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\copyrightinfo{2004}{Danijela Damjanovi\'{c} and Anatole Katok}
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\begin{document}
\title[Rigidity of actions of higher rank abelian groups and KAM]
{Local rigidity of actions of higher rank abelian groups and KAM method}
\author{Danijela Damjanovi\'{c}}
\address{Department of Mathematics,
The Pennsylvania State University,
University Park, PA 16802}        
\curraddr{Erwin Schroedinger Institute,
Boltzmanngasse 9,
A-1090 Vienna, Austria}
\email{damjanov@math.psu.edu}
\urladdr{http://www.math.psu.edu/damjanov}


\author{Anatole Katok}
\address{Department of Mathematics,
The Pennsylvania State University,
University Park, PA 16802}
\email{katok\_a@math.psu.edu}
\urladdr{http://www.math.psu.edu/katok\_a}
\thanks{Anatole Katok was partially supported by NSF grant DMS 0071339}
\date{September 19, 2004}
\subjclass[2000]{Primary 37C85, 37C15, 58C15}
\keywords{Local rigidity, group actions, KAM method, torus}
\commby{Gregory Margulis}
\begin{abstract}
We develop a new method for proving local differentiable rigidity for actions
of higher rank abelian groups. Unlike  earlier methods it does not  require 
previous knowledge of structural stability and instead uses a version of the
KAM (Kolmogorov-Arnold-Moser) iterative scheme. As an application we show
$\mathcal{C}^\infty$ local rigidity  for $\mathbb{Z}^k\ (k\ge 2)$ 
partially hyperbolic
actions by toral automorphisms. We also prove the existence of irreducible
genuinely partially hyperbolic higher rank actions by automorphisms on any
torus $\mathbb{T}^N$ for any even $N\ge 6$.
\end{abstract}
\maketitle

\section{Introduction}

\subsection{Old and new approaches to  rigidity} In this paper we introduce a
new method of proving rigidity  for smooth actions of higher rank abelian
groups (i.e.  $\Zk$ and $\Rk$ for $k\ge 2$) on compact manifolds. 

\subsubsection{The Katok-Spatzier method} The old method introduced in
\cite{KS} can be very briefly   described as follows.  One considers an action
with sufficiently strong hyperbolic properties. The {\em $C^0$  orbit
structure}  of such an action is  rigid (structural stability, Hirsch-Pugh-Shub
theory, \cite{HPS}).  Moreover, the  $C^0$ (orbit) conjugacy  is unique
transversally  to the orbits.  The  method then  consists in establishing {\em
a priori} regularity of the conjugacy. The central  idea is that {\em locally}
in the phase space there are invariant  geometric structures  on invariant
foliations for both the original action and  a perturbed action.  The simplest
example of such a structure is affine (local linearization); in   general  the
structure   still has a finite-dimensional Lie group of automorphisms.  Such  a
structure appears  due to  the fact that  some elements of the action contract
the foliations in question and this is true both in the rank one and  the
higher rank hyperbolic situation. Higher rank is crucial in showing that the 
conjugacy,  which is {\em a priori} only continuous, intertwines  the invariant
structures and hence is smooth in the direction of each foliation.  This in
turn relies on the fact that in certain  {\em critical} directions the
unperturbed action acts by isometries  on each leaf of the corresponding  
foliation.  This in particular  excludes the case of actions by automorphisms
of the torus or a nil-manifold  in the presence on nontrivial Jordan blocks.

This  method of establishing   regularity of a conjugacy  obtained from a 
different kind of reasoning  appears  as an  ingredient in the proofs of local
rigidity  for actions of  certain groups other than abelian \cite{MargulisQian,
FisherMargulis}. Notice that the earliest application of a version of this  
method can be found in \cite{KatokLewis} in the context of proving rigidity of 
some standard lattice actions on tori.

Both for the $\Rk$ actions and for the case of general lattice actions  such as
those considered in  \cite{FisherMargulis} there are certain  residual
directions left  where regularity (and often even the existence)  of the conjugacy 
has to be established separately. In the former case  this follows from the
cocycle rigidity which allows one to  straighten out the time change
\cite{KSfirst}.  Notice that cocycle rigidity  extends to some partially
hyperbolic actions \cite{KSpartial}.

If  one  follows this approach   for {\em partially hyperbolic} actions with
semisimple linear part, one can establish  rigidity  of the {\em neutral
foliation} for the unperturbed action  (Theorem~\ref{thmfoliations}) and hence 
show that every small perturbation of the original action  is differentiably
conjugate to a perturbation along the neutral foliation. Since the latter is 
bigger than the orbit foliation the cocycle rigidity is not sufficient to 
finish the proof.  On the other hand, there is no  local (in the space)
rigidity  in this case.    { \em A priori}  there is no geometric structure on
the neutral foliation   preserved by individual elements of  the perturbed
action. Furthermore  {\em locally}  the  higher rank  condition does not help
either. In other words,  one can perturb the  higher rank action locally (say,
in a neighborhood  of a periodic orbit) and change its topological structure in
a dramatic way.

\subsubsection{The new KAM-type method} In our new method  we do not start from
a conjugacy of low regularity. Instead we construct one of high regularity by 
an iterative process  as a  fixed point of a certain  nonlinear operator. We
use an iterative procedure, namely an adapted version of the KAM procedure
motivated by the procedure used by Moser in \cite{Moser} for commuting circle
diffeomorphisms. Moser first noticed that commutativity along with simultaneous
Diophantine condition was enough to provide a smooth solution to a certain
overdetermined system of equations.    Perhaps surprisingly, Moser's
philosophy turned out to be applicable to a very different class of
$\mathbb{Z}^k$ ($k\ge 2$) actions, namely actions by toral automorphisms. 

