}^{s,f} N (f^{-n+1}(x)) \cdots N(x)) \cr}
\EQ(distortion1)$$
Recall that $K(AB) \le K(A) K(B)$ and that if $A$ is diagonal $K(A)=1$.
Since $M,N$ are multiples of
the identity, we can estimate \equ(distortion1) by
$$
K\left(
(\Psi_{kf^{-n}(x)}^{s,g})^{-1} D_s k(f^{-n}(x)) \Psi_{f^{-n}(x)}^{s,f}
\right)
$$
which can be clearly estimated by a constant.
We recall that \clm(shub) guarantees that $h_n$ converges to a
homeomorphism.
Even if the pointwise limit of quasi-conformal maps may not be quasi-conformal,
when the limit is a homeomorphism it is.
(See e.g., Theorem 37.2 and Corollary 37.3 of \cite{V, p.125ff.}
or \cite{GP})
\QED
By standard properties of quasi-conformal mappings
(see e.g., \cite{V, \S32, p.109ff. and 32.3, p.109ff.}, \cite{GP}) we have:
\CLAIM Corollary(conformal2)
$h$ is differentiable almost everywhere,
it is absolutely continuous when restricted to almost all lines, the
partial derivatives are in $L_{\loc}^s$ and
$$\|D_sh(x)\| \le KJ(h,x)^{1/d_s}$$
where $J(h,x)$ denotes the Jacobian
and $d_s$ is the dimension of the stable manifold.
\SECTION Higher regularity of the conjugacy.
In this section, we want to present two proofs of the following result:
\CLAIM Theorem(bootstrap)
Assume that $f,g$ are $C^r$ diffeomorphisms $r=1,2,\ldots,\infty, \omega$ as before.
Assume that $h$ is a homeomorphism such that
$h\circ f= g\circ h$
and that $h$ is Lipschitz when restricted to the stable manifold,
Then, $h$ is $C_s^{r,f}$.
We note that the first proof is not a proof
of the theorem exactly as stated since it
does not cover the case
$r = \omega $. This will be obtained with the second proof.
The first part of the proof is common
to both approaches, we will show that the
conjugacy is $C^1$.
\REMARK
We point out that Theorem 6.1 of \cite{L2} shows
that, for
any high dimensional $C^r$ Anosov map,
all the conjugacies to another $C^r$ map
that are sufficently smooth are $C^r$
$r = 1, \cdots, \infty$.
The condition of this theorem depends on the metric
and it is advantageous to choose the conformal metrics
constructed in the previous results. In that case, we
obtain that the critical regularity is $2$. So that,
it follows from Theorem 6.1 in \cite{L2} that,
under the conditions $i)$,$ii)$ of \clm(main), any
$C^2$ conjugacy $h$ is $C^{r -\epsilon}$.
\SUBSECTION The conjugacy is $C^{1}$.
First, we will show that the mapping $h$ is $C^1$
and that the derivatives are H\"older along the manifold.
Again, it suffices to show it is in $C^1_s$ and a similar
argument will show that it is in $C^1_u$. For the case of
time one maps of flows, the differentiablility along the flow lines
is automatic.
We note that we will be assuming that $r \ge 2$,
so that in this step, we will actually
able to conclude $C^{1 + \alpha}$ for
some $\alpha > 0$.
We note that, since
we have shown that $h$
restricted to the stable leaf
is absolutely continuous,
it is differentiable almost everywhere.
Taking derivatives with respect to
the stable direction of $f$
the conjugacy equation
( $a)$ of \clm(main) ),
we have for almost all $x$:
$$
D_s^{(f)} f\circ h(x) D_s^{(f)} h(x) = D_s^{(f)} h \circ g(x) D_s^{(g)} g(x)
\EQ(derivativestable)
$$
Recall that we had used the notation $D_s^{(f)}, D_s^{(g)}$ to
denote the derivative along the stable leaves of $f$ and $g$.
Note that we have used the fact that $h$ maps stable leaves of
$g$ into stable leaves of $f$.
