# Shub’s example revisited

Chao Liang, Radu Saghin, Fan Yang and Jiagang Yang  School of Statistics and
Mathematics, Central University of Finance and Economics, Beijing, 100081,
China<EMAIL_ADDRESS>Instituto de Matemática, Pontificia Universidad
Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso-Chile.
<EMAIL_ADDRESS>Department of Mathematics, Michigan State University, MI,
USA<EMAIL_ADDRESS>Departamento de Geometria, Instituto de Matemática e
Estatística, Universidade Federal Fluminense, Niterói, Brazil<EMAIL_ADDRESS>
###### Abstract.

For a class of robustly transitive diffeomorphisms on ${\mathbb{T}}^{4}$
introduced by Shub in [24], satisfying an additional bunching condition, we
show that there exits a $C^{2}$ open and $C^{r}$ dense subset
${\mathcal{U}}^{r}$, $2\leq r\leq\infty$, such that any two hyperbolic points
of $g\in{\mathcal{U}}^{r}$ with stable index $2$ are homoclinically related.
As a consequence, every $g\in{\mathcal{U}}^{r}$ admits a unique homoclinic
class associated to the hyperbolic periodic points with index $2$, and this
homoclinic class coincides to the whole ambient manifold. Moreover, every
$g\in{\mathcal{U}}^{r}$ admits at most one measure with maximal entropy, and
every $g\in{\mathcal{U}}^{\infty}$ admits a unique measure of maximal entropy.

Chao Liang has been supported by NNSFC grants 12271538 and 12071328 and the
Disciplinary Funds in CUFE
Jiagang Yang was partially supported by CNPq, FAPERJ, PRONEX and NNSFC grants
11871487 and 12071202

###### Contents

  1. 1 Introduction and results
    1. 1.1 Shub class
    2. 1.2 Results
  2. 2 Preliminaries
    1. 2.1 Stable and unstable holonomies between center leaves.
    2. 2.2 Homoclinic holonomies
    3. 2.3 Hyperbolic measures
    4. 2.4 Criterion of uniqueness of equilibrium state
    5. 2.5 Hyperbolicity of equilibrium states
  3. 3 Proof of theorem A
    1. 3.1 Definition of $\mathcal{U}$ and plan of the proof
    2. 3.2 Proof of $C^{r}$ density
    3. 3.3 Proof of Proposition 3.5
  4. 4 Proof of Corollary B
  5. 5 Proof of Theorem C

###### Contents

  1. 1 Introduction and results
  2. 2 Preliminaries
  3. 3 Proof of theorem A
  4. 4 Proof of Corollary B
  5. 5 Proof of Theorem C

##  1\. Introduction and results

Shub introduced in [24] one of the first examples of robustly transitive
diffeomorphisms (on $\mathbb{T}^{4}$) which are not uniformly hyperbolic.
Although later Mañé [18] built a robustly transitive but non-hyperbolic
example on ${\mathbb{T}}^{3}$, the Shub’s example still has its special
interest, in particular because the center direction is two-dimensional and
may admit no further domination.

In this paper we will consider a slightly more general class than the original
setting of the Shub’s example, a precise definition is the following.

###  1.1. Shub class

######  Definition 1.1.

A diffeomorphism $f:M\to M$ is called _partially hyperbolic_ if the tangent
bundle admits a continuous $Df$-invariant splitting $TM=E^{s}\oplus
E^{c}\oplus E^{u}$ such that there exist continuous functions
$0<\lambda_{s}(x)<\lambda_{c}^{-}(x)\leq\lambda_{c}^{+}(x)<\lambda_{u}(x)$,
with $\lambda_{s}(x)<1<\lambda_{u}(x)$, satisfying the following conditions:

  * (1)

$\|Df(x)v^{s}\|\leq\lambda_{s}(x)$,

  * (2)

$\lambda_{c}^{-}(x)\leq\|Df(x)v^{c}\|\leq\lambda_{c}^{+}(x)$,

  * (3)

$\|Df(x)v^{u}\|\geq\lambda_{u}(x)$,

for every $x\in M$ and unit vectors $v^{i}\in E^{i}(x)$($i=s,c,u$).