The first key observation is that the linearized equation in this case is close
to a twisted cocycle equation over the unperturbed action. Thus the ``higher
rank trick'', which lies at the root of cocycle rigidity  \cite{KSfirst,
KSpartial}, adapted to the twisted case, allows for the existence of an
approximate solution of the linearized equation. 

The second key  observation  is as follows.  From the fact that the linearized
equation is close to being the twisted cohomology equation  we deduce  that it
is in fact close to a certain twisted cohomology equation for a cocycle over
the unperturbed action. We construct this cocycle explicitly by adapting and
developing a method from  \cite{KK}. 

In this paper we  outline how this new approach works in the case of actions on
the torus by automorphisms, affine maps and suspensions of such actions. We
give precise  formulation of the  results and detailed description of the
strategy of proofs.


As a byproduct of our approach we  remove the restriction of semisimplicity
for the unperturbed action. Our method works when the unperturbed action,
whether hyperbolic or partially hyperbolic, has nontrivial  Jordan blocks. A
disadvantage of our method is the requirement that the perturbation  is small
in $C^l$ topology for a  large $l$, while in the  method of \cite{KS} 
perturbations small in $C^1$ topology are allowed.



We  also discuss examples of partially hyperbolic but not hyperbolic $\mathbb{Z}^2$
actions on the torus $\mathbb{T}^N$. We show that  such genuinely partially
hyperbolic actions do not exist on tori of dimension less than 6. We also prove
the existence of irreducible genuinely partially hyperbolic  actions  of $\mathbb{Z}^2$
on the torus of any even dimension starting from 6, and in dimension 6 we give
an explicit example.  The existence of genuinely partially hyperbolic
irreducible actions in any odd dimension is ruled out by the fact that any
$A\in SL(N,\mathbb{Z})$ with irreducible characteristic polynomial and $N$ odd,
is hyperbolic \cite{RH}. We  also  present a much simpler proof of this fact.
However, reducible examples exist for all odd   $N\ge 9$.

Detailed proofs   can be found in \cite{DK}.


There are other important classes of  partially  hyperbolic algebraic actions,
for example restrictions  of Weyl chamber flows to  higher-rank subgroups or
compact group extensions of Weyl chamber flows. The use of our methods to
establish differentiable rigidity of  partially hyperbolic actions  of those
types   requires   a substitution of the explicit Fourier analysis methods of
the present  paper by   methods of the theory of unitary representations of
semisimple Lie groups. In order  to carry out constructions similar to those
of  special solutions of the second cohomology equation as in Section
\ref{twocoh} one  would need certain estimates of the decay  for
matrix coefficients, which are central for cocycle rigidity in \cite{KSfirst,
KS},  as well as more specialized information about the structure of
irreducible representations. This  work is in progress.

We would like to thank  Elon Lindenstrauss  who pointed out an inaccuracy in an
earlier version of \cite{DK}. 

\subsection{Statement of results}
\label{statements}

For  an action  of a finitely generated group the  $C^n$ topology is defined 
as  the $C^n$ topology for any finite set of generators. For a smooth  action
of a connected Lie group   it is defined similarly, with  the  vector fields 
generating the  action of the Lie algebra playing the role of the generators. 


For  actions of discrete groups the notions of rigidity  which we
 consider are summarized  as follows.

\begin{definition} An action $\alpha$ of a finitely generated discrete group 
$A$ on a manifold $M$ is $\mathcal{C}^{k,r,l}$ {\it locally rigid} if 
any sufficiently $\mathcal{C}^r$ small $\mathcal{C}^k$ 
perturbation $\tilde{\alpha}$ is $\mathcal{C}^l$ conjugate to $\alpha$, i.e., 
there exists a $\mathcal{C}^l$ close to identity diffeomorphism $\mathcal{H}$ 
of $M$ which conjugates $\tilde{\alpha}$ to $\alpha$: 
\begin{equation}\mathcal{H}\circ\alpha(g)=\tilde{\alpha}(g)\circ\mathcal{H}
\label{rigidityeq}
\end{equation}
for all $g \in A$.
The $\mathcal{C}^{\infty,1,\infty}$  case is usually referred to as  
$\mathcal{C}^\infty$ {\it local rigidity} and  the $\mathcal{C}^{1,1,0}$ 
case as $\mathcal{C}^1$ {\it structural stability}.
\end{definition}

In the definitions for continuous groups such as $\Rk$  one has to allow a
``time change'', i.e.  an automorphism $\rho$  of the  group close to $id$ such
that instead of \eqref{rigidityeq} one has 
\begin{equation}\mathcal{H}\circ\alpha(\rho(g))=
\tilde{\alpha}(g)\circ\mathcal{H}.\label{rigidityeqLie}
\end{equation}

A weaker notion in this case (and the one which  is used in the definition of
structural stability for $\R$ action) is   {\em foliation rigidity}: one
requires the diffeomorphism $\mathcal H$ to map an invariant foliation
$\mathcal{F}$ (e.g. the orbit foliation  $\mathcal O $ of the action  $\alpha$) 
to the perturbed foliation $\mathcal{F}'$ (e.g. the orbit foliation
$\tilde{\mathcal O}$ of  the action $\tilde{\alpha}$).

\begin{thm}\label{thm-main}   Let $\alpha :\Zk
\times\mathbb{T}^N\rightarrow\mathbb{T}^N$ be a $\mathcal{C}^\infty$partially
hyperbolic action of $\Zk$ $(k\ge 2)$ by toral automorphisms with no nontrivial
rank-one factors. Then there exists a constant $l=l(\alpha, N)\in \mathbb{N}$
such that for any $\mathcal{C}^l$ small $\mathcal{C}^\infty$ perturbation
$\tilde{\alpha }$ of $\alpha$ there exists a $\mathcal{C}^{\infty}$ map $H :
\mathbb{T}^N \rightarrow \mathbb{T}^N$ such that $\alpha \mycirc H = H
\mycirc\tilde{\alpha }$, i.e., $\alpha$ is $\mathcal{C}^{\infty,l,\infty}$
locally rigid.\end{thm}

