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\begin{document}
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\title{ Cocycles' stability for partially hyperbolic systems}
% Information for first author
\author{ A. Katok}
% Address of record for the research reported here
\address{ Department of Mathematics,
The Pennsylvania State University, University Park, PA 16802 }
% Current address
%\curraddr{Department of Mathematics and Statistics,
%Case Western Reserve University, Cleveland, Ohio 43403}
\email{$\rm katok \_ a$@math.psu.edu}
% \thanks will become a 1st page footnote.
\thanks{The first author is partially supported by the NSF Grant DMS
9404061. }
% Information for second author
\author{A. Kononenko}
\address{ Department of Mathematics,
The Pennsylvania State University, University Park, PA 16802}
\email{avk@math.psu.edu}
% General info
%\subjclass{Primary ; Secondary }
\begin{abstract}
In this paper we establish Livshitz-type theorems for partially
hyperbolic
systems. To be more precise, we prove that for a large class of
partially
hyperbolic transformations and flows the subspace of Hoelder
coboundaries is
closed and can be described by some natural geometric conditions. This
class
includes an open, in $C^2$ topology, neighborhood of the time-one
maps of contact Anosov flows (for example, the geodesic flows on
manifolds of negative
curvature).
Along the way we prove several results on the transitivity of the pair
of
stable and unstable foliations for partially hyperbolic systems. In
particular,
we establish the transitivity property for the time-one maps of contact
Anosow
flows and their small perturbations, which has important applications to
the
stable ergodicity of the time-one maps of geodesic flows on the
manifolds of
negative curvature.
\end{abstract}
\maketitle
\section{Stability of cocycle spaces.}
Let $f$ be a transformation of a space $X.$ The cohomological equation
corresponding to a fixed function $\phi$ (also called a cocycle) is the
following equation, with an
unknown function $h,$
$$ \phi(x)-h(x)+h(f(x))=0, x \in X.$$
If this equation has solutions then $\phi$ is called a coboundary, and
$h$ is
called a transfer function. Two functions are called cohomologous if
their difference is a coboundary. Depending on the classes of
regularity of cocycles and transfer functions allowed one associates
with the transformation $f$ various cohomology spaces
(\cite{katokhasselblat}, \cite{livsic1}).
A number of well known results describes the spaces of coboundaries
for different types of
maps $f.$ The most celebrated results of this kind are
the Livshitz-type theorems for hyperbolic diffeomorphisms and flows
(\cite{collet}, \cite{guilleminkazdan-onthe},
\cite{guilleminkazdan-some},
\cite{katokhasselblat} (Section 19.2), \cite{livsic1}, \cite{livsic2},
\cite{delalave}, \cite{delalave-preprint}). However some non-hyperbolic
systems also display certain regularity for cocycles. These include
Diophantine translations of a torus, affine maps (\cite{katokcocycles},
Section 10.5), partially hyperbolic toral automorphisms (\cite{veech}),
integrable systems, area-preserving flows on surfaces of high genus
(\cite{forni}).
It is a simple and very general fact that under some natural conditions
the closure of the space of coboundaries is equal to the intersection of
the kernels of all $f$-invariant linear functionals. It is true, for
example, for $C^{\infty}$-cocycles with $C^{\infty}$ transfer functions,
for Hoelder cocycles with Hoelder transfer functions, etc (the proof is
similar to the proof of Proposition 9.12 in \cite{katokcocycles}).
Therefore, if we want to describe a certain space of coboundaries $B$
for a map $f$ we naturally encounter the following two questions:
\begin{enumerate}
\item Is $B$ closed, and thus equal to the intersection of the kernels
of all $f$-invariant functionals?
\item What is the structure of the set $\cal{F}$ of all $f$-invariant
functionals? In particular, is it possible to find a dense or
generating subset of $\cal{F}$ that consists of the functionals which
have ``nice descriptions'' in terms of the dynamics of $f$?
\end{enumerate}
Livshitz theorems describe the space of coboundaries as being the common
zero set for a family of functionals corresponding to the periodic
points of $f,$ where $f$ is a hyperbolic diffeomorphism, thus giving a
positive answer to both questions.
In the present paper, we give an affirmative answer to the first
question for a large class of partially hyperbolic dynamical systems.
Moreover, we show that for such systems the subspace of Hoelder
coboundaries can be described as a common zero set of some natural
geometrically defined functionals -- the periodic cycles functionals
(see Definition~\ref{df-functionals}), thus answering the second
question.
Before we proceed to the formulations of our results we would like to
introduce some convenient terminology (which is a slight modification
of the terminology first suggested in \cite{katokcocycles}).
Let $\cal{P}$ be a space of functions on a space $X$ endowed
with some topology. Let $ \cal{P}_1 $ be
another space of functions (not necessarily equipped with
any topology).
\begin{df}
The space of $\cal{P}$-cocycles, i.e., cocycles which lie in $\cal{P},$
of
a transformation $f$ of $X$ is called {\bf $\cal{P}_1$-stable}
if the space of the cocycles cohomologous to a constant, with the
transfer functions from
$\cal{P}_1,$ is closed with respect to the topology of $\cal{P}.$
\end{df}
With the exception of Section~\ref{sec-smooth}, we will work with
Hoelder cocycles. To be more precise, the
role of $\cal{P}$ will be played by the space
$L_{\alpha}$ --- the space of Hoelder functions
with the fixed exponent $\alpha \in (0,1].$ The role of $\cal{P}_1$ will
be
played by $L_{\beta},$ $\beta \in (0,1]$ or the space of continuous
functions. We will refer to the functions from $L_{\alpha}$ as
$\alpha$-Hoelder functions, and we will refer to the functions from
$L_1$ as
Lipschitz functions.
\section{Transitivity of a pair of foliations.}
Let $M$ be a Riemannian manifold and $S$ its submanifold. For $x,y \in
M$
(correspondingly $x,y \in S$)
we will denote by $d_M(x,y)$ (correspondingly $d_S(x,y)$) the infinum
of the lengths of the smooth
curves in $M$ (correspondingly $S$) connecting $x$ and $y.$
\begin{df}
\label{df-ltf}
A pair of continuous foliations $\cal{F}_1$ and $\cal{F}_2$ with smooth
leaves
is called {\bf locally transitive} if, for any compact subset $M_1$ of
$M,$ there exists $N \in \Bbb{N},$ such
that for any $\epsilon >0$ there exists
$\delta >0$ such that for every $x, y \in M$
with $x \in M_1$ and $d_M(x,y) < \delta$
there are points $x_1, \ldots, x_N \in M,$ satisfying the following
conditions: \begin{enumerate} \item $x_1=x,$ $x_N=y;$
\item $x_{i+1} \in \cal{F}_j (x_i),$
$i=1,\ldots, N-1,$ $j=1$ or $2;$
\item $d_M(x_i,x) \leq \epsilon$ and
$d_{\cal{F}_j(x_i)}(x_{i+1},x_i)<2\epsilon,$
$i=1,\ldots, N-1,$ $j=1$ or $2.$
\end{enumerate}
\end{df}
The notion of local transitivity of a pair of foliations was introduced
by
Brin and Pesin in \cite{brinpesin-partially}. They studied extensively
the local
transitivity property for the pair of stable and unstable foliations
for
partially hyperbolic dynamical systems, its relations to other
properties of the system, and its stability under perturbations.
An important class of partially hyperbolic dynamical systems -- the
extensions of the hyperbolic ones -- was proved to posses the
transitivity property (slightly weaker property then the local
transitivity) generically, by Brin in \cite{brin}.
