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\begin{document}
\title{Analytic regularity of solutions of Livsic's
cohomology equation and some applications to
analytic conjugacy of hyperbolic dynamical systems}
\author{R. de la Llave\\
Department of Mathematics\\
The University of Texas at Austin\\
Austin, TX 78712-1802}
\date{}
\maketitle
\begin{abstract}
We study Livsic's problem of finding $\phi$
satisfying $X\phi=\eta$ where $\eta$ is a
given function and $X$ is a given Anosov
vector field.
We show that, if $\phi$ is a continuous solution
and $X,\eta$ are analytic, then $\phi$ is
analytic.
We use the previous result to show that if two
low-dimensional Anosov systems are topologically
conjugate and the Lyapunov exponents at
corresponding periodic points agree, the
conjugacy is analytic.
Analogous results hold for diffeomorphisms.
\end{abstract}
\section{Introduction and statement of results}\label{sec-one}
In this paper, we extend the regularity theory for
solutions of Livsic's cohomology equations to
analyticity. We also
discuss related results. We show that for
two dimensional analytic Anosov diffeomorphisms and
for three dimensional analytic Anosov flows, the
set of eigenvalues at periodic
orbits is a complete set of
invariants for analytic conjugacy.
The crux of the proof of analytic regularity
of solutions of Livsic equation is
a regularity lemma that states that functions that
are analytic restricted to the stable
and unstable foliations are analytic.
To prove that eigenvalues are a
complete set of invariants, we need the regularity lemma
established before as
well as an argument that bootstraps the regularity
along stable and unstable leaves from $C^1$ to
analytic. (Recall that the fact that eigenvalues are
a complete set of invariants for $C^1$ conjugacy
was already established in \cite{Ll}, \cite{LM})
More precisely, we prove
\begin{thm}\label{thm-one} %{Theorem 1}
Let $M$ be a compact manifold and $X$ a $C^\omega$
Anosov vector field on $M$. Then, if $\phi$ is a
continuous function so that $X\phi=\eta$ where
$\eta\in C^\omega(M)$ then $\phi\in C^\omega(M)$.
\end{thm}
Notice that Theorem 1 does not assume transitivity
of the flow.
If $X$ were transitive, we could apply the
fundamental Livsic's theorem \cite{Li1} to obtain
\begin{thm}\label{thm-one-prime} %{Theorem $1'$}
Let $M$ be a compact manifold and $X$ a
$C^\omega$ transitive Anosov vector field
on $M$. Then, if $\eta$ is a $C^\omega$
function on $M$ with the property that
$$
\int^{T(\gamma)}_0\eta\big(\gamma(t)\big)dt=0
$$
for all $\gamma$ periodic orbit of $X$ where
$T(\gamma)$ is the period of $\gamma$.
Then, there exist a function $\phi$ in $C^\omega$
satisfying $X\phi=\eta$.
Moreover, $\phi$ is unique up to constants.
\end{thm}
Analogous results hold for
diffeomorphisms.
\begin{thm}\label{thm-2} %{Theorem $2$}
If $f$ is
an analytic Anosov diffeomorphism
and $\phi$ is a continuous solution of
$\phi\circ f - \phi = \eta$ with $\eta$
analytic, then, $\Phi$ is analytic.
\end{thm}
\demo{Remarks}
For a transitive system as above,
the function $\phi$ solving $X\phi=\eta$ is unique
up to constants not only among $C^\omega$ functions
but also in much broader classes
such as $L^1(\mu_+)$, $L^1(\mu_-)$ where $\mu_+$ or
$\mu_-$ are the Sinai-Ruelle-Bowen measures or
$C^0$. This is because the SRB measures are
ergodic. Of course, since there is a dense orbit, the
solution is unique up to additive constants in $C^0$.
The most important shortcoming of these results with
respect to the $C^\infty$ analogues in \cite{LMM}
is that we do not know how to show analytic dependency on parameters.
This smooth dependence was essential in the discussion of conjugacy of
Hamiltonian vector fields which are Anosov in each energy
surface, since the energy was treated as an
external parameter.
Nevertheless, for the interesting case of geodesic
flows, the change in energy is just a rescaling in time.
Other results that do not involve dependence on parameters
but that only use the regularity lemma.
such as those of \cite{MM} on cocycle vector equations
or those of \cite{HK} can be improved to analytic conjugacy.
Besides these applications, we were told by A. Katok
that an analytic version of Livsic theorem could
have applications for the theory of automorphic
forms that a merely $C^\infty$ one would not have.
Unfortunately Theorem~\ref{thm-one} %Theorem 1
--- or the lemmas used in its proof --- do not seem to imply directly that
the results of \cite{Ll} \cite{LM} about smooth
conjugacy of low dimensional Anosov systems can
be improved to analyticity.
In a third section we give the extra arguments
needed to obtain such improvement. Namely we prove
\begin{thm}\label{thm-two} %% {Theorem 2}
Let $f$, $g$ be two analytic Anosov diffeomorphisms
of the two dimensional torus. If they are topologically
conjugate and, moreover the Lyapunov exponents of
corresponding periodic orbits are the same, then, the
conjugating homeomorphism is analytic.
\end{thm}
The same methods of proof suffice to prove an analogous
theorem for three-dimensional transitive Anosov flows.
We can prove
\begin{thm}\label{thm-two-prime} %% {Theorem $2'$}
Let $X$, $Y$ be two transitive Anosov vector
fields on a compact three dimensional manifold
without boundary. If their flows are topologically
conjugate and the Lyapunov exponents of
corresponding periodic orbits are
the same, then the conjugating homeomorphism is
analytic.