Now, multiplying \equ(derivativestable) on the right by
the trivialization map
$\Psi^{s,(f)}_{f\circ h(x) }
= \Psi^{s,(f)}_{h \circ g (x) }$
and on the left by $(\Psi^{s,(g)}_x)^{-1}$,
we have:
$$
\eqalign{
&\left[ \Psi^{s,(f)}_{f\circ h(x)} D_s^{(f)}
(\Psi^{s,(f)}_{h(x)})^{-1} \right] \cdot
\left[ \Psi^{s,(f)}_{h(x)} D_s^{(f)} h(x)
(\Psi^{s,(g)}_x)^{-1}
\right] \cr
&=
\left[ \Psi^{s,(f)}_{h \circ g (x) }
D_s^{(f)} h \circ g(x)
(\Psi^{s,(g)}_{g(x)})^{-1} \right] \cdot
\left[
\Psi^{s,(g)}_{g(x)}
D_s^{(g)} g(x)
(\Psi^{s,(g)}_x)^{-1}
\right]
}
\EQ(multiplied)
$$
We recall that, by hypothesis,
the first factor in the left and the last
factor in the right were
multiples of the identity.
Hence, introducting the function
$$
\Gamma(x) =
\Psi^{s,(f)}_{h (x) }
D_s^{(f)} h (x)
(\Psi^{s,(g)}_{x})^{-1}
\EQ(Gamma)
$$,
we see that \equ(multiplied) can be written as:
$$
\gamma_s^{(f)}\circ h(x) \Gamma(x) = \Gamma\circ g(x) \gamma_s^{(g)}(x)
$$
Or, using the fact that $\gamma_s^{(f)}, \gamma_s^{(g)} $
are just scalar functions
$$
\Gamma(x) \Gamma\circ g(x)^{-1} = \gamma_s^{(f)}(h(x))/\gamma_s^{(g)}(x) \Id
\EQ(transformed)
$$
This is an equation of the form considered in
\clm(Livsic).
Note also that, since by the structural stability theorem
we obtain that
$h$ is H\"older when restricted to the stable manifold,
the right of \equ(transformed) is H\"older.
More importantly, note that by the hypothesis on periodic orbits,
if $g^N(x) = x$
$$
\prod_{i= 0}^{N-1}
\left[ \gamma_s^{(f)}(h)/\gamma_s^{(g)}\right] \circ g^i(x) = 1
$$
We can apply the scalar Livsic theorem and find a H\"older function
$\sigma$ bounded away from zero such that
$$
\sigma(x)/\sigma\circ g(x) = \gamma_s^{(f)}(h(x))/\gamma_s^{(g)}(x)
$$
Nothat that
$\sigma(x) \Id$ is already a solution
of \equ(transformed) and
it is H\"older. So, our task
reduces to proving uniquess statements for
measurable
solutions of Livsic equations.
In the generality of the non-commutative case,
this results have been obtained recently
in \cite{PP},\cite{Wa}.
In, our case, however, we can imitate the proof
given in the commutative case in \cite{L1}.
Using \equ(transformed), we see
that
$$
[\sigma(x)]^{-1}\Gamma(x) =
[\sigma(g(x))]^{-1}\Gamma(g(x))
$$
Since $g$ is ergodic with respect to SRB measures, we
conclude that $\sigma^{-1} \Gamma$ is constant almost
everywhere with respect to the SRB measures.
We also recall that the SRB measures have density
on stable leaves. So that we can find a leaf
of the stable foliation for which
$\sigma(x)\Gamma(x)$ is constant almost everywhere
with respect to the Riemann measure on the leaf. In
particular, it is constant almost everywhere in an
open set.
Since for transitive Anosov maps, these open neighborhoods are
dense, using the invariance, we conclude that
$\Gamma$ agrees with a constant multiple of
$\sigma \Id$ on a dense set of stable leaves.
For the case of transitive Anosov flows, we need to observe that
if an invariant function is constant in an open neighborhood
in a stable manifold, it is also constant in the stable manifold.
Once we have that a continuous function is the derivative
in a dense set of leaves, we conclude it is the derivative
everywhere because we can justify
the finite increment formula
$$
h(x) - h(y) = \int_0^1 D_s h\circ \beta(t) \, dt,
$$
where $\beta$ is a path contained in a stable
leaf by approximating it by
paths contained in stable leafs for which the
derivative agrees with our H\"older function.
Of course the same argument can be applied
to obtain that the derivative along unstable
directions is also H\"older.
\QED
We note that, as a corollary of the proof,
we have obtained that
$$
\eqalign{
\Gamma^s(x) &=
\Psi^{s,(f)}_{h (x) }
D_s^{(f)} h (x)
(\Psi^{s,(g)}_{x})^{-1} \cr
\Gamma^u(x) &=
\Psi^{u,(f)}_{h (x) }
D_u^{(f)} h (x)
(\Psi^{u,(g)}_{x})^{-1}
}
$$
are multiples of the identity.