Brin's examples include, in particular, the frame flows
on the manifolds of negative curvature. Also, the transitivity
condition appears in the work of Grayson, Pugh and Shub
\cite{graysonpughshub} on stable ergodicity. Namely, it is one of the
conditions which they require for the system to be stably ergodic.
The following definition is the Hoelder version of local transitivity:
\begin{df}
\label{df-lht}
A pair of continuous foliations $\cal{F}_1$ and $\cal{F}_2$ with smooth
leaves
is called {\bf locally $\alpha$-Hoelder transitive} if there exists $N
\in \Bbb{N},$
$\delta >0,$ $C>0$ such that for every $x$ and $y \in M$
with $d_M(x,y) < \delta$
there are points $x_1, \ldots, x_N \in M,$ satisfying the following
conditions:
\begin{enumerate} \item $x_1=x,$ $x_N=y;$
\item $x_{i+1} \in \cal{F}_j (x_i)$ $i=1,\ldots, N-1,$ $j=1$ or $2;$
\item $d_{\cal{F}_j(x_i)}(x_{i+1},x_i) < Cd_M(x,y)^{\alpha},$
$i=1,\ldots, N-1,$ $j=1$ or $2.$
\end{enumerate}
\end{df}
The following definition is useful in questions related to Lipschitz
cocycles, in particular, to $C^{\infty}$-cocycles.
\begin{df}
\label{df-wlht}
A pair of continuous foliations $\cal{F}_1$ and $\cal{F}_2$ with smooth
leaves
is called {\bf weakly locally $\alpha$-Hoelder transitive}, where
$\alpha \in
(0,1],$ if there exists
$\delta >0$ and $C>0$ such that for every $x,y \in M$
with $d_M(x,y) < \delta$
there are points $x_1, \ldots, x_k \in M,$ satisfying the following
conditions: \begin{enumerate} \item $x_1=x,$ $x_k=y;$
\item $x_{i+1} \in \cal{F}_j (x_i),$
$i=1,\ldots, k-1,$ $j=1$ or $2;$
\item $\sum_{j=1}^{j=k-1}d_{\cal{F}_j(x_i)}(x_{i+1},x_i) <
Cd_M(x,y)^{\alpha}.$
\end{enumerate}
\end{df}
Clearly, local $\alpha$-Hoelder transitivity implies weak local
$\alpha$-Hoelder
transitivity and transitivity.
\section{Stability theorems for Hoelder cocycles.}
Recall the following standard definition due to Brin and Pesin
(\cite{brinpesin-partially}):
\begin{df}
\label{df-partially}
A diffeomorphism $g$ of a manifold
$M$ with a Riemannian norm $|| \cdot ||$ is called {\bf partially
hyperbolic}
if there exist real numbers $\lambda_1 > \mu_1 >0,$ $i=1,2$, $K,K'>0$
and a continuous
splitting of the tangent bundle
$$TM=E^+ \bigoplus E^0 \bigoplus E^-$$
such that for all $x \in M,$ for all $v \in E^+(x)$ ($v \in E^+(x)$
respectively) and $n >0$ ($n<0$ respectively) we have for the
differential
$g_*:TM \rightarrow TM$
$$||g_*(v)|| \leq K e^{-\lambda_1 n} ||v|| \, \, \, (||g_*(v)|| \leq K
e^{-\lambda_2 |n|} ||v||, {\text respectively}) $$
and for all $n \in \Bbb{Z}$ and $v \in E^0(x)$ we have
$$||g_*(v)|| \geq K' e^{-\mu_1 n} ||v||, \, n>0 \, {\text and} \,
||g_*(v)|| \geq K' e^{-\mu_2 |n|} ||v||, \, n<0.$$
Furthermore, we assume that the distribution $E^0$ is uniquely
integrable.
We will call $E^+$ and $E^-$ {\bf stable and unstable distributions}
respectively.
\end{df}
Note that if $M$ is compact these notions do not depend on the ambient
Riemannian
metric.
The following fact is a direct corollary of the Hadamard-Perron theorem
(see, for example, \cite{katokhasselblat}, Theorem 6.2.8).
\begin{thmA}
For a partially hyperbolic dynamical system, there are Hoelder
foliations
$W^s$ and $W^u$ tangent to the distributions $E^+$ and $E^-$
respectively.
We call this foliations {\bf stable and unstable foliations}. The
individual
leaves of these foliations are $\Ci$-immersed submanifolds of $M,$ and
are
called {\bf stable and unstable manifolds}.
\end{thmA}
For the basic theory of the partially hyperbolic systems see
\cite{brinpesin-partially} and \cite{hirshpughshub}.
\begin{df}
We will call a set $C$ of points $x_1,x_2, \ldots, x_{2n}, x_{2n+1}=x_1
\in M$
a {\bf periodic cycle} if $x_{2k} \in W^s(x_{2k-1})$ and $x_{2k+1} \in
W^u(x_{2k}),$ for $k=1, \ldots, n.$
\end{df}
Notice that if $y \in W^s(x)$ then $d_M(f^i (x),f^i(y))$ decreases
exponentially, thus $\phi(f^i (x))-\phi(f^i(y))$ also decreases
exponentially. Therefore, the series $$P^+(x,y)(\phi)=
\sum_{i=0}^{\infty} (\phi(f^i (x))-\phi(f^i(y)))$$ converges absolutely
for
any Hoelder function $\phi.$
Similarly, if $y \in W^u(x)$ then the series $$P^-(x,y)(\phi)=
-\sum_{i=-1}^{-\infty} (\phi(f^i (x))-\phi(f^i(y)))$$ converges
absolutely for any Hoelder function $\phi.$
Let us notice that if $\phi$ is $\Ci$ then $P^+(x,y)$ and $P^-(x,y)$
are infinitely differentiable with respect to $y$ along the stable and
unstable foliations, and the derivatives are continuous with respect to
$x.$
\begin{df}
\label{df-functionals}
For a periodic cycle $C,$ we will denote by $F(C)$ the following
continuous functional on the space $L$ of Hoelder functions:
$$F(C) (\phi)=P^+(x_1,x_2)(\phi)+P^-(x_2,x_3)(\phi)+\cdots $$ $$+
P^+(x_{2n-1},x_{2n})(\phi)+P^-(x_{2n},x_1)(\phi).$$
We will call this functional a {\bf periodic cycle functional}.
\end{df}
The following results give general criteria for cocycle's stability for
partially hyperbolic diffeomorphisms:
\begin{thm}
\label{thm-continuous}
If $f$ is a partially hyperbolic diffeomorphism such that the pair
$(W^s, W^u)$
is locally transitive, then, for any $\beta \in (0,1],$
the space of $\beta$-Hoelder cocycles of $f$ is $C^0$-stable, and the
subspace
of cocycles cohomologous to a constant is the common zero set of the
periodic
cycles functionals, i.e., $\phi \in L_{\beta}$ is cohomologous to a
constant,
with $C^0$ transfer function,
if and only if
$F(C)(\phi)=0$ for all periodic cycles $C.$ \end{thm}
{\bf Remark:} Notice that if the cocycle is $\Ci$ and the periodic
cycles
functionals vanish then the transfer function constructed in
Theorem~\ref{thm-continuous} has continuous derivatives of all orders
along the
stable and unstable foliations.
\begin{thm}
\label{thm-hoelder}
If $f$ is a partially hyperbolic diffeomorphism such that the pair
$(W^s,W^u)$ is locally $\alpha$-Hoelder transitive, then,
for any $\beta \in (0,1],$ the space of $\beta$-Hoelder cocycles is
both $\alpha
\beta$-Hoelder stable and $C^0$-stable.