A partially hyperbolic diffeomorphism is called _dynamically coherent_ if
there exist invariant foliations ${\mathcal{F}}^{cs}$ and ${\mathcal{F}}^{cu}$
tangent to $E^{cs}=E^{c}\oplus E^{s}$ and $E^{cu}=E^{c}\oplus E^{u}$. In this
case ${\mathcal{F}}^{cs}$ is subfoliated by the stable and central foliations
${\mathcal{F}}^{s}$ and ${\mathcal{F}}^{c}$, while ${\mathcal{F}}^{cu}$ is
subfoliated by the unstable and center foliations ${\mathcal{F}}^{u}$ and
${\mathcal{F}}^{c}$.

Let $A,B$ be two linear Anosov automormorphism on ${\mathbb{T}}^{2}$ such that
$1<|\lambda_{B}|<|\lambda_{A}|$, where $\lambda_{A}$ and $\lambda_{B}$ are the
unstable eigenvalues of $A$ and $B$. Then
$f_{A,B}(x,y):{\mathbb{T}}^{2}\times{\mathbb{T}}^{2}\to{\mathbb{T}}^{2}\times{\mathbb{T}}^{2}$

$f_{A,B}(x,y)=(A(x),B(y))$

is an Anosov automorphism, which can also be seen as a partially hyperbolic
diffeomorphism with two-dimensional center bundle and one-dimensional stable
and unstable bundles.

######  Definition 1.2.

Let $\operatorname{PH}_{A,B}$ be the set of partially hyperbolic
diffeomorphisms isotopic to $f_{A,B}$, all of them having the same dimension
(i.e., one dimension) of the stable and unstable bundle, and let
$\operatorname{PH}_{A,B}^{0}$ be the connected component of
$\operatorname{PH}_{A,B}$ containing $f_{A,B}$.

It is easy to see that $\operatorname{PH}_{A,B}^{0}$ is an open set of
diffeomorphisms of ${\mathbb{T}}^{4}$. It is known that:

######  Proposition 1.3.

[Fisher-Potrie-Sambarino [10]]. If $f\in\operatorname{PH}_{A,B}^{0}$, then $f$
is dynamically coherent and admits a center foliation where all central leaves
are $C^{1}$ two-dimensional tori, and $f$ is center leaf conjugate to
$f_{A,B}$.

######  Definition 1.4.

The _Shub class_
$\mathcal{SH}\subset\cup_{1<\lambda_{B}<\lambda_{A}}\operatorname{PH}_{A,B}^{0}$
is the set of partially hyperbolic diffeomorphisms $f$ of ${\mathbb{T}}^{4}$
such that $f$ belongs to some $\operatorname{PH}_{A,B}^{0}$ and there exists a
fixed point $p_{f}=f(p_{f})\in{\mathbb{T}}^{4}$, such that
$f\mid_{{\mathcal{F}}^{c}_{f}(p_{f})}$ is an Anosov diffeomorphism, where
${\mathcal{F}}^{c}_{f}(p_{f})$ is the (fixed) center leaf passing through
$p_{f}$. Also let

$\mathcal{SH}^{r}:=\\{f\in\mathcal{SH}:\ f\hbox{ is }C^{r}\\},\ \ r\geq 1.$

Although this part will not be used in the proof, through analyzing the
induced map on the fundamental group, it is easy to show that
$f\mid_{{\mathcal{F}}^{c}_{f}(p_{f})}$ is topological conjugate to $B$. Shub
proved the following.

######  Theorem 1.5.

[Shub [24]] $\mathcal{SH}$ is $C^{1}$ open and every $f\in\mathcal{SH}$ is
transitive.