The constant $l$ in the above theorem depends on the dimension of the torus and
the given linear action, and it will be precisely defined in the proof.
However, for a certain class of actions this result can be improved so that the
constant $l$ depend only on the dimension of the torus. By combining the
method used by Katok and Spatzier for Anosov  actions with semisimple linear
parts  \cite{KS} with the KAM procedure presented in this paper we obtain the
following:

\begin{thm} There exists a constant $l=l(N)$ such that any  partially
hyperbolic action by  automorphisms of the torus $T^N$  without rank one
factors and with semisimple hyperbolic parts is $\mathcal{C}^{\infty,l,\infty}$
locally rigid.\end{thm} 

\begin{rem} We note that the previous results extend to corresponding affine actions.
While any affine map  of the torus  without eigenvalue one in its linear part 
has  a fixed point and  hence is conjugate to its linear part, the case of
several commuting affine maps is more complicated (for examples of affine
actions without fixed points see \cite{Hurder}). However, by passing to a
subgroup of finite rank  one reduces this case to the case of actions  by
automorphisms. \end{rem} 

\begin{rem} Another  more or less direct extension of our  results concerns
suspensions of  $\Zk$ actions 
by toral automorphisms. Such a suspension is an $\Rk$ action and it allows  obvious modifications by 
an automorphism of $\Rk$. The rigidity  theorem in this case  states that any small perturbation is differentiably conjugate to such a modification.
\end{rem}

As for the existence of actions for which our method is essential we show:

\begin{thm}\label{thm:existence} Genuinely partially hyperbolic (not
hyperbolic) $\mathbb{Z}^2$ actions by ergodic toral automorphisms exist: on the
torus of  any even dimension $N\ge 6$ there are irreducible actions, while on
any torus of  any odd dimension $N\ge 9$ there are only reducible examples.
There are no such actions on tori of dimension $N\le 5$ and $N=7$. \end{thm}

\begin{rem} One can also give similar conditions for the existence of 
genuinely partially hyperbolic actions of $\Zk$ for $k>2$ by ergodic
automorphisms. For example,  an irreducible action of this kind exists in any
even dimension starting from $2k+2$. \end{rem}

In the rest of the paper  we describe the strategy  used for proving the above
results. Complete proofs are contained in \cite{DK}.


\section{Overview of KAM scheme and outline of the proof of Theorem 1}
\subsection{Setting}   Any matrix $A\in GL(N,\mathbb{Z})$ (i.e., a matrix with
integer entries and  determinant $\pm 1$) defines an automorphism of the torus
$\mathbb{T}^N=\mathbb{R}^N/\mathbb{Z}^N$, which we also denote by $A$. Any such
$A$ induces a decomposition of $\mathbb{R}^N$ into the  expanding, neutral and
contracting subspaces. The map $A$ of the torus induces a dual map given by the
transpose matrix, which we also denote by $A$.  Ergodicity of $A$ is equivalent
to  the fact that all nontrivial dual orbits are infinite, which in turn 
implies a well-known fact that $A$ is ergodic if and only if it has no roots of
unity among the eigenvalues.

An action $\alpha : \Zk \times\mathbb{T}^N\rightarrow \mathbb{T}^N$ by
automorphisms of $\mathbb{T}^N$ is given by an embedding $\rho_{\alpha}: \Zk
\to GL(N,\mathbb{Z})$ so that
$$\alpha(g, x)=\rho_{\alpha}(g)x$$
for each $g\in \Zk$ and $x\in \mathbb{T}^N$.

An action $\alpha '$ is a {\it rank-one factor} if it is an algebraic factor
(i.e., there exists an epimorphism $h:\mathbb{T}^N \to \mathbb{T}^N$ such that
$h\circ \alpha=\alpha ' \circ h$) and is  such that $\rho_{\alpha '}(\Zk)$
contains a cyclic subgroup of finite index.

An ergodic action $\alpha$ has no  nontrivial rank-one factors if and only if
$\rho_{\alpha }(\Zk)$ contains a subgroup isomorphic to $\mathbb{Z}^2$ which consists of
ergodic elements  (see for example \cite{Starkov}).

Since we are interested in actions with no rank-one factors, we assume that the
action $\alpha$ satisfies the following: 
\begin{itemize} 
\item There exist
$g_1, g_2 \in \mathbb{Z}^2$ with $A\df \rho_{\alpha}(g_1)$ and $B\df
\rho_{\alpha}(g_2)$ such that any $A^lB^k$ for a nonzero $(l, k) \in
\mathbb{Z}^2$ is ergodic. These will be referred to as {\it ergodic
generators}.  
\item The action is {\it genuinely partially hyperbolic}, i.e.,
one can choose ergodic generators $A$ and $B$ so that they have common
nontrivial neutral subspace and common hyperbolic subspace. We will assume that
such a choice is made for $A$ and $B$. If the choice is not possible, the
action is hyperbolic, and in that case we choose purely hyperbolic generators.
\item We choose ergodic generators so that they have the same root spaces.
\end{itemize}

Now let $\tilde{\alpha}$ be a small perturbation of $\alpha$. The topology in
which the perturbation is considered is a  $C^l$ topology, where the lower bound
for $l$ will be precisely defined in the proof. Our  goal is to prove the
existence of a $\mathcal{C}^\infty$ conjugacy, i.e., a map $H: \mathbb{T}^N
\rightarrow\mathbb{T}^N$ such that $\tilde{\alpha} \mycirc H = H
\mycirc\alpha$.