\end{thm}
Notice that as an immediate corollary, if two
flows as those considered in Theorem~\ref{thm-two-prime} are $C^1$
conjugate, the conjugacy is, then, analytic.
For geodesic flows in surfaces of negative curvature,
such $C^1$ conjugacy is implied by equality of the
lengths of closed geodesics in corresponding
homotopy classes as shown in Feldman and Ornstein
\cite{FO}.
\demo{Remark}
Transitivity does not appear in the hypothesis of
Theorem~\ref{thm-two} %Theorem 2
because it is automatic for Anosov
diffeomorphisms in two dimensional manifolds \cite{N}.
For tree dimensional flows, it is not automatic
as shown by the celebrated example \cite{FW}.
We also note that transitivity is not used
in the proof that $C^1$ conjugated flows are
analytically conjugate.
\section{Analytic Livsic's theorem}\label{sec-two} %% 2.
The proof of Theorem 1 will be obtained by
extending the methods of \cite{LMM}
and incorporating some ideas of \cite{HK}.
We will very often
resort to the notation of \cite{LMM}. So,
it will be convenient for the reader to have
a copy of that paper for reference.
The strategy of proof is very similar to that
of \cite{LMM}. There, p. 579 it was shown
that, under the conditions of Theorem~\ref{thm-one} %Theorem 1
or of Theorem~\ref{thm-2},
we have $\phi\in C^\omega_s\cap C^\omega_u$ .
(That is, the function $\phi$ is analytic
when restricted to the leaves of the stable and
unstable foliation.)
We refer to \cite{LMM} for the full proof
but we recall that the idea (explained here only
for diffeomorphisms)
is to notice that in
the stable manifold of a fixed point $p$
we have
$\phi(x) = \phi(p) \sum_n^\infty \eta(f^n(x))$.
Since $f$ restricted to the stable manifold is contractive,
the above sum converges in a complex neighborhood of
the stable manifold. A density argument shows that
we obtain uniform analyticity in all the
leaves of the stable foliation.
Similar formulas can be derived for flows.
Theorem~\ref{thm-one} %Theorem 1
follows once we show that
$C^\omega=C^\omega_s\cap C^\omega_u$.
Such a result was proved in \cite{LMM} under the
--- rather restrictive --- hypothesis that the
stable and unstable foliations are analytic.
In this paper we will establish the same result
for all analytic Anosov diffeomorphisms.
The technique we will use is a strengthening
of the technique used in \cite{HK} to prove
$C^\infty=C^\infty_s\cap C^\infty_u$.
An important ingredient of this proof is the
fact that the Jacobians of the stable and unstable
foliations are smooth when restricted to the leaves.
We will use the same argument as in \cite{LMM}
--- in turn this is related to the argument in \cite{A}
for absolute continuity --- to show that the
Jacobian of the stable foliation is in $C^\omega_s$ (and
that of the unstable foliation in $C^\omega_u$).
Another important ingredient will be a cut-off method
adapted to analytic functions. In \cite{LMM}
one such method is used (after the publication
of the paper, we learned from Alan Sokal that
the same method appears in \cite{AR} and
even there it is referred back to \cite{B}).
This method turns out not to be convenient for
the problems here --- even if it is possible to
use it --- but a slight improvement suffices.
We now start to adapt some definitions from
\cite{LMM} to analytic regularity. For the
definitions we use without any mention, we
refer to that paper.
We will take as standing convention that the
dimension of $M$ is $n$, $s$ is the dimension
of the stable foliation, $u$ the dimension of
the unstable direction. Hence, for flows,
$n=s+u+1$.
The notation for flows is always a little bit
contorted because of the neutral direction which
we can choose to lump with the stable, the unstable
or on a class of its own.
Putting a notation to cover all the cases is sort
of painful, particularly so because, for our
purposes, this direction does not matter.
(Regularity along the direction of the flow follows just by integrating.)
What \cite{LMM} did was to include regularity along
the flow --- as well as in the properly stable
directions --- in the definition of $C^j_s$. For
the purposes of this paper, this would complicate
the considerations in Lemma~\ref{lemma-two}. %Lemma 2.2.
So we have
decided to explicitly say whether the flow direction
is included or not. Therefore, many of the
objects that \cite{LMM} were denoted by
$s$ or $u$ subscripts will have now an overscore
``$\overline{\ }$'' denoting that the direction of the flow
is included.
Nevertheless, all these considerations, except for the
desire of having a perfectly consistent notation,
can be safely ignored since the flow direction is
trivial to deal with anyway.
The following is an adaptation of Definition 2.1 of
\cite{LMM} for analytic regularity. Of course
part 1
goes back to \cite{A}.
\begin{defin}\label{defin-one} %% {Definition 1}
We say that $\Lambda_s:U\times
V\rightarrow M$
(where $U\subset\real^s $,
$V\subset \real^{u+1}$ are balls)
is a local parameterization of the stable foliation
\begin{description}
\item[1.] $\Lambda$ is a homeomorphism from $U\times V$ to an
open set of $M$, (which we will assume is contained
in an analytic chart, so we can consider
$\Lambda_s(x,y)$ as a mapping from a ball in
$\real^{s+u+1}$ to itself).
\item[2.] For each $y\in V, \Lambda_{s,y}:
U\rightarrow M$ given by
$\Lambda_{s,y}(x)=\Lambda_s(x,y)$ is an
analytic immersion whose image is an open
set of a stable leaf.