\REMARK
We point out that the only place
where we use that $r$ is at least two is in the
uniqueness theorem for $L^\infty$ solutions of
cohomology equations.
The proof we presented depends on
the use of ergodic theory --
the invariant measures --
which is not well controlled
for $C^1$ maps.
The arguments in \cite{Wa}
establish uniqueness when the flow is
$C^1$ and the solution is almost everywhere
with respect to an equilibrium state
corresponding to a $C^\alpha$ function.
It seems plausible that, for area preserving maps,
one can assume only $C^1$ and push through
a similar argument.
\SUBSECTION Proof of \clm(bootstrap) using the Smooth Livsic theorem.
The method presented in this subsection cannot obtain the result
$r = \omega$.
We note that
we have shown that the
stable deriviative
of $h$ exists
and that,
in the previously introduced notations,
$\Gamma$,(see \equ(Gamma)) the trivialization of the
stable derivative of $h$ can be written
as $\sigma(x) \Id$
and, hence,
\equ(transformed)
can be written as
$$
\sigma(x)/\sigma(g(x)) = \gamma_s^((f))(h(x))/\gamma_s^{(g)}(x).
$$
We note that if $h \in C^{\ell,(g)}_s$, the theorem
\clm(smoothlivsic) implies that
$\sigma \in C^{\ell,(g)}_s$,
which in turn implies
$h \in C^{\ell +1, (g)}_s $.
Since, from previous results we
have $h \in C^{1,(g)}_s$,
we can start the induction and
use that the induction stops only
when we cannot
obtain more regularity of
the RHS of \equ(transformed).
Recall that the RHS of \equ(transformed)
is formed out of the derivatives of
the maps $f$ and $g$ and the trivializations.
If the maps are $C^r$, obviously the
derivatives are $C^{r-1}$ and
we showed that the
trivializations are in $C^{r-1}_s$.
Hence, using this argument, we can
conclude that $h \in C^r_s$,
when $s = 1, 2, \infty$.
The reason we cannot decide
the case $C^\omega$ is because
we have a very poor control of
the size of derivatives
of increasing order. This involves not
only substituting, but also obtaining
good estimates in
\clm(smoothlivsic) depending on the order.
\SUBSECTION Proof of \clm(bootstrap) using SRB measures.
As it turns out this proof does not obtain the full regularity
in \clm(bootstrap).
We recall that transitive Anosov systems carry some distinguished
invariant measures (which we will denote by $\mu_+ (f),\mu_-(f)$)
which are equilibrium states corresponding to the unstable or stable
Jacobians of the system.
In our context, what makes them particularly useful is that they can be
approximated by periodic orbits --- so that hypothesis at periodic orbits
gives us control over them --- and that they have absolutely continuous
--- in fact with a smooth density --- restrictions to stable and unstable
foliations.
We refer to \cite{Si}, \cite{Ne}, \cite{Ru} for properties of SRB measures.
\CLAIM Lemma(transport)
Let $f,g$ be transitive $C^r$ Anosov diffeomorphisms
(resp.\ $X_t$ $Y_t$ transitive Anosov flows).
Let $h$ be a homeomorphism.
Assume that $h\circ f= g\circ h$ (resp.\ $h\circ X_t = Y_t\circ h$).
If the stable Jacobian at corresponding periodic orbits of $f$ and $g$
(resp.\ of $X$ and $Y$) then $h^*\mu_- (f) = \mu_- (g)$
(resp.\ $h^* \mu_-(X) = \mu_-(Y)$).
In particular, denoting by $\omega_{-x}^{(f}(y)$ the density of $\mu_-$
on $W_x^s$, we have
$$\int_A \omega_{-x}^{(f} (y)\,dy
= \int_{h(A)} \omega_{-h(x)}^{(g} (y)\,dy$$
(analogously for flows).
\PROOF
We will only give the proof for diffeomorphisms.
Recall that, by results in \cite{Si}, \cite{Ru}, \cite{Ne} we have:
$$\mu_- (f) = \lim_{n\to\infty} {1\over Z_n} \sum_{o\in \Per [n]}
J_o^{u,(f)} \delta_o$$
where $J_o^{u,(f)}$ is the unstable Jacobian
of the return map along the periodic orbit $o$,
$\delta_0$ is the usual Dirac measure at the point $o$
$\Per [n]$ denotes the set of periodic orbits of period $n$
and $Z_n$ is a normalization factor to have the measure have mass~1.