\subsection{KAM scheme} The strategy we take is usual for KAM proofs. The unusual part is that while this strategy is known to have worked previously   for elliptic situations,  here we manage to carry it through in a  global hyperbolic setting. The problem of finding a conjugacy  is equivalent  to solving the following nonlinear functional equation:
$$\mathcal{N}(\tilde\alpha, H)=\tilde{\alpha}\circ H-H \circ \alpha=0.$$
 Assuming  the existence of a linear structure in the neighborhood of $id$, we may look at the linearization of the operator $\mathcal{N}$ at $(\alpha, id)$:
\begin{align*}
 \mathcal{N}(\tilde\alpha, H)&= \mathcal{N}(\alpha, id) +D_1\mathcal{N}(\alpha, id)(\tilde\alpha-\alpha)+D_2\mathcal{N}(\alpha, id)(\Omega) +\mbox{Res}(\tilde\alpha-\alpha,\Omega)\\
&=\tilde\alpha-\alpha +\alpha(\Omega)-(\Omega)\circ\alpha + \mbox{Res}
(\tilde\alpha-\alpha,\Omega),
\end{align*}
where $\Omega =H-id$ and $\mbox{Res}(\tilde\alpha-\alpha,\Omega)$  is quadratically small with respect to $\tilde\alpha-\alpha$ and $\Omega$.   
If one can find $H$ so that the linear part of the  above expression vanish, 
i.e.,
$$\alpha(H-id)-(H-id)\circ\alpha=-(\tilde\alpha-\alpha ),$$
then such an $H$ is a much better  approximation to a solution to the equation
$\mathcal{N}(\tilde\alpha,H)=0$ than the  identity. After conjugating by $H$
and thus obtaining a smaller perturbation of the linear action, the
linearization procedure and solving the linearized equation may be repeated for
the new perturbation leading to an even better approximate solution.

In particular, on the torus $\T^N$ any map can be lifted to the universal cover
$\mathbb{R}^N$. For every $g\in \Zk$, the lift of $\alpha(g)$ is a linear map
of $\mathbb{R}^N$, i.e., a matrix with integer entries and with determinant $\pm
1$, which is also denoted by $\alpha(g)$ . The lift of $\tilde{\alpha}(g)$ is
$\alpha(g)+\mathcal{R}(g)$, where $\mathcal{R}(g)$ is an $N$-periodic function
for every $g$, i.e. $ \mathcal{R}(g)(x+m)=\mathcal{R}(g)(x)$ for $m\in
\mathbb{Z}^N$. The lift of $H$ has the form  $id+\Omega$ with an $N$-periodic
$\Omega$.
 
In terms of $\Omega $ the conjugacy equation takes the  following form:
\begin{equation}
\label{nonlinear}
 \alpha \Omega - \Omega \mycirc\alpha = - \mathcal{R} \mycirc(\rm{id} 
+ \Omega ). 
\end{equation}
If $\Omega$ is  an approximate solution  for the corresponding 
linearized equation
\begin{equation}
\label{linear}
\alpha \Omega - \Omega \mycirc \alpha = - \mathcal{R},
\end{equation}
i.e., if  it solves this  equation with an error which is quadratically small
with respect to $\mathcal{R}$, then  the new perturbation defined
as
$$\tilde{\alpha }^{(1)} \df H^{-1} \mycirc \tilde{\alpha } \mycirc H$$
should
be much closer to $\alpha $  than $\tilde{\alpha}$, i.e., the new
error
$$\mathcal{R}^{(1)} \df \tilde{\alpha }^{(1)} - \alpha $$ 
should
be quadratically small with respect to the old error $\mathcal{R}$. The
comparison of the two errors usually cannot be made in the same function
space. The norm of the old error in some function space may only be comparable
to the norm of the new error in some larger space. This is the ``loss of
regularity''. In the proof that we present here there will be no actual loss of
regularity at the end, namely, all the functions involved will be
$\mathcal{C}^\infty$. We do however have a loss in the sense described above,
and thus we work with the family of function spaces $\mathcal{C}^r$ ($r\in
\mathbb{N}$). Since we manage to obtain the loss of regularity which is fixed
and since the family of spaces used admits smoothing operators, the iterative
procedure can be set to converge to a solution of the nonlinear equation. 
\subsection{Individual step---the error estimate}\label{twocoh} The new error
is
$$\mathcal{R}^{(1)} = \tilde{\alpha }^{(1)} - \alpha = \left[\Omega \mycirc
\tilde{\alpha }^{(1)} - \Omega \mycirc \alpha +\mathcal{R}(\rm{id}+ \Omega ) -
\mathcal{R} \right]  + \left[\mathcal{R}- ( \alpha \mycirc \Omega - \Omega
\mycirc \alpha) \vphantom{\tilde{\alpha }^{(1)}}\right].$$
The error term in the
first bracket comes from the linearization of the problem and is easy to
estimate provided $\Omega$ is as small as $\mathcal{R}$. The difficulty lies
in estimating the part of the error in the second bracket, namely solving the
linearized equation \eqref{linear} approximately, with an error quadratically
small with respect to $\mathcal{R}$. 

The linearized equation \eqref{linear} actually consists of infinitely many
equations corresponding to different elements of the action:  $$\alpha
(g_i)\Omega - \Omega \mycirc \alpha (g_i) = - \mathcal{R}(g_i),$$  and we need
a common (at least approximate)  solution $\Omega$ to all the equations above. 
At this point two problems arise. The first is that $\mathcal{R}$ is not a 
cocycle
over $\alpha$ and therefore this equation cannot be viewed as a cohomology
equation over the linear action (thus one cannot immediately try to apply
cocycle rigidity results for linear actions), and the second is that this 
is clearly an overdetermined problem in any
case.

