Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

%\controldates{14-JAN-1997,14-JAN-1997,14-JAN-1997,16-JAN-1997}
\documentstyle[newlfont]{era-l}
%\issueinfo{2}{3}{December}{1996}
\pagespan{124}{133}
\PII{S 1079-6762(96)00016-9}
\copyrightinfo{1997}{Anatole Katok and Ralf J. Spatzier}



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\title{ Nonstationary normal forms and rigidity of group actions }
\author{A. Katok and R. J. Spatzier}

\thanks{The first author was partially supported by  NSF grant DMS 9404061}
\thanks{The second author was partially supported by  NSF grant DMS 9626173}

\address{Department of Mathematics,
The Pennsylvania State University, University Park, PA 16802}
\email{katok\_a@@math.psu.edu}

\address{Department of 
Mathematics, University of Michigan, Ann Arbor, MI 48103}
\email{spatzier@@math.lsa.umich.edu}



\commby{Gregory Margulis}

\date{September 28, 1996}

\subjclass{Primary 58Fxx; Secondary 22E40, 28Dxx}

\begin{document}



\begin{abstract} We develop a proper ``nonstationary'' generalization
of the classical theory of normal forms for local
 contractions. In particular, it is
shown under some assumptions that the centralizer of a contraction in
an extension is a particular Lie group, determined by the spectrum of
the linear part of the contractions.  We  show that
most homogeneous Anosov actions of higher rank abelian groups are
locally $C^{\infty}$ rigid (up to an automorphism). This result is the
main part in the proof of local $C^{\infty}$ rigidity for two very
different types of   algebraic
 actions of irreducible   lattices in higher rank semisimple Lie
groups: (i)  the actions of cocompact lattices  on  Furstenberg
boundaries, in particular projective spaces, and (ii) the actions by
automorphisms of tori and nilmanifolds. The main new technical
ingredient in the proofs is the centralizer result mentioned above.
\end{abstract}
\maketitle



\section{Normal forms for extensions of dynamical systems by contractions}

In this section we announce results of \cite{K2} which generalize
 certain aspects of the classical theory of local normal forms
 \cite{Sternberg,Chen}.

Let  $X$ be a compact
metric space and  $f: X\to X$ a homeomorphism (continuous  dynamical
system), $V$ a vector bundle over $X$ with projection $\pi:V\to
X$, and  $F: V\to V$ a continuous  invertible linear extension of $f$.
Let us fix a continuous family of Riemannian metrics in the fibers. 
An extension
$F$ of $f$  is called {\it a contraction} if  $\|DF\|<1$. Consider the induced
operator
$F^{\ast}$ in the Banach space of continuous sections of $V$ provided
with the uniform norm, i.e.  
\begin{equation*} F^{\ast}v(x)=F(v(f^{-1}x)).
\end{equation*} Under the very mild assumption that nonperiodic points of 
$f$ are
dense the  spectrum of the operator $F^{\ast}$ is the union of
finitely many closed annuli centered at the origin.  Hence the {\it
characteristic set} of $F$,
\[ \chi (F)= \{\la\in\Bbb R_+:\exp\la\in  sp \:  F^{\ast}\}, \]
 is the union of finitely many intervals.  We denote these
intervals by
$\Delta_1,\dots,\Delta_l$. Let $\Delta_i=[\la_i,\mu_i]$ and assume that
the intervals  are ordered in increasing order so that
$\la_{i+1}>\mu_i$. Let us assume that the space $X$ is connected or
that the map $f$ is topologically transitive. Then the bundle $V$
splits into the direct sum of
$F$-invariant subbundles $V_1,\dots,V_l$  such that 
$\chi (F)\mid _ {V_i}=\Delta_i,\,\,\,\,i=1,\dots,l$ (cf.  \cite{K1}). Let
$m_i$ be the dimension of the subbundle
$V_i$.

\begin{Definition} The extension $F$ has {\it   narrow band
spectrum} if
 \begin{equation*}\mu_i+\mu_l<\la_i\end{equation*} for $i=1,\dots,l$.
\end{Definition}