If the pair $(W^s,W^u)$ is weakly locally $\alpha$-Hoelder transitive
then the
space of Lipschitz cocycles is $C^0$-stable and $\alpha$-Hoelder stable.
In all cases, the subspace
of cocycles cohomologous to a constant is the common zero set of the
periodic
cycles functionals.
\end{thm}
{\bf Proof of Theorems~\ref{thm-continuous} and ~\ref{thm-hoelder}.} Let
$\phi$ be a $\beta$-Hoelder function. Then there exists a constant
$L$ such that $|\phi(a)-\phi(b)|\leq Ld_M(a,b)^{\beta},$ for all $a, b
\in M.$
Assume that $F(C)(\phi)=0$ for any periodic cycle $C.$
We will call a set $S(x,y)$ of points $x_1=x,x_2, \ldots, x_{k}=y \in
M$
a {\bf broken path} from $x$ to $y,$ if $x_{i+1} \in \cal{F}_j (x_i),$
$i=1,\ldots, k-1,$ $j=1$ or $2.$
We will call $\sum_{j=1}^{j=k-1}d_{\cal{F}_j(x_i)}(x_{i+1},x_i)$ the
{\bf length of the broken path} $S.$ We will call the point $x_i$ the
{\bf turning points} of $S$ and the points $x$ and $y$ the end points of
$S.$ We will also say that $S$ {\bf connects} $x$ with $y.$
Let $$F_1(S(x,y))(\phi)=\sum_{i=1}^{k-1} P^*(x_i,x_{i+1})(\phi),$$ where
$*$ is equal to $+$ or $-$ depending on whether the $x_{i+1}$ belong to,
correspondingly, the stable or unstable manifolds of $x_i.$
Fix an arbitrary point $x \in M,$ and define a function
$h(y)=F_1(S(x,y))(\phi),$ where $S$ is some broken path from $x$ to $y.$
The function $h(y)$ is defined for all points $y \in M$ due to the
transitivity of the pair $(W^s,W^u),$ and $h(y)$ does not depend on the
choice of the path $S,$ due to the fact that $F(C)(\phi)=0,$ for any
periodic cycle $C.$
Now we will prove that due to the local transitivity condition
(correspondingly
local $\alpha$-Hoelder transitivity condition) on the pair of stable
and
unstable foliations, $h(y)$ is continuous, (correspondingly
$\alpha\beta$-Hoelder).
Indeed, there exists $\delta_1>0$ such that for $a$ and $b$ belonging to
the same
leaf $W$ of the stable foliation with $d_{W}(a,b)<\delta_1,$
$$P^+(a,b)(\phi) =
\sum_{i=0}^{\infty} (\phi(f^i (a))-\phi(f^i(b))) \leq
\sum_{i=0}^{\infty}
L d_M(f^i(a), f^i(b))^{\beta} \leq$$ $$ \sum_{i=0}^{\infty} L
\lambda^{i \beta}
d_M(a,b)^{\beta}=K_1 d_M(a,b)^{\beta},$$ where $\lambda$ is a
contracting exponent of the stable foliation,
and $L$ and $K_1=\sum_{i=0}^{\infty} L \lambda^{i \beta}$ are constants
that depend on $\phi$ and $f$ but not $a$
and $b.$
Similarly, there exists $\delta_2 >0$ such that for $a$ and $b$
belonging to the
same leaf $W$ of the unstable foliation with $d_{W}(a,b)<\delta_2,$
$$P^-(a,b)(\phi) \leq K_2 d_M(a,b)^{\beta},$$ for some constant $K_2$
that depends on $\phi$ and $f$ but not $a$
and $b.$
Let $K=\max(K_1,K_2).$ Then we have
$$P^*(a,b)(\phi) \leq K d_M(a,b)^{\beta},$$
for all $a$ and $b$ which belong to the same leaf of either the stable
or unstable foliation and such that $d_M(a,b) <
\min(\delta_1,\delta_2).$
Suppose the pair $(W^s,W^u)$ is locally transitive. Fix $\epsilon >0$
and
let $\delta$ and $N \in \Bbb{N}$ be as in Definition~\ref{df-ltf}. Also,
assume that $\delta < \min(\delta_1,\delta_2).$ Let $z_1$
and $z_2$ be arbitrary points in $M$ such that $d_M(z_1,z_2)< \delta,$
and
let $Z=\{x_1=z_1, x_2, \ldots, x_N=z_2\}$ be a broken path from $z_1$ to
$z_2$ as in
Definition~\ref{df-ltf}. Then,
$$h(z_1)-h(z_2)=F_1(Z)(\phi)\leq \sum_{i=1}^{N-1} K
d_M(x_i,x_{i+1})^{\beta}
\leq (N-1)K(2\epsilon)^{\beta},$$
which proves that $h$ is continuous.
If the pair $(W^s,W^u)$ is locally $\alpha$-Hoelder transitive then we
have
$$h(z_1)-h(z_2)=F_1(Z)(\phi)\leq \sum_{i=1}^{N-1} K
d_M(x_i,x_{i+1})^{\beta}
\leq (N-1)KC^{\beta}d_M(z_1,z_2)^{\alpha \beta},$$ where $C$ is as in
Definition~\ref{df-lht}, which shows that
$h$ is $\alpha \beta$-Hoelder.
If $\phi$ is Lipschitz, and the pair $W^s,$ $W^u$ is weakly locally
$\alpha$-Hoelder transitive then we still have
$$h(z_1)-h(z_2)=F_1(Z)(\phi)\leq \sum_{i=1}^{k-1} K d_M(x_i,x_{i+1})
\leq KCd_M(z_1,z_2)^{\alpha},$$ which shows that
$h$ is $\alpha$-Hoelder.
Now we will show that $-h(y)$ solves the cohomological equation for
$\phi.$ Indeed,
let $S(x,y)=\{x_1=x,x_2, \ldots, x_{k}=y\}$
be some broken path from point $x$ to point $y,$ and let $S(f(x),f(y))$
be a broken path from $f(x)$ to $f(y)$ consisting of points
$f(x_1)=f(x),f(x_2), \ldots,f(x_k)=f(y).$
Then
$$h(f(y))=h(f(x))+F_1(S(f(x),f(y))(\phi)=$$ $$h(f(x))+\sum_{i=1}^{k-1}
P^*(f(x_i),f(x_{i+1}))(\phi)=$$ $$h(f(x))+\sum_{i=1}^{k-1}
P^*(x_i,x_{i+1})(\phi)-\sum_{i=1}^{k-1}(\phi(x_i)-\phi(x_{i+1}))=$$
$$h(f(x))+\sum_{i=1}^{k-1}
P^*(x_i,x_{i+1})(\phi)-\phi(x)+\phi(y)=h(f(x))+
h(y)-\phi(x)+\phi(y).$$
If we denote $h(f(x))+\phi(x)$ by $c,$ then we have
$$\phi(y)+h(y)-h(f(y))=c.$$
This shows that if $F(C)(\phi)=0$ for any periodic cycle $C$ then $\phi$
is cohomologous to a
constant cocycle.
Now assume that $\phi(y)-h(y)+h(f(y))=c$ is cohomologous to a constant
with
the continuous transfer function $h.$ Let $C=\{x_1,\ldots,x_{2k},
x_{2k+1}=x_1\}$
be some periodic cycle.
