# Regularity of conjugacies of algebraic actions of Zariski dense groups

Alexander Gorodnik, Theron Hitchman, Ralf Spatzier  School of Mathematics
University of Bristol
Bristol BS8 1TW, U.K<EMAIL_ADDRESS>Department of Mathematics
University of Northern Iowa
Cedar Falls, IA 50614-0506<EMAIL_ADDRESS>Department of Mathematics
University of Michigan
Ann Arbor, MI 48109-1043<EMAIL_ADDRESS>
###### Abstract.

Let $\alpha_{0}$ be an affine action of a discrete group $\Gamma$ on a compact
homogeneous space $X$ and $\alpha_{1}$ a smooth action of $\Gamma$ on $X$
which is $C^{1}$-close to $\alpha_{0}$. We show that under some conditions,
every topological conjugacy between $\alpha_{0}$ and $\alpha_{1}$ is smooth.
In particular, our results apply to Zariski dense subgroups of
$\hbox{SL}_{d}(\mathbb{Z})$ acting on the torus $\mathbb{T}^{d}$ and Zariski
dense subgroups of a simple noncompact Lie group $G$ acting on a compact
homogeneous space $X$ of $G$ with an invariant measure.

A.G. was supported by NSF grants DMS 0400631, 0654413 and RCUK Fellowship,
R.S. was supported by NSF Grant DMS 0604857

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgement
  2. 2 Main result
  3. 3 Proof of the Main Theorem
    1. 3.1 $C^{0}$ implies Hölder
    2. 3.2 Invariance of fast stable manifolds
    3. 3.3 Convergence of the sequences $f^{-n}gf^{n}$
    4. 3.4 Hölder implies $C^{1}$ along fast stable manifolds
    5. 3.5 Completion of the proof of the main theorem
  4. 4 Existence of good pairs
    1. 4.1 Tori
    2. 4.2 Semisimple groups

##  1\. Introduction

The investigation of rigidity properties has been at the forefront of research
in dynamics in the past two decades. Of particular interest has been the study
of higher rank abelian groups and local rigidity of their actions by Hurder,
Katok, Lewis, and the last author amongst others. Remarkably, many such
actions cannot be perturbed at all, in the sense that any $C^{1}$-close
perturbation is $C^{\infty}$-conjugate to the original action. Critically,
these groups contain higher rank abelian groups. Similar results were found
for higher rank semisimple Lie groups and their lattices by Hurder, Lewis,
Fisher, Margulis, Qian and others. We refer to [5] for a more extensive survey
of these developments.

Smoothness of the conjugacy for these actions came as quite a surprise.
Classically, in fact, the stability results of Anosov and later Hirsch, Pugh
and Shub guaranteed a continuous conjugacy or orbit equivalence between a
single Anosov diffeomorphism or flow and their perturbations [13]. Simple
examples however show that such a conjugacy cannot be even $C^{1}$ in general.

In the present paper, we investigate similar regularity phenomena for affine
actions of a large class of groups. Notably, our results do not require the
presence of higher rank subgroups or any assumptions on the structure of the
group. In particular they hold for discrete subgroups of rank one semisimple
groups. We recall that a group acts affinely on a homogeneous space
$H/\Lambda$ for $H$ a Lie group and $\Lambda$ a discrete subgroup if every
element acts by an affine diffeomorphism i.e. one which lifts to a composite
of a translation and an automorphism on $H$. We denote by $\hbox{\rm Diff}(X)$
the group of $C^{\infty}$-diffeomorphisms of a space $X$.

For simplicity let us mention two corollaries of our main theorem in Section
2.

######  Theorem 1.1.

Let $\Gamma\subset\hbox{\rm SL}_{d}(\mathbb{Z})$ for $d\geq 2$ be a finitely
generated Zariski dense subgroup in $\hbox{\rm SL}_{d}({\mathbb{R}})$, and
$\alpha_{0}$ the associated action on the $d$-torus $\mathbb{T}^{d}$. If a
perturbation $\alpha_{1}:\Gamma\to\hbox{\rm Diff}(\mathbb{T}^{d})$ is
sufficiently $C^{1}$-close to $\alpha_{0}$, then any $C^{0}$-conjugacy
$\Phi:\mathbb{T}^{d}\mapsto\mathbb{T}^{d}$ between $\alpha_{0}$ and
$\alpha_{1}$ is a $C^{\infty}$-diffeomorphism.

For $d=2$, E. Cawley found a $C^{1+\alpha}$-regularity result for Zariski-
dense subgroups of $\hbox{\rm SL}_{2}(\mathbb{Z})$ acting on the 2-torus in
[4] in the early 1990’s. Her techniques however are restricted to the 2-torus
due to the use of $C^{1+\alpha}$-regularity of stable foliations.
Subsequently, the second author obtained a general $C^{\infty}$-regularity
theorem for groups acting on general tori in his thesis [14].

A second application of our main theorem to actions on homogeneous spaces of
semisimple groups is novel.

######  Theorem 1.2.

Let $G$ be a connected simple noncompact Lie group, $\Lambda$ a cocompact
lattice in $G$, and $\Gamma$ a finitely generated Zariski dense subgroup of
$G$. Let $\alpha_{0}$ be the affine action of $\Gamma$ on $G/\Lambda$. If a
$C^{\infty}$-action $\alpha_{1}$ is sufficiently $C^{1}$-close to
$\alpha_{0}$, then any $C^{0}$-conjugacy $\Phi:G/\Lambda\mapsto G/\Lambda$ is
a $C^{\infty}$-diffeomorphism.

Let us note that our techniques are based on certain mixing properties of the
actions and do not allow the treatment of actions on general nilmanifolds.

Fisher and Hitchman recently proved a local rigidity theorem for actions of
lattices with the Kazhdan property [8]. We recall that an action $\alpha$ is
called $C^{k,l}$-rigid if any $C^{k}$-close perturbation of the action is
$C^{l}$-conjugate to $\alpha$.

######  Theorem 1.3 (Fisher-Hitchman).

Let $\Gamma$ be a lattice in a semisimple Lie group without compact factors
which satisfies Kazhdan’s property. Then any affine action $\alpha$ of
$\Gamma$ is $C^{3,0}$-locally rigid.

Fisher and Hitchman actually prove this for _quasi-affine_ actions, which are
extensions of affine actions by isometries. Their technique is based on a type
of heat flow. If $\alpha$ does not admit a common neutral direction, then
Fisher and Hitchman’s proof yields $C^{1,0}$-local rigidity. Using our
regularity result, we immediately obtain

######  Corollary 1.4.

Let $G$ be a simple noncompact Lie group which satisfies Kazhdan’s property,
$\Gamma$ a lattice in $G$, and $X$ a compact homogeneous space of $G$
supporting an invariant measure. Then the affine action of $\Gamma$ on $X$ is
$C^{1,\infty}$-locally rigid.

######  Remark 1.5.

We can also deduce $C^{1,\infty}$-local rigidity for the action of a Kazhdan
lattice $\Gamma$, embedded in $\hbox{SL}_{d}(\mathbb{Z})$, on the torus
$\mathbb{T}^{d}$ under the assumption that $\Gamma\times\Gamma$ is not
contained in the subvarieties $\det([X^{\ell},Y]-id)=0$, $\ell\geq 1$,
$\phi(\ell)\leq d^{2}$, where $\phi$ is the Euler totient function (see Lemma
4.2). This assumption is needed to construct good pairs in $\Gamma$ (see
Definition 2.1).

Fisher and Hitchman proved $C^{\infty,\infty}$-local rigidity for a more
general class of actions of cocompact lattices in the same groups [8]. In
particular their approach works on nilmanifolds.

At the heart of our argument lies the investigation of sequences of the form
$\gamma^{-n}\delta\gamma^{n}$ for two hyperbolic elements $\gamma$ and
$\delta$ in “general position”. Such elements always exist in Zariski-dense
groups. The behavior of these sequences is badly divergent in directions
transverse to the fast stable direction of $\gamma$, and cannot be controlled.
However, these sequences do converge along the fast stable manifolds of
$\gamma$. This is elementary for an affine action. We prove
$C^{1}$-convergence for the perturbed action. These limiting maps along the
fast stable foliation of $\gamma$ form a rich system which acts transitively
along the fast stable leaves under suitable conditions. Moreover, the
conjugacy $\Phi$ between the actions will also intertwine these limiting maps
along fast stables. It follows that $\Phi$ has to be $C^{1}$ along each of
these fast stable manifolds. We prove smoothness in a separate argument.

The proof of $C^{1}$-convergence is technically the most difficult piece of
the argument. It requires careful estimates which are an adaptation of the
proof of Livsic’ theorem for cocycles with non-abelian targets.

The use of sequences of the form $\gamma^{-n}\delta\gamma^{n}$ was introduced
by Hitchman in his thesis [14]. His argument relied on the idea that the
resulting limit maps along fast stable leaves often exhibit higher rank
abelian behavior which could then be used to prove regularity similar to the
case of actions by higher rank abelian groups.

Let us comment that our arguments seem to be of rather general nature. In the
weakly hyperbolic setting, the hard part in proving local rigidity results
lies in getting a $C^{0}$-conjugacy. Indeed, the common strategy for most of
the known local rigidity results has been to show existence of a
$C^{0}$-conjugacy and then improve the regularity. Margulis–Qian in higher
rank and Fisher-Hitchman for all Kazhdan Lie groups have the most extensive
results [19, 9]. The current paper shows regularity under rather general
conditions, reducing smooth local rigidity to continuous local rigidity. To
pinpoint precisely when local rigidity holds appears difficult. On the one
hand, we have the results above for actions of lattices in the Kazhdan rank
one groups. On the other hand, Fisher found non-trivial affine deformations of
actions of lattices in $SO(n,1)$ resulting from “bending lattices” [6, 7].
Finally, if the action has isometric directions, even regularity becomes
difficult as evidenced even in higher rank by the works of Fisher and Margulis
[10] and Fisher and Hitchman [8].

###  1.1. Acknowledgement

We would like to thank R. Feres, A. Gogolev, J. Heinonen, B. Kalinin, and B.
Schmidt for useful discussions. We also would like to thank a referee for
careful reading and for pointing out some deficiencies in our original proofs.
A.G. would like to express his thanks for hospitality to Princeton University,
where part of this work was completed.

##  2\. Main result

Let $G$ be a connected Lie group, $\Lambda$ a cocompact lattice in $G$, and
$X=G/\Lambda$. The space $X$ is equipped with a finite invariant Radon
measure. The group $\hbox{\rm Aff}(X)$ of affine transformations of $X$
consists of maps of the form

$f:x\mapsto L_{g}\circ a(x),\quad x\in X,$

where $L_{g}$ denotes the left mutiplication action of $g\in G$ and $a$ is an
automorphism of $G$ preserving $\Lambda$. Every such map $f$ preserves the
measure and defines an automorphism $Df$ of $\hbox{Lie}(G)\simeq
T_{e\Lambda}(X)$ given by

$Df:=\hbox{Ad}(g)\circ D(a)_{e}.$

We denote by $W^{min}_{f}$ the sum of the generalized eigenspaces of $Df$ with
eigenvalues of minimal modulus and by $P^{min}_{f}:\hbox{Lie}(G)\to
W^{min}_{f}$ the projection map along the other generalized eigenspaces.

######  Definition 2.1.

We call a pair $f,g\in\hbox{\rm Aff}(X)$ good if the following conditions are
satisfied:

  1. (i)

The map $Df$ is partially hyperbolic.

  2. (ii)

The map $Df:W^{min}_{f}\to W^{min}_{f}$ is semisimple.

  3. (iii)

The map $P^{min}_{f}\circ Dg:W^{min}_{f}\to W^{min}_{f}$ is nondegenerate.

  4. (iv)

For every subsequence $\\{n_{i}\\}$ , the sequence
$\\{f^{-n_{i}}gf^{n_{i}}(x)\\}$ is dense in $X$ for $x$ in a set of full
measure.

If for $f\in\hbox{\rm Aff}(X)$, there exists $g\in\hbox{\rm Aff}(X)$ so that
the pair $f,g$ is good, we say $f$ is a good mapping.

######  Remark 2.2.

In the case when the map $Df:W^{min}_{f}\to W^{min}_{f}$ does not have a
rotation component of infinite order (e.g., when $\dim W^{min}_{f}=1$), it
suffices to assume that the sequence $\\{f^{-n}gf^{n}(x)\\}$ is dense in $X$
for $x$ in a set of full measure. In general, we have to pass to a subsequence
to guarantee that the maps $f^{-n}gf^{n}$ converge along the fast stable
leaves as $n\to\infty$ (see Proposition 3.13).

The theorems stated in the introduction will be deduced from the following
general result:

Main Theorem. Let $\Gamma$ be a finitely generated discrete group and
$\alpha_{0}:\Gamma\to\hbox{\rm Aff}(X)$ an affine action of $\Gamma$ such that

  * •

$(D\alpha_{0})(\Gamma)$ acts irreducibly on $\hbox{\rm Lie}(G)$,

  * •

$\alpha_{0}(\Gamma)$ contains a good pair.

Let $\alpha_{1}:\Gamma\to\hbox{\rm Diff}(X)$ be a $C^{\infty}$-action of
$\Gamma$ which is sufficiently $C^{1}$-close to $\alpha_{0}$. Then every
homeomorphism $\Phi:X\to X$ satisfying

$\Phi\circ\alpha_{0}(\gamma)=\alpha_{1}(\gamma)\circ\Phi\quad\hbox{for all
$\gamma\in\Gamma$}$

is a $C^{\infty}$-diffeomorphism.

######  Remark 2.3.

Irreducibility of the action of $\Gamma$ on $\hbox{Lie}(G)$ is used in the
following places:

  * •

In Section 3.1, to deduce weak hyperbolicity (see (1)),

  * •

In Section 3.2, to construct essential sets (see Lemma 3.9),

  * •

In Section 3.5, to deduce that $\Phi$ is $C^{\infty}$ from smoothness on
subspaces of the fast stable leaves (see (51)).

Existence of good pairs for some classes of affine actions will be proved in
Section 4. In particular, Theorem 1.1 follows from the Main Theorem and
Proposition 4.1, and Theorem 1.2 follows from the Main Theorem and Proposition
4.4.

###### Outline of the proof of the Main Theorem.

Irreducibility of $\Gamma$-action and property (i) of a good pair are used to
prove that $\Phi$ is bi-Hölder (Section 3.1). Next, irreducibility of the
$\Gamma$-action and property (ii) of a good pair are used to show that $\Phi$
maps fast stable manifolds to fast stable manifolds (Section 3.2). Property
(ii) is also used to show that a subsequence of maps $f^{-n}gf^{n}$ restricted
to fast stable manifolds is precompact in the $C^{0}$-topology and, in fact,
in the $C^{1}$-topology (Section 3.3). Then one utilizes property (iii) of a
good pair to deduce that the limits of these maps are homeomorphisms and
property (iv) of a good pair to deduce that these limits generate transitive
$C^{1}$-action on fast stable manifolds. Using that $\Phi$ is a conjugacy
between the constructed $C^{1}$-actions, we show that $\Phi$ is $C^{1}$ along
the fast stable leaves (Section 3.4). A more elaborate argument, which is
based on the nonstationary Sternberg linearization [16, 12, 11], shows that
$\Phi$ is $C^{\infty}$ along some subspaces of fast stable leaves. Finally, we
deduce that $\Phi$ is $C^{\infty}$ on $X$ from elliptic regularity using
irreducibility of the $\Gamma$-action (Section 3.5). ∎

##  3\. Proof of the Main Theorem

We continue with the notation that $X=G/\Lambda$ is a compact quotient of a
connected Lie group $G$ by a discrete subgroup $\Lambda\subset G$.