From now on we assume that this condition is satisfied. Represent
$\Rm$ as the direct sum of the spaces  $\Bbb R^{m_1},\dots, \Bbb
R^{m_l}$ and let $(t_1,\dots,t_l)$ be the corresponding coordinate
representation of a vector $t\in\Rm$. Let $P:
\Rm\to\Rm;\,\,(t_1,\dots,t_l)\mapsto
(P_1(t_1,\dots,t_l),\dots,P_l(t_1,\dots,t_l))$ be a polynomial map
preserving the origin. We will say that the map $P$ is of {\it 
subresonance type} if it has   nonzero homogeneous terms in
$P_i(t_1,\dots,t_l)$ with degree of homogeneity $s_j$ in the
coordinates of
$t_j,\,\,i=1,\dots,l$, only for
\begin{equation}
\la_i\le\sum_{j\neq i}s_j\mu_j.
\tag{$*$}\end{equation} 
We remark that 
the notions of a homogeneous polynomial and
degree of homogeneity in $\Rm$ are invariant under the action of the
group $\Gm$. Thus the notion of a map of subresonance type depends
only on the decomposition $\Rm=\bigoplus_{i=1}^l\Bbb R^{m_i}$, but not
on a choice of coordinates in each component of this decomposition.


We will call any  inequality  of type $(*)$ {\it a subresonance
relation}. There are always subresonance relations of the form
$\la_i\le\mu_j$ for $j=i,\dots,l$. They correspond to the  linear
terms of the polynomial. We will call such relations {\it trivial}. 
The narrow band condition  guarantees that, for any nontrivial
subresonance relation, $s_j=0$ for $j=1,\dots,i$.

\begin{Proposition} Polynomial maps of the subresonance type with
invertible derivative at the origin are invertible and  form a group, 
which we will
denote by $G_{\la,\mu}$.
\end{Proposition}


 Let $U$  be a neighborhood of the zero
section in $V$. We consider an extension map ${\cal F}: U\to V$, which
is continuous, smooth (usually $\CI$) along the fibers and preserves
the zero section. We will denote  by $D{\cal F} _0$ 
the derivative of $\Cal F$ in the
fiber direction at the zero section. It is a linear
extension of $f$. As before, fix a continuous family of Riemannian
metrics in the fibers.  The extension
$\Cal F$ is called {\it a contraction} if  $D{\cal F}_0$ is a
contraction.

 Two extensions are conjugate if there exists
a continuous family of local  $\CI$  diffeomorphisms of the fibers
$V(x)$, preserving the origin, which transforms one extension into the
other. 


\begin{Theorem} [subresonance normal form] Suppose the
extension $\Cal F$  is a contraction and the linear extension $D{\cal
F}_0$ has  narrow band spectrum determined by the vectors
$\la=(\la_1,\dots,\la_l)$ and $\mu=(\mu_1,\dots,\mu_l)$.
 
Then there exists an extension $\tilde{\Cal F}$ equivalent to $\Cal F$ 
such that for every $x\in X$,
\begin{equation*}
\tilde{\Cal F}\mid _{ V(x)}:\,\,\, \bigoplus_{i=1}^lV_i(x)\to
\bigoplus_{i=1}^lV_i(f(x))
\end{equation*} 
is given by a polynomial map of subresonance type, i.e an element
of the group $ G_{\la,\mu}$. 
\end{Theorem}


 The proof of  this theorem  \cite{K2} follows one of the usual schemes in the
normal form theory. In particular, it is parallel to the proof of the 
nonstationary normal form theorem \cite[Theorem 2.1]{K1} for
extensions of measure preserving transformations. It includes three
steps:
\vspace{.2em} 

\begin{step1} Finding a  {\it formal} coordinate change, i.e. the
Taylor series  at the zero section for the desired coordinate change.
Naturally, the coefficients should depend continuously on the base
point. This is the most essential step.
\end{step1}


\begin{step2} Constructing a continuous family of smooth ($\CI$)
coordinate changes in $V(x), \,\,x\in X$, for which  the formal power
series  found at Step 1 is the Taylor series at the zero section. This
follows directly from a parametric version of \cite[Proposition
6.6.3]{KH}.
\end{step2}


\begin{step3} The coordinate change constructed at Step 2 conjugates
our extension with an extension which is $\CI$ {\it tangent} to the
derivative extension. We show that any two  $\CI$ tangent contracting
extensions  are conjugate via a $\CI$  coordinate change  $\CI$
tangent to the identity. This is achieved via a parametric version of the
``homotopy trick''. See \cite[Theorem 6.6.5]{KH}.
\end{step3}