The answer to the second problem is that it is not necessary to find a common solution to all the equations above. Indeed, we can show  that commutativity and ergodicity assumptions  imply that it is enough to produce conjugacy $H$ for one ergodic generator. The same conjugacy will work for 
all other elements of the action. The key observation used here is discreteness of the centralizer  for an ergodic partially hyperbolic  automorphism of a torus.
However, in general, it is not possible to produce a $\mathcal{C}^\infty$
conjugacy for a single map, since a single genuinely partially hyperbolic toral
automorphism is not even structurally stable. In fact, we show that there are
infinitely many obstructions to solving the  linearized equation for one
generator. In the  case of  an automorphism of a torus  these obstructions may
be represented as sums of Fourier coefficients  along the dual orbits of the 
map with some weights added at each point, since our equations are twisted.
Therefore, we consider two ergodic generators and reduce the problem of
solving the linearized equation \eqref{linear} to solving simultaneously the
following system: 
\begin{equation}\label{linearizedeq}
\begin{aligned}A \Omega
- \Omega \mycirc A &= - \mathcal{R}_A, \\
B \Omega - \Omega \mycirc B &= -
\mathcal{R}_B,\end{aligned} \end{equation}
where $A$ and $B$ are ergodic
generators. The system above splits further into several simpler systems using
the fact that $A$ and $B$ commute and have common root spaces, i.e.,
\eqref{linearizedeq} is reduced to several equations of the form
\begin{equation}
\label{Jordanlineq}\begin{aligned}J_A\Omega - \Omega \mycirc A &= \Theta, \\
J_B \Omega - \Omega \mycirc B &= \Psi,\end{aligned} \end{equation}
where $J_A$ and $J_B$ are corresponding Jordan blocks. In particular, 
if $\lambda, \mu$ are simple eigenvalues of $A, B$ respectively, than we have
\begin{equation}
\label{simplelineq}\begin{aligned}\lambda\omega - \omega \mycirc A &= \theta, 
\\
\mu \omega - \omega \mycirc B &= \psi,\end{aligned}\end{equation}
where $\theta$ and $\psi$ are $C^\infty$ functions on the torus.

\subsubsection{Conditions for solvability of \eqref{linearizedeq}} Using the
higher rank assumption, i.e. the existence of two elements of the action which
generate an action by ergodic toral automorphisms, we show that all the
obstructions to solving each individual equation above actually vanish
provided that $\mathcal{R}$  satisfies a certain condition, which is satisfied
if $\mathcal{R}$ is a twisted cocycle over the \emph{unperturbed} action. The
cocycle condition is  \eqref{solvability} below.

The vanishing of the obstructions allows one to  construct a distribution  
solution
of the system \eqref{simplelineq}, which can be proven to be $C^\infty$.
Moreover, using the cone method of \cite{Veech} it is possible to obtain an
estimate for the $C^r$ norm of this solution with respect to the $C^{r+\sigma}$
norm of $\theta$ and $\psi$ with only a fixed loss of regularity. This loss
$\sigma$ depends on the dimension $N$ and eigenvalues of $A$ and $B$.  Obtaining
estimates is not difficult when ergodic generators are hyperbolic. The
existence of a nontrivial neutral direction requires the use of the fact that
integer vectors cannot remain mostly in the neutral direction for too long. In
fact, the time  for which they remain mostly in the neutral direction  is 
at most approximately logarithmic   with respect to the norm of the vector.
The precise statement which is used  is that of  Lemma 3 from Katznelson's paper
\cite{Katz}. Thus the decay of Fourier coefficients along the dual orbits of
the action can be well estimated for all the orbits, and this implies the norm
estimate of the solution.


More generally, allowing the existence of Jordan blocks, we show that there exists a $\C^{\infty}$ solution to \eqref{linearizedeq} if the following cocycle condition is satisfied by $\mathcal{R}$:
\begin{equation}
\label{solvability}
 L(\mathcal{R}_A, \mathcal{R}_B) \df  \Delta ^B
\mathcal{R}_A-\Delta^A \mathcal{R}_B \equiv 0,
\end{equation}
where
$\Delta ^B\mathcal{R}_A\df\mathcal{R}_A \mycirc B - B 
\mathcal{R}_B$ and $\Delta^A \mathcal{R}_B\df \mathcal{R}_B\mycirc A-A\mathcal{R}_B$.

The precise statement is the following lemma proven in \cite{DK}.


\begin{alemma} 
\label{HRtricklemma}
If $L(\mathcal{R}_A, \mathcal{R}_B)
\equiv 0$, where $\mathcal{R}_A, \mathcal{R}_B$ are $\mathcal{C}^{\infty}$
maps described before, then the equations \eqref{linearizedeq},
\begin{equation*}
\begin{aligned}
        \Delta^A \Omega &= \mathcal{R}_A, \\
        \Delta^B \Omega &= \mathcal{R}_B
    \end{aligned} 
\end{equation*}
have a common $\mathcal{C}^{\infty}$ solution satisfying
\begin{equation*}
    {\|\Omega \|}_{\mathcal{C}^{r}} \le C_r{\|\mathcal{R}_A,
        \mathcal{R}_B\|}_{\mathcal{C}^{r+\sigma}},
\end{equation*}
for any $r>0$ and $\sigma>M_1=\max\{N+1, M_0(A,B)\}$, where $M_0$ is the
logarithm of the spectrum width for $A$ or for $B$, whichever is greater.
In particular, if the perturbation $\tilde\alpha$ is in the neutral direction
only, then $M_0=0$.  \end{alemma}