###  3.1. $C^{0}$ implies Hölder

In this section, we will prove that the conjugacy map $\Phi:X\to X$ in the
Main Theorem is bi-Hölder. The proof is similar to Proposition 5.7 of [19]. As
they do not show that their map is Hölder, and also use somewhat different
hypotheses, we will give a proof here for simplicity.

Following [19], we say that a $C^{1}$-action $\alpha$ of a discrete group
$\Gamma$ on a compact manifold $M$ is _weakly hyperbolic_ when there is a
choice of finitely many elements $\gamma_{1},\ldots,\gamma_{k}$ in $\Gamma$
such that each diffeomorphism $\alpha(\gamma_{i})$ is partially hyperbolic
and, for each point $x\in M$,

(1) $\sum_{i=1}^{k}T_{x}W^{s}_{\alpha(\gamma_{i})}(x)=T_{x}M,$

where $W^{s}_{\alpha(\gamma_{i})}(x)$ denotes the stable manifold of
$\alpha(\gamma_{i})$ through $x$.

######  Theorem 3.1.

Let $\Gamma$ be a finitely generated discrete group,
$\alpha_{0}:\Gamma\to\hbox{\rm Aff}(X)$ be an affine weakly hyperbolic action,
and $\alpha_{1}:\Gamma\to\hbox{\rm Diff}^{1}(X)$ a smooth action which is
sufficiently $C^{1}$-close to $\alpha_{0}$. Then every homeomorphism
$\Phi:X\to X$ such that

$\Phi\circ\alpha_{0}(\gamma)=\alpha_{1}(\gamma)\circ\Phi\quad\hbox{for all
$\gamma\in\Gamma$}$

is bi-Hölder.

The proof is divided into several lemmas.

######  Lemma 3.2.

Let $f_{1},\ldots,f_{k}$ be partially hyperbolic diffeomorphisms of $X$ such
that

$\sum_{i=1}^{k}T_{x}W^{s}_{f_{i}}(x)=T_{x}M\quad\hbox{for all $x\in X$,}$

and $g_{1},\ldots,g_{k}$ are $C^{1}$-close $C^{1}$-diffeomorphisms. Then
$g_{i}$’s are partially hyperbolic and

$\sum_{i=1}^{k}T_{x}W^{s}_{g_{i}}(x)=T_{x}M\quad\hbox{for all $x\in X$}.$

Lemma 3.2 follows from stability of partial hyperbolicity under perturbations
(see, for example, [20, Lemma 3.5]).

######  Lemma 3.3.

Let $\Phi$ be a continuous conjugacy between two partially hyperbolic
diffeomorphisms of a compact manifold. Then $\Phi$ is bi-Hölder continuous
along the stable manifolds of these mappings.

Lemma 3.3 follows from the standard argument as in [15, Theorem 19.1.2].

######  Lemma 3.4.

Let $\alpha:\Gamma\to\hbox{\rm Diff}^{1}(X)$ be a smooth weakly hyperbolic
action and $\gamma_{1},\ldots,\gamma_{k}\in\Gamma$ satisfy (1). Then there
exist $c,\epsilon>0$ such that for every $x,y\in X$ satisfying
$d(x,y)<\epsilon$, there exists a path $\ell$ from $x$ to $y$ which consists
of $2k$ pieces contained in stable manifolds of
$\alpha(\gamma_{1}),\ldots,\alpha(\gamma_{k})$, and $L(\ell)\leq c\,d(x,y)$.

###### Proof.

We will use an argument similar to [22, Lemma 3.1].

Let $d=\dim X$. There exists a family of (global) continuous unit vector
fields $v_{1},\ldots,v_{d}$ that span the tangent space at every point and for
some $1=d_{0}\leq d_{1}\leq\cdots\leq d_{k}=d+1$ and every $i=1,\ldots,k$, the
vectors $v_{d_{i-1}},\ldots,v_{d_{i}-1}$ are contained in the stable
distribution of $\alpha(\gamma_{i})$. Let $\delta>0$. There exists
$\delta^{\prime}>0$ such that $d(u,w)<\delta^{\prime}$ implies that
$d(v_{i}(u),v_{i}(w))<\delta$ for all $i$. By [18, Corollary 4.5], for every
$x\in X$, there exists $\epsilon(x)>0$ such that every $y\in
B_{\epsilon(x)}(x)$ can be connected to $x$ by a path $\ell$ of length at most
$\delta^{\prime}/2$, and for some $0=t_{0}\leq t_{1}\leq\cdots\leq
t_{d}=L(\ell)$, we have $\ell^{\prime}(t)=v_{i}(\ell(t))$ when
$t\in[t_{i-1},t_{i})$. Let $\epsilon>0$ be the Lebesgue number of the cover
$\\{B_{\epsilon(x)}(x)\\}$. Then every $y_{1},y_{2}\in X$ such that
$d(y_{1},y_{2})<\epsilon$ are connected by a path $\ell$ which consists of
$2k$ pieces tangent to $v_{j}$’s and $L(\ell)<\delta^{\prime}$. To estimate
the distance $d(y_{1},y_{2})$, we may assume, without loss of generality, that
we work in an open neighborhood of $\mathbb{R}^{d}$ equipped with the standard
metric. By the triangle inequality,

$\displaystyle\|y_{1}-y_{2}\|$
$\displaystyle=\left\|\sum_{i}\int_{t_{i-1}}^{t_{i}}v_{i}(\ell(t))dt\right\|$
$\displaystyle\geq\left\|\sum_{i}(t_{i}-t_{i-1})v_{i}(y_{1})\right\|-\sum_{i}\int_{t_{i-1}}^{t_{i}}\|v_{i}(\ell(t))-v_{i}(y_{1})\|dt$
$\displaystyle\geq(c-\delta)L(\ell)$

where

$c=\min\left\\{\left\|\sum_{i}s_{i}v_{i}(y)\right\|:\,\sum_{i}|s_{i}|=1,\,y\in
X\right\\}>0.$

Taking $\delta$ sufficiently small, this implies the estimate for $L(\ell)$.
Since the stable distributions are uniquely integrable,
$\ell([t_{i-1},t_{i}))$ is contained in the stable manifold of
$\alpha(\gamma_{i})$. ∎

###### Proof of Theorem 3.1.

Let $\gamma_{1},\ldots,\gamma_{k}\in\Gamma$ be elements satisfying (1). By
Lemma 3.3, the map $\Phi$ is bi-Hölder restricted to the stable manifolds of
$\alpha_{0}(\gamma_{i})$’s. By Lemma 3.4, for sufficiently close $x,y\in X$,
there exist points $x_{0}=x,x_{1},\ldots,x_{2k}=y$ such that $x_{j-1}$ and
$x_{j}$ are on the same stable manifold of some $\alpha_{0}(\gamma_{i_{j}})$,
and $d(x_{j-1},x_{j})\leq c\,d(x,y)$. Then

$\displaystyle d(\Phi(x),\Phi(y))$
$\displaystyle\leq\sum_{j=1}^{2k}d(\Phi(x_{j-1}),\Phi(x_{j}))\leq\sum_{j=1}^{2k}c_{j}d(x_{j-1},x_{j})^{\theta_{j}}$
$\displaystyle\leq\left(\sum_{j=1}^{2k}c_{j}c^{\theta_{j}}\right)d(x,y)^{\theta}$

where $\theta=\min\theta_{j}$.

By Lemma 3.2, the action $\alpha_{1}$ is also weakly hyperbolic. Then the
proof that $\Phi^{-1}$ is Hölder follows the same argument. ∎

###  3.2. Invariance of fast stable manifolds

Let $f\in\hbox{\rm Diff}(X)$, and the tangent bundle $TX$ has continuous
$f$-invariant splitting

(2) $TX=E^{-}\oplus E^{+}$

such that for some $\lambda\in(0,1)$ and $\mu>\lambda$,111The notation $A\ll
B$ means that there exists $c>0$, independent of other parameters, such that
$A\leq c\,B$.

(3) $\displaystyle\|D(f^{n})_{x}v\|$
$\displaystyle\ll\lambda^{n}\|v\|\quad\hbox{for all $n\geq 0$, $x\in X$, and
$v\in E_{x}^{-}$},$ $\displaystyle\|D(f^{n})_{x}v\|$
$\displaystyle\gg\mu^{n}\|v\|\quad\hbox{for all $n\geq 0$, $x\in X$, and $v\in
E_{x}^{+}$}.$

We recall (see, for example, [20, Theorem 4.1]) that the distribution $E^{-}$
is integrable to the fast stable foliation $\\{W^{fs}_{f}(x)\\}_{x\in X}$, and
this foliation is Hölder continuous with $C^{\infty}$-leaves. We denote by
$d^{fs}$ the induced metrics on the leaves of this foliation. For
$\rho>\lambda$ and $x,y\in X$ such that $y\in W^{fs}_{f}(x)$,

$d^{fs}(f^{n}(x),f^{n}(y))\ll\rho^{n}d^{fs}(x,y).$

There exists $\epsilon_{0}>0$ such that for every $z,w\in X$ satisfying $w\in
W^{fs}_{f}(z)$ and $d^{fs}(z,w)<\epsilon_{0}$, we have

(4) $d^{fs}(z,w)\ll d(z,w)\leq d^{fs}(z,w).$

Let $f_{0}\in\hbox{\rm Aff}(X)$ be such that $Df_{0}$ is partially hyperbolic,
and $\lambda_{0}<\mu_{0}$ denote the least two absolute values of the
eigenvalues of $Df_{0}$. If $f\in\hbox{\rm Diff}(X)$ is a $C^{1}$-small
perturbation of $f_{0}$, then we have a splitting as above with
$\lambda=\lambda_{0}+\epsilon$ and $\mu=\mu_{0}-\epsilon$ for some small
$\epsilon>0$, depending on $d_{C^{1}}(f,f_{0})$ (see [20, Lemma 3.5]). The
fast stable manifolds $W^{fs}_{f}(x)$ are defined with respect to this
splitting. Note that

$W^{fs}_{f_{0}}(x)=\exp(W^{min}_{f_{0}})x$

where $\exp$ is the Lie exponential map, and $W^{min}_{f_{0}}$ is defined as
on page 2.

The aim of this section is to prove the following theorem.

######  Theorem 3.5.

Let $\alpha_{0}:\Gamma\to\hbox{\rm Aff}(X)$ and $\alpha_{1}:\Gamma\to\hbox{\rm
Diff}(X)$ be $C^{1}$-close actions of a finitely generated discrete group
$\Gamma$, and let $\Phi:X\to X$ be a homeomorphism such that

$\Phi\circ\alpha_{0}(\gamma)=\alpha_{1}(\gamma)\circ\Phi\quad\hbox{for all
$\gamma\in\Gamma$}.$

Assume that $(D\alpha_{0})(\Gamma)$ acts irreducibly on $\hbox{\rm Lie}(G)$.
Then for every partially hyperbolic $f_{0}:=\alpha_{0}(\gamma)$ and
$f:=\alpha_{1}(\gamma)$, $\gamma\in\Gamma$, such that $Df_{0}$ is semisimple
on $W^{min}_{f_{0}}$,

$\Phi(W^{fs}_{f_{0}}(z))=W^{fs}_{f}(\Phi(z))\quad\hbox{for all $z\in X$.}$

Moreover, the map $\Phi$ is bi-Hölder with respect to the induced metrics on
fast stable leaves of $f_{0}$ and $f$.

Let us start with some preliminary reductions. We will prove that

(5) $\Phi^{-1}(W^{fs}_{f}(z))\subset
W^{fs}_{f_{0}}(\Phi^{-1}(z))\quad\hbox{for all $z\in X$.}$

This also implies that the equality. Indeed, it follows from (5) that every
leaf $W^{fs}_{f_{0}}(\Phi^{-1}(z))$ is a disjoint union of sets of the form
$\Phi^{-1}(W^{fs}_{f}(y))$ for some $y\in X$. By [20, Lemma 3.5], the fast
stable leaves of $f_{0}$ and $f$ have the same dimension. Hence, by the
invariance of domain, every set $\Phi^{-1}(W^{fs}_{f}(y))$ is open in
$W^{fs}_{f_{0}}(\Phi^{-1}(z))$. Since $W^{fs}_{f_{0}}(\Phi^{-1}(z))$ is
connected, we deduce that

$\Phi^{-1}(W^{fs}_{f}(z))=W^{fs}_{f_{0}}(\Phi^{-1}(z)).$

Let

(6) $\mathcal{S}_{\epsilon^{\prime}}(x)=\\{\Phi^{-1}(z):\;z\in
W^{fs}_{f}(\Phi(x)),\,d^{fs}(z,\Phi(x))<\epsilon^{\prime}\\}.$

We will show that there exists $\epsilon^{\prime}\in(0,\epsilon_{0})$ such
that for every $x\in X$,

$\mathcal{S}_{\epsilon^{\prime}}(x)\subset W^{fs}_{f_{0}}(x).$

This will imply the theorem.

First, we observe the following property of points lying on the same fast
stable leaf for affine actions:

######  Proposition 3.6.

Let $f_{0},g_{0}\in\hbox{\rm Aff}(X)$ be such that
$(Df_{0})|_{W^{min}_{f_{0}}}$ is semisimple. Then there exists $c>0$ such that
for every $z,w\in X$ satisfying $w\in W^{fs}_{f_{0}}(z)$ and $n\geq k\geq 0$,

$d(f_{0}^{-k}g_{0}f_{0}^{n}(z),f_{0}^{-k}g_{0}f_{0}^{n}(w))\leq
c\,\lambda_{0}^{n-k}d^{fs}(z,w)$

where $\lambda_{0}$ is the least absolute value of the eigenvalues of
$Df_{0}$.