Notice that $$ P^+(x_i,x_{i+1})(\phi) =\sum_{j=0}^{\infty}
(\phi(f^j(x_i))-\phi(f^j(x_{i+1})))=$$ $$\sum_{j=0}^{\infty}
(h(f^j(x_i))-h(f^{j+1}(x_i))+c
-(h(f^j(x_{i+1}))-h(f^{j+1}(x_{i+1}))+c))=$$
$$\sum_{j=0}^{\infty}
((h(f^j(x_i))-h(f^j(x_{i+1})))-(h(f^{j+1}(x_i))-h(f^{j+1}(x_{i+1})))=$$
$$
h(x_i)-h(x_{i+1})-\lim_{j \rightarrow
\infty}(h(f^j(x_i))-h(f^j(x_{i+1})))=
h(x_i)-h(x_{i+1}),$$ if $i$ is odd.
Similarly, $$ P^-(x_i,x_{i+1})(\phi)=h(x_i)-h(x_{i+1}),$$ if $i$ is
even.
Therefore, $$F(C)(\phi)=\sum_{i=1}^{i=2k}(h(x_i)-h(x_{i+1})) =0.$$
Theorems~\ref{thm-continuous} and ~\ref{thm-hoelder} are proved.
Notice, that from the last part of the proof we get the following
\begin{cor}
\label{cor-necessary}
For any partially hyperbolic diffeomorphism, if a Hoelder function
$\phi$
is cohomologous to a constant with a continuous transfer function then
it belongs to the intersection of the kernels of all periodic cycles
functionals.
\end{cor}
\section{Transformations coming from the Weyl \newline chamber flows.}
Recall the following standard
\begin{df}
A pair of smooth foliations $\cal{F}_1$ or $\cal{F}_2$ is called {\bf
totally non-integrable} with index $p \in \Bbb{N}$ if for any
$x \in M$ the Lie brackets of degree at most $p$ of the vector fields
tangent to
either $\cal{F}_1$ or $\cal{F}_2$ span the whole $T_xM.$
We will call a smooth distribution $E$ on $M$ {\bf totally
non-integrable} with
index $p \in \Bbb{N}$ if for any $x \in M$ the Lie brackets of degree
at most
$p$ of the vector fields tangent to $E$ span the whole $T_xM.$
\end{df}
The following simple statement opens up a way to construct a large
class of examples of
transformations with Hoelder stable and $C^0$-stable spaces of Hoelder
cocycles:
\begin{prop}
\label{prop-integrability}
If a pair of $C^{\infty}$ foliations on $M$ is totally non-integrable
with index $p,$ then it is locally transitive and, if $M$ is compact it
is
locally $(1/2^p)$-Hoelder transitive.
\end{prop}
We omit the proof of Proposition~\ref{prop-integrability}, since it is
standard and is unrelated to our main subject here. (A proof of a
similar result that can be easily modified to prove
Proposition~\ref{prop-integrability} can be found in
\cite{brinpesin-partially}.)
>From Proposition~\ref{prop-integrability} and Theorem~\ref{thm-hoelder}
we have the following
\begin{cor}
\label{cor-weyl}
Let $G$ be a semisimple group of non-compact type, $A$ its maximal split
Cartan
subgroup, $\Gamma$ an irreducible cocompact lattice, and $K$ any compact
subgroup of $G$ that commutes with $A.$ Then, for any $\beta \in
(0,1],$
for any regular element $a \in A$
acting on $M=K\backslash G/\Gamma,$ the space of $\beta$-Hoelder
cocycles is both
$\beta/2^p$-Hoelder stable and $C^0$-stable, where $p$ depends only on
$G$
and $K,$ but
not on $a,$ $\Gamma$ or $\beta.$
In particular, if $K$ is maximal such compact group as described above
then
the space of $\beta$-Hoelder cocycles is $\beta/2$-Hoelder stable.
If $a \in A$ is not regular but $\log(a)$ still has non-trivial
projections to
all simple components of the Lie algebra $\cal{G}$ of $G,$ then
the space of $\beta$-Hoelder cocycles is both
$\beta/2^{p(a)}$-Hoelder stable and $C^0$-stable, where now
$p(a)$ depends only on $G,$
$K$ and $a,$ but
not on $\Gamma$ or $\beta.$
Moreover, the subspace
of cocycles cohomologous to a constant is the common zero set of the
periodic cycles functionals.
\end{cor}
In particular, if $K$ is the maximal compact subgroup commuting with
$A,$
the above action of $a \in A$ is a part of the Weyl chamber flow. And
if $G$ is rank one the Weyl chamber flow coincides with the geodesic
flow on the corresponding
compact rank-one locally symmetric space of non-compact type.
Also notice, that if $\Gamma$ is not cocompact then the $C^0$ part of
Corollary~\ref{cor-weyl} still holds.
Theorem~\ref{thm-continuous} together with the results of Brin and Pesin
(\cite{brinpesin-partially}, Theorem 4.3) imply the
following
\begin{cor}
\label{cor-stable}
For any transformation $f$ described in Corollary~\ref{cor-weyl} there
is
an open neighborhood $U(f)$ in ${\rm Diff}^2(M)$ such that for every
$f_1 \in U(f)$
the space of $\beta$-Hoelder cocycles of $f_1$ is $C^0$-stable, and the
subspace
of cocycles cohomologous to a constant is the common zero set of the
periodic cycles functionals.
\end{cor}
\section{Time-one maps of contact Anosov flows.}
\label{sec-contact}
We will call a continuous distribution $E$ on a manifold $Q$ {\bf
locally transitive} ({\bf $\alpha$-Hoelder locally transitive}
respectively) if for
any $\epsilon>0$ there exists $\delta >0$ such that for any $x,y \in Q$
with $d_Q(x,y) < \delta$ there exists a broken piecewise smooth curve
$\gamma$ tangent
to $E,$ on its intervals of smoothness, and such that its length is less
then
$\epsilon$ ($C\delta^{\alpha},$ for some constant $C,$ respectively).
Then we prove the following:
\begin{prop}
\label{prop-2}
If for an Anosov flow $g^t$ on a manifold $Q$ the distribution $E^+
\bigoplus E^-$ is locally transitive
($\alpha$-Hoelder locally transitive, respectively) then the pair
$(W^s,W^u)$ is locally transitive (weakly $\alpha$-Hoelder locally
transitive, respectively).
\end{prop}
\begin{proof}
For $y \in Q$ we will denote the ball of radius $a$ around $y$ on $Q$
by $B_{a}(y).$
For a point $y \in Q$ denote the orbit passing through $y$ by $O_y.$
Fix $\epsilon.$ We will find numbers $\delta$ and $N$ satisfying
Definition~\ref{df-ltf} in several steps.
Let $x$ be an arbitrary point of $Q.$ In steps $1-7$ $x$ is the same
point.