\subsubsection{Approximating an almost solvable system by a solvable one} Even
though the error term $\mathcal{R}$ may not satisfy the cocycle  condition
\eqref{solvability} over the unperturbed action, the fact that it satisfies the
twisted cocycle condition over the \emph{perturbed} action implies that it
satisfies the cocycle condition for the unperturbed action up to an error which is
quadratically small with respect to $\mathcal{R}$. More precisely, it is easy
to show that if $\tilde{\alpha}=\alpha+\mathcal{R}$ is a commutative action,
then $L(\mathcal{R}_A, \mathcal{R}_B)$ is quadratically small in the 
$\mathcal{C}^0$
norm with respect to $\mathcal{R}_A,\mathcal{R}_B$. This leads us to the key
step: we prove that  any almost  cocycle over $\alpha$ is close to a cocycle
over $\alpha$. This requires finding a projection of $\mR$ to the space of
cocycles over $\alpha$. What makes this part of the problem difficult is  the
need to find a projection such that the $C^r$ norm of the difference is
quadratically small with respect to some $C^{r+\sigma}$ norm of $\mR$, where
$\sigma$ is a fixed constant independent of individual steps of the scheme. We 
construct this projection effectively. By reducing the problem again to simpler
equations the core of the problem is to obtain \emph{tame} solutions
$\tilde\theta$ and $\tilde\psi$ to an equation of the kind
$$(\mu\tilde\theta-\tilde\theta\circ B)-(\lambda\tilde\psi-\tilde\psi\circ
A)=\varphi$$ 
assuming that $\theta$ and $\psi$ satisfy the same equation with
$\varphi$ on the right hand side, i.e. assuming that the ``double'' obstructions
for $\varphi$ vanish.

The approach to this part of the problem is as follows: the fact that $\psi$,
$\theta$ and $\varphi$ satisfy the equation above implies that the obstructions
for $\psi$ with respect to $B$, i.e., sums of the Fourier coefficients of $\psi$
along the dual orbits of $B$ (even if they do not vanish) are actually small
of the order of $\varphi$. So the strategy is to construct $\tilde\psi$ by
defining its Fourier coefficients to be zero on each  dual  orbit except at the
minimal point on the orbit where we assign the Fourier coefficient of
$\tilde\psi$ to be the obstruction of $\psi$ along that orbit. Thus
$\tilde\psi$ and $\psi$ have  the same obstructions and their difference has
vanishing obstructions so that the linearized equation for the difference
$\psi-\tilde\psi$ can be solved. The crucial part is that the estimates are
needed for the Fourier coefficients of $\tilde\psi$, i.e., for the obstructions
of $\psi$ with respect to $\varphi$. This involves estimating the decay of
Fourier coefficients for $\varphi$ along the ``double'' dual orbits, i.e., the
dual orbits of the action, rather than for only one generator. At first
glance this may pose a problem, since the expanding and contracting directions
for two generators may mix obstructing exponential decay. In fact, there may be
a situation where some (infinitely many) elements have eigenvalues outside 
the unit circle but still very close to it. However, due to the higher rank
assumption if this happens, there must be an extra direction which grows.  For,
otherwise one would have integer matrices  close to the identity, which is not
possible unless they are the identity. The existence of this  extra direction, along
with the fact that it ought to be irrational,  implies (using Katznelson's
lemma again) that there is a sufficient exponential decay, enough to imply
norm estimates for the ``double'' obstructions for $\varphi$ and thus for
``single'' obstructions for $\psi$.

We further show that this procedure can be carried out in the case where  there are nontrivial Jordan blocks and that the solution is tame. More precisely, we obtain:
\begin{alemma} 
\label{twocohglobal}
For two
$\mathcal{C}^{\infty }$ maps $\mR_A$, $\mR_B$ with
$L(\mR_A, \mR_B) = \Phi$ there exist splittings 
$\mR_A=\dot{\mR}_A+\tilde{\mR}_A$ and $\mR_B=\dot{\mR}_B+\tilde{\mR}_B$  
such that
$$L(\dot{\mR}_A, \dot{\mR}_B) = 0,\qquad 
L(\tilde{\mR_A}, \tilde{\mR_B}) = \Phi,$$
$$ \|\dot{\mathcal{R}_A}, \dot{\mathcal{R}_B}\|_{\mathcal{C}^{r}}\le C_r\|\mathcal{R}\|_{\mathcal{C}^{r+\sigma}}$$
for any $r>0$ and $\sigma>M^1=(N+1)\max\{1, M_0\}$. Furthermore, 
$$ \|\tilde{\mathcal{R}_A}, \tilde{\mathcal{R}_B}\|_{\mathcal{C}^r}\le 
C_r\|\Phi\|_{\mathcal{C}^{r+\sigma}}$$ 
for any $r>0$ and $\sigma>M$, where 
the constant $M$, similarly to $M_1$ and $M^1$, depends only on the dimension of the torus and the action generated by $A$ and $B$, and also reduces to $N+1$ in the case when the  action is  perturbed in the neutral direction only.
\end{alemma}

\begin{rem} The constant $M$ is precisely defined in the proof of the above
lemma in Section 3.3 of \cite{DK}. It involves a constant which depends on the
action generated by $A$ and $B$ and characterizes the exponential growth along dual
orbits of the action with respect to certain projections. \end{rem}

The above discussion implies that there exists a  tame (with respect to $\mR$),
and tamely approximate (the error is quadratically small with respect to $\mR$)
solution to the linearized equation \eqref{linearizedeq}.