\begin{Theorem} [Centralizer for subresonance maps] Suppose
$g$ is a homeomorphism of the space $X$ commuting with $f$, and
$\tilde{\Cal G}$ is an extension of $g$ by a $\CI$ diffeomorphism of
the fibers (not necessarily contraction) commuting with an extension
$\tilde{\Cal F}$  satisfying the assertion of Theorem 3. Then
$\tilde{\Cal G}$ has a similar form:
\begin{equation*}
\tilde{\Cal G}\mid _{ V(x)}:\,\,\,\bigoplus_{i=1}^lV_i(x)\to
\bigoplus_{i=1}^lV_i(g(x))
\end{equation*} is also a polynomial map of subresonance type. 
\end{Theorem}


 The proof of this theorem \cite{K2} consists of two steps:


\begin{step1} Proving that the Taylor series of $\tilde{\Cal G}$ at 
0 are
polynomial maps of subresonance type.
\end{step1} 


\begin{step2} Proving that the map $\tilde{\Cal G}$ coincides with
its  Taylor series.
\end{step2}


Combining these two theorems we see that a local action of an abelian group by
extensions which contain a contraction with  narrow
 band spectrum can be simultaneously brought to a normal form. 

\begin{Corollary}        \label{cor-jointnormal}
 Let $\rho$ be a continuous action of
$\Rk$ on a compact connected metric space $X$. Let $V$ be a vector bundle 
over $X$.
Suppose that $\sigma$ is a local action of  $\Rk$ in  a neighborhood of the 
zero
section of $V$ such that $\sigma$ covers
$\rho$, $\sigma$ is differentiable along the leaves and each 
$\sigma (a) \mid _{V_x}$ depends continuously on the base point $x$ in the
$C^{\infty}$ topology. Suppose further that for some $a \in \Rk$, $\sigma 
(a)$ is a
contraction and that the induced linear operator on $C^0$ sections of $V$ 
has narrow
band spectrum.   Then there exist  $C^{\infty}$ changes of coordinates in 
the fibers
$V_x$, depending continuously on $x$, such that for all $b \in
\Rk$, $\sigma (b)$ is a polynomial map of subresonance type.
\end{Corollary}


\section {Rigidity of Anosov group actions: formulation of results}


In this section and Section 4 we announce the results of \cite{KS5}.


\subsection{Anosov actions of $\Zk$}
 A \ci action $\rho$ of a finitely generated discrete group 
$\Gamma$ on a compact manifold $M$ is called  \ci {\em locally rigid} if 
any \ci action
$\tilde{\rho}$ of $\Gamma$ on $M$  that is $C^1$ 
close to $\rho $ on a fixed finite set of
generators is \ci conjugate to $\rho$. 

We recall that an action of a group $G$ on  a compact manifold is {\em Anosov}
if some element $g \in G$ acts normally hyperbolically with respect to the 
orbit foliation (cf. \cite{KS1} for more details). If $G$ is discrete this 
simply 
means that the action contains an Anosov diffeomorphism. Since the only 
manifolds 
known to carry an Anosov diffeomorphism are infranilmanifolds (in 
particular, tori 
and nilmanifolds), Anosov actions of any discrete groups, including $\Zk$ or 
lattices
 in semisimple Lie groups are also known to exist only on such manifolds. 
Any Anosov 
 diffeomorphism $g$ of an infranilmanifold is topologically  conjugate to  an 
automorphism $f$  (\cite{Franks,Manning}) and hence has a fixed point. Any 
such map
 is a $\pi _1$-diffeomorphism in the sense of Franks \cite{Franks}. Hence any 
 commuting homeomorphism is unique in the homotopy class. Moreover, any 
automorphism  
 of $\pi _1$ which commutes with $f_*$ can be realized by another  
automorphism 
 $h$ commuting with $f$ by Maltsev's theorems. The meaning of our next result 
is 
 that, under certain assumptions on $f$ and $h$,  unless the conjugacy 
between 
 $g$ and $f$ is smooth, the centralizer of $g$ in the diffeomorphism group 
reduces
  to a finite extension of the powers of $g$. The assumptions on $f$ and $h$ 
are
   of two kinds. The first condition on $f$ and $h$ is essential and 
guarantees  
   irreducibility of the $\Z ^2$ action generated by $f$ and $h$. The 
condition 
   is that every nontrivial element of this $\Z ^2$ is 
weak mixing. By a theorem of Parry, it is equivalent to an algebraic 
condition on
 the eigenvalues \cite{Parry}. The second condition is that $f$ and $h$ are 
 semisimple, which is only of technical nature. 