###### Proof.

It suffices to prove the proposition when $d^{fs}(z,w)$ small. Write
$w=\exp(v)z$ for $v\in W^{min}_{f_{0}}$. Then

$w=\exp(D(f_{0}^{-k}g_{0}f_{0}^{n})v)f_{0}^{-k}g_{0}f_{0}^{n}(z),$

and it suffices to show that for a norm on $\hbox{Lie}(G)$,

$\|D(f_{0}^{-k}g_{0}f_{0}^{n})v\|\ll\lambda_{0}^{n-k}\|v\|,$

which is easy to check. ∎

A similar but weaker property also holds for small perturbations of affine
actions:

######  Proposition 3.7.

Let $f_{0}\in\hbox{\rm Aff}(X)$, $g\in\hbox{\rm Diff}(X)$, and $\nu>1$. Then
there exists $c>0$ such that for any sufficiently $C^{1}$-small perturbations
$f\in\hbox{\rm Diff}(X)$ of $f_{0}$, $z,w\in X$ satisfying $w\in
W^{fs}_{f}(z)$, and $n\geq 0$,

(7) $d(f^{-n}gf^{n}(z),f^{-n}gf^{n}(w))\leq c\,\nu^{n}d^{fs}(z,w).$

###### Proof.

Let $\lambda_{0}$ denote the least absolute value of the eigenvalues of
$Df_{0}$. Take $\lambda_{-}<\lambda_{0}<\lambda_{+}$ such that
$\frac{\lambda_{+}}{\lambda_{-}}<\nu$. For $f$ sufficiently $C^{1}$-close to
$f_{0}$, we have

$\|D(f^{-n})_{u}\|\ll\lambda_{-}^{-n}\quad\hbox{for all $u\in X$ and $n\geq
0$,}$

and

$\left\|D(f^{n})|_{T_{u}(W^{fs}_{f}(u))}\right\|\ll\lambda_{+}^{n}\quad\hbox{for
all $u\in X$ and $n\geq 0$.}$

This implies that

$\left\|D(f^{-n}gf^{n})|_{T_{u}(W^{fs}_{f}(u))}\right\|\ll\left(\frac{\lambda_{+}}{\lambda_{-}}\right)^{n}\quad\hbox{for
all $u\in X$ and $n\geq 0$.}$

Let $\ell$ be a smooth curve in $W^{fs}_{f}(z)$ from $z$ to $w$ such that
$L(\ell)=d^{fs}(z,w)$. Then

$L(f^{-n}gf^{n}(\ell))\ll\left(\frac{\lambda_{+}}{\lambda_{-}}\right)^{n}L(\ell)<\nu^{n}d^{fs}(z,w)$

for all $n\geq 0$. This proves the proposition. ∎

It turns out that property (7) characterizes points lying on the same fast
stable leaves. This observation is crucial for the proof of Theorem 3.5 and is
the main point of Theorem 3.10 below. Since the proof of Theorem 3.10 is quite
involved, we first present its linear analogue – Proposition 3.8. Although the
argument in the proof of Theorem 3.10 follows the same idea, it requires more
delicate quantitative estimates because we have to work in injectivity
neighborhoods of the exponential map.

Let $A\in\hbox{GL}_{l}(\mathbb{R})$. We denote by
$\lambda_{1}<\cdots<\lambda_{d}$ the absolute values of the eigenvalues of
$A$, and $P_{i}$ denote the projection to the sum of the generalized
eigenspaces of $A$ corresponding to $\lambda_{i}$ along the other eigenspaces.

######  Proposition 3.8.

Let $B_{1},\ldots,B_{k}\in\hbox{\rm GL}_{l}(\mathbb{R})$ be such that for some
$\eta>0$

(8) $\max_{k}\|P_{1}B_{k}v\|>\eta\|v\|\quad\hbox{for all
$v\in\mathbb{R}^{l}$.}$

Then there exists $\nu>1$ such that

$W^{min}_{A}=\\{v:\,\max_{k}\|A^{-n}B_{k}A^{n}\,v\|=O(\nu^{n})\quad\hbox{as
$n\to\infty$}\\}.$

###### Proof.

For every small $\rho>0$ there exists a norm on $\mathbb{R}^{l}$ (see [15,
Proposition 1.2.2]) such that $\|v_{1}+v_{2}\|=\|v_{1}\|+\|v_{2}\|$ for
$v_{1}$ and $v_{2}$ in different generalized eigenspaces and

$\displaystyle(\lambda_{i}-\rho)\|v\|$
$\displaystyle\leq\|Av\|\leq(\lambda_{i}+\rho)\|v\|,\quad
v\in\hbox{im}(P_{i}).$

The parameter $\rho$ is fixed, but has to be chosen sufficiently small so that

$(\lambda_{1}-\rho)^{-1}(\lambda_{1}+\rho)<\min_{i>1}(\lambda_{1}+\rho)^{-1}(\lambda_{i}-\rho).$

It follows from (8) that

$\displaystyle\max_{k}\|A^{-n}B_{k}A^{n}v\|$
$\displaystyle\geq\max_{k}(\lambda_{1}+\rho)^{-n}\|P_{1}B_{k}A^{n}v\|$
$\displaystyle\geq(\lambda_{1}+\rho)^{-n}\eta\|A^{n}v\|$
$\displaystyle\geq(\lambda_{1}+\rho)^{-n}\eta\sum_{i}(\lambda_{i}-\rho)^{n}\|P_{i}v\|.$

We take $\nu>1$ such that

$\nu<(\lambda_{1}+\rho)^{-1}(\lambda_{i}-\rho)$ for $i>1$ and
$\nu>(\lambda_{1}-\rho)^{-1}(\lambda_{1}+\rho)$.

Then $\max_{k}\|A^{-n}B_{k}A^{n}v\|=O(\nu^{n})$ implies that $P_{i}v=0$ for
$i>1$. Also, for $v\in W^{min}_{A}$,

$\displaystyle\max_{k}\|A^{-n}B_{k}A^{n}v\|$
$\displaystyle\leq(\lambda_{1}-\rho)^{-n}\left(\max_{k}\|B_{k}\|\right)(\lambda_{1}+\rho)^{n}=O(\nu^{n}).$

This proves the proposition. ∎

Proposition 3.7 and (4) imply that uniformly on $z,w\in X$, satisfying $w\in
W^{fs}_{f}(z)$ and $d^{fs}(z,w)<\epsilon_{0}$, and $n\geq 0$, we have

$d(f^{-n}gf^{n}(z),f^{-n}gf^{n}(w))\ll\nu^{n}d(z,w).$

Now we take $g=\alpha_{1}(\delta)$ and $g_{0}=\alpha_{0}(\delta)$ for some
$\delta\in\Gamma$. Since the action of $\Gamma$ on $\hbox{Lie}(G)$ is
irreducible, $\alpha_{0}$ is weakly hyperbolic. Hence, by Theorem 3.1, the
conjugacy map $\Phi$ and its inverse are Hölder with some exponent $\theta>0$.
It follows that uniformly on $x,y\in X$, satisfying
$y\in\mathcal{S}_{\epsilon^{\prime}}(x)$, and $n\geq 0$,

(9) $\displaystyle d(f_{0}^{-n}g_{0}f_{0}^{n}(x),f_{0}^{-n}g_{0}f_{0}^{n}(y))$
$\displaystyle\ll d(f^{-n}gf^{n}(\Phi(x)),f^{-n}gf^{n}(\Phi(y)))^{\theta}$
$\displaystyle\ll\nu^{\theta n}d(\Phi(x),\Phi(y))^{\theta}$
$\displaystyle\ll\nu^{\theta n}d(x,y)^{\theta^{2}}.$

Let $\lambda_{1}<\cdots<\lambda_{d}$ be the absolute values of the eigenvalues
of $Df_{0}$ and $P_{i}$ denote the projection from $\hbox{Lie}(G)$ to the sum
of the generalized eigenspaces of $Df_{0}$ corresponding to $\lambda_{i}$
along the other generalized eigenspaces.

We say that a set $\\{g_{1},\ldots,g_{l}\\}\subset\hbox{\rm Aff}(X)$ is
essential for $f_{0}$ if for some $\eta>0$ and every $v\in\hbox{Lie}(G)$,

$\displaystyle\max_{k}\|P_{1}(Dg_{k})v\|>\eta\|v\|.$

Note this definition does not depend on a choice of the norm. Existence of
essential sets follows from the following lemma:

######  Lemma 3.9.

A set $g_{1},\ldots,g_{l}\in\hbox{\rm Aff}(X)$ is essential if and only if

(10) $\bigcap_{k=1}^{l}(Dg_{k})^{-1}\hbox{\rm ker}(P_{1})=0.$

In particular, every subgroup $\Gamma\subset\hbox{\rm Aff}(X)$ such that
$D\Gamma$ acts irreducibly on $\hbox{\rm Lie}(G)$ contains an essential set.

Although the group $\Gamma$ in the Main Theorem needs to be finitely
generated, this assumption is not needed in Lemma 3.9.

###### Proof.

Since the map

$v\mapsto(P_{1}(Dg_{k})v:\,k=1,\ldots l):\hbox{Lie}(G)\to\hbox{Lie}(G)^{l}$

is injective when (10) holds, one can take

$\eta=\min\\{\max_{k}\|P_{1}(Dg_{k})v\|:\|v\|=1\\}>0.$

The converse is also clear.

To prove the second claim, we observe that there exists a subset
$\\{g_{1},\ldots,g_{l}\\}\subset\Gamma$ such that

$\bigcap_{k=1}^{l}(Dg_{k})^{-1}\hbox{ker}(P_{1})=\bigcap_{g\in\Gamma}(Dg)^{-1}\hbox{ker}(P_{1}),$

and this space is zero by irreducibility. ∎

The following theorem is the main ingredient of the proof of Theorem 3.5:

######  Theorem 3.10.

There exists $\nu=\nu(\vartheta,f_{0})>1$ such that given constants
$a,\vartheta>0$, a map $f_{0}\in\hbox{\rm Aff}(X)$ such that $Df_{0}$ is
semisimple on $W^{min}_{f_{0}}$, an essential set
$g_{1},\ldots,g_{l}\in\alpha_{0}(\Gamma)$, and a family of subsets
$\mathcal{L}_{\epsilon}(x)$, $x\in X$, of $X$ that satisfy

  1. (i)

$x\in\mathcal{L}_{\epsilon}(x)\subset B_{\epsilon}(x)$,

  2. (ii)

$f_{0}^{-1}(\mathcal{L}_{\epsilon}(x))\supset\mathcal{L}_{\epsilon}(f_{0}^{-1}(x))$,

  3. (iii)

for every $y\in\mathcal{L}_{\epsilon}(x)$ and $n\geq 0$,

(11) $\max_{k}d(f_{0}^{-n}g_{k}f_{0}^{n}(x),f_{0}^{-n}g_{k}f_{0}^{n}(y))\leq
a\nu^{n}d(x,y)^{\vartheta},$

one can choose $\epsilon>0$ such that

$\mathcal{L}_{\epsilon}(x)\subset W^{fs}_{f_{0}}(x)\quad\hbox{for every $x\in
X$}.$

###### Outline of the proof of Theorem 3.10.

We first observe that the sets $\mathcal{L}_{\epsilon}(x)$ lie in “cones”
around $W^{fs}_{f_{0}}(x)$ where the size of the cones is controlled by $\nu$
and can be made sufficiently small (Lemma 3.11). Note that this argument is
analogous to the proof of Proposition 3.8, but we can only derive a weaker
conclusion because one has to work in injectivity neighborhoods of the
exponential map. In the next step, we show that applying the map $f_{0}^{-1}$,
the size of the cones can be made arbitrary small (Lemma 3.12). This implies
the theorem. ∎

We fix a norm on $\hbox{Lie}(G)$, depending on parameter $\rho>0$, as in the
proof of Proposition 3.8 with $A=Df_{0}$. The parameter $\rho$ has to be
chosen sufficiently small. It controls the size of the cone in Lemma 3.11. We
always take $\rho>0$ so that

$\displaystyle\lambda_{i}<\lambda_{j}-\rho\quad\hbox{when
$\lambda_{i}<\lambda_{j}$,}$ $\displaystyle\lambda_{i}-\rho>1\quad\hbox{when
$\lambda_{i}>1$,}$ $\displaystyle\lambda_{i}+\rho<1\quad\hbox{when
$\lambda_{i}<1$.}$

Note that since $(Df_{0})|_{W^{min}_{f_{0}}}$ is semisimple, we also have

$\displaystyle\|(Df_{0})v\|=\lambda_{1}\|v\|,\quad v\in\hbox{im}(P_{1}),$

and

$\|(Df_{0})^{-n}\|\leq\lambda_{1}^{-n}.$

By the assumption on $g_{k}$’s, there exists $\eta>0$ such that

(12) $\displaystyle\max_{k}\|P_{1}(Dg_{k})v\|>\eta\|v\|,\quad
v\in\hbox{Lie}(G).$

Let $\mu_{i}=\lambda_{1}^{-1}(\lambda_{i}+\rho)$ and
$\sigma_{i}=\frac{\log\mu_{i}}{\log\mu_{d}}$. For $v\in\hbox{Lie}(G)$, we
define

$N(v)=\max_{i>1}\left\\{\|P_{i}v\|^{\sigma_{i}^{-1}}\right\\}.$

For $\beta,s>0$, we define

$C(\beta,s)=\\{v\in\hbox{Lie}(G):\,N(v)\leq\beta\|v\|^{s}\\}.$

######  Lemma 3.11.

There exist $\epsilon,\beta>0$ such that for every $x,y\in X$ satisfying
$d(x,y)<\epsilon$ and (11),

$y\in\exp(C(\beta,s))x.$

where $s=s(\nu,\rho,\vartheta,f_{0})>0$ is such that $s\to\infty$ as $\nu\to
1^{+}$ and $\rho\to 0^{+}$.

###### Proof.

Let $c_{1}=\max_{k}\|Dg_{k}\|$.

There exist $\delta_{0}>0$ and $c_{0}>1$ such that for every $x\in X$ and
$v\in\hbox{Lie}(G)$ satisfying $\|v\|<\delta_{0}$, we have

(13) $c_{0}^{-1}\|v\|\leq d(x,\exp(v)x)\leq c_{0}\|v\|.$

Let $b>0$ such that $\sum_{j>1}b^{\sigma_{j}}=\delta_{0}/(2c_{1})$. We choose
$\epsilon>0$ so that $d(x,y)<\epsilon$ implies that $y=\exp(v)x$ where

$N(v)<\min\\{1,b\\}\quad\hbox{and}\quad\|v\|<\min\left\\{\delta_{0},\delta_{0}/2c_{1}\right\\}.$

Assuming that the claim fails, we will show that there exists $n\geq 0$ such
that

(14) $\displaystyle
ac_{0}^{\vartheta+1}\nu^{n}\|v\|^{\vartheta}<\max_{k}\|D(f_{0}^{-n}g_{k}f_{0}^{n})v\|<\delta_{0}.$

Since

$d(f_{0}^{-n}g_{k}f_{0}^{n}(x),f_{0}^{-n}g_{k}f_{0}^{n}(y))=d(f_{0}^{-n}g_{k}f_{0}^{n}(x),\exp(D(f_{0}^{-n}g_{k}f_{0}^{n})v)f_{0}^{-n}g_{k}f_{0}^{n}(x)),$

we deduce from (13) and (14) that

$ac_{0}^{\vartheta+1}\nu^{n}(c_{0}^{-1}\,d(x,y))^{\vartheta}<\max_{k}c_{0}\,d(f_{0}^{-n}g_{k}f_{0}^{n}(x),f_{0}^{-n}g_{k}f_{0}^{n}(y)),$

which contradicts (11).