\item[3.] there is a $\xi>0$ such that all
The $\{\Lambda_{s,y}\}_{y\in V}$ extend to a
domain $$D_\xi=\{(z_1,\dots,z_s)\in
\complex^s/\vert \hbox{\rm Im\,}\ z_i\vert\leq\xi,\
(\hbox{\rm Re\,}z_1\dots \hbox{\rm Re\,}z_s)\in V\}$$
\item[4.] If we give
analytic functions the topology of the sup in
$D_\xi$, the mapping that to each $y$ associates
$\Lambda_{s,y}$ is continuous and
$$\sup_{\textstyle y\in\overline V\atop\textstyle \underline z\in D_\xi}
\vert\Lambda_{s,y}(z)\vert\leq K<\infty .$$
\end{description}
\end{defin}
There is an analogous definition for unstable
foliations and center stable and center unstable.
The local parameterizations of center-stable we
will denote by $\overline\Lambda_s$ and those of
the center unstable by $\overline\Lambda_u$.
It is a consequence of the standard regularity theory
for invariant foliations \cite{HPS} (see also
\cite{HP}, \cite{S}) that the stable foliations of analytic
Anosov vector fields admit local parameterizations in
the sense of Definition~\ref{defin-one} %Definition 1
around every point.
One just has to modify their proof making the
contractive operators they consider act in
appropriate spaces of analytic functions
endowed with the supremum norm. All
the papers mentioned above present detailed
sketches, so we will not repeat them here.
It also follows from this proof that the mapping
that to each $y$ associates $\Lambda_{s,y}$ is
H\"older. This later property will not be used,
but see the remarks after the statement of Lemma~\ref{lemma-two}. %Lemma 2.
\begin{defin}\label{defin-two} %% {Definition 2}
If we fix a finite set of local parameterizations
covering $M$, we say that a function
$\phi:M\rightarrow\real$ is in
$C^\omega_s$ when $\phi\big(\Lambda_s(\cdot,y)\big)$
are uniformly analytic (that is to say they extend
to a domain $D_\xi$ as above with $\xi$ independent
of $y$ and there is a bound for the absolute value
of this extension in $D_\xi$) and, moreover
$y\rightarrow \phi\big(\Lambda_s(\cdot,y)\big)$
is a continuous mapping to the appropriate
of analytic functions in $D_\xi$.
Given a fixed set of local parameterization and
a number $\xi>0$ we give a Banach space structure
to $C^{\omega,\xi}_s\subset C^\omega_s$ by the sup
of the absolute value of all the complex extensions
of the functions $\Phi\big(\Lambda(\cdot,y)\big)$.
We, moreover, say that a family of functions
$\phi_\varepsilon$ is uniformly in $C^\omega_s$
when there exist a $\xi>0$ so that all
$\phi_\varepsilon\in C^{\omega,\xi}_s$, and are
uniformly bounded and, the mapping that to
$\varepsilon$, $y$ associates
$\phi_\varepsilon\big(\Lambda_s(\cdot,y)\big)$
is uniformly continuous when the range is
given the sup norm on $D_\xi$.
\end{defin}
Even if these definitions have been given
in terms of a fixed covering set of local
parameterizations it is easy to check that any
two such covering sets will give rise to
the same concept of $C^\omega_s$ and uniformly
in $C^\omega_s$ --- even if $C^{\omega,\xi}_s$
change ---
Analogous definitions hold for the unstable,
center stable, center unstable foliations and
we will call them
$C^\omega_uC^\omega_{\overline s}$,
$C^\omega_{\overline u}$. If we denote by
$C^\omega_X$ the functions analytic when
restricted to the integral lines of
the flow.
It will turn out that
$$
C^\omega_{\overline u}=
C^\omega_u\cap C^\omega_X,\quad
C^\omega_{\overline s}=
C^\omega_s\cap C^\omega_X .
\label{intermediate}
$$
and this will follow from our main regularity lemma.
Analogous definitions of $C^\omega$ regularity
hold for geometric objects which
can be reduced locally to functions, e.g.,
vector fields, forms, etc.
The reason why this result is easy is
because the
\begin{lemma}\label{lemma-one} %{Lemma 1}
For an Anosov diffeomorphism or the time $t$
map of an Anosov flow, it
is possible to choose a local parameterization of the
stable foliation so that the Jacobian of the stable foliation
agrees
with a $C^\omega_s$ function.
\end{lemma}
\begin{proof}
We follow the argument in \cite{LMM}
p.~582 and following. Roughly, we follow the strategy of
\cite{A} to prove absolute continuity, but we pay
considerable attention to the way that the limits are reached.
To the stable foliation of an Anosov system
we can associate a $(n-s)$ form $\psi$ such
that $\psi$ contracted with vectors tangent
to the stable foliation is zero.
Moreover, there is a continuous l-form $\alpha$,
whose contraction with the flow or the vectors
tangent to the unstable foliation vanishes, such that
$$
d\psi=\alpha\Lambda\psi
$$
where the derivative is understood in a weak sense.
The key for the proof of the regularity
of the Jacobians in \cite{LMM} is the
following prescription to compute
$\alpha:\hbox{\rm \,Fix any\ }C^\infty$ metric. If
we split $TM=E^s\oplus \widehat{\overline{E^u}}$
where $\widehat{\overline{E^u}}$ is a
$C^\infty$ bundle transverse to $E^s$
(approximately $\overline{E^u}$ --- the center
unstable bundle). Then, for a fixed $t$, we
have that $\Phi_t$ --- the time $t$ map ---
has a derivative of the form
$$\left(
\begin{array}{cc}
A_t(x) & B_t(x)\\
0 & D_t(x)
\end{array}\right)
$$
Call ${\cal D}_t(x)=\log\ \det\big(D_t(x)\big)$.