\subsubsection{Smoothing} The  loss of  a fixed number of derivatives at each
step of the iteration process can be overcome by introducing a family of
smoothing operators: $\{S_J, J\in \mathbb{N}\}$. Then instead of 
approximately solving \eqref{linearizedeq} we approximately solve 
the following system:
\begin{equation}
\label{lineqsmoothing}
\begin{aligned}    A \Omega -\Omega \mycirc A &= -S_{J} \mathcal{R}_A,  \\    
B\Omega - \Omega \mycirc B &= - S_J \mathcal{R}_B.\end{aligned}\end{equation}
For a $\mathcal{C}^\infty$ function $f$ with Fourier series $f=\sum_n \hat{f}_n \chi_n$ we define
$S_J f$ as
\begin{equation*}    S_J f\df \sum_{|n|b>0$ and $\sigma >N+1$. Also
$$    {\| (I-S_J) f \|}_{\mathcal{C}^{a-b}} \le C J^{-b+\sigma}{\| f 
\|}_{\mathcal{C}^a}$$ 
for $a>b>\sigma>N+1$. These simple smoothing operators are convenient in our
setting since they behave well with respect to the operator $L$. Namely, we
have
$$L(S_Jf,S_Jg)=S_{\frac{J}{\xi}}(L(f,g))+ \mathcal{F}_{>\frac{J}{\xi}},$$
where $\xi$ is a constant depending on $A$ and $B$ and the last term in 
the expression consists of the part of the Fourier series for $f$ and $g$ with terms $|n|>\frac{J}{\xi}$. 

 This implies the following estimate:
$$\|L(S_Jf,S_Jg)\|_{\mathcal{C}^{r}}\le J^{2\sigma}\|L(f,g)\|_{\mathcal{C}^{r}}+CJ^{-b+\sigma}\|f,g\|_{\mathcal{C}^{r+b}}
$$
for any $b>\sigma>N+1$.


\subsubsection{Estimate for the error} At each step of the iterative scheme we
first choose an appropriate smoothing operator $S_J$. In order to approximately 
solve
\eqref{lineqsmoothing} we use Lemma \ref{twocohglobal} to obtain
the splitting
\begin{align*}    
S_J \mR_{A} & = \overline{S_J \mR_{A}}+    \widetilde{S_J \mR_{A}},\\
S_J \mR_{B} & = \overline{S_J \mR_{B}}+    \widetilde{S_J \mR_{B}},
\end{align*}
so that $L(\overline{S_J \mR_A},\overline{S_J \mR_B})=0$. Here, we use the bar
notation $\overline{f}$  instead  of the dot  notation $\dot{f}$ of Lemma
\ref{twocohglobal}.  Now  Lemma \ref{HRtricklemma} implies that  the system
\begin{align*}    A \Omega -\Omega \mycirc A &= -\overline{S_{J} \mR_A },\\   
B\Omega - \Omega \mycirc B &= - \overline{S_J \mR_B}\end{align*}
has an
approximate solution $\Omega$ such that
\begin{equation} \label{solbound}   
{\| \Omega  \|}_{\mathcal{C}^r} \le C_r J^{3\sigma }{\| \mR_A, \mR_B
\|}_{\mathcal{C}^{r }}.\end{equation} 
Then we define $H\df id +\Omega$ (since
$\Omega $ is made small throughout the iteration, $H$ is invertible) and
$$\tilde{\alpha }^{(1)} \df H^{-1}\mycirc \tilde{\alpha } \mycirc H.$$
The new
error is equal to  
\begin{equation*}    \mathcal{R}^{(1)}\df\tilde{\alpha
}^{(1)}-\alpha\end{equation*}
and it has two parts:
\begin{itemize}    
\item The
error coming from solving the linearized equation only approximately,
\begin{equation*}        E_1 = \widetilde{S_J \mathcal{R}} + (I-S_J)
\mathcal{R}.
\end{equation*}    
\item The standard error coming from the
linearization,
\begin{equation*}        E_2 = \Omega \mycirc \tilde{\alpha
}^{(1)} - \Omega \mycirc \alpha            + \mathcal{R}(id+\Omega
)-\mathcal{R}.   
\end{equation*}
\end{itemize}
We estimate both parts of the
error using the estimates obtained after approximately solving the linearized
equation and bounds for the smoothing operators. The new error can be estimated
as
\begin{equation} \label{newerror}{\| \mathcal{R}^{(1)} \|}_{\mathcal{C}^{0}}\le   
C J^{3\sigma}{\| \mathcal{R} \|}_{\mathcal{C}^1}{\| \mathcal{R}
\|}_{{\mathcal{C}^{0}}}    + C J^{-l+2\sigma }{\| \mathcal{R}
\|}_{\mathcal{C}^l}
\end{equation}
for any $l>2\sigma$. From $\displaystyle
\tilde{\alpha }^{(1)}=H^{-1}\mycirc \tilde{\alpha }\mycirc H$, using the fact
that $\Omega$ is $N$-periodic and satisfies the estimate \eqref{solbound}, we
have
\begin{equation}
\label{linearbound} 
{\| \mathcal{R}^{(1)} \|}_{\mathcal{C}^l} \le C_l J^{3\sigma }        \left( 1
+ {\| \mathcal{R} \|}_{\mathcal{C}^l} \right).\end{equation}