{\bf Step 1.} There exists $0<\delta_1<\epsilon$ such that for any
$0<\delta<
\delta_1$ and any $y \in B_{\delta}(x)$ there is a broken piecewise
smooth curve
$\gamma$ tangent to $E^+\bigoplus E^-$ and connecting $x$ and $y.$
Moreover, $\gamma$ is inside $B_{\epsilon_1/2}(x),$ (where
$\epsilon_1/2=\epsilon/2 $ or
$\epsilon_1/2=C_1\delta^{\alpha}, $ for some constant $C_1.$ Therefore,
approximating $\gamma$ by broken paths we see that for any $\epsilon_2$
there exists
a broken path inside $B_{\epsilon_1}(x)$ connecting $x$ with some point
$y_1 \in
B_{\epsilon_2}(y).$ (Such an approximation always exists due to the
following
simple argument. Project the curve $\gamma$ to the transverse section
$T$ constructed in Step 2. Let $\Pi(\gamma)$ be the projection on $T.$
Then due to the product structure on $T$ (see Step 3), for small enough
$\delta_1,$ the curve
$\Pi(\gamma)$ can be arbitrarily well approximated by a broken path on
$T,$ (see Step 2) and
the lifts (see Step 3) of such approximations to $Q,$ with initial point
$x,$ give approximations of $\gamma$ by
broken paths on $Q.$)
{\bf Step 2.} Let $0<\delta_2$ be so small that there exists a piece
$T$ of smooth sub-manifold of $Q$ of codimension one, such that, for $
y \in B_{\delta_2}(x),$ the maximal connected piece of $O_y$ that
contains $y$ and belongs entirely to $B_{\delta_2}(x),$ intersects $T$
transversely in exactly one point $\Pi(x) \in T.$ From now on, for $
y \in B_{\delta_2}(x),$ we will denote by $O_y$ not the whole orbit
$O_y$ but the maximal connected piece of the orbit that contains $y$
and belongs entirely to $B_{\delta_2}(x).$
We have a well defined projection $\Pi: B_{\delta_2}(x) \rightarrow
T$ along the orbits. Two points $z_1$ and $z_2$ are mapped by $\Pi$ into
one point $n \in T$ if and only if $O_{z_1}=O_{z_2}.$ Moreover, their
stable and unstable manifolds are mapped into the same sub-manifolds of
$T.$ We will denote these sub-manifolds by $F^s(n)$ and $F^u(n).$
Therefore, we obtain two local foliations $F^s$ and $F^u$ in the
$\delta_2$-neighborhood of $x$ in $T.$ We will call broken paths on
$T$ defined with respect to $F^s$ and $F^u$ {\bf broken paths on $T.$}
{\bf Step 3.} Locally, there is a product structure on $T.$ Namely,
there exists $0<\delta_3<\delta_2$ such that for
$n_1, n_2 \in T_1=(B_{\delta_3}(x) \bigcap T)$
there are unique points $I_1$ and $I_2$ of intersections of $F^s(n_1)$
and $F^u(n_2)$ and
of $F^u(n_1)$ and $F^s(n_2)$ correspondingly.
Moreover, for any $\delta < \delta_3$ there exists $\delta'$ such that
if $n_1,n_2 \in B_{\delta'}(x)$ then $I_1,I_2 \in B_{\delta}(x).$
Now, notice that for any broken path $S_{T_1}=\{X, n_1, \ldots, n_k \}
\in T_1$ on $T$ and for any $\tilde{X} \in O_X$ there is a unique {\bf
lift to $Q$}, which is a broken path $S_{Q}(\tilde{X})=\{\tilde{X},
x_1,\ldots, x_k\} \in Q$ such that $\Pi(x_i)=n_i, i=1, \ldots,k.$
Indeed, there is a unique point $x_1$ of intersection of
$W^*(\tilde{X})$ with $O_{n_1}.$ Then there is a unique point $x_2$ of
intersection of $W^*(x_1)$ with $O_{n_2},$ and so on, where $*$ is equal
to $s$ or $u$ in accordance with the sequence of $s$ and $u$ in the
broken path $S_{T_1}$ on $T.$
{\bf Step 4.} Since, $T$ is transversal to the orbits of $g^t,$ there
exist $0<\delta_4<\delta_3$ such that for any $y \in B_{\delta_4}(x)$
the norm of the differential $D(\Pi |_{W^*(y)})$ is bounded away from
zero by some constant independent of $y$ and $*=s$ or $u.$
{\bf Step 5.} Therefore, due to the choice of $\delta_4$ in Step 4,
there exists $0<\delta_5<\delta_4$ and a constant $K$ such that for any
$\delta < \delta_5$ a lift to $Q$ of any broken path on $T$ inside a
$\delta$-neighborhood of $x$ belongs to a $K\delta$-neighborhood of
$\tilde{x}$ in $Q,$ if $\tilde{x} \in O_x \bigcap B_{\delta_5}(x).$
{\bf Step 6.} Therefore, due to the local product structure (Step 3),
the lifting procedure (Step 3) and the restriction on the norm of the
differential (Step 4), there exist $0<\delta_6 <\delta_5$ such that for
any $y \in B_{\delta_6}(x)$ there is a broken path consisting of $3$
turning points inside $B_{\delta_5}(x)$ that connects $y$ with some
point $z \in O_x$ such that $d_Q(y,z) < Kd_Q(\Pi(y),x).$
{\bf Step 7.} Now we will prove that there exists
$0<\delta_7<\delta_6,$
$0<\delta_7<\delta_1,$ such that for any point of $O_x \bigcap
B_{\delta_7}(x)$ there is a broken path inside $B_{\delta_5}$ with no
more then some fixed number $L$ of turning points that connects this
point with $x.$
First of all notice that if there is a path $S$ inside $B_{\delta_6}(x)$
connecting $x$ and $p \in O_x$ then for any $p' \in O_x$ between $x$ and
$p$ on $O_x$ there is a path $S'$ connecting $x$ and $p',$ and such
that $S'$ has as many turning points as $S$ does. Indeed, consider,
the closed broken path $\Pi(S)$ in $T.$ Due to the local product
structure it is easy to construct a continuous family $S^t$ of closed
broken paths in $T,$ with the same number of turning points as $S$ has,
and such that $S^0=\Pi(S)$ and $S^1=\{x\},$ where $\{x\}$ is a trivial
broken path that consists of several copies of the point $x,$ and the
end points for all $S^t$ are all equal to the point $x.$ Then the end
point $p^t$ of their lifts $S^t_{Q}(x)$ changes continuously on $O_x$
and $p^0=p$ and $p^1=x.$
Let us prove that there are broken paths $S_1(x,p_1)$
and $S_2(x,p_2)$ such that $p_1$ precedes $x$ on $O_x$ and $p_2$ follows
it.
Let us construct $S_1,$ the construction of $S_2$ is absolutely
parallel.