Shub proved this result for some specific examples, but the proof can be
adapted for the Shub class of diffeomorphisms with minor modifications. In
this article we consider the class of Shub diffeomorphisms which also satisfy
some bunching conditions.

######  Definition 1.6.

The _bunched Shub class_ $\mathcal{SH}^{r}_{b}$ is the set of partially
hyperbolic diffeomorphisms $f\in\mathcal{SH}^{r}$ which also satisfy the
following bunching conditions:

  1. (a)

Global bunching:

(1)
$\lambda_{s}(x)<\frac{\lambda_{c}^{-}(x)}{\lambda_{c}^{+}(x)}\leq\frac{\lambda_{c}^{+}(x)}{\lambda_{c}^{-}(x)}<\lambda_{u}(x),\
\ \forall x\in{\mathbb{T}}^{4};$

  2. (b)

Stronger local bunching at the fixed center leaf
${\mathcal{F}}^{c}_{f}(p_{f})$:

(2)
$\lambda_{s}(x)<\left(\lambda_{c}^{-}(x)\right)^{2}\leq\left(\lambda_{c}^{+}(x)\right)^{2}<\lambda_{u}(x),\
\ \forall x\in{\mathcal{F}}^{c}_{f}(p_{f}),$

and

(3)
$\lambda_{s}(x)<\frac{\lambda_{c}^{-}(x)}{\left(\lambda_{c}^{+}(x)\right)^{2}}\leq\frac{\lambda_{c}^{+}(x)}{\left(\lambda_{c}^{-}(x)\right)^{2}}<\lambda_{u}(x),\
\ \forall x\in{\mathcal{F}}^{c}_{f}(p_{f}).$

Clearly $\mathcal{SH}^{r}_{b}$ is a $C^{1}$ open set.

######  Remark 1.7.

The condition (1) implies (see [22]) that if $f$ is $C^{2}$ then the stable
and unstable bundles are $C^{1}$ when restricted to the center-stable and
center-unstable leaves, and as a consequence the strong stable and strong
unstable holonomies between the center leaves are of class $C^{1}$ (when
restricted to the center-stable respectively center-unstable leaves). We will
see later that in fact these holonomies depend continuously in the $C^{1}$
topology with respect to the points (or the center leaves) and with respect to
the map $f$ (in the $C^{2}$ topology).

######  Remark 1.8.

The condition (2) is the standard 2-bunching condition, and [12] implies that
if $f$ is $C^{2}$ then ${\mathcal{F}}^{c}_{f}(p_{f})$,
${\mathcal{F}}^{cs}_{f}(p_{f})$ and ${\mathcal{F}}^{cu}_{f}(p_{f})$ are of
class $C^{2}$. If the central bounds are symmetric, of
$\lambda_{c}^{-}\lambda_{c}^{+}=1$, then it is equivalent to the global
bunching condition (1).

The condition (3) implies that if $f$ is $C^{3}$ then the strong stable and
strong unstable bundles and (sub)foliations are of class $C^{2}$ when
restricted inside the center-stable respectively center-unstable leaves of the
fixed point $p_{f}$ (see [22]).

###  1.2. Results

The homoclinic intersections between hyperbolic periodic points were first
observed by Poincaré, and since then, they play an important role in the
theory of dynamical systems. Smale [25] used them in order to define
homoclinic classes.

######  Definition 1.9.

Given two hyperbolic periodic points $p,q$ of the diffeomorphism $f$, with the
same stable index, we say that they are _homoclinically related_ if their
stable and unstable manifolds intersect transversally:

(4) $W^{s}(p)\pitchfork W^{u}(q)\neq\emptyset\;\;\text{ and
}\;\;W^{s}(q)\pitchfork W^{u}(p)\neq\emptyset.$

We say that ${\rm Orb}(p)$ and ${\rm Orb}(q)$ are _homoclinically related_ if:

(5) $W^{s}({\rm Orb}(p))\pitchfork W^{u}({\rm Orb}(q))\neq\emptyset\;\;\text{
and }\;\;W^{s}({\rm Orb}(q))\pitchfork W^{u}({\rm Orb}(p))\neq\emptyset.$

This is an equivalence relation between hyperbolic periodic orbits. The
_homoclinic class of ${\rm Orb}(p)$_, $HC({\rm Orb}(p))$, is the closure of
the equivalence class of ${\rm Orb}(p)$.