\subsection{Setting the iteration and convergence} 

To set up the iterative process let
\begin{equation*}   
\mathcal{R}^{(0)}=\mathcal{R}; \qquad \tilde{\alpha }^{(0)}=\tilde{\alpha };
\qquad    H^{(0)}=id.\end{equation*}
Construct $\mathcal{R}^{(n)}$ inductively
for every $n$: for $\mathcal{R}^{(n)}$ we choose an appropriate $J_n$ to
obtain $S_{J_n}\mathcal{R}^{(n)}$, which produces a  new $\Omega^{(n)}$ 
after 
approximately solving the linearized equation. Then we have
\begin{equation*}    H^{(n)}=id+\Omega ^{(n)}, \quad   
\tilde{\alpha}^{(n+1)}={( H^{(n)} )}^{-1}\mycirc \tilde{\alpha
}^{(n)}\mycirc H^{(n)},    \quad \mathcal{R}^{(n+1)}=\tilde{\alpha
}^{(n+1)}-\alpha\end{equation*}
\begin{align*}    \tilde{\alpha }^{(n+1)} &= {( H^{(n)} )}^{-1}
\mycirc {( H^{(n-1)} )}^{-1}\mycirc        \cdots \mycirc {(
H^{(0)} )}^{-1} \mycirc \tilde{\alpha }\mycirc        H \mycirc \cdots
\mycirc H^{(n)} \\    &= \mathcal{H}_n^{-1} \mycirc \tilde{\alpha }\mycirc
\mathcal{H}_n.\end{align*}
To ensure the convergence of
the process we set
\begin{equation*}    {\| \mathcal{R}^{(n)}
\|}_{\mathcal{C}^0}<\varepsilon_n=\varepsilon^{(k^n)},    \qquad    {\|
\mathcal{R}^{(n)} \|}_{\mathcal{C}^l} <
\varepsilon_n^{-1},\end{equation*}\begin{equation*}    {\| \Omega ^{(n)}
\|}_{\mathcal{C}^1}<\varepsilon_n^{{1/2}},    \qquad   
J_n=\varepsilon_n^{-\frac{1}{3(3\sigma +2)}},\end{equation*}
where $ 
k=\frac{4}{3}$. At this point we fix $l$: $l=23\sigma+15$, where
$\sigma=\sigma(A,B,N)=[\max\{M,M^1,M_1\}]+1$ is a constant for which all the
estimates obtained above hold.

Convergence of the process is then proven by induction, using the estimates
already obtained. Namely, for sufficiently small ${\| \mathcal{R}
\|}_{\mathcal{C}^0}$ and ${\| \mathcal{R} \|}_{\mathcal{C}^1}$ the process
converges to a solution $\Omega \in \mathcal{C}^1$ with $ {\|
\Omega  \|}_{\mathcal{C}^1}<\frac{1}{4}$. Then, using interpolation
inequalities, we show that $\Omega$ is $\mathcal{C}^{\infty }$.

\section{Outline of the proof of Theorem 2}

\begin{thm}[Foliation rigidity]\label{thmfoliations} The neutral foliation 
$\mathcal N$ of a $C^{\infty}$ action  $\alpha$  by partially hyperbolic  
automorphisms of $\ \mathbb T^N$ without rank one factors  and with semisimple
hyperbolic parts is $C^{\infty}$ rigid, i.e., for  any $C^{\infty}$ foliation
$\mathcal N'$ sufficiently close to $\mathcal N$ in the $C^1$ topology there exists
a $C^{\infty}$ diffeomorphism  $h:\mathbb T^N\to\mathbb T^N$ such that
$h\mathcal N=\mathcal N'$. \end{thm}

This theorem is proven exactly like  Theorem 1 and  Theorem 6 in \cite{KS}. In
fact, a more general statement is true concerning foliation rigidity for a
broad class of homogeneous  partially hyperbolic  actions \cite{KS}. 

This reduces the local rigidity problem for an action with the semisimple
linear part to showing that a sufficiently small perturbation along the leaves
of the neutral foliation of a given linear action is smoothly conjugate to the
original action. This is a special case of Theorem~\ref{thm-main}. But in this
special situation the estimates obtained for the solution of the corresponding
linearized equation are much better. Namely, since the action is perturbed only 
in the neutral direction,  the linearized equation splits into several
equations of the form  \eqref{simplelineq} or \eqref{Jordanlineq}  with
$|\lambda|=|\mu|=1$. The constants that describe the loss of regularity turn
out to be at most $ N+2$ in this case.  Therefore, in the iterative proof  for
perturbations in the neutral direction it is sufficient to choose
$l=l(N)=23N+61$ for the convergence of the successive approximations to a
$\mathcal{C}^\infty$ conjugacy.

Combined with the foliation rigidity result where the perturbation needs to be
only $\mathcal{C}^1$ small, this implies $\mathcal{C}^{\infty, l, \infty}$
local rigidity for an action with the semisimple linear part, where  $l$ is
defined  as above and hence depends  only of the dimension of the torus.
  
 
\section{Existence of genuinely partially hyperbolic actions} To prove the
statement of Theorem~\ref{thm:existence} we first show case by case that there
are no examples of genuinely partially hyperbolic $\Zk$ actions (irreducible or
reducible) for $N\le 5$. In \cite{RH} it is proven  that there are no
irreducible automorphisms in odd dimensions with nontrivial neutral direction.
In Section 6.2 of \cite{DK} we give a simple proof of this fact. Then we show
that there are irreducible examples of such actions in any even dimension $N\ge
6$.  For that  we use matrices with recurrent characteristic polynomials. By
taking  direct products of these examples with hyperbolic actions we obtain
reducible examples in any odd dimension $N\ge 9$.  For example on the
9-dimensional torus, in the construction above take $n_1=3$, $n_2=6$, let
$A_{1,2}$ and $A_{2,2}$ be commuting matrices that give an irreducible action
in even dimension (we give explicit examples of those), and for $A_{1,1}$ and
$A_{2,1}$ choose any two $3 \times 3$ commuting hyperbolic totally real integer
matrices (for various examples see \cite{KKS}). 

      
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\end{document}