To obtain the upper estimate in (14), we observe that

$\displaystyle\max_{k}\|D(f_{0}^{-n}g_{k}f_{0}^{n})v\|$
$\displaystyle\leq\lambda_{1}^{-n}\max_{k}\|D(g_{k}f_{0}^{n})v\|\leq\lambda_{1}^{-n}c_{1}\|D(f_{0}^{n})v\|$
$\displaystyle\leq
c_{1}\|P_{1}v\|+\lambda_{1}^{-n}c_{1}\sum_{j>1}(\lambda_{j}+\rho)^{n}\|P_{j}v\|$
$\displaystyle\leq c_{1}\|v\|+c_{1}\sum_{j>1}\mu_{j}^{n}\|P_{j}v\|.$

We choose $n\geq 0$ so that

(15) $\mu_{d}^{-1}\frac{b}{N(v)}<\mu_{d}^{n}\leq\frac{b}{N(v)}.$

Then

$\mu_{d}^{-\sigma_{j}}\frac{b^{\sigma_{j}}}{N(v)^{\sigma_{j}}}<\mu_{j}^{n}\leq\frac{b^{\sigma_{j}}}{N(v)^{\sigma_{j}}}$

and

$\displaystyle\max_{k}\|D(f_{0}^{-n}g_{k}f_{0}^{n})v\|\leq
c_{1}\|v\|+c_{1}\sum_{j>1}b^{\sigma_{j}}\frac{\|P_{j}v\|}{N(v)^{\sigma_{j}}}<\delta_{0}.$

The lower estimate in (14) is proved similarly using that $g_{1},\ldots,g_{l}$
is essential (see (12)). Let $\gamma_{j}>0$ be such that
$\lambda_{1}^{-1}(\lambda_{j}-\rho)=\mu_{j}^{1-\gamma_{j}}$. We have

$\displaystyle\max_{k}\|D(f_{0}^{-n}g_{k}f_{0}^{n})v\|$
$\displaystyle\geq\max_{k}\lambda_{1}^{-n}\|P_{1}D(g_{k}f_{0}^{n})v\|\geq\lambda_{1}^{-n}\eta\|D(f_{0}^{n})v\|$
$\displaystyle\geq\lambda_{1}^{-n}\eta\left(\lambda_{1}^{n}\|P_{1}v\|+\sum_{j>1}(\lambda_{j}-\rho)^{n}\|P_{j}v\|\right)$
$\displaystyle\geq\eta\sum_{j>1}\mu_{j}^{n(1-\gamma_{j})}\left\|P_{j}v\right\|$
$\displaystyle\geq\eta\sum_{j>1}(\mu_{d}^{-1}b)^{\sigma_{j}(1-\gamma_{j})}N(v)^{\sigma_{j}\gamma_{j}}\frac{\|P_{j}v\|}{N(v)^{\sigma_{j}}}$
$\displaystyle\geq\eta(\mu_{d}^{-1}b)^{\sigma_{j_{0}}(1-\gamma_{j_{0}})}N(v)^{\sigma_{j_{0}}\gamma_{j_{0}}}.$

where $j_{0}>1$ is such that $\|P_{j_{0}}v\|^{1/\sigma_{j_{0}}}=N(v)$. This
implies that

$\max_{k}\|D(f_{0}^{-n}g_{k}f_{0}^{n})v\|\geq\min_{j>1}\eta(\mu_{d}^{-1}b)^{\sigma_{j}(1-\gamma_{j})}N(v)^{\sigma_{j}\gamma_{j}}.$

Let $\omega=\frac{\log\nu}{\log\mu_{d}}$. It follows from (15) that the first
inequality in (14) is satisfied provided that

$ac_{0}^{\vartheta+1}N(v)^{-\omega}b^{\omega}\|v\|^{\vartheta}<\min_{j>1}\eta(\mu_{d}^{-1}b)^{\sigma_{j}(1-\gamma_{j})}N(v)^{\sigma_{j}\gamma_{j}}.$

Since this gives a contradiction, we deduce that

$ac_{0}^{\vartheta+1}b^{\omega}\|v\|^{\vartheta}\geq\min_{j>1}\eta(\mu_{d}^{-1}b)^{\sigma_{j}(1-\gamma_{j})}N(v)^{\omega+\sigma_{j}\gamma_{j}}.$

Hence,

$N(v)\leq\beta\|v\|^{s}$

with explicit $\beta>0$ and
$s=\vartheta/(\omega+\max_{j>1}(\sigma_{j}\gamma_{j}))$. Clearly, $s\to\infty$
as $\nu\to 1^{+}$ and $\rho\to 0^{+}$. This completes the proof. ∎

For $i=1,\ldots,d$ and $\delta,\beta,s>0$, we define

$C^{i}_{\delta}(\beta,s)=\\{v\in\hbox{Lie}(G):\,\|v\|<\delta;\,\|P_{i}v\|^{\sigma_{i}^{-1}}\leq\beta\|v\|^{s};\,P_{j}v=0,j>i\\}.$

######  Lemma 3.12.

For every $\delta,\beta,s>0$,

$(Df_{0})^{-1}(C^{i}_{\delta}(\beta,s))\subset
C^{i}_{\xi\delta}(\rho_{i}\beta,s).$

where $\xi=\max\\{1,\|(Df_{0})^{-1}\|\\}$ and
$\rho_{i}=(\lambda_{i}-\rho)^{-{\sigma_{i}^{-1}}}(\lambda_{i}+\rho)^{s}$.

###### Proof.

Let $v\in(Df_{0})^{-1}(C^{i}_{\delta}(\beta,s))$. Then

$\displaystyle(\lambda_{i}-\rho)^{\sigma_{i}^{-1}}\|P_{i}v\|^{\sigma_{i}^{-1}}\leq\beta\left(\sum_{j\leq
i}(\lambda_{j}+\rho)\|P_{j}v\|\right)^{s}\leq\beta(\lambda_{i}+\rho)^{s}\|v\|^{s}.$

This implies the lemma. ∎

###### Proof of Theorem 3.10.

We start by setting up notation for the Jordan form of $Df_{0}$ for
$\lambda_{i}=1$. It follows from our choice of the norm that there exist
linear maps $Q_{1},\ldots,Q_{j_{0}}$ such that

(16)
$\|(Df_{0}^{k})v\|=\left\|\sum_{j=0}^{j_{0}}k^{j}Q_{j}v\right\|\quad\hbox{for
$k\geq 0$ and $v\in\hbox{im}(P_{i})$.}$

Let $s>0$ be as in Lemma 3.11. Recall that $s\to\infty$ as $\nu\to 1^{+}$ and
$\rho\to 0^{+}$. We choose $\rho>0$ and $\nu>1$ so that

$\displaystyle s-\sigma_{i}^{-1}>0\quad\hbox{when $\lambda_{i}=1$},$
$\displaystyle\rho_{i}:=(\lambda_{i}-\rho)^{-{\sigma_{i}^{-1}}}(\lambda_{i}+\rho)^{s}<1\quad\quad\hbox{when
$\lambda_{i}<1$.}$

Let $\xi\geq 1$ be as in Lemma 3.12 and $\beta,\epsilon>0$ as in Lemma 3.11.
Take $\delta\in(0,1)$ such that for $\|v\|<\xi\delta$, the exponential
coordinates $v\mapsto\exp(v)z$, $z\in X$, are one-to-one, and

(17) $\|Q_{j}P_{i}v\|<\beta^{-(s-\sigma_{i}^{-1})^{-1}}$ when $\lambda_{i}=1$
and $j=0,\ldots,j_{0}$.

In addition, we assume that $\epsilon$ is sufficiently small so that

$B_{\epsilon}(x)\subset\exp(\\{\|v\|<\delta\\})x\quad\hbox{for all $x\in X$.}$

Then by Lemma 3.11,

(18)
$\mathcal{L}_{\epsilon}(x)\subset\exp(C(\beta,s)\cap\\{\|v\|<\delta\\})x\quad\hbox{for
every $x\in X$.}$

In particular,

(19) $\mathcal{L}_{\epsilon}(x)\subset\exp(C^{d}_{\delta}(\beta,s))x.$

If $\lambda_{d}\leq 1$, we argue as in the following paragraph. Otherwise, we
observe that since $\delta<1$, we have

$C^{d}_{\delta}(\beta,s_{1})\subset C^{d}_{\delta}(\beta,s_{2})\quad\hbox{for
$s_{1}>s_{2}$},$

and hence inclusion (18) also holds for $s>0$ such that
$\rho_{d}=(\lambda_{d}-\rho)^{-{\sigma_{d}^{-1}}}(\lambda_{d}+\rho)^{s}<1$.
Applying $f_{0}^{-1}$ to (19), we deduce from Lemma 3.12 that

(20)
$\mathcal{L}_{\epsilon}(x)\subset\exp(C^{d}_{\xi\delta}(\rho_{d}\beta,s))x$

for every $x\in X$. Using that the exponential coordinates are one-to-one, we
obtain from (20) and (18) that

$\mathcal{L}_{\epsilon}(x)\subset\exp(C^{d}_{\delta}(\rho_{d}\beta,s))x.$

Repeating this argument, we conclude that

$\mathcal{L}_{\epsilon}(x)\subset\bigcap_{k\geq
1}\exp(C^{d}_{\delta}(\rho^{k}_{d}\beta,s))x=\exp(C^{d}_{\delta}(0,s))x.$

Now (18) implies that

$\mathcal{L}_{\epsilon}(x)\subset\exp(C^{d-1}_{\delta}(\beta,s))x.$

Applying the same reasoning inductively on $i$, we deduce that

$\mathcal{L}_{\epsilon}(x)\subset\exp(C^{i}_{\delta}(0,s))x$

provided that $\lambda_{i}>1$. It follows from (18) that
$\mathcal{L}_{\epsilon}(x)\subset\exp(C^{i-1}_{\delta}(\beta,s))x$.

Suppose $\lambda_{i}=1$ and
$\mathcal{L}_{\epsilon}(x)\subset\exp(C^{i}_{\delta}(\beta,s))x$ for some
$\beta>0$. We will show that

$\mathcal{L}_{\epsilon}(x)\subset\exp(C^{i}_{\delta}(0,s))x.$

Applying $f_{0}^{-1}$, we deduce that for
$y=\exp(v)x\in\mathcal{L}_{\epsilon}(x)$, $\|v\|<\delta$, and $k\geq 0$, we
have

$\|(Df_{0}^{k})P_{i}v\|^{\sigma_{i}^{-1}}\leq\beta\left(\sum_{j<i}(\lambda_{j}+\rho)^{k}\|P_{j}v\|+\|(Df_{0}^{k})P_{i}v\|\right)^{s}.$

Using that $\lambda_{j}+\rho<1$ for $j<i$ and taking $k\to\infty$, we deduce
from (16) that

$\|Q_{j_{0}}P_{i}v\|^{\sigma_{i}^{-1}}\leq\beta\|Q_{j_{0}}P_{i}v\|^{s}.$

By the choice of $\delta$ (see (17)), $\|Q_{j_{0}}P_{i}v\|=0$. Similar
arguments imply that $\|Q_{j}P_{i}v\|=0$ for all $j=0,\ldots,j_{0}$. Hence,
$P_{i}v=0$ and $\mathcal{L}_{\epsilon}(x)\subset\exp(C^{i}_{\delta}(0,s))x$.
Combining this estimate with (18), we deduce that
$\mathcal{L}_{\epsilon}(x)\subset\exp(C^{i-1}_{\delta}(\beta,s))x$.

Now we consider the case when
$\mathcal{L}_{\epsilon}(x)\subset\exp(C^{i}_{\delta}(\beta,s))$ for some $i$
such that $\lambda_{i}<1$ and $\beta>0$. Applying $f_{0}^{-1}$, it follows
from Lemma 3.12 that

$\mathcal{L}_{\epsilon}(x)\subset\exp(C^{i}_{\xi\delta}(\rho_{i}\beta,s))x\quad\hbox{for
every $x\in X$.}$

Then it follows from (18) that

$\mathcal{L}_{\epsilon}(x)\subset\exp(C^{i}_{\delta}(\rho_{i}\beta,s))x,$

and repeating this argument, we deduce that

$\mathcal{L}_{\epsilon}(x)\subset\bigcap_{k\geq
1}\exp(C^{i}_{\delta}(\rho^{k}_{i}\beta,s))x=\exp(C^{i}_{\delta}(0,s))x.$

Since the above argument can be applied inductively on $i$, and we conclude
that $\mathcal{L}_{\epsilon}(x)\subset\exp(C^{2}_{\delta}(0,s))x$. This
completes the proof. ∎

###### Proof of Theorem 3.5.

The first claim of Theorem 3.5 follows from Theorem 3.10 with
$\mathcal{L}_{\epsilon}(x)=\mathcal{S}_{\epsilon^{\prime}}(x)$ where
$\mathcal{S}_{\epsilon^{\prime}}(x)$ is as in (6) with sufficiently small
$\epsilon^{\prime}>0$. Note that $\alpha_{0}(\Gamma)$ contains an essential
subset by Lemma 3.9, and (11) follows from (9) where the parameter $\nu$ is
close to one if $f$ and $f_{0}$ are $C^{1}$-close.

It remains to show that $\Phi$ is bi-Hölder with respect to the metrics
$d^{fs}$. There exists $\epsilon>0$ such that for every $x\in X$, any points
$z,w\in X$ lying on the same local leaf of $W^{fs}_{f}$ in $B(x,\epsilon)$
satisfy (4). Let $\delta>0$ be such that $\Phi(B_{\delta}(y))\subset
B_{\epsilon}(\Phi(y))$ for every $y\in X$. Consider points $z_{0},w_{0}\in X$
lying on the same leaf of $W^{fs}_{f_{0}}$ such that
$d^{fs}(z_{0},w_{0})<\delta$. Let $\ell$ be a curve from $z_{0}$ to $w_{0}$
contained in $W^{fs}_{f_{0}}(z_{0})$ such that $L(\ell)=d^{fs}(z_{0},w_{0})$.
Then $\Phi(\ell)$ is contained in $B_{\epsilon}(\Phi(z_{0}))\cap
W^{fs}_{f}(\Phi(z_{0}))$. Moreover, since $\Phi(\ell)$ is connected,
$\Phi(\ell)$ is contained in a single local leaf of $W^{fs}_{f}$ in
$B_{\epsilon}(\Phi(z_{0}))$. Hence,

$d^{fs}(\Phi(z_{0}),\Phi(w_{0}))\ll d(\Phi(z_{0}),\Phi(w_{0})).$

Since $\Phi$ is Hölder with respect to $d$, this implies that $\Phi$ is Hölder
with respect to $d^{fs}$ as well. The proof that $\Phi^{-1}$ is Hölder with
respect to $d^{fs}$ is similar. ∎

###  3.3. Convergence of the sequences $f^{-n}gf^{n}$

In this section, we study convergence of the sequence of maps $f^{-n}gf^{n}$
as $n\to\infty$.

First, we consider the algebraic setting:

######  Proposition 3.13.

Let $f_{0},g_{0}\in\hbox{\rm Aff}(X)$ be such that $Df_{0}:W_{f_{0}}^{min}\to
W_{f_{0}}^{min}$ is semisimple. Then

  1. (1)

Given a sequence $\\{m_{i}\\}$ such that

$(f_{0}^{-m_{i}}g_{0}f_{0}^{m_{i}})(x)\to y\quad\hbox{as $i\to\infty$}$

for some $x,y\in X$, the sequence of maps
$f_{0}^{-m_{i}}g_{0}f_{0}^{m_{i}}:{W^{fs}_{f_{0}}(x)}\to X$ is precompact in
the $C^{0}$-topology.

  2. (2)

There exist a sequence $\\{n_{i}\\}$ and a linear map $A:W_{f_{0}}^{min}\to
W_{f_{0}}^{min}$ such that if for some $x,y\in X$ and a subsequence
$\\{n_{i_{j}}\\}$,

$(f_{0}^{-n_{i_{j}}}g_{0}f_{0}^{n_{i_{j}}})(x)\to y\quad\hbox{as
$j\to\infty$,}$

then uniformly on $v\in W^{min}_{f}$ in compact sets,

$(f_{0}^{-n_{i_{j}}}g_{0}f_{0}^{n_{i_{j}}})\exp(v)x\to\exp(Av)y\quad\hbox{as
$j\to\infty$}.$

The map $A$ is nondegenerate provided that
$P_{f_{0}}^{min}Dg_{0}:W_{f_{0}}^{min}\to W_{f_{0}}^{min}$ is nondegenerate.