Note that, under the hypothesis about the Jacobians at corresponding
periodic orbit~1,
$$\eqalign{h^* \sum_{o\in \Per_f [n]} (J_o^{s(f)})^{-1} \delta_o
& = \sum_{o\in \Per_f [n]} J_o^{s(f)^{-1}} \delta_{h(o)} =\cr
&= \sum_{p\in \Per_g[n]} J_p^{s(g)^{-1}} \delta_{po}\cr},$$
Taking limits, we obtain the first result in the claim.
The result about the densities is just a consequence of the definition
of the densities.
\QED
Note that if $h$ is differentiable along the stable directions at a
point $x$ this gives us immediately
$$J(D_s h(x)) = {\omega_{-h(x)}^g (h(x)) \over
\omega_{-x}^f (x)}
\EQ(Jacobian)$$
where the Jacobian is understood with respect to the induced Riemann metrics
on the stable and unstable manifolds.
We just observe that, in particular, the Jacobian is bounded for any
Riemann metric.
The control over the Jacobian complements very nicely the information we have
using quasi-conformal theory.
We obtain a formula for the determinant of
the function $\Gamma$ introduced in
\equ(Gamma) as the expression of
$D_s h$ on the trivializations.
Its determinant can also be computed as
$$\det \Gamma (x) = J(\Psi_{h(x)}^{s,g}) J(D_s h(x)) J(\Psi_x^{s,f})^{-1}$$
where we denote by $J$ the Jacobian of a linear mapping when the metrics
in the domain and the range are understood.
We also recall that by \equ(Jacobian) we can write
$$
\det \Gamma (x) = F(x,h(x))
\EQ(Jacobian2)
$$
where
$$
F(x,y) = {\omega_{-y}^{s,g} (y) \over \omega_{-x}^{s,g}(x)}.
$$
We emphasize that the function $F$ is $C^{r-1}$ in its arguments.
(In this case, $r$ can be $\omega$.)
The reason for that is that the densities $\omega^{s,(f)}(x)$
are in
in $C^{r-1,(f)}_s$ because, since they are invariant,
they satisfy
$$
\omega^{s,(f)}( f(x) ) = J_(f,x) \omega^{s,(f)}(x)
$$
Since we know that $\omega^{s,(f)}(x) $ is
continuous, we can apply \clm(smoothlivsic).
We can write \equ(Jacobian2) as
$$
D_s h(x) = G(x,h(x))
\EQ(diffeq)
$$
where
$$G(x,y) = (\Psi_y^{s,g})^{-1} \Psi_x^{s,f} (F(x,y))^{1/s}$$
Note that the function $G(x,y)$ is $C^{r-1}$ in each of its arguments and, in
this case $r$ can be $\omega$.
We consider \equ(diffeq) as a first order
partial differential equation satisfied by $h$.
The meaning of this is that if we
fix a smooth one-dimensional curve contained in the
stable manifold,
\equ(diffeq) reduces to an ODE.
Using standard theory of
ODE's we obtain that $h$ is
$C^{r}$ in the parameter of
the curve. (Since the parameter
is the independent variable, we
gain one derivative.)
>From that, it is standard in ODE theory to
show that the function is $C^r$ restricted to
the manifold.
(The standard dependence on parameters for the
ODE gives that the function is $C^{r-1}$, but then,
looking at the differential equation we see that
it implies that the partial derivatives are $C^{r-1}$
and, therefore, the function is $C^r$.
\QED
\REMARK
The differential equation
\equ(diffeq) was derived in \cite{L4}
by a very different method.
\SECTION Proof of \clm(almostLip)
The idea of the proof is that we will show that
the map $h$ restricted to a leaf of the stable foliation
is $C^{1-\epsilon}$ and that the
Lipschitz seminorm is bounded uniformly independenly of
the leaf. The same argument will establish a similar result for the
unstable manifold.
Since the stable and unstable foliations are transversal,
the angle between the tangent spaces of the leaves is bounded
uniformly from below and the leaves are uniformly $C^1$
we have that for $x,y$ such that
$d(x,y) \le \delta$ (where $\delta > 0$ is
a number that only depends on the map)
$$
d(x,y) \ge K \big( d_s([x,y],x) + d_u([x,y], y) \big)
$$
where $[x,y]$ is the usual construction in
the local product structure of hyperbolic theory.