We call an action of a group on an infranilmanifold {\em algebraic} or {\em 
affine} if the
lift of every  element of the action  to the universal cover is affine 
w.r.t. the
connection given by right invariant vector fields.


\begin{Theorem}          \label{cor-nilautos}
 Let $\rho$ be an algebraic Anosov action of $\Zk$, $k \geq 2$,
  on an infranilmanifold $M$. Suppose that the linearization is semisimple 
and that
 no nontrivial element of the group   has roots of unity
  as eigenvalues in the induced representation on $H_1 (M,\R)$.  Then $\rho$ 
is \ci  locally rigid. Moreover, the conjugacy 
  can be chosen $C^1$ close to the 
identity, and is unique amongst  conjugacies close to the identity.
\end{Theorem}

This theorem is really a corollary of the corresponding result
for $\Rk$ actions applying it to a suspension of the $\Zk$ action (cf. 
Section 2.3 and 
\cite{KS5}). The weak mixing condition then guarantees that ergodic 
components of any
one-parameter subgroup of the suspension consist of entire nilmanifold 
fibers. This is 
sufficient to guarantee the technical hypothesis on ergodic components 
of our main result on $\Rk$ actions. 



\subsection{Anosov actions of lattices in higher rank semisimple Lie 
groups}
Our results on smooth local rigidity of $\Zk$ actions can be used to study 
Anosov 
actions of irreducible lattices in higher rank semisimple Lie groups of the 
noncompact type on tori and nilmanifolds by automorphisms. Such a lattice 
$\Gamma$ always contains a $\Zk$ subgroup, $k \geq 2$, a so-called {\em 
Cartan subgroup}, such that the restriction of the action to $\Zk$ has 
semisimple linear part, and satisfies the eigenvalue condition of 
Theorem~\ref{cor-nilautos}.
The approach taken here uses R. Zimmer's cocycle superrigidity theory 
\cite{Zimmer,Qian-Zimmer} to provide us with a measurable 
$\Gamma$-equivariant framing,
which is then shown to be smooth using the rigidity of the action of the 
Cartan subgroup.  The first paper using this approach  \cite{KLZ} was based 
on  more limited 
results on the rigidity of abelian Anosov actions from \cite{KL1}. This 
approach was developed by N. Qian and C. Yue in \cite{Q3,Q4,Qian-Yue}. With 
the help of our
more powerful result, Theorem~\ref{cor-nilautos}, we obtain the following  
definitive 
 result.


\begin{Theorem}   \label{thm-lattice-anosov}
 Let $G$ be  a linear semisimple  Lie 
group all of whose simple factors have real rank at 
least 2. Let $\Gamma$ be an irreducible lattice in $G$. Then a 
sufficiently small (in the  $C^1$ topology) perturbation of an    
Anosov action of $\Gamma$  by affine maps
 on a nilmanifold is \ci conjugate 
to the original action  by a conjugacy that is 
$C^1$ close to the identity. 
\end{Theorem}



\subsection{Anosov actions of $\Rk$ and reductive groups}

For actions of a continuous group $G$, there are two aspects of local 
rigidity. 
First one can consider  rigidity of the orbit foliation as a foliation,
 i.e. when a \ci small perturbation is not necessarily the orbit foliation 
 of an action of the same group. Secondly, there is local rigidity of an 
action
  which in this case must allow
a change of the action by an automorphism of the group $G$ close to the 
identity.
Local rigidity of such an  action can be obtained from the foliation 
rigidity and 
 certain rigidity results for cocycles over the action taking values in $G$ 
 (time changes). 


We define  algebraic $\Rk$ actions as follows. Suppose 
$\Rk \subset H$ is a subgroup of a connected Lie group $H$. Let $\Rk$ act on 
a compact quotient $H/\Lambda$ by left translations where $\Lambda$ is a 
lattice in $H$. Suppose $C$ is a compact subgroup of 
$H$ which commutes with $\Rk$. Then the $\Rk$ action on $H/ \Lambda$ descends 
to an action on $ C \setminus H / \Lambda$. The general algebraic 
$\Rk$ action $\rho$ is a finite factor of such an action. Let ${\frak c}$ 
be the Lie algebra of $C$. The {\em linear part} of $\rho$  is the 
representation of $\Rk$ on ${\frak c} \setminus {\frak h}$ induced by 
the adjoint representation
of $\Rk$ on the Lie algebra ${\frak h}$ of $H$. 