######  Remark 3.14.

If $\dim W_{f_{0}}^{min}=1$, one can take $n_{i}=i$ and
$A=P_{f_{0}}^{min}Dg_{0}$. In general,
$A=\lim_{i\to\infty}\omega^{-n_{i}}P^{min}_{f_{0}}(Dg_{0})\omega^{n_{i}}$ for
some $\omega\in\hbox{Isom}(W^{min}_{f_{0}})$.

###### Proof.

We have

$(f_{0}^{-n}g_{0}f_{0}^{n})\exp(v)x=\exp(D(f_{0}^{-n}g_{0}f_{0}^{n})v)(f_{0}^{-n}g_{0}f_{0}^{n})x.$

It follows from the assumption on $f_{0}$ that

$Df_{0}|_{W_{f_{0}}^{min}}=\lambda\cdot\omega$

where $\lambda>0$ and $\omega$ is an isometry of $W_{f_{0}}^{min}$. Then

$\displaystyle
D(f_{0}^{-n}g_{0}f_{0}^{n})v=(\omega^{-n}P^{min}_{f_{0}}(Dg_{0})\omega^{n})v+(Df_{0})^{-n}P^{max}_{f_{0}}(Dg_{0})\lambda^{n}\omega^{n}v$

where $P^{min}_{f_{0}}$ denotes the projection on $W^{min}_{f_{0}}$ and
$P^{max}_{f_{0}}$ denotes the projection on the sum of eigenspaces
complimentary to $W^{min}_{f_{0}}$. Since $\omega$ is an isometry, and

$(Df_{0})^{-n}P^{max}_{f_{0}}(Dg_{0})\lambda^{n}\omega^{n}v\to 0,$

it is clear that the sequence of maps $v\mapsto D(f_{0}^{-n}g_{0}f_{0}^{n})v$
is precompact in $C^{0}$-topology. This implies that the sequence
$f_{0}^{-m_{i}}g_{0}f_{0}^{m_{i}}|_{W^{fs}_{f_{0}}(x)}$ is precompact in
$C^{0}$-topology as well.

To prove (2), it suffices to choose the sequence $\\{n_{i}\\}$ so that
$\\{\omega^{n_{i}}\\}$ converges. This proves the proposition. ∎

We show that the convergence of
$f_{0}^{-n}g_{0}f_{0}^{n}|_{W^{fs}_{f_{0}}(x)}$ persists under small
perturbations:

######  Theorem 3.15.

Let $f_{0},g_{0}\in\hbox{\rm Aff}(X)$ satisfy

  1. (i)

The map $f_{0}$ is partially hyperbolic,

  2. (ii)

The map $Df_{0}:W^{min}_{f_{0}}\to W^{min}_{f_{0}}$ is semisimple.

Let $f,g\in\hbox{\rm Diff}(X)$ be $C^{1}$-small perturbations of $f_{0}$ and
$g_{0}$ and $\Phi:X\to X$ a Hölder isomorphism such that

$\Phi\circ f_{0}=f\circ\Phi\quad\hbox{and}\quad\Phi\circ g_{0}=g\circ\Phi$

and

$\Phi(W^{fs}_{f_{0}}(x))=W^{fs}_{f}(\Phi(x))\quad\hbox{for every $x\in X$.}$

Then for every $x\in X$ and a sequence $\\{m_{i}\\}$ as in Proposition
3.13(1), the sequence of maps

$f^{-m_{i}}gf^{m_{i}}:W^{fs}_{f}(x)\to X,\quad i\geq 0,$

is precompact in the $C^{1}$-topology.

Throughout this section, we assume that $X$ is a submanifold of
$\mathbb{R}^{N}$, which allows us to identify tangent spaces at different
points.

We have a Hölder continuous decomposition (cf. (2))

(21) $T_{x}X=E^{-}_{x}\oplus E^{+}_{x},\quad x\in X,$

where $E^{-}_{x}=T_{x}W^{fs}_{f}(x)$. Let

$P_{x}:T_{x}X\to E^{-}_{x}\quad\hbox{and}\quad P^{+}_{x}:T_{x}X\to E^{+}_{x}$

denote the corresponding projections.

The following proposition is the main ingredient of the proof of Theorem 3.15.

######  Proposition 3.16.

Let $r>0$. Then under the assumptions of Theorem 3.15, for every $x,y\in X$
satisfying $y\in W^{fs}_{f}(x)$ and $d^{fs}(x,y)\leq r$,

$\|D(f^{-n}gf^{n})_{x}P_{x}-D(f^{-n}gf^{n})_{y}P_{y}\|\ll
d^{fs}(x,y)^{\vartheta}\|D(f^{-n}gf^{n})_{x}P_{x}\|+\delta_{n}$

where $\vartheta>0$ and $\delta_{n}\to 0$.

###### Proof.

Note that $\Phi$ and $\Phi^{-1}$ are also Hölder with respect to the metrics
$d^{fs}$ on the fast stable leaves of $f_{0}$ and $f$ (see proof of Theorem
3.5). By Proposition 3.6,

$\displaystyle
d(f_{0}^{-k}g_{0}f_{0}^{n}(\Phi^{-1}(x)),f_{0}^{-k}g_{0}f_{0}^{n}(\Phi^{-1}(y)))$
$\displaystyle\ll\lambda_{0}^{n-k}d^{fs}(\Phi^{-1}(x),\Phi^{-1}(y))$
$\displaystyle\ll\lambda_{0}^{n-k}d^{fs}(x,y)^{\omega_{0}}$

where $\omega_{0}>0$ is the Hölder exponent of $\Phi^{-1}$ with respect to
$d^{fs}$. Then it follows that we have the estimate

(22)
$d(f^{-k}gf^{n}(x),f^{-k}gf^{n}(y))\ll\lambda_{0}^{\omega(n-k)}d^{fs}(x,y)^{\omega_{0}\omega}$

where $\omega>0$ is the Hölder exponent of $\Phi$ with respect to $d$.

Since the decomposition (21) is $f$-invariant, we have

$P_{f(x)}D(f)_{x}P_{x}=D(f)_{x}P_{x}\quad\hbox{and}\quad
P_{f^{-1}(x)}D(f^{-1})_{x}P_{x}=D(f^{-1})_{x}P_{x}.$

By (3), there exist $\lambda\in(0,1)$ and $\mu>\lambda$ such that

(23)
$\|D(f^{n})_{x}P_{x}\|\ll\lambda^{n}\quad\hbox{and}\quad\|D(f^{-n})_{x}P^{+}_{x}\|\ll\mu^{-n}$

uniformly on $x\in X$ and $n\geq 0$. It is crucial for the proof that the map
$D(f)_{x}P_{x}$ is approximately conformal (cf. assumption (ii) on $f_{0}$).
Namely, for some small $\epsilon>0$,

(24) $\|D(f^{-n})_{x}P_{x}\|\ll(\lambda-\epsilon)^{-n}$

uniformly on $x\in X$ and $n\geq 0$. We also recall that for $\rho>\lambda$
and $x,y\in X$ such that $y\in W^{fs}_{f}(x)$,

(25) $d^{fs}(f^{n}(x),f^{n}(y))\ll\rho^{n}d^{fs}(x,y).$

Note that the parameter $\epsilon$ in (24) satisfies $\epsilon\to 0$ as
$d_{C^{1}}(f_{0},f)\to 0$. We assume $f$ is sufficiently close to $f_{0}$ so
that

$\zeta:=(\lambda-\epsilon)^{-1}\lambda\rho^{\theta}<1\quad\hbox{and}\quad\nu:=(\lambda-\epsilon)^{-1}\lambda\lambda_{0}^{\omega}<1$

where $\theta$ is the Hölder exponent of the map $x\mapsto P_{x}$.

We have

$\displaystyle D(f^{-n}gf^{n})_{x}P_{x}=$ $\displaystyle
D(f^{-n})_{gf^{n}(x)}P_{gf^{n}(x)}D(g)_{f^{n}(x)}D(f^{n})_{x}P_{x}$
$\displaystyle+D(f^{-n})_{gf^{n}(x)}P^{+}_{gf^{n}(x)}D(g)_{f^{n}(x)}D(f^{n})_{x}P_{x}.$

It follows from (23) that

$\|D(f^{-n})_{gf^{n}(x)}P^{+}_{gf^{n}(x)}D(g)_{f^{n}(x)}D(f^{n})_{x}P_{x}\|\ll\lambda^{n}\mu^{-n}\to
0.$

Hence, to prove the theorem, it suffices to show that for

$A_{n}(x):=\left(\prod_{i=n-1}^{0}D(f^{-1})_{f^{-i}gf^{n}(x)}\right)P_{gf^{n}(x)}D(g)_{f^{n}(x)}\left(\prod_{i=n-1}^{0}D(f)_{f^{i}(x)}\right)P_{x},$

we have

$\|A_{n}(x)-A_{n}(y)\|\ll d^{fs}(x,y)^{\kappa}\|A_{n}(x)\|.$

We consider the operators

$\displaystyle A_{n,k}(x,y):=$
$\displaystyle\left(\prod_{i=n-1}^{0}D(f^{-1})_{f^{-i}gf^{n}(x)}\right)P_{gf^{n}(x)}D(g)_{f^{n}(x)}$
$\displaystyle\times\left(\prod_{i=n-1}^{k+1}D(f)_{f^{i}(x)}\right)P_{f^{k+1}(x)}\left(\prod_{i=k}^{0}D(f)_{f^{i}(y)}\right)P_{y}.$

Note that

(26)
$\displaystyle\|A_{n}(x)-A_{n,-1}(x,y)\|\leq\|A_{n}(x)\|\cdot\|P_{x}-P_{x}P_{y}\|\ll\|A_{n}(x)\|d(x,y)^{\theta}.$

Now we estimate $\|A_{n,n-1}(x,y)-A_{n,-1}(x,y)\|$. We use that

$\displaystyle A_{n,k}(x,y)-A_{n,k-1}(x,y)=A_{n}(x)B_{n,k}(x,y)$

where

$\displaystyle B_{n,k}(x,y):=$
$\displaystyle\left(\prod_{i=0}^{k}D(f)^{-1}_{f^{i}(x)}\right)P_{f^{k+1}(x)}\left(D(f)_{f^{k}(y)}P_{f^{k}(y)}-D(f)_{f^{k}(x)}P_{f^{k}(x)}\right)$
$\displaystyle\times\left(\prod_{i=k-1}^{0}D(f)_{f^{i}(y)}\right)P_{y}.$

By (25), we have

$\|D(f)_{f^{k}(y)}P_{f^{k}(y)}-D(f)_{f^{k}(x)}P_{f^{k}(x)}\|\ll
d(f^{k}(x),f^{k}(y))^{\theta}\ll\rho^{\theta k}d^{fs}(x,y)^{\theta},$

and by (23) and (24),

$\displaystyle\left\|\left(\prod_{i=k-1}^{0}D(f)_{f^{i}(y)}\right)P_{y}\right\|$
$\displaystyle\ll\lambda^{k},$
$\displaystyle\left\|\left(\prod_{i=0}^{k}D(f)^{-1}_{f^{i}(x)}\right)P_{f^{k+1}(x)}\right\|$
$\displaystyle\ll(\lambda-\epsilon)^{-k-1}.$

Hence,

$\|B_{n,k}(x,y)\|\ll\zeta^{k}d^{fs}(x,y)^{\theta}$

Since $\zeta<1$, it follows that

(27) $\displaystyle\|A_{n,n-1}(x,y)-A_{n,-1}(x,y)\|$
$\displaystyle\leq\sum_{k=0}^{n-1}\|A_{n,k}(x,y)-A_{n,k-1}(x,y)\|$
$\displaystyle\ll\|A_{n}(x)\|d^{fs}(x,y)^{\theta}.$

We claim that for some $c>0$ and all $k=-1,\ldots,n-1$,

(28) $\|A_{n,k}(x,y)\|\ll(1+c\,d^{fs}(x,y)^{\theta})\cdot\|A_{n}(x)\|.$

Setting

$C_{k}(x,y):=\left(\prod_{i=0}^{k}D(f)^{-1}_{f^{i}(x)}\right)P_{f^{k+1}(x)}\left(\prod_{i=k}^{0}D(f)_{f^{i}(y)}\right)P_{y},$

we have

$A_{n,k}(x,y)=A_{n}(x)C_{k}(x,y).$

Now equation (28) will follow from the estimate

$\|C_{k}(x,y)\|\ll 1+c\,d^{fs}(x,y)^{\theta}.$

In fact, we will show that

(29) $\|C_{k}(x,y)-P_{x}P_{y}\|\ll d^{fs}(x,y)^{\theta}.$

Using (23) and (24), we deduce that

$\displaystyle\|C_{k}(x,y)-C_{k-1}(x,y)\|$ $\displaystyle=$
$\displaystyle\left\|\left(\prod_{i=0}^{k-1}D(f)^{-1}_{f^{i}(x)}\right)P_{f^{k}(x)}\left(D(f)^{-1}_{f^{k}(x)}D(f)_{f^{k}(y)}-id\right)\right.$
$\displaystyle\quad\quad\left.\times\left(\prod_{i=k-1}^{0}D(f)_{f^{i}(y)}\right)P_{y}\right\|$
$\displaystyle\ll$
$\displaystyle(\lambda-\epsilon)^{-k}d(f^{k}(x),f^{k}(y))^{\theta}\lambda^{k}\ll\zeta^{k}d^{fs}(x,y)^{\theta}.$

Since $C_{-1}(x,y)=P_{x}P_{y}$ and $\zeta<1$, the last estimate implies (29)
and (28).