Fix point any point $y \in O_x \bigcap B_{\delta_6}(x)$ such that it
precedes $x$ and $d_Q (y,x) <\left( \frac{\delta_5}{2C} \right)^2.$ As
in Step 1,
let $\gamma$ be a curve tangent to $E^+\bigoplus E^-$ connecting $x$
and $y.$ Let $\epsilon_2 < \frac{d_Q(x,y)}{10K}.$ Then, due to the
Step 1, there exists a broken path $S$ in $Q$ inside $B_{\delta_5}$
connecting $x$ with some point $y_1 \in B_{\epsilon_2}(y).$ Then, we
can add two more points to $S$ in order to connect $y_1$ with some point
$p_1 \in O_x.$ Denote the resulting path by $S_1,$ then the point $p_1$
and the path $S_1$ satisfy our requirements. Indeed, $S_1$ is inside
$B_{\delta_5}$ and $d_Q(p_1,x) > d_Q(y,x) -Kd_Q(y_1,y)>0,$ thus $p_1$
precedes $x$ on $O_x.$
Now, we will show that it is possible to choose the broken path $S_1$
with no more than $6$ turning points. Let $L>6$ be the minimal possible
number of turning points and let $$S_1=\{x_1=x, x_2,x_3,x_4,x_5, x_6,
\ldots, x_L =p_1\}$$ be the path with exactly $L$ turning points. Let
$\{x_4, y_5, y_6 \in O_x\}$ be a
path like in Step 3, connecting $x_4$ with some point $y_6 \in O_x.$
Then,
1) If $y_6 $ precedes $x$ on $O_x,$ then the broken path
$\{x_1,x_2,x_3,x_4,y_5,y_6\}$ contradicts to the choice of $S_1.$
2) If $y_6 =x$ then the broken path $$\{y_6=x, y_5, x_4,x_5, x_6,
\ldots, x_L =p_1\}$$ contradicts to the choice of $S_1.$
3) Assume that $y_6 $ follows $x$ on $O_x.$ Then let $$\tilde{S_1} = \{
p_1=x_L, \ldots, x_6,x_5,x_4,y_5,y_6 \}$$ be the path connecting $p_1$
and $y_6.$ Then we consider $\Pi(\tilde{S_1})$ and construct the family
$S^t$ of broken paths on $T,$ just like before, i.e.,
$S^0=\Pi(\tilde{S_1})$ and
$S^1=\{x\}.$ Then consider the lifts $S^t_{Q}(p_1).$ Denote the end
point of
$S^t_{Q}(p_1)$ by $p^t.$ Then, $p^0=y_6,$ and $p^1=p_1.$
Therefore, there exists $t_0$ such that $p^{t_0}=x.$ Then, the broken
path
$S^{t_0}_{Q}(p_1)$ contradicts to the choice of $S_1.$
Similarly, construct $S_2.$ And let $$\delta_7(x)=\delta_7
=\min\{d_Q(x,p_1), d_Q(x,p_2)\}.$$
Then we see that for any $y \in B_{\delta_7}(x)$ there is a broken
path inside $B_{\delta_5}(x)$ with no more then $8$ turning points that
connects $x$ and $y.$ Therefore, for any $y_1, y_2 \in
B_{\delta_7}(x)$ there is a broken path inside $B_{\delta_5}(x)$ with no
more then $16$ turning points that connects $y_1$ and $y_2.$
{\bf Step 8.} Cover $Q$ by the balls $U_x= B_{\delta_7(x)}(x).$ Let
$U_1,\ldots U_m$ be some finite sub-cover. Then there exists $\delta_8$
such that for any $y \in Q$ there is $i \in \{1,\ldots,m\}$ such that
$B_{\delta_8}(y) \subset U_i.$
Therefore, the conditions of Definition~\ref{df-ltf} are satisfied with
$\delta=\delta(\epsilon) =\delta_8$ and $N=16,$
where $x_i$ are the centers of $U_i.$ (Actually, for the case of
contact Anosov flows $N=7$ is enough (easily follows from the arguments
in the proof of Theorem 18.3.6 from \cite{katokhasselblat}), but we do
not need the estimates on $N$ in this work.)
The weak local $\alpha$-Hoelder transitivity follows easily from the
local transitivity of $(W^s,W^u)$ and the local $\alpha$-Hoelder
transitivity of $E^+\bigoplus E^-.$ Indeed, there exists $\delta >0$
such that
for $x,y$ with $d_Q(x,y) <\delta$ there is a curve $\gamma$ connecting
$x$ and $y$ as in Step 1. Then its length $l$ is less then $C_2
d_Q(x,y)^{\alpha},$ where $C_2$ is some constant that does not depend on
$x$ and $y.$ Let $\epsilon < \min \{l/10, \delta \}.$ Then due to the
local transitivity there exists $\delta'$ such that every point of
$B_{\delta'}(y)$ can be connected with $y$ by a broken path of length
less then $\epsilon.$
Then, like in Steps 1 and 7, we can approximate $\gamma$ by a broken
path $S'$ in such a way that the length of $S'$ is less then
$l+\epsilon,$ and $S'$ connects $x$ with some point $y' \in
B_{\delta'}(y).$ Then connecting $y'$ with $y$ we can extend $S'$ to a
broken path $S$ that connects $x$ and $y$ and has length less then
$l+2\epsilon <2C_2 d_Q(x,y)^{\alpha}.$
\end{proof}
Then we have
\begin{thm}
\label{thm-negativetr}
For a contact Anosov flow $g^t$ on a compact manifold the pair
$(W^s,W^u)$ is locally
transitive and weakly locally 1/2-Hoelder transitive.
\end{thm}
(The local $1/2$-Hoelder transitivity of $E^+\bigoplus E^-$ follows from
the fact that $E^+\bigoplus E^-$ is smooth (Lemma 18.3.7,
\cite{katokhasselblat}), its total non-integrability (Theorem 18.3.6,
\cite{katokhasselblat}) with index $2$ and an obvious analog of
Proposition~\ref{prop-integrability} for a single totally non-integrable
distribution.)
Let $M$ be a compact Riemannian manifold of negative curvature, and
$g^t$ be the
geodesic flow on $SM.$ It is well known that $g^t$ is a contact Anosov
flow,
and thus Theorem~\ref{thm-negativetr} applies to $g^t.$
As an immediate corollary of Theorems~\ref{thm-continuous},
~\ref{thm-hoelder}
and ~\ref{thm-negativetr} we obtain the following
\begin{cor}
\label{cor-geodesicshift} For the time-one map of a contact Anosov flow
on a compact
manifold the space of $\beta$-Hoelder cocycles is $C^0$-stable, and the
subspace of cocycles cohomologous to a constant is the common zero set
of the
periodic cycles functionals.
Moreover, the spaces of Lipschitz cocycles and $C^{\infty}$-cocycles
are
$1/2$-Hoelder stable, and the subspaces of cocycles cohomologous to a
constant are the common zero
sets of the periodic cycles functionals. \end{cor}
Now, we will show that small perturbations of the time-one maps
of contact Anosov flows on compact manifolds
have transitive but not necessarily locally transitive pairs of stable
and
unstable foliations. Therefore, Theorem~\ref{thm-continuous} is not
immediately
applicable. Nevertheless, we will show
that for small enough perturbations the conclusion of
Theorem~\ref{thm-continuous} still holds, i.e., the space of Hoelder
cocycles
cohomologous to a constant is the common zero set of the periodic
cycles
functionals.
{\bf Step 1.}
Cover the manifold $Q$ by a finite number of open sets $U_i$ such that
inside
each $U_i$ the conditions described in Steps 2,3 and 4 of the proof of
Theorem~\ref{prop-2} are satisfied. Namely, there are projections
$\Pi_i:U_i \rightarrow T_i,$ where $T_i$ are codimension one
submanifolds
transversal to the neutral foliation of $g=g^1.$ Moreover, the images
of
the foliations $W^s$ and $W^u$ are foliations $F_i^s$ and $F_i^u$ of
$T_i,$
which define a product structure on $T_i$ and such that their
derivatives
satisfy the conditions of Step 4.
{\bf Step 2.} Let $g'$ be a small perturbation of $g.$ Then, by
Hirsh-Pugh-Shub
``Fundamental Theorem of Normally Hyperbolic Invariant Manifolds,'' if
$g'$ is close enough to $g$ %in $C^2$ topology,
then it is also partially
hyperbolic, with one-dimentional neutral foliation $O',$ and foliations
$W^{s'},$ $W^{u'},$ which are $C^0$ close to the neutral foliation of
$g$ and
$W^s,$ $W^u,$ respectively. Therefore, if the perturbation was small
enough,
then we will have projections $\Pi': U_i \rightarrow T_i$
along the neutral foliation for $g'.$ Moreover, the images of
the foliations $W^{s'}$ and $W^{u'}$ are foliations $F_i^{s'}$ and
$F_i^{u'}$ of $T_i,$
which define a product structure on $T_i.$
%and such that their derivatives
%satisfy the conditions of Step 4, may be with a new constant $K.$
Also, any broken path on $T$ with respect to $F_i^{s'}$ and $F_i^{u'}$
may be
lifted to a broken path on $Q$ with respect to $W^{s'}$ and $W^{u'},$
and
the lifting is unique up to the choice of the initial point.