Let us note that the suspension of an affine $\Zk$ action  is an algebraic 
$\Rk$ action (cf. \cite[2.2]{KS1}).

The next theorem is our principal technical result for algebraic 
Anosov $\Rk$ actions.
We  denote the strong stable foliation 
of $a \in \Rk$ by ${\cal W} ^- _a$,  the strong stable distribution by $E^- 
_a$, and the
0-Lyapunov space by $E^0 _a$. Note that $E^0 _a$ is always integrable for
algebraic actions. Denote the corresponding foliation by ${\cal W} ^0 _a$. 
Also note that any algebraic action leaves a Haar measure $\mu$ on the 
quotient 
invariant. Given a collection of subspaces of a vector space, we call a 
nontrivial intersection {\em maximal} if it does not contain any other 
nontrivial intersection of these subspaces.

\begin{Theorem}   	\label{thm-main}
Let $\rho$ be an algebraic Anosov action of $\Rk$, 
for $ k \geq 2$, such that the linear part of $\rho$ is semisimple.
Assume that for any maximal nontrivial intersection 
$\bigcap _{i=1 \ldots r} {\cal W} ^- _{b_i}$
of stable manifolds of elements $b_1, \dots, b_r \in \Rk$ there is an 
element 
$a \in \Rk$ such that for a.e. $x \in M$, 
$\bigcap _{i=1 \ldots r} E ^- _{b_i}  (x) \subset E^0 _a (x)$ and such that
a.e. leaf of the intersection $\bigcap _{i=1 \ldots r} {\cal W} ^- _{b_i}$ is
contained in an ergodic component of the one-parameter subgroup $t a$ of
$\Rk$ (w.r.t. Haar measure).
Then the orbit foliation of
   $\rho$ is locally $C^{\infty}$ rigid. In fact, the orbit equivalence
can be chosen $C^1$ close to the identity. Moreover, 
the orbit equivalence is transversally unique, i.e. for any two different 
orbit equivalences close to the identity, 
the induced maps on  the set of leaves  agree.
\end{Theorem}

We immediately get the following corollary. 


\begin{Corollary}   	
Let $\rho$ be an algebraic Anosov action of $\Rk$, 
for $ k \geq 2$, such that the linear part of $\rho$ is semisimple.
Assume that  every one-parameter subgroup of $\Rk$ acts ergodically 
with respect to the Haar measure $\mu$. 
 Then the orbit foliation of
   $\rho$ is locally $C^{\infty}$ rigid. In fact, the orbit equivalence
can be chosen $C^1$ close to the identity. Moreover, 
the orbit equivalence is transversally unique i.e. for any two different 
orbit equivalences close to the identity, 
the induced maps on  the set of leaves  agree.
\end{Corollary}


All known weakly mixing Anosov $\Rk$ actions belong to and almost exhaust the 
list of {\em standard} $\Rk$ actions, introduced  in \cite{KS1}.  They 
essentially consist of suspensions of actions by toral automorphisms, Weyl 
chamber flows, twisted 
Weyl chamber flows and some further extensions
(cf. \cite{KS1} for more details and for the definition of twisted Weyl 
chamber flows). 


 For the standard Anosov 
actions, we showed that every smooth cocycle is smooth\-ly cohomologous 
to a constant cocycle \cite[Theorem 2.9]{KS1}. As a consequence, all 
smooth time changes are smoothly conjugate to the original action
 (possibly composed with an automorphism of $\Rk$). Combining this 
 with Theorem~\ref{thm-main}, we obtain the following corollary.

\begin{Corollary}
The standard algebraic Anosov actions of $\Rk$ for $k \geq 2$ with 
semi\-simple linear part are  locally $C^{\infty}$ rigid. 
Moreover, the $C^{\infty}$ 
conjugacy $\phi$ between the action composed with 
an automorphism $\rho$ and a 
perturbation can be chosen $C^1$ close to the 
identity.
The automorphism $\rho$ is unique and also close to the identity. Finally,
 $\phi$  is  unique amongst  conjugacies close to the identity modulo
  translations in the acting group. 
\end{Corollary}