Next, we consider the operators

$\displaystyle D_{n,k}(x,y):=$
$\displaystyle\left(\prod_{i=n-1}^{k}D(f^{-1})_{f^{-i}gf^{n}(y)}\right)P_{f^{-k}gf^{n}(y)}\left(\prod_{i=k-1}^{0}D(f^{-1})_{f^{-i}gf^{n}(x)}\right)$
$\displaystyle\times
P_{gf^{n}(x)}D(g)_{f^{n}(x)}P_{f^{n}(x)}\left(\prod_{i=n-1}^{0}D(f)_{f^{i}(y)}\right)P_{y}.$

Using (22), we deduce that

$\displaystyle\|A_{n,n-1}(x,y)-D_{n,n}(x,y)\|$
$\displaystyle\leq\|P_{f^{-n}gf^{n}(x)}-P_{f^{-n}gf^{n}(y)}P_{f^{-n}gf^{n}(x)}\|\cdot\|A_{n,n-1}(x,y)\|$
$\displaystyle\ll
d(f^{-n}gf^{n}(x),f^{-n}gf^{n}(y))^{\theta}\|A_{n,n-1}(x,y)\|$
$\displaystyle\ll d^{fs}(x,y)^{\theta\omega_{0}\omega}\|A_{n,n-1}(x,y)\|$
$\displaystyle\ll d^{fs}(x,y)^{\theta\omega_{0}\omega}\|A_{n}(x)\|.$

To estimate $\|D_{n,n}(x,y)-D_{n,0}(x,y)\|$, we use the argument similar to
the proof of (27). We have

$D_{n,k}(x,y)-D_{n,k-1}(x,y)=E_{n,k}(x,y)A_{n,n-1}(x,y)$

where

$\displaystyle E_{n,k}(x,y):=$
$\displaystyle\left(\prod_{i=n-1}^{k}D(f^{-1})_{f^{-i}gf^{n}(y)}\right)P_{f^{-k}gf^{n}(y)}$
$\displaystyle\times\left(D(f^{-1})_{f^{-(k-1)}gf^{n}(x)}P_{f^{-(k-1)}gf^{n}(x)}-D(f^{-1})_{f^{-(k-1)}gf^{n}(y)}P_{f^{-(k-1)}gf^{n}(y)}\right)$
$\displaystyle\times\left(\prod_{i=k-1}^{n-1}D(f^{-1})^{-1}_{f^{-i}gf^{n}(x)}\right)P_{f^{-n}gf^{n}(x)}$

Applying (24), (22), and (23), we deduce that

$\|E_{n,k}(x,y)\|\ll\nu^{n-k}d^{fs}(x,y)^{\theta\omega_{0}\omega}.$

Since $\nu<1$, it follows that

(30) $\displaystyle\|D_{n,n}(x,y)-D_{n,0}(x,y)\|$
$\displaystyle\leq\sum^{n}_{k=1}\|D_{n,k}(x,y)-D_{n,k-1}(x,y)\|$ (31)
$\displaystyle\ll d^{fs}(x,y)^{\theta\omega_{0}\omega}\|A_{n,n-1}(x,y)\|\ll
d^{fs}(x,y)^{\theta\omega_{0}\omega}\|A_{n}(x)\|.$

Next, we compare the maps $A_{n}(y)$ and $D_{n,0}(x,y)$:

$\displaystyle\|A_{n}(y)-D_{n,0}(x,y)\|=$
$\displaystyle\left\|\left(\prod_{i=n-1}^{0}D(f^{-1})_{f^{-i}gf^{n}(y)}\right)P_{gf^{n}(y)}\right.$
$\displaystyle\quad\quad\times\left(P_{gf^{n}(y)}D(g)_{f^{n}(y)}P_{f^{n}(y)}-P_{gf^{n}(x)}D(g)_{f^{n}(x)}P_{f^{n}(x)}\right)$
$\displaystyle\left.\quad\quad\times\left(\prod_{i=n-1}^{0}D(f)_{f^{i}(y)}\right)P_{y}\right\|.$

We have

$\displaystyle\left\|P_{gf^{n}(y)}D(g)_{f^{n}(y)}P_{f^{n}(y)}-P_{gf^{n}(x)}D(g)_{f^{n}(x)}P_{f^{n}(x)}\right\|$
$\displaystyle\ll d(f^{n}(x),f^{n}(y))^{\theta}$ $\displaystyle\ll\rho^{\theta
n}d^{fs}(x,y)^{\theta}.$

Combining this estimate with (23) and (24), we deduce that

$\|A_{n}(y)-D_{n,0}(x,y)\|\ll\zeta^{n}d^{fs}(x,y)^{\theta}.$

Finally, the proposition follows from the estimate

$\displaystyle\|A_{n}(x)-A_{n}(y)\|\leq$
$\displaystyle\|A_{n}(x)-A_{n,-1}(x,y)\|+\|A_{n,-1}(x,y)-A_{n,n-1}(x,y)\|$
$\displaystyle+\|A_{n,n-1}(x,y)-D_{n,n}(x,y)\|+\|D_{n,n}(x,y)-D_{n,0}(x,y)\|$
$\displaystyle+\|D_{n,0}(x,y)-A_{n}(y)\|.$

This completes the proof. ∎

######  Proposition 3.17.

Let $x_{0}\in X$ and $r>0$. Then under the assumptions of Theorem 3.15,

$\sup\\{\|D(f^{-n}gf^{n})_{x}P_{x}\|:\,\,x\in
W^{fs}_{f}(x_{0}),\,d^{fs}(x,x_{0})\leq r,\,\,n\in\mathbb{N}\\}<\infty.$

###### Proof.

Suppose that the claim fails, i.e., there exist sequences $x_{i}\in
W^{fs}_{f}(x_{0})$, $d^{fs}(x_{i},x_{0})\leq r$, and $n_{i}\in\mathbb{N}$,
$n_{i}\to\infty$, such that

$\|D(f^{-n_{i}}gf^{n_{i}})_{x_{i}}P_{x_{i}}\|\to\infty.$

Passing to a subsequence, we may assume that $x_{i}\to x_{\infty}$ for some
$x_{\infty}\in W^{fs}_{f}(x_{0})$ such that $d^{fs}(x_{\infty},x_{0})\leq r$.
It follows from Proposition 3.16 that

$\|D(f^{-n_{i}}gf^{n_{i}})_{x_{\infty}}P_{x_{\infty}}\|\geq(1-c\cdot
d^{fs}(x_{i},x_{\infty})^{\kappa})\|D(f^{-n_{i}}gf^{n_{i}})_{x_{i}}P_{x_{i}}\|-\delta_{n_{i}}\to\infty.$

Let $v_{i}\in T_{x_{\infty}}(W^{fs}_{f}(x_{\infty}))$ with $\|v_{i}\|=1$ be
such that

$\|D(f^{-n_{i}}gf^{n_{i}})_{x_{\infty}}P_{x_{\infty}}\|=\|D(f^{-n_{i}}gf^{n_{i}})_{x_{\infty}}v_{i}\|.$

Passing to a subsequence, we may assume that $v_{i}\to v_{\infty}$. We have

$\displaystyle\|D(f^{-n_{i}}gf^{n_{i}})_{x_{\infty}}v_{\infty}\|$
$\displaystyle\geq\|D(f^{-n_{i}}gf^{n_{i}})_{x_{\infty}}v_{i}\|-\|D(f^{-n_{i}}gf^{n_{i}})_{x_{\infty}}P_{x_{\infty}}(v_{\infty}-v_{i})\|$
$\displaystyle\geq\|D(f^{-n_{i}}gf^{n_{i}})_{x_{\infty}}P_{x_{\infty}}\|\cdot(1-\|v_{\infty}-v_{i}\|).$

Hence, for sufficiently large $i$, we have

$\displaystyle\|D(f^{-n_{i}}gf^{n_{i}})_{x_{\infty}}v_{\infty}\|\geq\frac{1}{2}\|D(f^{-n_{i}}gf^{n_{i}})_{x_{\infty}}P_{x_{\infty}}\|.$

Let $\alpha_{n}=\|D(f^{-n}gf^{n})_{x_{\infty}}v_{\infty}\|$. Note that
$\alpha_{n_{i}}\to\infty$.

Fix small $\epsilon>0$. Let $x\in W^{fs}_{f}(x_{\infty})$ be such that
$d^{fs}(x,x_{\infty})<\epsilon$ and $v\in T_{x}W^{fs}_{f}(x)$ such that
$\|v-v_{\infty}\|<\epsilon$. We have

$\displaystyle\|D(f^{-n}gf^{n})_{x}v-D(f^{-n}gf^{n})_{x_{\infty}}v_{\infty}\|\leq$
$\displaystyle\|D(f^{-n}gf^{n})_{x}P_{x}v-D(f^{-n}gf^{n})_{x_{\infty}}P_{x_{\infty}}v\|$
$\displaystyle+\|D(f^{-n}gf^{n})_{x_{\infty}}P_{x_{\infty}}v-D(f^{-n}gf^{n})_{x_{\infty}}P_{x_{\infty}}v_{\infty}\|$
$\displaystyle\ll$ $\displaystyle
d^{fs}(x,x_{\infty})^{\kappa}\|D(f^{-n}gf^{n})_{x_{\infty}}P_{x_{\infty}}\|+\delta_{n}$
$\displaystyle+\|D(f^{-n}gf^{n})_{x_{\infty}}P_{x_{\infty}}\|\cdot\|v-v_{\infty}\|$
$\displaystyle\ll$
$\displaystyle(\epsilon^{\kappa}\alpha_{n}+\delta_{n})+\epsilon\alpha_{n}.$

For some $\rho=\rho(\epsilon)>0$, there exists a smooth curve
$\ell:[0,\rho]\to W^{fs}_{f}(x_{\infty})$ such that

$\displaystyle\ell(0)=x_{\infty},\quad\ell^{\prime}(0)=v_{\infty},\quad\ell^{\prime}(t)\in
T_{\ell(t)}W^{fs}_{f}(\ell(t)),$
$\displaystyle\hbox{diam}(\ell([0,\rho]))<\epsilon,\quad\quad\|\ell^{\prime}(t)-\ell^{\prime}(0)\|<\epsilon.$

We consider the sequence of curves $\ell_{n}=(f^{-n}gf^{n})\ell$. Note that
$\|\ell_{n}^{\prime}(0)\|=\alpha_{n}$, and it follows from the previous
computation that, choosing $\epsilon$ sufficiently small,

$\|\ell_{n_{i}}^{\prime}(t)-\ell_{n_{i}}^{\prime}(0)\|\leq\frac{1}{3}\|\ell_{n_{i}}^{\prime}(0)\|$

for all $t\in[0,\rho]$ and sufficiently large $i$. Since
$\|\ell^{\prime}_{n_{i}}(0)\|\to\infty$, it follows that the distance between
$\ell_{n_{i}}(0)$ and $\ell_{n_{i}}(\rho)$ in the ambient Eucledean space goes
to infinity as $i\to\infty$. This contradiction proves the proposition. ∎

###### Proof of Theorem 3.15.

By Proposition 3.13(1), the maps $\\{f^{-m_{i}}gf^{m_{i}}|_{W^{fs}_{f}(x)}\\}$
are precompact in $C^{0}$-topology. Then it follows from Proposition 3.17 that
the maps $\\{f^{-m_{i}}gf^{m_{i}}|_{W^{fs}_{f}(x)}\\}$ are uniformly bounded
in the $C^{1}$-topology. Also, combining Proposition 3.16 and Proposition
3.17, we obtain that for every $z$ and $w$ in a compact neighborhood of $x$ in
$W^{fs}_{f}(x)$,

$\|D(f^{-m_{i}}gf^{m_{i}})_{z}P_{z}-D(f^{-m_{i}}gf^{m_{i}})_{w}P_{w}\|\ll
d^{fs}(z,w)^{\kappa}+\delta_{m_{i}}.$

Since $\delta_{m_{i}}\to 0$, it follows that the maps
$\\{f^{-m_{i}}gf^{m_{i}}|_{W^{fs}_{f}(x)}\\}$ are equicontinuous in the
$C^{1}$-topology. This implies the theorem. ∎

###  3.4. Hölder implies $C^{1}$ along fast stable manifolds

######  Theorem 3.18.

Let $f_{0},g_{0}\in\hbox{\rm Aff}(X)$ be a good pair, $f,g\in\hbox{\rm
Diff}(X)$ $C^{1}$-small perturbations of $f_{0},g_{0}$, and $\Phi:X\to X$ a
Hölder isomorphism such that

$\Phi\circ f_{0}=f\circ\Phi\quad\hbox{and}\quad\Phi\circ g_{0}=g\circ\Phi,$

and

$\Phi(W^{fs}_{f_{0}}(x))=W^{fs}_{f}(\Phi(x))\quad\hbox{for all $x\in X$.}$

Then for a.e. $x\in X$, the maps $\Phi|_{W^{fs}_{f_{0}}(x)}$ and
$\Phi^{-1}|_{W^{fs}_{f}(\Phi(x))}$ are $C^{1}$-diffeomorphisms.

###### Proof.

Fix a sequence $\\{n_{i}\\}$ and $A\in\hbox{GL}(W^{min}_{f_{0}})$ as in
Proposition 3.13(2). For a set of $x\in X$ of full measure, the sequence
$\\{f_{0}^{-n_{i}}g_{0}f_{0}^{n_{i}}(x)\\}$ is dense in $X$. In particular,
for a.e. $x\in X$ and every $y\in W^{fs}_{f_{0}}(x)$, there exists a
subsequence $\\{n_{i_{j}}\\}$ such that
$f_{0}^{-n_{i_{j}}}g_{0}f_{0}^{n_{i_{j}}}(x)\to y$. Then by Proposition
3.13(2), for every $v\in W^{min}_{f_{0}}$,

(32) $f_{0}^{-n_{i_{j}}}g_{0}f_{0}^{n_{i_{j}}}(\exp(v)x)\to\exp(Av)y$

uniformly on compact sets.

For $k\in\mathbb{N}$ and $y\in W^{fs}_{f_{0}}(x)$, we consider maps

$\displaystyle\rho^{0}_{k,y}:W^{fs}_{f_{0}}(x)\to
W^{fs}_{f_{0}}(x):\exp(v)x\mapsto\exp(A^{k}v)y,$
$\displaystyle\rho^{1}_{k,y}:W^{fs}_{f}(\Phi(x))\to
W^{fs}_{f}(\Phi(x)):\Phi(\exp(v)x)\mapsto\Phi\left(\exp(A^{k}v)y\right),$

where $v\in W^{min}_{f_{0}}$. Note that

(33) $\rho^{0}_{k,y}=\Phi^{-1}\circ\rho^{1}_{k,y}\circ\Phi.$

In particular, it follows that $\rho^{1}_{k,y}$ is a homeomorphism, and by
(32),

$\rho^{1}_{1,y}=\lim_{j\to\infty}(f^{-n_{i_{j}}}gf^{n_{i_{j}}})|_{W^{fs}_{f}(\Phi(x))}.$

in the $C^{0}$-topology. By Theorem 3.15, the sequence of maps
$(f^{-n_{i_{j}}}gf^{n_{i_{j}}})|_{W^{fs}_{f}(\Phi(x))}$ is precompact in the
$C^{1}$-topology. Hence, there exists a subsequence which converges in the
$C^{1}$-topology, and $\rho^{1}_{1,y}$ is a $C^{1}$-map for every $y\in
W^{fs}_{f_{0}}(x)$. Since each map $\rho_{k,y}^{1}$, $k\geq 1$, is a
composition of maps $\rho_{1,z}^{1}$, it is also $C^{1}$.