Namely,
$ [x,y] = W^{s,{\rm loc} }_x \cap W^{u,{\rm loc} }_y $
(that is the only point of the intersection of the local
stable manifold of $x$ and the local unstable manifold of
$y$, where we choose the segments of the local
stable manifold small enough that for
points at a distance $\delta$ there is not more
than one point in the intersection and big enough so that all points
at a distance less that $\delta$ indeed have such a point
\cite{KH} p. 268 Prop. 6.4.13 ).
Since we, obviously have
$$
d(x,y) \le d_s([x,y],x) + d_u([x,y], y)
$$
the metrics $d(x,y),
\big( d_s([x,y],x) + d_u([x,y], y)$ are
equivalent.
It then follows that for an Anosov system and
for $0 \le \alpha \Lip $,
$C^\alpha_s \cap C^\alpha_u = C^\alpha$.
(Noting that a function is continuously differentiable if and
only if it has continuous partial derivatives, we
note that the case for $\alpha = 1 $ is also true.)
For flows, the corresponding result is
$C^\alpha_s \cap C^\alpha_u \cap C^\alpha_X = C^\alpha$
for $o \le \alpha \le 1$.
Hence, to prove \clm(almostLip) it suffices to
establish that $h \in C^{1-\epsilon}_s$.
The same argument will then establish that
$h \in C^{1-\epsilon}_u$ and this establishes the
claimed result for Anosov diffeomorphisms.
For Anosov flows we need to supplement this argument
with the observation that $X_t\circ h = h \circ Y_t$
implies that $h$ is differentiable along the flow lines.
To prove that $h \in C^{1-\epsilon}_s$ we proceed as
in \clm(shub). We will consider
$h$ as a fixed point of
$\Tau(k) = g\circ k \circ f^{-1}$
acting on a space of mappings $k$ which send the stable foliation of
$f$ into the stable foliation of $g$.
Similarly, we can consider
$h^{-1}$ as a fixed point of
$\N(\ell) = f \circ \ell \circ g^{-1}$
acting on functions that send the stable
foliation of $g$ into the stable foliation of $f$.
Note that, as a consequence of the usual results in
structural stability, by assuming that $g$ is
$C^1$ close to $f$, we have that $h, h^{-1}$ are
in an arbitrarily small $C^0$ neighborhod
of the identity.
We observe that, as in the proof of \clm(shub) this is a
contraction in $C^0_s$. Hence, if $g$ is close to
$f$, and we take $k^0$ to be one of the functions produced
in \clm(smoothing) smoothing $h$
and $\ell^0$
one obtained smoothing $h^{-1}$ -- note in special point $d)$ --
we have that $\Tau^n(k^0)$ does not leave a $C^0$ neighborhood
of the identity in $C_s^{0,(f)}$ and
$\N^n(\ell^0)$ does not leave a $C_s^{0,(g)}$ neighborhood
of the identity. The size of these neighborhoods can be nade as small
as desired by assuming that $f$,$g$ are are sufficiently $C^1$ close.
\CLAIM Proposition (Holdercontraction)
Given $\epsilon > 0$
we can find $\eta > 0$ and
fix a $C^2$ neighborhood of $f$
so that
$\Tau^n(k^0)$
remains bounded in the seminorm
$$
||k||_{\alpha,eta} = \sup_x \sup_{y,z \in W^{s,(g)}_x \atop 0 < d_s(y,z);
d_s(x,y), d_s(x,z) \le \eta }
d(k(y), k(z))/d_s(y,z)^{\alpha}
$$
for $\alpha = 1 - \epsilon$.
\PROOF
Since this is a local result -- notice that we are only
considering an $\eta$ neighborood -- we will take coordinates
so that we can use the more transparent notation
$|y -z|$ rather that $d(x,z)$.
We denote by $k_n = \Tau^n (k_0)$
Note that
$k_{n+1} = g\circ k_n \circ f^{-1}$
and that $||k_n - \Id|| \le \sigma$,
where we can make $\sigma$ as small
as we want by assuming $f,g$ are $C^1$ close.