Given a $C^\infty_s$ function $h$ define
$d_sh$ as the continuous l-form that
vanishes on vectors tangent to the center
unstable bundle and moreover
$(d_sh)\vert_{W^s_x}=d(h\vert_{W^s_x})$
(where the last $d$ is on the sense of the
exterior derivative of
$W^s_x$).
Then \cite{LMM} proves (eq. 2.18)
$\alpha=\lim_{t\to\infty}d_s\log{\cal D}_t$
and that the convergence is uniform along with
that of all the stable derivatives, (and,
of course, that the limit is independent of the
choices of bundle and of metric since it has
an geometric meaning that is independent of
the choice of coordinates).
\end{proof}
Here we prove.
\begin{prop}\label{prop-one} %% {Proposition}
With the notations above:
$$
\alpha\in C^\omega_s
$$
\end{prop}
\pf{Proof of the proposition}
The proof consists essentially in walking through
the proof in \cite{LMM} but checking that we
have uniform convergence in a complex extension
of the stable manifold. Since we only have to
estimate suprema of functions, the proofs
are even technically easier
than those in \cite{LMM} for finitely
many derivatives.
First we claim that we can choose our approximation to
the center-unstable bundle to be analytic.
This follows from the fact, mentioned
above that subbundles of the tangent
bundle can be identified with
forms and forms can be approximated
by analytic ones.
We refer to \cite{Gr} for a discussion of what objects in
an analytic manifold can be approximated by analytic ones.
(Forms certainly can because we can for example use an analytic
version or Whitney's embedding theorem --- this is what is proved in
\cite{Gr} --- , then, extend the forms using
Whitney extension theorem, then use the customary
smoothing by convolving with approximations of
identity and restricting.) We emphasize that
we only need to approximate the center-stable bundle
by an analytic one and
not the center-stable foliation.
Approximating foliations is very
delicate even with finite differentiability. (See \cite{EHP}
for examples of foliations that cannot be approximated by smoother ones)
Hence, we have that ${\cal D}_t\in D^\omega_s$
uniformly for $t\in[0,1]$.
By the chain rule we have ( See \cite{LMM} (2.17))
$$
\log{\cal D}_{t+u}=\log {\cal D}_u+
\log{\cal D}_t\circ\Phi_u
$$
If we take the coordinates along the stable
manifolds as done in \cite{LMM} p. 573 we
get calling
$$
\widetilde{\cal D}_{x,t}={\cal D}_t\circ w^s_x
$$
where the $w^s_x$ is a coordinate system that
maps a ball around $0$ in
$\real^s$ into neighborhood of $x$
into its center stable manifold.
After we take coordinates in this fashion,
as pointed out in \cite{LMM} we can think
as the $x$'s only as parameters and the
arguments of the functions are just
vectors in $\real^s$
\begin{eqnarray*}
d_s(\log\ \widetilde{\cal D}_{x,t})=& \sum^{N-1}_{j=0}
\big [d_s[\log\widetilde{\cal D}_{x_j,1}]
\circ\overline{\Phi}^s_{x,j}\big]
\cdot(\Phi^s_{x,j})'\\
%%
&+\big[d_s\log\ {\cal D}_{x_N,(t-N)}\circ
\overline{\Phi}^s_{x,N}\big]\cdot
(\Phi^s_{x,N})'
\end{eqnarray*}
where $N-1$ is the integer part of $t$,
$x_j$ is $\Phi_jx$ and
$\overline{\Phi}^s_t$ is the
version in coordinates of $\Phi^s_t$,
the flow restricted to the stable bundle.
(In the corresponding formula
of \cite{LMM} there is a misprint,
the last $N$ in the first line should be a
$j$.)
In order to prove that this is an l-form in
$C^\omega_s$ we just observe that all the
functions involved extend to a domain of
the form
$
D=\{(z_1\dots z_s)\in\complex^s\
\vert\ \vert z_i\vert\leq\xi\}
$
where $\xi>0$ can be picked uniformly for
$x\in M$, $t\in[0,1]$.
Moreover, in this complex extension we have that
all the functions are uniformly bounded and
$\Phi '$ satisfies (2.7) of \cite{LMM} which tells
it decreases exponentially fast in $N$, so that the
sums converge uniformly by Weierstrass $M$ test. This
finishes the proof of the proposition.
Once that we have that $\alpha$ is in $C^\omega_s$ we
can invoke a functional equation for the Jacobian
$\rho$ derived in \cite{LMM} p.~587 following
Anosov \cite{A} p.~134-135.
$$
\frac{d\rho_{\nu,v}^{(w)}}{d\nu}=
<\alpha\big(\Lambda(\tau v,t,w)\big)
>\rho_{\nu,v}
$$
out of which we can obtain that the Jacobian is
indeed in $C^\omega_s$. This finishes the
proof of Lemma~\ref{lemma-one}. %Lemma 1.
The corresponding result for diffeomorphisms can
be proved by invoking the usual suspension trick.
This concludes the proof of Proposition~\ref{prop-one} %Proposition 1.
and of Lemma~\ref{lemma-one}. %Lemma 1.
\endpf
\demo{Remark}
We believe that this result can also be obtained from
the methods of \cite{PS} who consider the holonomy
maps of the transversals, but have not checked all
the details nor written a coherent notation for it.
Analogous proofs hold, of course, for the unstable
and center stable and center unstable foliation.
The unstable is the stable for $-X$ and the
introduction of the extra direction of the flow
is trivial since the vector field is $C^\omega$
by assumption.
Now we finish the proof that
$C^\omega=C^\omega_s\cap C^\omega_{\overline{u}}$.
\begin{lemma}\label{lemma-two} %% {Lemma 2}
If the stable and center unstable foliations admit
local parameterizations in the sense of Definition~\ref{defin-one} %Definition 1
and, if their Jacobians are in
$C^\omega_s$, $C^\omega_{\overline{u}}$
respectively, then
$C^\omega=C^\omega_s\cap C^\omega_{\overline{u}}$.