Next, we show that

(34) $D(\rho^{1}_{1,y})_{z}\neq 0\quad\hbox{for every $y\in W^{fs}_{f_{0}}(x)$
and $z\in W^{fs}_{f}(\Phi(x))$.}$

Suppose that, to the contrary, $D(\rho^{1}_{1,y_{0}})_{z_{0}}=0$ for some
$y_{0}\in W^{fs}_{f_{0}}(x)$ and $z_{0}\in W^{fs}_{f}(\Phi(x))$. We claim that
for every $y\in W^{fs}_{f_{0}}(x)$, there exists $y_{1}\in W^{fs}_{f_{0}}(x)$
such that

(35) $\rho^{0}_{2,y}=\rho^{0}_{1,y_{1}}\rho^{0}_{1,y_{0}}.$

Indeed, if we write $y=\exp(v)x$, $y_{1}=\exp(v_{1})x$, $y_{0}=\exp(v_{0})x$
for some $v,v_{1},v_{0}\in W^{min}_{f_{0}}$, then (35) is equivalent to

$A^{2}w+v=A(Aw+v_{0})+v_{1},\quad w\in W^{min}_{f_{0}},$

and we can take $v_{1}=v-Av_{0}$. Now by (33) and (35),

$\rho^{1}_{2,y}=\rho^{1}_{1,y_{1}}\rho^{1}_{1,y_{0}}.$

Hence, $D(\rho^{1}_{2,y})_{z_{0}}=0$ for every $y\in W^{fs}_{f_{0}}(x)$.
Similarly, using (33), we deduce that for every $z\in W^{fs}_{f}(\Phi(x))$,
there exists $y_{z}\in W^{fs}_{f_{0}}(x)$ such that
$\rho^{1}_{1,y_{z}}(z)=z_{0}$. If we fix $y_{2}\in W^{fs}_{f_{0}}(x)$, there
exists $y_{z}^{\prime}\in W^{fs}_{f_{0}}(x)$ such that

$\rho^{1}_{3,y_{2}}=\rho^{1}_{2,y_{z}^{\prime}}\rho^{1}_{1,y_{z}}.$

Then we have

$D(\rho^{1}_{3,y_{2}})_{z}=0\quad\hbox{for every $z\in W^{fs}_{f}(\Phi(x))$.}$

This contradicts the map $\rho^{1}_{3,y_{2}}$ being a homeomorphism, and (34)
follows. We have proved that $\rho^{1}_{1,y}$ is a $C^{1}$-diffeomorphism for
every $y\in W^{fs}_{f_{0}}(x)$. This implies that the map $\rho^{1}_{0,y}$,
which can be represented as a composition of $\rho^{1}_{1,z_{1}}$ and
$(\rho^{1}_{1,z_{2}})^{-1}$, is also a $C^{1}$-diffeomorphism for every $y\in
W^{fs}_{f_{0}}(x)$.

We have a free transitive action of $W^{min}_{f_{0}}$ on $W^{fs}_{f}(\Phi(x))$
defined by

(36) $s(v,\Phi(\exp(w)x))=\Phi(\exp(v+w)x)$

where $v,w\in W^{min}_{f_{0}}$. Note that the action $s:W^{min}_{f_{0}}\times
W^{fs}_{f}(\Phi(x))\to W^{fs}_{f}(\Phi(x))$ is continuous, and since

$s(v,\Phi(\exp(w)x))=\rho^{1}_{0,\exp(v)x}(\Phi(\exp(w)x)),$

the map $s(v,\cdot)$ is a $C^{1}$-diffeomorphism for every $v\in
W^{min}_{f_{0}}$. Hence, by the Bochner–Montgomery theorem [3], the map $s$ is
$C^{1}$. Now it follows from (36) that the map $\Phi_{x}(v):=\Phi(\exp(v)x)$
is $C^{1}$.

Suppose that for some $v\in W^{min}_{f_{0}}$, we have $\phi^{\prime}(0)=0$
where $\phi(t)=\Phi(\exp(tv)x)$. Then since
$\phi(t_{1}+t)=\rho^{1}_{0,\exp(tv)x}(\Phi(\exp(t_{1}v)x))$, it follows that
$\phi^{\prime}(t)=0$ for every $t\in\mathbb{R}$. This contradicts the action
$s$ being free. Hence, we conclude that $D(\Phi_{x})_{0}$ is nondegenerate,
and because

$\Phi(\exp(v+w)x)=\rho^{1}_{0,\exp(v)x}(\Phi(\exp(w)x)),$

$D(\Phi_{x})_{v}$ is nondegenerate for every $v\in W^{min}_{f_{0}}$. This
shows that $\Phi|_{W^{fs}_{f_{0}}(x)}$ is a $C^{1}$-diffeomorphism for a.e.
$x\in X$. ∎

###  3.5. Completion of the proof of the main theorem

Let $\\{f_{0},g_{0}\\}\subset\alpha_{0}(\Gamma)$ be a good pair and
$\\{f,g\\}\subset\alpha_{1}(\Gamma)$ its conjugate under $\Phi$. We use
notation $A$, $\\{n_{i}\\}$, $\omega$ as in Proposition 3.13 and Remark 3.14.
Recall that

$A=\lim_{i\to\infty}\omega^{-n_{i}}P^{min}_{f_{0}}(Dg_{0})\omega^{n_{i}}.$

Hence, replacing the pair $\\{f_{0},g_{0}\\}$ by the pair
$\\{f_{0},f_{0}^{l}g_{0}\\}$ for some $l\geq 1$, we can get a good pair with
$A$ satisfying $\|A\|<1$, which we now assume.

By Theorem 3.5, $\Phi(W^{fs}_{f_{0}}(x))=W^{fs}_{f}(\Phi(x))$ for all $x\in
X$, so we consider the maps

$\displaystyle\alpha^{0}_{x}:W^{fs}_{f_{0}}(x)\to
W^{fs}_{f_{0}}(x):\exp(v)x\mapsto\exp(Av)x,$
$\displaystyle\alpha_{x}:W^{fs}_{f}(\Phi(x))\to
W^{fs}_{f}(\Phi(x)):\Phi(\exp(v)x)\mapsto\Phi\left(\exp(Av)x\right),$

where $v\in W^{min}_{f_{0}}$. Note that

(37) $\Phi\circ\alpha^{0}_{x}=\alpha_{x}\circ\Phi.$

In particular, it follows that $\alpha_{x}$ is a homeomorphism. For a.e. $x\in
X$, there exists a subsequence $\\{n_{i_{j}}\\}$ such that
$f_{0}^{-n_{i_{j}}}g_{0}f_{0}^{n_{i_{j}}}(x)\to x$ as $j\to\infty$. Then by
Proposition 3.13,

$\alpha^{0}_{x}=\lim_{j\to\infty}(f_{0}^{-n_{i_{j}}}g_{0}f_{0}^{n_{i_{j}}})|_{W^{fs}_{f_{0}}(x)},$

and by (37),

$\alpha_{x}=\lim_{j\to\infty}(f^{-n_{i_{j}}}gf^{n_{i_{j}}})|_{W^{fs}_{f}(\Phi(x))}$

in the $C^{0}$-topology. It follows from Theorem 3.15 that the sequence of
maps $(f^{-n_{i_{j}}}gf^{n_{i_{j}}})|_{W^{fs}_{f}(\Phi(x))}$ is precompact in
the $C^{1}$-topology. Hence, it contains a subsequence which converges in the
$C^{1}$-topology, and $\alpha_{x}$ is a $C^{1}$-map for a.e. $x\in X$.

Let $W^{min}_{f_{0},g_{0}}$ be the sum of eigenspaces of $A$ with eigenvalues
of minimal modulus. Our aim is to show that the map $\Phi$ restricted to the
leaves $\exp(W^{min}_{f_{0},g_{0}})x$ is linear in suitable
$C^{\infty}$-coordinate systems which depend continuously on $x$. The first
step is to show that the maps $\alpha_{x}$ are linear in suitable coordinates
on the fast stable leaves. Consider the measurable function

$\sigma(x)=\sup\\{\|D(f^{-n}gf^{n})_{\Phi(x)}P_{\Phi(x)}\|:\,\,n\in\mathbb{N}\\},$

which is well defined by Proposition 3.17. For $c>0$, let $X(c)$ be the subset
of $x\in X$ such that $\sigma(x)\leq c$ and the sequence
$\\{f_{0}^{-n_{i}}g_{0}f_{0}^{n_{i}}(x)\\}$ has $x$ as an accumulation point.
By property (iv) of good pair and Proposition 3.17, the set $\cup_{c>0}X(c)$
has full measure in $X$.

Let $Df_{0}|_{W^{min}_{f_{0}}}=\lambda\cdot\omega$ where
$\lambda\in\mathbb{R}$, $|\lambda|<1$, and
$\omega\in\hbox{Isom}(W^{min}_{f_{0}})$. Using the Poincare recurrence
theorem, for a.e. $(x,\omega^{\prime})\in
X(c)\times\hbox{Isom}(W^{min}_{f_{0}})$, one can construct a sequence
$\\{k_{j}\\}$, $k_{0}=0$, such that

$\displaystyle f_{0}^{k_{j}}(x)\in X(c)\quad\hbox{for every $j\geq
1$}\quad\hbox{and}\quad\omega^{k_{j}}\omega^{\prime}\to\omega^{\prime}\quad\hbox{as
$j\to\infty$.}$

Then $\omega^{k_{j}}\to id$. Hence, by the Fubini theorem, for a.e. $x\in
X(c)$, there exists a sequence $\\{k_{j}\\}$, $k_{0}=0$, such that

(38) $f_{0}^{k_{j}}(x)\in X(c)\quad\hbox{for every $j\geq
1$}\quad\hbox{and}\quad\omega^{k_{j}}\to id\quad\hbox{as $j\to\infty$.}$

Now we assume that $x\in X(c)$ satisfies (38). Let $\\{n^{(j)}_{i}\\}$ be a
subsequence such that

$(f_{0}^{-n^{(j)}_{i}}g_{0}f_{0}^{n^{(j)}_{i}})f_{0}^{k_{j}}(x)\to
f_{0}^{k_{j}}(x)\quad\hbox{as $i\to\infty$,}$

Then by Proposition 3.13(2),

(39)
$(f_{0}^{-n^{(j)}_{i}}g_{0}f_{0}^{n^{(j)}_{i}})|_{W^{fs}_{f_{0}}(f_{0}^{k_{j}}(x))}\to\alpha^{0}_{f_{0}^{k_{j}}(x)}\quad\hbox{as
$i\to\infty$}$

in the $C^{0}$-topology. A direct computation shows that

$\alpha_{x,j}^{0}=f_{0}^{-k_{j}}\circ\alpha^{0}_{f_{0}^{k_{j}}(x)}\circ
f_{0}^{k_{j}}$

where

$\alpha^{0}_{x,j}:W^{fs}_{f_{0}}(x)\to
W^{fs}_{f_{0}}(x):\exp(v)x\mapsto\exp((\omega^{-k_{j}}A\omega^{k_{j}})v)x,\quad
v\in W^{min}_{f}.$

Clearly, $\alpha^{0}_{x,j}\to\alpha^{0}_{x}$ as $j\to\infty$ in the
$C^{0}$-topology. It follows from (37) that

(40) $\alpha_{x,j}=f^{-k_{j}}\circ\alpha_{f_{0}^{k_{j}}(x)}\circ f^{k_{j}}$

where

$\alpha_{x,j}=\Phi\circ\alpha^{0}_{x,j}\circ\Phi^{-1}\to\alpha_{x}\quad\hbox{as
$j\to\infty$}$

in the $C^{0}$-topology.

Since $f$ is $C^{1}$-close to the algebraic map $f_{0}$, its Mather spectrum
on fast stable leaves is contained in a small interval, and by the
nonstationary Sternberg linearization [12, 11], $f|_{W^{fs}_{f}(z)}$ is linear
in suitable coordinate systems. Namely, there exists a family of
$C^{\infty}$-diffeomorphisms

$L_{z}:W^{min}_{f_{0}}\to W^{fs}_{f}(z),\quad z\in X,$

such that the map $z\mapsto L_{z}$ is continuous in the $C^{\infty}$-topology,
$L_{z}(0)=z$, $D(L_{z})_{0}=id$, and

(41) $(L_{f(z)}^{-1}\circ f\circ L_{z})(v)=\rho(z)v,\quad v\in
W^{min}_{f_{0}},$

with $\rho(z)\in\hbox{GL}(W^{min}_{f_{0}})$, $\|\rho(z)\|<1$. Consider the
sequence of maps

$G_{k}=L_{f^{k}(\Phi(x))}^{-1}\circ\alpha_{f_{0}^{k}(x)}\circ
L_{f^{k}(\Phi(x))}:W^{min}_{f_{0}}\to W^{min}_{f_{0}}.$

We claim that the sequence of maps $G_{k_{j}}$ restricted to compact sets is
uniformly bounded and equicontinuous in the $C^{1}$-topology. This is
equivalent to the sequence $\\{\alpha_{f_{0}^{k_{j}}(x)}\\}$ being uniformly
bounded and equicontinuous in the $C^{1}$-topology. It follows from (37),
(39), and (40) that

$F_{i,j}:=(f^{-k_{j}-n^{(0)}_{i}}gf^{n^{(0)}_{i}+k_{j}})|_{W^{fs}_{f}(\Phi(x))}\to\alpha_{x,j}\quad\hbox{as
$i\to\infty$}$

in the $C^{0}$-topology, and by Theorem 3.15, we may assume, after passing to
a subsequence, that convergence also holds in the $C^{1}$-topology. By (40),

$\displaystyle\alpha_{f_{0}^{k_{j}}(x)}=$
$\displaystyle(f^{k_{j}}\circ\alpha_{x,j}\circ
f^{-k_{j}})|_{W^{fs}_{f}(f^{k_{j}}(\Phi(x)))}.$

We observe that $(f^{k_{j}}\circ F_{i,j}\circ
f^{-k_{j}})|_{W^{fs}_{f}(f^{k_{j}}(\Phi(x)))}$ converges in the
$C^{1}$-topology to $\alpha_{f_{0}^{k_{j}}(x)}$ as $i\to\infty$, and since
$f_{0}^{k_{j}}(x)\in X(c)$ for all $j$, the derivative of

$(f^{k_{j}}\circ F_{i,j}\circ
f^{-k_{j}})|_{W^{fs}_{f}(f^{k_{j}}(\Phi(x)))}=(f^{-n_{i}^{(0)}}gf^{n_{i}^{(0)}})|_{W^{fs}_{f}(f^{k_{j}}(\Phi(x)))}$

is uniformly bounded over compact sets and $i\in\mathbb{N}$. This implies that
the sequence $\\{\alpha_{f_{0}^{k_{j}}(x)}\\}$ is uniformly bounded in the
$C^{1}$-topology. To prove equicontinuity, we observe that for $z,w\in
W^{fs}_{f}(f^{k_{j}}(\Phi(x)))$,

$\displaystyle\left\|D(\alpha_{f_{0}^{k_{j}}(x)})_{z}P_{z}-D(\alpha_{f_{0}^{k_{j}}(x)})_{w}P_{w}\right\|$
$\displaystyle\leq\|D(f^{k_{j}}\circ(\alpha_{x,j}-F_{i,j})\circ
f^{-k_{j}})_{z}P_{z}\|$
$\displaystyle+\|D(f^{k_{j}}F_{i,j}f^{-k_{j}})_{z}P_{z}-D(f^{k_{j}}F_{i,j}f^{-k_{j}})_{w}P_{w}\|$
$\displaystyle+\|D(f^{k_{j}}\circ(F_{i,j}-\alpha_{x,j})\circ
f^{-k_{j}})_{w}P_{w}\|.$

Since $F_{i,j}\to\alpha_{x,j}$ as $i\to\infty$ in the $C^{1}$-topology, taking
$i=i(j)$ sufficiently large, we can make the first and the last terms
arbitrary small. To estimate the middle term, we use that $f_{0}^{k_{j}}(x)\in
X(c)$ for all $j$ and Proposition 3.16. We get

$\|D(f^{k_{j}}F_{i,j}f^{-k_{j}})_{z}P_{z}-D(f^{k_{j}}F_{i,j}f^{-k_{j}})_{w}P_{w}\|\ll
d^{fs}(z,w)^{\kappa}+\delta_{n_{i}^{(0)}},$

where $\delta_{n}\to 0$ as $n\to\infty$. This proves equicontinuity, and in
fact, the stronger conclusion:

(42)
$\left\|D(\alpha_{f_{0}^{k_{j}}(x)})_{z}P_{z}-D(\alpha_{f_{0}^{k_{j}}(x)})_{w}P_{w}\right\|\ll
d^{fs}(z,w)^{\kappa}.$

Let $\rho_{k}=\prod_{s=k-1}^{0}\rho(f^{s}(\Phi(x)))$ with $\rho$ defined as in
(41). Since $f$ is $C^{1}$-close to the map $f_{0}$, which is conformal on the
fast stable leaves, it follows that for some $\lambda<1$ and small
$\epsilon>0$, we have

(43)
$\|\rho_{k}(x)\|\ll(\lambda+\epsilon)^{k}\quad\hbox{and}\quad\|\rho_{k}(x)^{-1}\|\ll(\lambda-\epsilon)^{-k}$

uniformly on $x\in X$ and $k\in\mathbb{N}$. We deduce from (40) and (41) that

(44) $G_{0,j}=\rho_{k_{j}}^{-1}G_{k_{j}}(\rho_{k_{j}}v),\quad v\in
W^{min}_{f_{0}},$

where $G_{0,j}=L_{x}^{-1}\circ\alpha_{x,j}\circ L_{x}\to G_{0}$ as
$j\to\infty$. Fix a basis $\\{e_{\ell}\\}$ of $W^{min}_{f_{0}}$ and write

$G_{k}(v)=\sum_{\ell}G_{k,\ell}(v)e_{\ell}.$

Applying the mean value theorem to the functions $t\mapsto
G_{k_{j},\ell}(t\rho_{k_{j}}v)$, $t\in[0,1]$, we deduce that

(45) $G_{k_{j},\ell}(\rho_{k_{j}}v)=\sum_{s}\frac{\partial
G_{k_{j},\ell}}{\partial
x_{s}}(t_{j,\ell}(v)\rho_{k_{j}}v)(\rho_{k_{j}}v)_{s}$

for some $t_{j,\ell}(v)\in[0,1]$. Hence, by (44),

(46) $G_{0,j}(v)=(\rho_{k_{j}}^{-1}B_{j}(v)\rho_{k_{j}})v$

where $B_{j}(v)$ is the linear map of $W^{min}_{f_{0}}$ with coefficients
coming from (45). Let $B_{j}=D(G_{k_{j}})_{0}$. From (42), we deduce that

$\|B_{j}(v)-B_{j}\|\ll\|\rho_{k_{j}}v\|^{\kappa},$

and it follows from (43) that

(47)
$\|\rho_{k_{j}}^{-1}B_{j}(v)\rho_{k_{j}}-\rho_{k_{j}}^{-1}B_{j}\rho_{k_{j}}\|\to
0\quad\hbox{as $j\to\infty$.}$

Let $B=B(x)$ be a limit point of the sequence of maps
$\rho_{k_{j}}^{-1}B_{j}(v)\rho_{k_{j}}$. The crucial point of our argument is
that $B(x)$ is independent of $v$ because of the equicontinuity estimate.
Taking $j\to\infty$, we deduce from (46) that $G_{0}(v)=B\,v$, and by the
definition of $G_{0}$,

(48) $L_{x}^{-1}(\Phi(\exp(Av)x))=B(x)\,L_{x}^{-1}(\Phi(\exp(v)x))).$

This equality holds for a.e. $x\in X(c)$ with $c>0$ and hence, for a.e. $x\in
X$. Note that since $\Phi$ is a homeomorphism, the linear map $B(x)$ is
nondegenerate. Although $B(x)$ is defined only for a.e. $x\in X$, it follows
from (48) that it can be extended continuously to the whole space so that (48)
holds everywhere.

Now we consider the maps

$\Phi_{x}(v)=L_{\Phi(x)}^{-1}(\Phi(\exp(v))x),\quad x\in X,\;v\in
W^{min}_{f_{0}},$

which satisfy the equivariance relation $\Phi_{x}\circ A=B(x)\circ\Phi_{x}$.
Recall that by Theorem 3.18, $\Phi_{x}$ is a $C^{1}$-diffeomorphism for a.e.
$x\in X$. Hence, we have $D(\Phi_{x})_{0}A=B(x)D(\Phi_{x})_{0}$, and it
follows that the map $\Psi_{x}:=D(\Phi_{x})_{0}^{-1}\circ\Phi_{x}$ commutes
with the contraction $A$. We write
$A|_{W^{min}_{f_{0},g_{0}}}=\rho\cdot\theta$ where $\rho\in\mathbb{R}$,
$|\rho|<1$, and $\theta\in\hbox{Isom}(W^{min}_{f_{0},g_{0}})$. Since

$W^{min}_{f_{0},g_{0}}=\\{v\in
W^{min}_{f_{0}}:\,\|A^{n}v\|=O(\rho^{n})\quad\hbox{as $n\to\infty$}\\},$

and $\Psi_{x}$ is a $C^{1}$-map, we deduce that

$\Psi_{x}(W^{min}_{f_{0},g_{0}})\subset W^{min}_{f_{0},g_{0}}.$

We claim that $\Psi_{x}|_{W^{min}_{f_{0},g_{0}}}$ is linear. We fix a basis
$\\{e_{\ell}\\}$ of $W^{min}_{f_{0},g_{0}}$ and write $\Psi:=\Psi_{x}$ as

$\Psi(v)=\sum_{\ell}\Psi_{\ell}(v)e_{\ell},\quad v\in W^{min}_{f_{0},g_{0}}.$

By the mean value theorem,

$\Psi_{\ell}(A^{n}v)=\sum_{s}\frac{\partial\Psi_{\ell}}{\partial
x_{s}}(t_{\ell}(v)A^{n}v)(A^{n}v)_{s}$

for some $t_{\ell}(v)\in[0,1]$. Hence,

(49)
$\Psi(v)=A^{-n}\Psi(A^{n}v)=(A^{-n}C_{n}(v)A^{n})v=(\theta^{-n}C_{n}(v)\theta^{n})v$

where $C_{n}(v)$ is the matrix with coefficients
$\frac{\partial\Psi_{\ell}}{\partial x_{s}}(t_{\ell}(v)A^{n}v)$. Since $\Psi$
is a $C^{1}$-map and $\|A\|<1$, it follows that $C_{n}(v)\to D(\Psi)_{0}$ as
$n\to\infty$. Passing to a subsequence, we may assume that the sequence of
isometries $\theta^{n}$ also converges. Then it follows from (49) that $\Psi$
is linear. We conclude that for a.e. $x\in X$, there exists a linear map
$C(x):W^{min}_{f_{0},g_{0}}\to W^{min}_{f_{0},g_{0}}$ such that

(50) $\Phi(\exp(v)x)=L_{\Phi(x)}(C(x)v),\quad v\in W^{min}_{f_{0},g_{0}}.$

It follows from this relation that $C(x)$ is nondegenerate. Moreover, by
continuity, we may assume that (50) holds for all $x\in X$, and $C(x)$ depends
continuously on $x$. Now relation (50) also implies that $\Phi$ is a
$C^{\infty}$-diffeomorphism along the leaves $\exp(W^{min}_{f_{0},g_{0}})x$,
and the partial derivatives along this leaves depend continuously on $x\in X$.

Note that if $\\{f_{0},g_{0}\\}$ is a good pair, then
$\\{h^{-1}f_{0}h,h^{-1}f_{0}h\\}$ is good as well for every
$h\in\alpha_{0}(\Gamma)$, and we have
$W^{min}_{h^{-1}f_{0}h,h^{-1}g_{0}h}=(Dh)^{-1}W^{min}_{f_{0},g_{0}}$. Hence,
it follows from the irreducibility of the $\Gamma$-action on $\hbox{Lie}(G)$
that

(51) $\sum_{\\{f_{0},g_{0}\\}\subset\alpha_{0}(\Gamma)\hbox{\tiny-
good}}W^{min}_{f_{0},g_{0}}=\hbox{\rm Lie}(G).$

Now we consider the elliptic differential operator
$\mathcal{D}^{s}=\sum_{i}\frac{\partial^{2s}}{\partial x_{i}^{2s}}$ where the
partial derivatives $\frac{\partial}{\partial x_{i}}$ span the tangent space
and are taken in directions of $W^{min}_{f_{0},g_{0}}$ for some choice of good
pairs $\\{f_{0},g_{0}\\}$. It follows from the previous paragraph that
$\mathcal{D}^{s}\Psi$ is continuous for every $s\geq 2$. Hence, by the
regularity of solutions of elliptic PDE, $\Psi$ is $C^{\infty}$. Since
$D(\Phi)_{x}$ is onto when restricted to fast stable distributions of good
$f_{0}$ and its conjugate $f$, it follows that $D(\Phi)_{x}$ is onto as well.
Thus, $\Psi^{-1}$ is $C^{\infty}$ by the inverse function theorem.

##  4\. Existence of good pairs

###  4.1. Tori

In this section, we set $X=\mathbb{T}^{d}$, $d\geq 2$, and prove

######  Proposition 4.1.

Let $\Gamma$ be a subgroup of $\hbox{\rm Aff}(X)$ such that the Zariski
closure of $D\Gamma$ contains $\hbox{\rm SL}_{d}$. Then $\Gamma$ contains a
good pair.

We will use the following lemma, which is easy to prove using Fourier analysis
(see, for example, [2, Corollary 1.6 and Remark 1.8]). Let $\phi$ be the Euler
totient function.

######  Lemma 4.2.

Let $f_{1},f_{2}\in\hbox{\rm Aff}(X)$ be such that for every $l\geq 1$
satisfying $\phi(l)\leq d^{2}$, the map $Df_{1}^{-l}Df_{2}^{l}$ does not have
eigenvalue 1. Then for every $\phi_{1},\phi_{2}\in L^{2}(X)$,

$\int_{X}\phi_{1}(f_{1}^{n}(x))\phi_{2}(f_{2}^{n}(x))d\mu(x)\to\left(\int_{X}\phi_{1}\,d\mu\right)\left(\int_{X}\phi_{2}\,d\mu\right)\quad\hbox{as
$n\to\infty$.}$

If the conclusion of Lemma 4.2 holds, then we call the pair
$\\{f_{1},f_{2}\\}$ mixing. Mixing pairs can be used to construct affine maps
satisfying property (iv) of good pairs.

######  Lemma 4.3.

Let $f,g\in\hbox{\rm Aff}(X)$ and suppose the pair
$\\{f^{-1},gf^{-1}g^{-1}\\}$ is mixing. Then for every subsequence
$\\{n_{i}\\}$ and for a.e. $x\in X$, the sequence
$\\{f^{-n_{i}}gf^{n_{i}}(x)\\}_{n\geq 0}$ is dense in $X$.

###### Proof.

We have

$\int_{X}\phi_{1}(gf^{-n}g^{-1}(x))\phi_{2}(f^{-n}(x))d\mu(x)\to\left(\int_{X}\phi_{1}\,d\mu\right)\left(\int_{X}\phi_{2}\,d\mu\right)\quad\hbox{as
$n\to\infty$}$

for every $\phi_{1},\phi_{2}\in L^{2}(X)$. By invariance of the measure, this
also implies that

$\int_{X}\phi_{1}(x)\phi_{2}(f^{-n}gf^{n}(x))d\mu(x)\to\left(\int_{X}\phi_{1}\,d\mu\right)\left(\int_{X}\phi_{2}\,d\mu\right)\quad\hbox{as
$n\to\infty$}$

for every $\phi_{1},\phi_{2}\in L^{2}(X)$.

Now we show that for $\delta_{n}=f^{-n}gf^{n}$, the sequence
$\\{\delta_{n_{i}}x\\}$ is dense in $X$ for a.e. $x\in X$. Let $U$ be a
nonempty open subset of $X$ and $A=\cup_{i\geq 0}\delta_{n_{i}}^{-1}(U)$. We
have

$0=\int_{X}\chi_{U}(\delta_{n_{i}}(x))\chi_{A^{c}}(x)\,d\mu(x)\to\mu(U)\mu(A^{c}).$

This implies that $\mu(A^{c})=0$, i.e. for a.e. $x\in X$,

$\\{\delta_{n_{i}}x\\}_{i\geq 0}\cap U\neq\emptyset.$

Since $X$ has countable base of topology, this proves the lemma. ∎

###### Proof of Proposition 4.1.

Since $D\Gamma$ is Zariski dense, there is $f\in\Gamma$ such that $Df$ is
$\mathbb{R}$-regular (see [1, 21]). In particular, $Df$ is semisimple and
hyperbolic. Because of Lemmas 4.2 and 4.3, it suffices to find $g\in\Gamma$
such that $Dg$ belongs to the set

$\displaystyle\left\\{X\in\hbox{SL}_{d}:\;\;\det(P^{min}_{f}X|_{W^{min}_{f}})\neq
0,\quad\det([Df^{l},X]-id)\neq 0\;\;\hbox{for $\phi(l)\leq d^{2}$}\right\\}.$

One can check that this is a nonempty Zariski open subset of $\hbox{SL}_{d}$.
Hence, existence of such $g\in\Gamma$ follows from Zariski density. ∎

###  4.2. Semisimple groups

Let $G$ be a connected semisimple Lie groups with no compact factors,
$\Lambda$ a lattice in $G$, and $X=G/\Lambda$.

######  Proposition 4.4.

Let $\Gamma$ be a subgroup of $\hbox{\rm Aff}(X)$ such that the Zariski
closure of $D\Gamma$ contains $\hbox{\rm Ad}(G)$. Then $\Gamma$ contains a
good pair.

###### Proof.

Since $D\Gamma$ contains a finite index subgroup consisting of inner
automorphisms, we may assume without loss of generality that $D\Gamma$ is a
subgroup of $\hbox{Ad}(G)$. It follows from Zariski density [1, 21] that
$\Gamma$ contains an element $f$ such that $Df$ is $\mathbb{R}$-regular. In
particular, it is partially hyperbolic and semisimple, and hence it satisfies
properties (i)–(ii) of the definition of a good pair. If we choose
$g\in\Gamma$ so that the pair $\\{f^{-1},gf^{-1}g^{-1}\\}$ is mixing, then by
Lemma 4.3, $f$ and $g$ will satisfy property (iv) of the definition of a good
pair. By the Howe–Moore theorem, the pair $\\{f^{-1},gf^{-1}g^{-1}\\}$ is
mixing provided that for all projections
$\pi_{i}:\hbox{Ad}(G)\to\hbox{Ad}(G_{i})$ on simple factors of $\hbox{Ad}(G)$,
the sequence $\\{\pi_{i}(Dg(Df)^{-n}(Dg)^{-1}(Df)^{n})\\}$ is divergent. Since
$\pi_{i}(Df)$ is also $\mathbb{R}$-regular,

$P_{i}=\\{g\in G_{i}:\,\pi_{i}(Df)^{-n}\cdot g\cdot\pi_{i}(Df)^{n}\hbox{ is
nondivergent}\\}$

is a proper parabolic subgroup of $G_{i}$. By Zariski density, there exists
$g\in\Gamma$ such that $\pi_{i}(Dg)\notin P_{i}$ for all $i$, and
$P_{f}^{min}(Dg):W^{min}_{f}\to W^{min}_{f}$ is nondegenerate. Such $f$ and
$g$ provide a good pair. ∎

## References

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