Note also that,
if $\eta' > \eta$, we
can bound
$$
\eqalign{
\sup_{y,z \in B_{\eta'}(x) \atop x \ne y}
| k(y) - k(z)|/|y-z|^\alpha
&\le
\max(
\sup_{y,z \in B_{\eta'}(x) \atop 0 < | y-z| \le \eta}
| k(y) - k(z)|/|y-z|^\alpha, \cr
&\qquad\qquad\sup_{y,z \in B_{\eta'}(x) \atop \eta < | y-z| }
| k(y) - k(z)|/|y-z|^\alpha
) \cr
& \le
\max\left(
\sup_{y,z \in B_{\eta'}(x) \atop 0 < | y-z| \le \eta}
| k(y) - k(z)|/|y-z|^\alpha,
2 * \eta^{-\alpha} ||k||_{C_0}
\right)
}
\EQ(auxiliary)
$$
If we fix $x$ and take $y,z$ such that
$d_s(x,y), d_s(x,z) \le \eta$
we have
$$
\eqalign{
|k_{n+1} (y) - k_{n+1}(z)| &\le
\sup_{\xi \in B_{\eta + \sigma} (x) } |g'(\xi)| \,
|k_n \circ f^{-1} (y) - k_n \circ f^{-1}(z)| \cr
&\le
\sup_{\xi \in B_{\eta + \sigma} (x) } |g'(\xi)| \cdot \cr
&\qquad
\cdot \sup_{a,b \in B_{||D f^{-1}||_{C^0} \eta}(f^{-1}(x)) }
|k_n(a) - k_n(b)|/| a -b|^\alpha \,
|f^{-1}(y) - f^{-1}(z) |^\alpha
}
$$
If in this inequality we bound the second factor as
in \equ(auxiliary)
and also use
$
|f^{-1}(y) - f^{-1}(z) |^\alpha \le
\sup_{\xi \in B_\eta(x) } |D f^{-1}(\xi)|^\alpha | y - z|^\alpha$,
we obtain:
$$
\eqalign{
||k_{n+1}||_{\alpha,\eta}
& \le
\max \big(
\sup_{\xi \in B_{\eta + \sigma} (x) } |g'(\xi)| \,
\sup_{\xi \in B_\eta(x) } |Df^{-1}(\xi)|^\alpha ||k_n||_{\alpha,\eta}
, \cr
&\qquad
\sup_{\xi \in B_{\eta + \sigma} (x) } |g'(\xi)| \,
\sup_{\xi \in B_\eta(x) } |D f^{-1}(\xi)|^\alpha \eta^{-\alpha}||k_n||_{C^0}
\big)
}
\EQ(auxiliary2)
$$
We note that if $f$ is 1-conformal,
for any $\epsilon > 0$, if
$\eta$, $\sigma$, are sufficiently small
and $g$ is sufficiently $C^1$ close to $f$
then
$$
\sup_{\xi \in B_{\eta + \sigma} (x) } |D g(\xi)| \,
\sup_{\xi \in B_\eta(x) } |D f^{-1}(\xi)|^{1 - \epsilon} \le \kappa < 1
\EQ(contraction)
$$
where $\kappa$ is independent of $x$.
Since $\eta^{-1 + \epsilon}$, $||k_n||_{C^0}$ are bounded
independently of $n$, we obtain that \equ(auxiliary2)
can be written as
$$
||k_{n+1}||_{\alpha,\eta}
\le \max \kappa
||k_{n}||_{\alpha,\eta} , A)
$$
where $\kappa$ and $A$ are independent of $n$ and $\kappa < 1$.
It then follows that $|| k_n||_{1 - \epsilon, \eta}$
remains bounded.
Since $k_n$ is converging uniformly to $h$, we obtain that
$h$ is in $C^{1-\epsilon}_s$.
\QED
The argument for $\N$ is exactly the same
with $f$ and $g$ interchanged,
except for a minor modification in the
paragraph leading to \equ(contraction).
An identical argument works for the unstable foliation
both for $\Tau$ and $\N$.
For flows, the regularity in the direction of the flow
is automatic.
This finishes the proof of
\clm(almostLip)
\QED
\SECTION Some examples
In view of the fact that one can have a Livsic
theorem for nilpotent groups, one could have
hopes that, when the normal forms are
just Jordan blocks, one could have rigidity.
This turns out not to be true as shown
by the following examples.
Let $A$ be an Anosov linear automorphism of
$\torus^2$.