\end{lemma}
Lemma~\ref{lemma-two} is really a regularity result independent of
the theory of Anosov systems.
It should -- perhaps -- have been stated as saying that
given two transverse absolutely continuous foliations
with uniformly analytic leaves such
that the Jacobians are also uniformly analytic restricted to the leaves
have the property that functions that uniformly analytic
restricted to the leaves are analytic.
We also point out that the restriction to two foliations is not essential.
Similar regularity questions have been considered
in harmonic analysis, specially in the
theory of $\Lambda_\alpha$ spaces
and it is related to regularity of integral transforms
such as Riesz transform that play a fundamental role
in elliptic regularity theory.
(See \cite{Ste} \cite{Kr}.)
The $C^\omega$ regularity relates to
``edge of the wedge theorems'' \cite{AR}.
In the analytic category this theorem is
somewhat reminiscent of the Hartoggs theorem
that establishes that separate analyticity
implies analyticity. The hardest part of
Hartoggs theorem is to recover the a-priori
bounds that we are already given.
Under the hypothesis of uniform
analyticity, this result
was established in \cite{B}.
We also call attention to the fact that
similar questions when regularity is
assumed only on sets of full measure have been
considered in \cite{Ll2} for finite regularity
and motivated by non-uniformly hyperbolic systems and
in the analytic case in \cite{UY} Section~6.
With
$C^\infty$ regularity it was proved by \cite{LMM},
\cite{Jo1},\cite{Jo2}, \cite{HK}.
Some improvements on the method of \cite{Jo1},
\cite{Jo2} was introduced in \cite{Ll2}.
The proofs of \cite{LMM} and \cite{HK} use
the regularity of the Jacobian but only
continuity of the foliation. \cite{Jo1},
on the other hand, uses H\"older continuity
of the foliation but makes no assumptions
on the Jacobian. The paper \cite{Jo2} only
needs to assume that the foliations are continuous.
In this paper we will pattern our proof along the lines
of \cite{HK} since it seems to be the shortest.
This will have the inconvenient that the
estimates that we obtain will not be tame.
This problem is related to the fact that
the proof in \cite{HK} relies only on the
size of the Fourier coefficients. The
proof of \cite{LMM}, bases on elliptic regularity theory
makes a more efficient use of the coefficients.
Since elliptic regularity is somewhat cumbersome
for the case of analyticity, we thought it best to
try the faster \cite{HK}. The proofs of
\cite{Jo1}, \cite{Jo2} seem to run into
problems with analyticity.
Note that as a simple corollary of \ref{lemma-two}
we have \ref{intermediate}. If we restrict ourselves to
a leaf of the stable foliation, the strong stable
foliation and the direction of the flow have
are smooth along the leaves of the corresponding
foliations.
\pf{Proof of Lemma 2.5}
Taking analytic coordinates we can always reduce
to proving the theorem in an open set in
$\real^n$.
Given $x_0\in\real^n,\ \{v_i\}^n_{i=1}$
a basis in $\real^n\delta\in\real^+$ we will
construct a mapping $\tau:T^n\rightarrow\real^n$
($T^n$ is the standard torus) and a function
$\Omega:T^n\rightarrow\real$ by
$$
\tau(\theta_1\dots\theta_n)=
x_0+\delta\sum^n_{i=1}v_i
\sin\theta_1;\ \Omega(\theta_1\dots\theta_n)=
(\cos\theta_1\cdot\dots\cdot\cos\theta_n)^n\ .
$$
We will prove that, if $\phi\in C^\omega_s
\cap C^\omega_{\overline{u}}$ then,
$(\Phi\circ\tau)\Omega$ will be analytic
provided that $x_0$, $\delta$, $\{v_i\}$
satisfy some mild conditions. Clearly, if
$(\phi\circ\tau)\Omega$ is analytic then
$\Phi$ itself is analytic in a subset of
Range $(\tau)$. The conditions of
smallness will be such that we can cover
the whole of $M$ with a finite number of
those subsets, hence establishing the
theorem.
We will pick any arbitrary point
$x_0$ and
$v_{-1}\dots v_{-s}, v_{-s+1}$
spanning the tangent space to the
center stable manifold through
$x_0$. The vectors $v_{-s+2}\dots v_{-n}$
will span the tangent space to
the center unstable manifold.
We study the Fourier coefficients
$$
\hat\phi_{\underline {k}}=
\int_{T^n}e^{i\underline{k}
\underline{\theta}}\phi\circ
\tau(\underline{\theta})\Omega
(\theta)d\theta_1
\dots d\theta_n
$$
we can rewrite those coefficients as
$$
\hat\phi_{\underline {k}}=
\int_{T^n}e^{i\underline{k}
\underline{\theta}}\phi\circ
\Lambda_s \circ\Lambda^{-1}_s\tau
(\theta)\Omega(\theta)d\theta.
$$
Now, if $\underline{\psi}$ is a system of coordinates in the range of
$\Lambda^{-1}_s\tau(\theta)$ in such a
way that $\psi_1\dots\psi_s$ are the
coordinates corresponding to the stable directions.