In Section 4 we discuss  local rigidity results for projective lattice
actions. Those results are based on  similar foliation rigidity results for 
Anosov actions of certain reductive groups.
 Let $G$ be a connected semisimple Lie group with  finite center and 
without
compact factors. Let $\Gamma \subset G$  be an irreducible cocompact lattice. 
Let $P$ be 
a parabolic subgroup of $G$, and let $H$ be its Levi subgroup. Thus $P= H \,
U^+$, where $U^+$ is the unipotent radical of $P$. 
Then $H$ acts on $G/\Gamma$ by left translations. These actions are
Anosov.


\begin{Theorem} \label{thm-reductive}
If the real rank of $G$ is at least 2, then the orbit foliation
${\cal O}$ of $H$ is $C^{\infty}$ locally rigid. Moreover, the orbit 
equivalence
can be chosen $C^1$ close to the identity.
\end{Theorem}

 We will actually use the following corollary of the proof of the  theorem. 
 Let $P=L\, C\, U^+$ be the Langlands  decomposition of $P$ (with respect to 
some Iwasawa
 decomposition $G=K\, A\, N$ of $G$), and let $M_P$ be the
 centralizer of $C$ in $K$. Then the orbit foliation
of $H$ on $\Gamma \setminus G $ descends to a foliation ${\cal R}$ on 
$\Gamma \setminus G /M_P$. 

\begin{Corollary} \label{cor-reductive}
 If the real rank of $G$ is at least 2, then the 
foliation ${\cal R}$ on $\Gamma \setminus G /M_P$ is $C^{\infty}$ locally 
rigid.
Moreover, the orbit equivalence
can be chosen $C^1$ close to the identity.
\end{Corollary}


\section{Anosov actions: sketches  of proofs}


\subsection{Rigidity of orbit foliations of $\Rk$ actions via normal forms}


We present an outline of the proof of Theorem~\ref{thm-main}, emphasizing 
the crucial
new step involving Corollary~\ref{cor-jointnormal}  from the theory of  
normal forms.
Theorem~\ref{cor-nilautos} follows immediately from  Theorem~\ref{thm-main}
applied to the suspension of the $\Zk$ action once the technical condition 
on ergodic
components is checked.
The proofs of Theorem~\ref{thm-reductive} and its corollary are obtained by 
minor
modifications of this approach. 

Consider an algebraic $\Rk$ action $\rho$ with semisimple linear part 
$\sigma$. Define
the {\em Lyapunov exponents} of
$\rho$ as the logarithms of the absolute values of the eigenvalues of 
$\sigma$. We
get linear functionals $\chi : \Rk \rightarrow \R$. There is a splitting 
of the
tangent bundle into $\Rk$-invariant subbundles 
$TM = \bigoplus _{\chi} E_{\chi}$ such that the Lyapunov exponent of $v \in
E_{\chi}$ with respect to
$\rho (a)$ is given by $\chi (a)$. We call $E_{\chi}$ a {\em Lyapunov space} 
or
{\em Lyapunov distribution} for the action. Then the strong stable  
distribution
$E^- _a$ of $a \in \Rk$ is given by $E^- _a = \sum _{\chi (a) <0}  E_{\chi}$.
The sum 
$E^{\chi} =\bigoplus E_{\lambda}$, where $\lambda$ ranges over all Lyapunov
functionals which
 are positive multiples of a given Lyapunov functional $\chi$, is always 
integrable.
 In fact, set
$H=\{a \in \Rk \mid \chi (a) \leq 0\}$, and call it a {\em Lyapunov
half-space}.
Denote $E^{ \chi}$ by $E_H$. 
Then we get a decomposition 
\[ TM = \bigoplus E_H  \oplus T{\cal O},\] 
where $H$ runs over all Lyapunov 
half-spaces and $T {\cal O}$ is the tangent bundle to the 
orbits of the action. We
call this decomposition  the {\em coarse Lyapunov decomposition} of $TM$. 
Denote by ${\cal W} _H$ the integral foliation of the distribution $E_H$. 