{\bf Step 3.} Now, find $\epsilon$ so small that for any $x \in Q$ there
is an $i=i(x)$ such that
$B_{2\epsilon}(x) \subset U_i.$
Since $g$ is locally transitive we can find $N$ and $\delta (\epsilon)$
such that
every two points $x$ and $y$ such that $d_Q(x,y) \leq \delta
(\epsilon)$ can
be joined by a broken path on $Q$ with respect to $W^s$ and $W^u,$
which is inside $B_{\epsilon}(x)$ and has no more than $N$ turning
points.
{\bf Step 4.} For any $\epsilon_1,$
there is a neighborhood $V$ of $g$ in $C^2$ topology, such that for
any
$g' \in V,$
the pair $(F_i^{s'},F_i^{u'})$ is so close to
$(F_i^s ,F_i^u)$ and $(W^{s'},W^{u'})$ is so close to
$(W^s ,W^u)$ that, for any $i,$ and for
any broken path $P$ on $T_i, $ with no more than $N$ turning points,
with respect to
$F_i^s$ and $F_i^u$ there is a broken path $P'$ on $T_i$ with
respect to
$F_i^{s'}$ and $F_i^{u'}$, with no more than $N$ turning points and
such that the distance between the end points of the lifts of $P$
and
$P'$ is not bigger than $\epsilon_1,$ if the lifts are with the
same initial point.
{\bf Step 5.}
Let us prove that if $g'$ is close enough to $g$ then
every two points $x$ and $y$ such that $d_Q(x,y) < \delta (\epsilon)/2$
can
be joined by a broken path on $Q$ with respect to $W^{s'}$ and
$W^{u'},$
which is inside $B_{2\epsilon}(x)$ and has no more than $N+4$
turning points. Due to the arguments similar to the ones
in the proof of Proposition~\ref{prop-2} it is enough to show that for
$x$ and $y$ from one leaf of $O'$ (with no more than $N+2$ turning
points, in this case).
Indeed, let $x$ be an arbitrary point of $Q.$ And let $y \in O'(x)$
and $d_Q(x,y) < \delta (\epsilon)/2.$ Let $z \in O'(x)$ be such that
$d_Q(x,z) = \delta (\epsilon)$ and $y$ lies between $x$ and $z$ on
$O'(x).$
There is a broken path $S$ on $Q$ with respect
to $W^s$ and $W^u,$
which is inside $B_{\epsilon}(x),$ has no more than $N$
turning points, and joins $x$ and $z.$
Now, let $i=i(x).$ Consider $P=\Pi_i (S).$ Let $P'$ be as in Step 4
above.
Consider the lift $S'$ of $P'$ with respect to $W^{s'}$ and
$W^{u'}$ and initial
point $x.$ Then the distance between the end point $p$ of the lift
of $P'$ and $z$ is less than $\epsilon_1.$ Choosing $\epsilon_1$
small enough we can guarantee, just like in Step 7 of the proof of
Proposition~\ref{prop-2}, that if we add two more turning points to
$S'$ we will get a broken path on $Q$ with respect to $W^{s'}$ and
$W^{u'}$
connecting $x$ with some point $y_1$ on $O'(x),$ such that
$d_Q(y_1, z)< \delta (\epsilon)/2.$ Therefore contracting the path
$P'$ as in Step 7 of the proof of
Proposition~\ref{prop-2} we can connect $x$ and $y$ by a broken path
with
respect to $W^{s'}$ and $W^{u'},$ with no more than $N+2$ turning
points.
Also, it is easy to se that if $S$ was inside $B_{\epsilon}(x),$ than
for $g$ and $g'$ close enough, $S'$ is inside $B_{2\epsilon}(x).$
{\bf Step 6.} Choose a neighborhood $U$ of $g$ in $\rm Diff^2 (Q)$
such that it would satisfy all ``close enough'' conditions above. We
can
always do that since $\epsilon,$ $\delta(\epsilon),$ $U_i,$ $T_i$ and
$\Pi_i$ are
fixed throughout all the steps in our argument.
Thus, we have the following
\begin{prop}
\label{prop-3}
Every small enough perturbation $g'$ of a time-one map $g$ of a
contact Anosov flow
on a compact manifold $Q$ has a transitive pair of stable and unstable
foliations
$W^{s'}$ and $W^{u'},$ i.e., there exists $\epsilon >0$ and $N' \in
\Bbb{N}$
such that every $x,y \in Q$ can be joined by a broken path
with no more then $N'$ turning points $\{x_i\}$ and such that
$d_{W^*(x_i)}(x_i,x_{i+1}) < \epsilon,$ $i=1, \ldots N-1.$
\end{prop}
Notice that it is impossible to claim local transitivity in
Proposition~\ref{prop-3}, since $g$ can always be perturbed so that in
a neighborhood of some point $x$
the pair $(W^{s'},W^{u'})$ will become locally
completely integrable (actually
that is a generic situation). Therefore, for small enough
$\epsilon,$
the local transitivity condition will fail.
Transitivity of the pair $(W^{s'},W^{u'})$ allows us to define a
solution
$h$ for the cohomological equation as in the proof of
Theorem~\ref{thm-continuous}. (The local transitivity was used only
to prove
the continuity of the solution.) This solution is uniformly
continuous along
$W^{s'}$ and $W^{u'}.$ Moreover, it is uniformly continuous with
respect to
the leaves. To be more precise, for any $\epsilon >0$ there exists
$\delta >0$
such that for any $x,y \in Q$ if $y \in W^*(x)$ and $d_{W*(x)}(x,y) <
\delta$ then $|h(x)-h(y)|< \epsilon.$
Let us prove that it is also continuous along $O'.$
Fix a point $x.$ As we showed above,
there is a broken path $S$ that connects $x$ with a point
$y$ belonging to $O'(x)$ and such that $S$ can be contracted.
Therefore, we have a continuous family of paths $S^t$ connecting $x$
with all the points of $O'(x)$ from some neighborhood of $x.$
But then $F_1(S^t)(\phi)$ also changes continuously, for any Hoelder
function $\phi.$ Thus the solution $h$ is continuous along $O'.$
This, together with the uniform continuity of $h$ on the leaves of
$W^s$ and $W^u,$ proves
that $h$ is continuous on $Q.$
Thus we have the following:
\begin{thm}
\label{thm-perturbation}
Let $Q$ be a compact manifold that admits a contact Anosov flow.
Then there is an open set $U$ in ${\rm
Diff}^2(M)$ such that for every $f \in U$ and for any $\beta \in
(0,1],$
the space of $\beta$-Hoelder cocycles of $f$ is $C^0$-stable, and the
subspace
of cocycles cohomologous to a constant is the common zero set of the
periodic
cycles functionals, i.e., $\phi \in L_{\beta}$ is cohomologous to a
constant,
with $C^0$ transfer function,
if and only if
$F(C)(\phi)=0$ for all periodic cycles $C.$
Moreover, $U$ contains all time-one maps for contact Anosov flows on
$M.$
\end{thm}
Actually, it follows easily from the arguments above that:
\begin{prop}
\label{prop-4}
Proposition~\ref{prop-3} and Theorem~\ref{thm-perturbation} are true in
a slightly more general form. Namely, they hold for small enough
perturbations
of any partially hyperbolic diffeomorphism $g$ of a compact manifold
such that
\begin{enumerate}
\item the neutral foliation of $g$ is smooth and one-dimensional;
\item the pair $(W^s,W^u)$ is locally transitive.