Since the multiplicity of the mapping $\tau$
is $4^n$ except in a set of zero measure we have
$$
\hat\phi_{\underline {k}}=4^n
\int_{\hbox{\rm Range of\ }\Lambda^{-1}_s\circ\tau}
e^{\underline{i}\underline{k}\underline{\theta}
(\underline\psi)}
\phi\circ
\Lambda_s (\underline{\psi})\underline{\Omega}
(\underline{\theta})
\frac{\partial(\underline{\theta})}
{\partial(\underline{\psi})}d\psi .
$$
Notice that, by assumption, the Jacobian
$\frac{\partial(\psi)}
{\partial(\theta)}$ exists and its only zeros
are those of the Jacobian of $\tau$, which
are less than $n$. Hence,
$\Omega(\theta)\frac{\partial(\theta)}
{\partial(\psi)}$ is an analytic function.
Moreover, we have that $\theta$,
$\phi\circ\Lambda_s$ are analytic functions of
$\psi_1\dots\psi_s$ and, by making $\delta$ small
enough we can assume $\theta_1\dots\theta_s$ are
very close to $\psi_1\dots\psi_s$ respectively.
All these statements are uniform in the remaining
variables.
In order to perform the integration in
$\psi_1\dots\psi_s$ we deform the
contour of integration and, using the fact
that, for sufficiently small $\delta$,
$\underline{\theta}(\psi_1)\approx
\psi_i\ \ i\leq s+1$ in an analytic extension
of the domain. We can get:
\begin{equation}
\vert \hat \phi_{\underline{k}}\vert\leq M e^
{-\xi\big(\vert k_1\vert+\dots+\vert k_s\vert\big)
+\varepsilon(\delta)\vert\underline{k}\vert} .
\label{bounds}
\end{equation}
Where $\xi$ is the width of the strip of
analyticity, $M$, a uniform bound of the
functions in this complex strip --- times
$4^n$ --- and $\varepsilon$ a number which
can be made as small as desired by taking
$\delta$ small enough.
An analogous argument using the unstable
foliation will show that
\begin{equation}
\vert \hat \phi_{\underline{k}}\vert\leq M e^
{-\xi\big(\vert k_{s+1}\vert+\dots+\vert k_n\vert\big)
+\varepsilon(\delta)\vert k\vert} .
\label{boundu}
\end{equation}
Hence $\vert\hat \phi_k\vert\leq\sqrt
{{\vert\hat \phi_{\underline{k}}\vert}^2}
\leq Me^{-\big(\xi/2-\varepsilon(\delta)\big)
\vert\underline{k}\vert}$ and hence, the lemma
is established.
\endpf
\demo{Remark}
Note that the proof is not restricted to considering just two
foliations which are transversal.
If we had several foliations -- with uniformly analytic leaves
and Jacobians analytic along the leaves -- such that the
tangent spaces of the leaves span the whole tangent space,
we could have decomposed the tangent space
in several pieces and obtain a finite number of
bounds analogues to (\ref{bounds}), (\ref{boundu})
which would also lead to the conclusion of
exponentially fast decay.
\demo{Remark}
Unfortunately, the bounds we obtained are not tame
in the sense of \cite{H} \cite{Ze1},\cite{Ze2}.
(Note that the parameter $\xi$ that measures the
analyticity domain for the solution is roughly
half of what it was for the data.)
It is possible to choose other norms (for example
$\Vert\quad\Vert_{C^\omega_s} +
\Vert\quad\Vert_{C^\omega_u} +
\Vert\quad\Vert_{C^\omega_X}$)
in which the estimates are indeed tame, but
with the choices we could come up with, analytic
differential operators do not act nicely except in
the case that the foliations are smooth.
This is unfortunate, because one of the original
motivations to study regularity of Livsic's
equation was its application in Nash-Moser
type theorems. Except in the case where the
foliations are analytic, already covered in
\cite{LMM}, we do not know how to do that.
Using the strategy of \cite{LMM}, proving smoothness in
parameters would allow to avoid hard implicit
function theorems.
\demo{Remark}
We learned the idea of using two different
changes of coordinates to estimate the Fourier
coefficients from \cite{HK} where it is
attributed to C. Toll (In \cite{LMM} only one
was used, which required the extra assumption
that the foliations were analytic.)
\demo{Remark}
Composing the functions with a map of the torus
is used in \cite{B} (see \cite{AR}) much before
its use in \cite{LMM}. The further introduction of
the $\Omega$ function done here, simplifies matters
but it is not quite essential.
\section{Analytic conjugacy of
low dimensional Anosov systems}\label{sec-three} %% 3.
In view of the results of the previous section, to
prove Theorem~\ref{thm-two} it suffices to show that the
conjugating homeomorphism is in
$C^\omega_s\cap C^\omega_u$.
In order to show it belongs to $C^\omega_s$
it suffices to show that the restriction to
the stable manifold of a fixed point is
uniformly analytic because a closed graph
argument will conclude that, since it is
possible to extend continuously to all stable
leaves, the extension will be uniformly
analytic in the stable direction (every stable
leaf is dense because of transitivity \cite{Ka}).
In order to prove that it is uniformly analytic
on this stable manifold we will show that the
restriction of the conjugacy to the stable manifold
satisfies a certain ordinary differential equation
that, under the hypothesis of equality of the Lyapunov
exponents, has uniformly analytic coefficients.
\begin{lemma}\label{lemma-three-one} %{Lemma 3.1}
Let $f,g$ be contractive diffeomorphisms
of class $C^kk\geq 2$ of the real line
$f(0)=g(0)=0$; $f'(0)=g'(0)=\lambda \quad 0
<\vert\lambda\vert<1\quad\lambda$ and $h$ a
$C^\ell$,$\ell>1$ diffeomorphism satisfying
$f\circ h=h\circ g$. Then,
$$
\Gamma(x,y)=
\lim_{n\to\infty}
\frac
{g'(x)g'\big(g(x)\big)\dots g'\big(g^n(x)\big)}
{f'(y)f'\big(f(y)\big)\dots
f'\big(f^n(y)\big)}
$$
exists and it is a $C^{k-1}$ function.