Now  we start with the description of the proof of Theorem~\ref{thm-main} 
proper.
Let ${\cal F}$ and $\tilde{\cal F}$ denote the orbit foliation and a 
$C^1$ small
perturbation.  First, by Hirsch-Pugh-Shub's structural stability theorem, 
there exists a
H\"{o}lder orbit equivalence $\phi$  between ${\cal F}$ and $\tilde{\cal F}$ 
which is
close to the identity and transversally unique \cite{HPS}. It can be chosen 
\ci  along
 ${\cal F}$. Thus, it is sufficient to show that $\phi$ is \ci 
transversally. This
in turn follows from standard elliptic theory once we establish the 
smoothness of
$\phi$  along each foliation ${\cal W} _H$ via standard elliptic theory. 
This is the
central part of the proof.

To prepare for the main argument, we consider the  continuous action 
$\tilde{\rho}$  of
$\Rk$ on $M$  obtained by conjugating $\rho$ by $\phi$. Structural stability 
also
implies that $\phi$ carries over the hull of the ${\cal W} _H$-foliation and 
${\cal F}$
to a H\"{o}lder foliation $\tilde{\cal U} _H$ with smooth leaves, 
which is thus invariant
under the holonomy of ${\tilde{\cal F}}$.  This allows us to construct a 
natural
extension of $\tilde{\rho}$ via holonomy in $\tilde{\cal U} _H$, smooth 
along the
leaves. Since
$\rho$ is Anosov, this extension includes a contraction. Since the algebraic 
action
has point spectrum, it follows that this natural extension satisfies the 
narrow band
condition. Thus we can apply the results of Section 1, and in particular
Corollary~\ref{cor-jointnormal}. 

The conjugacy $\phi$ induces conjugacies $\psi _x$ between the action of 
$\rho$ on
${\cal W} _H (x)$ and the fibers $T_x$ of the  natural extension of 
$\tilde{\rho}$ at
$x$ described above. Smoothness of $\psi _x$ will follow once we see that 
$\psi _x$
intertwines smooth transitive actions of a certain Lie group $G$ on the 
fibers. 


The action of the group $G$ on ${\cal W} _H (x)$ is obtained by taking 
limits of 
returns restricted to ${\cal W} _H (x)$ of some one-parameter subgroup  
$a_t$ on the boundary of $H$ to points in ${\cal
W} _H (x)$. The semisimplicity of $\rho$ implies that
these returns are isometries along ${\cal W} _H$, and thus converge to an 
isometry.
Our condition on ergodic components of  $a_t$ shows that this group of 
limits is 
transitive for generic $x$. 
A similar device was used in \cite{KS3} to study invariant measures for 
Anosov actions 
of higher rank abelian groups. 

The action of $G$ on $T_x$ is obtained by conjugating the returns of $a_t$ 
by $\phi$.
Since the $\psi _x$ form a uniformly continuous family of homeomorphisms, we 
get
convergence in the $C^0$ topology of the conjugated returns any time that 
the returns
converge themselves. The  key point in the argument is that these conjugated 
returns
belong to a Lie group by Corollary~\ref{cor-jointnormal}. Hence 
$C^0$ convergence 
implies \ci convergence. 


\subsection{From rigidity of $\Zk$ actions and cocycle superrigidity to 
rigidity of 
 lattice actions}

Our proof of Theorem~\ref{thm-lattice-anosov} is based on the local rigidity 
of the 
action of a  Cartan subgroup $\Delta$
of
$\Gamma$  and Zimmer's measurable cocycle superrigidity. The former provides 
a smooth
equivariant framing $\tau$ for the perturbed action of $\Delta$, and the 
latter an a
priori measurable   framing, equivariant modulo a commuting compact group, 
for the
extension of the perturbed action of
$\Gamma$ to a certain finite measurable cover of $M$. The main work consists 
in 
showing that the measurable framing can be projected back to $M$, and in 
fact is a
constant translate of $\tau$.  The difference with previous approaches along 
these
lines is that before, e.g. in \cite{KLZ}, the smooth framing was moved 
around by
elements of the perturbed action to produce enough smooth data to force the
superrigidity framing to conform with a smooth one. This is in the spirit of 
topological
superrigidity and   requires  more detailed structural information about the
lattice and its representations than is available in the present setup.  

Here instead we directly  compare the measurable superrigidity framing with 
$\tau$.
Their difference produces certain cocycles, which are shown to be constant 
modulo a
compact group by ergodicity type arguments. Along the way, to get rid of the 
finite
cover, we consider the superrigidity framing on this cover as a multivalued 
framing on
$M$, and apply the ergodicity arguments to it. 