\end{enumerate}
\end{prop}
Also, we would like to mention that the transitivity of stable and
unstable
foliations is a key property in the study of stable ergodicity of
partially
hyperbolic diffeomorphisms. The stable ergodicity was established for
the
time-one map of the geodesic flow on a surface of constant (Grayson,
Pugh, Shub
in \cite{graysonpughshub}) and variable (Wilkinson in \cite{wilkinson})
negative
curvature. Pugh and Shub (\cite{pughshub-ergodicity}) proved the stable
ergodicity for a class of algebraic diffeomorphisms of homogeneous
spaces.
Brezin and Shub (\cite{brezinshub}) proved the stable ergodicity of time
one map
of the geodesic flow on a compact manifold of constant negative
curvature.
Our results, together with the results of Pugh and Shub
(\cite{pughshub-ergodicity}), imply the stable ergodicity of the
time-one maps
of the geodesic flows on the compact manifolds of variable negative
curvature
(with suitable restrictions on the pinching of the
curvature).
\section{Stability of $C^{\infty}$-cocycles.}
\label{sec-smooth}
As we have pointed out if the cocycle is $\Ci$ and the periodic cycles
functionals vanish then the transfer function constructed in
Theorem~\ref{thm-continuous} has continuous derivatives of all orders
along the
stable and unstable foliations.
Therefore, to prove the regularity of the transfer function we need some
result
which would guarantee the smoothness of functions which are smooth
along
``sufficiently many directions.'' We use the following powerful
Hermander-type
theorem (see \cite{katokspatzier-cocycles}, Theorem 2.1, where it is
deduced
from a number of results published elsewhere).
\begin{thmB}
%\label{thm-distributions}
Let $\cal{D}_1,\ldots , \cal{D}_k$ be $\Ci$ distributions on a manifold
$M$
such that their sum is totally non-integrable. Also, assume that
for each $j$, the dimension of the space spanned by the commutators of
length at most $j$ at each point is constant in a neighborhood.
Let $P$ be a function, or even a generalized function or distribution,
on $M$. Suppose that for any positive integer $p$ and $\Ci$ vectorfield
$X$ tangent to any $\cal{D}_j$, the $p$-th partial derivative
$X^{p}(P)$ exists as a continuous or a local $L^2$ function, then
$P$ is $\Ci$ on $M$.
\end{thmB}
Thus, we immediately have (from Theorems~\ref{thm-continuous} and
B) the following:
\begin{thm}
\label{thm-smooth} If $f$ is a partially hyperbolic
diffeomorphism of a compact manifold such that the foliations $W^s$
and $W^u$ are $C^{\infty}$ and
the pair $(W^s,W^u)$ is totally non-integrable, then the space of
$C^{\infty}$-cocycles of $f$ is $C^{\infty}$-stable, and the subspace
of
cocycles cohomologous to a constant is the common zero set of the
periodic cycles functionals.
\end{thm}
In particular we have the following
\begin{cor}
\label{cor-weylsmooth}
For all transformations described in Corollary~\ref{cor-weyl}
the spaces of $C^{\infty}$-cocycles are $C^{\infty}$-stable, and the
subspaces
of cocycles cohomologous to a constant are the common zero sets of the
periodic cycles functionals.
\end{cor}
Recently R.de le Llave (private communication) was able to extend
Theorem B
to certain cases when the distributions themselves are not necessarily
smooth,
but their sum is still smooth. In particular, his results apply in the
setting
of Section~\ref{sec-contact}, so one has the following \begin{cor}
For the time-one map of a contact Anosov flow (in particular, for a
geodesic
shift $g^1$ on a compact manifold of negative curvature) the space of
$C^{\infty}$-cocycles is $C^{\infty}$-stable,and the subspace of
cocycles
cohomologous to a constant is the common zero set of the periodic
cycles
functionals. \end{cor}
\section{Concluding remarks.}
Theorems~\ref{thm-continuous}, ~\ref{thm-hoelder} and ~\ref{thm-smooth}
can
easily be reformulated and reproved for partially hyperbolic flows.
Corollaries~\ref{cor-necessary}, ~\ref{cor-weyl},~\ref{cor-stable},
and ~\ref{cor-weylsmooth} remain true for flows. (Of course,
in the statements of Corollaries~\ref{cor-weyl} and
~\ref{cor-weylsmooth} we
need to replace the single transformation generated by $a \in A$ by the
flow
generated by the one parameter subgroup of $A$ that contains $a.$)
An even more interesting generalization is to twisted cocycles. Namely,
we can
prove the analogs of Theorems~\ref{thm-continuous} and
~\ref{thm-hoelder} for
twisted cocycles with coefficients in $\Bbb{R}^k,$ twisted by
``slow'' cocycles with coefficients in $GL(k,\Bbb{R}).$ (A cocycle
$\alpha$
is called slow if for all $x$ and $n,$ $||\alpha(f^n(x))|| \leq
Ce^{|n| \lambda},$ where $C$ is a constant and $\lambda < \min
\{\lambda_1,\lambda_2 \}.$ Here
$\lambda_1$ and $\lambda_2$ are as in Definition~\ref{df-partially}.)
Such twisted cocycles arise in connection with the infinitesimal
conjugacy
problems.
In all likelihood, Theorems~\ref{thm-continuous} and ~\ref{thm-hoelder}
can be
extended to cocycles with much more general coefficients. Namely, to Lie
groups
with the two-sided metrics (like in \cite{livsic1}) and other finite and
even
infinite dimensional Lie groups (like in \cite{niticatorok}).
Also, notice that our results for Weyl chamber flows are complementary
to the
results of Katok and Spatzier in \cite{katokspatzier-cocycles}. To be
more
precise, the results from \cite{katokspatzier-cocycles} describe the
cocycles
for the actions of subgroups of $A,$ of dimension bigger than one, on
$K\backslash G/\Gamma,$ while Corollaries~\ref{cor-weyl} and
~\ref{cor-weylsmooth} give us a description of the cocycle spaces for
the
actions of one-parameter subgroups and individual elements $a \in A.$
Let us point out the importance of the transitivity conditions in
Theorems~\ref{thm-continuous}, ~\ref{thm-hoelder} and
~\ref{thm-smooth}. Veech
(\cite{veech}) shows that for partially hyperbolic automorphisms of a
torus the
periodic conditions form a generating set of cohomological
obstructions. In
particular, it implies that if the periodic conditions vanish then our
periodic
cycles functionals vanish as well. Veech has shown (\cite{veech},
Proposition
1.5) that for partially hyperbolic, but not hyperbolic, automorphisms
of a
torus the spaces of $C^1$-cocycles are not $C^1$-stable . Most likely
one
can construct examples of partially hyperbolic automorphisms of a
torus with
$C^0$-non-stable spaces of Hoelder cocycles.
And finally we would like to formulate two open problems:
{\em 1. Will Theorems~\ref{thm-continuous} and/or ~\ref{thm-hoelder} be
true if we replace in their assumptions the periodic cycles functionals
by $f$-invariant measures?
2. Will Corollaries~\ref{cor-weyl} and ~\ref{cor-geodesicshift} be true
if we
replace in their assumptions the periodic cycles functionals by,
respectively, the $g^t$-invariant and $A$-invariant measures?}
\centerline {\bf Acknowledgments.}
We would like to express out sincere gratitude to M.Brin and Ya.Pesin
for helpful discussions of the transitivity property for various
partially hyperbolic systems. We also would like to
thank A.Wilkinson who read a preliminary version of the paper and made
several
useful comments.
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