Moreover, $h$ satisfies
$$
h'(x)=\Gamma\big(x,h(x)\big)h'(0) .
$$
\end{lemma}
\begin{proof}
Once we know that the limit exists the fact
that $h'(x)$ satisfies the differential equation
is easy because out of the equation
$f^n\circ h=h\circ g^n$ we get by taking derivatives
$$
h'(x)=
\frac
{g'(x)g'\big(g(x)\big)\dots g'\big(g^n(x)\big)
h\big(g^{n+1}(x)\big)}
{f'\big(h(x)\big)f'\big(fh(x)\big)\dots
f'\big(f^nh(x)\big)}
$$
To prove the existence of the limit it is easier
to study the logarithms (we will assume for the sake
of notation that $f,g$ preserve orientation). We
can also assume that $x,y$ are small.
To prove that the limit exists, it suffices to show
$$
\lim_{n\to\infty}
\sum^n\big(\log g'\big(g^i(x)\big)-
\log f'\big( f^i(y)\big)\big)
$$
converges and is differentiable.
Convergence follows from the fact that
$\log g'(0)=\log f'(0)$ and that we
have
$$
\log g'\big(g^i(x)\big)=
\log g'(0) +k\vert g'(0)\vert^nx+
\circ(\vert g'(0)\vert^n)\vert x\vert
$$
and likewise for $f$. Then, the series
converges absolutely and uniformly in a
compact interval around zero because it
can be dominated by a geometric series.
To prove existence of higher derivatives we
just have to take formal derivatives and
check they are uniformly bounded by numerical
series that converge absolutely.
When we take derivatives with respect to
$x$ we get factors $(g^i)'$ in each term
involving $g$ and the ones containing $y$
drop. So we are left again with a geometric
series.
Getting higher derivatives proceeds along
the same lines but we have to cope with
the extra terms coming from higher derivatives
of $g$. Since we only have to study those
derivatives in the interval $[0,x]$ they are
uniformly bounded and the argument of \cite{LMM}
p.~574 shows these extra factors can be
dominated by a geometric series times a
polynomial. Notice that all mixed derivatives
are zero.
Notice also that, even when $f$ and $g$ are
uniformly close over the whole real line,
it is not clear that $\Gamma(x,y)$ would
be uniformly bounded on the whole real line.
In fact, for the diffeomorphisms of the line
obtained by restricting two dimensional Anosov
diffeomorphism to a stable manifold, the proof
of smoothness of the conjugacy we will present,
amounts to showing that $\Gamma(x,y)$ is indeed
uniformly bounded. So, any proof of uniform
boundedness for $\Gamma(x,y)$ will require
extra hypothesis; in our case, they will be
implied by equality of Lyapunov exponents
of the two-dimensional diffeomorphisms.
In order to complete the proof of Theorem~\ref{thm-two} we
will resort to the notation of \cite{Ll} which
is better adapted to this technique rather
than that of \cite{LM} and later on we will
explain which modifications are necessary to
prove the result for flows on three dimensional
manifolds.
We will henceforth call $f,g$ the two Anosov
diffeomorphisms satisfying the hypothesis of
theorem 2 and $h$ the conjugacy between then
$f\circ h=h\circ g$.
First of all we observe that it suffices to
prove the theorem under the assumption that
the conjugating homeomorphism is $C^k$-close
to the identity $k=1,\dots,\infty$.
In effect by \cite{LM} we know $h$ is a
$C^\infty$ diffeomorphism and hence we
can approximate it in $C^{k+1}$ by an
analytic diffeomorphism $h^*$. It suffices
to prove the theorem for $f$ and
$g^*=h^*gh^{*-1}$.
Now, following \cite{Ll} we observe that it
suffices to study what happens in the
stable manifolds of the fixed point, which
we will assume is at $0$.
We will take coordinates in these stable manifolds
and call $\tilde f$, $\tilde g$ and
$\tilde h$, the expression in coordinates of $f$,
$g$ and $h$ respectively.
We will require that the systems of coordinates satisfy:
\begin{description}
\item[1)] Derivatives with respect to the coordinates extend to
$C^\omega_{s(f)}\ C^\omega_{s(g)}$
vector fields respectively.
\item[2)] $\sup_{t\in\real}\vert\tilde h(t)-t\vert<\infty$.
\end{description}
The construction in \cite{Ll} works without
modification, but there is a more elegant
construction due to Moriy\'on \cite{MM}, which
we will now describe.
For one of the stable manifolds, we choose
arc-length as our parameterization. For the
second manifold, we say that the coordinate of a
point in the first manifold closest to the given point.
(Later on, we will see yet another construction.)
Notice that due to assumption 1), and standard
facts about the stable manifolds the derivatives
$\tilde f'$, $\tilde g'$, $\tilde h'$, extend
to functions in the manifold which are
$C^\omega_{s(f)}$, $C^\omega_{s(g)}$,
$C^\omega_{s(h)}$ respectively. Moreover, there
is a complex extension to a domain
$D_\xi =\{z\vert\ \vert\hbox{\rm Im}\ z\vert\leq\xi\}$
in such a way that $\tilde f$, $\tilde g$ are
defined there and are uniform contractions.
Notice that the function in parenthesis extends
to the whole manifold and that, the sum over
periodic orbits is zero because of the assumption
on Lyapunov exponents. We can, then, apply
Livsic's theorem \cite{Ll} to conclude that
there is a bounded function $\psi$ in such
a way that
$$
[\log \tilde{g}'-\log \tilde{f}'\circ
\tilde h](x)
=\psi(x)-\psi\big(\tilde g(x)\big)
$$
so that all the sums telescope. This concludes
the proof when $y=\tilde h(x)$, $x\in\real$.