Therefore both actions transform a translation invariant framing according 
to a 
homomorphism (modulo a commuting compact group). Hence  stable and unstable
 foliations of the original and the perturbed action are homogeneous, and 
also 
 $C^0$ close. Lift these foliations to the universal cover. Since two 
distinct 
 closed subgroups of a nilpotent group cannot 
stay a bounded distance apart, it 
 follows that the stable (and unstable) foliations of the original and the 
 perturbed action coincide.  It follows quickly that the actions coincide on
  a subgroup of finite index since the actions agree on intersections of 
stable 
  and unstable manifolds of suitable periodic points. 

Finally, a detailed analysis of  the previous arguments shows that we only 
get local
rigidity on the subgroup of $\Gamma$ generated by $\Delta$ and all its
conjugates. This subgroup has finite index  in $\Gamma$ by Margulis' 
finiteness
theorem. To obtain local rigidity for $\Gamma$ itself, we again use our 
analysis of
superrigidity framings and the fine structure of linear representations of 
arithmetic
lattices discovered by Margulis. 




  







\section{Rigidity of lattice actions on Furstenberg boundaries}

 A class of algebraic actions of lattices on compact manifolds very 
different from the
affine actions  discussed in Section 2.2 are given by the ``projective 
actions'' on
Furstenberg boundaries of the ambient Lie group. These actions include the 
classical
actions on projective spaces and Grassmannians. These actions do not 
preserve any
measure,  are highly dissipative and do not possess the robust orbit 
structures of
Anosov diffeomorphisms. In fact, while many elements of these actions are 
structurally
stable (they are Morse-Smale), the conjugacies are highly nonunique and 
thus do not
lend themselves to provide a common conjugacy for the whole group. However, 
these
projective actions are ``dual'' to the actions of reductive groups discussed 
in Section
2.3. We use the local foliation rigidity of the latter actions to obtain 
rigidity of
the projective actions. This approach is not specific to the higher rank 
case. E. Ghys
first used this duality for  smooth classification of boundary actions of 
Fuchsian
groups \cite{Ghys}. Later, C. Yue obtained partial results of this nature 
for rank 1
symmetric spaces \cite{Yue}.


\begin{Theorem} Let $G$ be a connected semisimple Lie group with  finite 
center and
without compact factors. Suppose that the real rank of $G$ is at least 2.  
Let $\Gamma
\subset G$  be a cocompact  irreducible lattice and $P$ a parabolic subgroup 
of $G$.
Then the action of $\Gamma$ on $G/P$ by left translations is locally
$C^{\infty}$ rigid. 
\end{Theorem}


The boundary $G/P$ can be thought of as a transversal to the weak stable 
foliation
${\cal W} ^+$ of
the action of a certain element  $c \in P$ on $M_P \setminus G/\Gamma$, 
where $M_P$ is
a suitable compact subgroup of $P$. Then the action $\rho$ of the lattice 
$\Gamma$ on
the boundary $G/P$ is the holonomy of  ${\cal W} ^+$. 
 A $C^1$ close smooth perturbation $\tilde{\rho}$
of the boundary $\Gamma$ action is a perturbation of this holonomy,
which in fact appears as the holonomy of a perturbed foliation $\tilde{{\cal 
W}} ^+$ on
$M_P \setminus  G/\Gamma$. Similarly, the holonomy of the weak unstable 
foliation ${\cal
W} ^-$ of $c$  is also given by the $\Gamma$ action on $G/P$. Again, 
$\tilde{\rho}$
gives rise to a perturbed foliation $\tilde{{\cal W}} ^-$ on $M_P \setminus 
G/\Gamma$. Then the intersection of $\tilde{{\cal W}} ^-$ with $\tilde{{\cal 
W}} ^+$
is a perturbation of the neutral foliation of $c$ which is actually the orbit
foliation of a suitable reductive subgroup of $P$ on $M_P \setminus  
G/\Gamma$. Now
Corollary~\ref{cor-reductive} provides a \ci orbit equivalence of 
this orbit foliation
and its perturbation, which in turn yields a \ci conjugacy between $\rho$ and
$\tilde{\rho}$. 

 
\begin{remark} This approach does not directly 
apply to  projective actions of
nonuniform lattices since the duality breaks down due to noncompactness. 
Thus the 
 local rigidity of projective actions of nonuniform lattices remains an 
open problem.
\end{remark}
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\end{document}