Consider the automorphism $B$ of $\torus^4 = \torus^2 \times \torus^2$
defined by:
$$
B(x,y) = (Ax+y, Ay)
$$
where $x \in \torus^2$, $y \in \torus^2$.
Consider perturbations of the form
$$
F(x,y) = (Ax + y + e_s \Phi(y), Ay)
$$
where $e_s$ is the stable eigenvalue of $A$
and $\Phi$ is a small analytic
function.
Denote by $\lambda_s$, $\lambda_u$
the eigenvalues of $A$.
The matrix of the derivative of $F$ in the basis
$$(e_s,0), (0,e_s), (e_u,0),(0,e_u)$$
is
$$ \left( \matrix{\lambda_s & 1+\alpha (y)&0&\beta (y)\cr
0&\lambda_s &0&0\cr
0&0&\lambda_u&1\cr
0&0&0&\lambda_u\cr}
\right)$$
where $\alpha,\beta$ are real valued functions
that involve only derivatives of $\Phi$.
We note that if $\Phi$ is $C^1$ small, then
the Jordan normal form of
$F$ is exactly $B$.
Hence, we can find an infinite dimensional
manifold around $B$ in which the Jordan
form around periodic orbits does not change.
Now, we can compute what is the
change of variables
satisfying
$$
h\circ B = F\circ h
\EQ(structural)
$$
given by
structural stability.
Since the structural
stability theorem includes
uniqueness conclussions we can
make an ansatz for the form
of the solution and then, provided
that we find a solution of this
form, it will be the unique solution.
If we use the ansatz
$$h(x,y) = (x+e_u\Gamma (y),y)
\EQ(ansatz)
$$
equation \equ(structural)
becomes
$$(Ax+y+e_u\Gamma (Ay),y)
= (Ax +Ae_u \Gamma (y) + y +e_u \Phi (y),y)$$
which is equivalent
to:
$$
\Gamma (Ay) - \lambda_u \Gamma (y) = \Phi (y)
\EQ(tosolve)
$$
If $\Phi$ is $C^1$-small,
we can get a $C^0$ small
solution of \equ(tosolve) by
$$
\Gamma (y) = - \sum_{i=1} \lambda_u^{-i} \Phi (A^{i-1} y)
\EQ(solution)
$$
which converges uniformly.
These series are quite well studied
since they are variants of
the Riemann-Weierstrass function.
We note that in general -- even for
trigonometric polynomials -- the sum
will not correspond to a $C^1$ function.
Note that if we take $\Phi(x) = e^{2\pi i k\cdot x}$
then
$\hat \Gamma_{(A^t)^{i-1} k } = \lambda_u^{-i}$.
Since $|(A^t)^{i-1} k| \approx \lambda_u^i$
we have
that
$$
\hat \Gamma_{(A^t)^{i-1} k } |(A^t)^{i-1} k|
$$
does not tend to zero.
We recall that Riemann-Lebesgue theorem
says that if $\varphi$ is $C^1$, then,
$|k| \hat \varphi_k$ tends to zero.
Hence, the function
constructed in
\equ(solution) for this
$\Phi$ is not even $C^1$.
There are arguments not based on
Fourier series that show that for many
choices of $\Phi$, the function
given in \equ(solution) is not $C^1$.
Of course, trivial modifications of this
argument lead to infinite dimensional parameters
of familes of examples of
real valued functions which preserve the
normal form at periodic orbits but
for which the conjugacy given by structural
stability is not even $C^1$.
There are many variants to this
example, for
example, one could do similar
constructions using automorphisms
of nil-manifolds rather than
automorphisms of the torus
or construct more complicated Jordan
normal forms.
For example:
$$\eqalign{
&B(x_1,\cdots,x_n) = (Ax_1 + x_2, Ax_2+ x_3,\cdots, Ax_n) \cr
&B(x,y,z) = (Ax + y, Ay, Az) \cr
}
$$
which contain longer Jordan blocks or
several Jordan blocks, some trivial and some not.
Simple variants of the methods given here allow
to show that they are not rigid in a very strong
sense and that they allow infinite dimensional
families of deformations.
We, however, point out that
Theorem 6.1 in \cite{L2} shows that
given $\Phi \in C^r$
$r = 2, \cdots, \infty$
sufficiently small,
any
$C^{2}$ conjugacy is $C^{r-\epsilon}$.
\SECTION Acknowledgements
The work of the author has been supported
by NSF grants.
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