If $x, y\in D_\xi$ we proceed as follows
\begin{eqnarray*}
\log \Gamma_n(x,y)&=&\log \Gamma_n(x,y)=
\log \Gamma_n\big(Re(x), \tilde h \big(Re(x)\big)\\
%%
&&-\big[\log \Gamma_n\big(Re(x),\tilde h\big(Re(x)\big)\big)
-\log \Gamma_n\big(x,\tilde h\big(Re(x)\big)\big]\\
%%
&&-\big[\log \Gamma_n\big(x,
\tilde h\big(Re(x)\big)\big)-
\log \Gamma_n\big(x,y)\big]
\end{eqnarray*}
and we can bound each one of the terms --- using
the uniform contractivity estimates --- independently
of $n$. (This is done in \cite{Ll} in more detail.)
Clearly the $\Gamma_n$ are analytic and since they
are uniformly bounded on a complex domain, they
converge pointwise on the real line. Hence it
follows that they converge uniformly in the
domain $D_\xi$.
It then, follows that $\Gamma(x,y)$ is not only
defined in the real line but also extends to a
complex neighborhood.
Since $\tilde h'(x)=\Gamma\big(x,\tilde h(x)\big)$
in the real line, it follows that $\tilde h$ extends
to a domain $D_{\xi'}$ and in that domain
$\tilde h(t)$ --- $t$ is uniformly bounded.
To prove $h\in C^\omega_s$, we just observe that
any stable leaf is dense, so we can find
segments of the stable leaf of zero approaching any
leaf and that $\tilde h'$ restricted to these
segments is uniformly analytic and
converges in the real line. It follows it
converges in the complex extension.
Since there is an identical argument to show
$h\in C^\omega_u$, an invocation to our
Lemma~\ref{lemma-two} finishes the proof of Theorem~\ref{thm-two} as
stated.
The proof of the Theorem~\ref{thm-two-prime} for three dimensional
flows goes along similar lines. Hence, we just
sketch the differences.
First, we can assume, by the same argument as
before, that the two vector fields, $X$, $Y$ are
$C^k$ close and the conjugating
homeomorphism $\Phi$ is $C^k$ close to
the identity.
We now pick a periodic orbit $\gamma$ for $X$ of
period $T$ --- which we will assume to be 1 ---
$W^{s(X)}_\gamma$ can be mapped analytically to
$S^1\times\real$, in such a way that the angle
coordinate represents the phase of the orbit
and the real coordinate is a parameterization
of the stable manifold. (In other words, the
lines given by ``angle = cte'' are stable
manifolds.)
In $W^{s(y)}_{\Phi(\gamma)}$ we can introduce
a similar system of coordinates. We can either
use Moriy\'on's device or we can realize that
the smoothed out homeomorphism of \cite{LM}
induces a $C^\infty$ coordinate system. We
can further smooth out the restriction to
$S^1\times\real$ to obtain analyticity.
The same properties as before about the
possibility of extending the
derivatives in coordinates hold. If we
now look to the time $T$ map in these
coordinates, we see that it preserves the
vertical directions. We just restrict to
one stable leaf and obtain a pair of one
dimensional maps to which we can apply the
same trick with the differential equation
and the $\Gamma(x,y)$ function.
The reason why the functions $\Gamma_n(x,y)$
corresponding to the restrictions to a
one dimensional stable manifold are bounded
is the following: In \cite{LM} equation (2)
gives a formula for the derivative of such
a mapping as the exponential of the integral
along an orbit of a function in the manifold.
$\Gamma_n(x,y)$ can then be expressed as
the exponential of an integral along an
piece of an orbit running from $0$ to $n$.
The integrand (called $\eta$ in \cite{LM},
has zero integral along any periodic orbit,
if the Lyapunov exponents are to be the
same along corresponding periodic orbits.
$\eta$ is also a H\"older function.
Hence, by Livsic's theorem, it is the
derivative along the flow of a
H\"older and function $\psi$
hence $\Gamma_n(x,y)$ is the exponential
of the difference of the function $\psi$
at two different points, hence bounded
independently of $n$. The rest of the
proof is as the previous one.
\end{proof}
\demo{Remark}
The use of the function $\Gamma$ to improve
smoothness of the conjugacy shows that in
one dimension, when the conjugacy is
$C^1$ is actually as smooth as the function.
This should be compared with the results
of \cite{St} --- where one derivative
is lost --- and with his Example~9 which
shows that the assumption $h\in C^1$ in
this remark cannot be removed. (See also the
discussion in \cite{Ne}.)
This approach also can be used to prove the
results of \cite{Pr}, which were
an inspiration for us.
We also point out that after this paper was written,
there appeared two papers that clarify the role of $\Gamma$.
In \cite{Po} it was shown that another way of
considering the bootstrap of regularity is noticing that
the SRB measures have to be transported into each other.
Hence the function $\Gamma$ has the dynamical interpretation of
quotient of densities. This method was also used in
\cite{Ll2}. We point out that the approach described here,
even if less geometric.
makes sense even outside of the support of the SRB measure
but in the basin.
\section{Acknowledgments}
I thank S.~Hurder and T.~Katok for showing
me a preprint of \cite{HK} prior to
publication and R.~Moriy\'on for
telling me the contents of reference
\cite{MM}.
Two visits to Caltech in the Fall semester
of 1986 were instrumental in getting me
to write this paper and I thank
A.~Katok and B.~Simon for
organizing them.
The inspiration and encouragement of
A.~Katok is particularly acknowledged.
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\end{document}