If a diffeomorphism has no dominated splitting of index 2, it seems unexpected
that any two hyperbolic points of stable index $2$ are homoclinically related
to each other. Indeed, the sizes of stable and unstable manifolds of the
hyperbolic periodic points are nonuniform, and the intersection in (5) can be
empty. On the other hand, even if the intersection is non-empty, the
intersection may not be transverse, because of the lack of domination (see
Pujals-Sambarino [23], Lan Wen [27]).

The main result of this paper is the following.

######  Theorem A.

For any $2\leq r\leq\infty$, there exists a $C^{2}$ open and $C^{r}$ dense
subset ${\mathcal{U}}^{r}\subset\mathcal{SH}^{r}_{b}$, such that for any
$f\in{\mathcal{U}}^{r}$, any two hyperbolic points of $f$ with stable index
$2$ are homoclinically related.

As a consequence, any diffeomorphism $f\in{\mathcal{U}}^{r}$ admits a unique
homoclinic class associated to the hyperbolic periodic points of index $2$.
Denote by $p_{f}$ a hyperbolic fixed point of $f\in{\mathcal{U}}^{r}$.

######  Corollary B.

For any $f\in{\mathcal{U}}^{r}$, $f$ admits a unique homoclinic class
$H(p_{f},f)$ associate to the hyperbolic periodic points of index $2$, and the
homoclinic class coincides to the ambient manifold.

For a continuous potential $\phi$ and a continuous map $f$, an $f$-invariant
probability measure $\mu$ is called an _equilibrium measure_ for the potential
$\phi$, if

$h_{\mu}(f)+\int\phi d\mu=P_{top}(\phi),$

where $P_{top}(\phi):=\sup_{\nu\in{\mathcal{M}}_{e}(f)}\\{h_{\nu}(f)+\int\phi
d\nu\\}$.

The equilibrium states do not necessarily exist. Assuming entropy
expansiveness, Bowen [4] proved the equilibrium states do exist. It was shown
by Liao, Viana and Yang [17] that any diffeomorphism away from homoclinic
tangencies is entropy expansive. Yomdin [29], (see also Buzzi [6]) proved also
that for any $C^{\infty}$ diffeomorphism, equilibrium states always exist.

The uniqueness of equilibrium states is a more subtle problem. Recently,
Climenhaga and Thompson [9] (see also [21]) gave a criterion based on Bowen
property and specification. Another method used by Buzzi, Crovisier and Sarig
[7] (see also Ben Ovadia [2, 3]) is based on the use of the homoclinic class
of measures:

######  Definition 1.10.

Suppose $f$ is a $C^{r}$ diffeomorphism for some $r>1$. For two ergodic
hypperbolic measures $\mu_{1}$ and $\mu_{2}$ of $f$, we write
$\mu_{1}\preceq\mu_{2}$ iff there exist measurable sets $A_{1},A_{2}\subset M$
with $\mu_{i}(A_{i})>0$ such that for any $x_{1}\in A_{1}$ and $x_{2}\in
A_{2}$, the manifolds $W^{u}(x_{1})$ and $W^{s}(x_{2})$ have a point of
transverse intersection.

$\mu_{1}$, $\mu_{2}$ are _homoclinically related_ if $\mu_{1}\preceq\mu_{2}$
and $\mu_{2}\preceq\mu_{1}$. We write $\mu_{1}\overset{h}{\sim}\mu_{2}$. The
set of ergodic measures homoclinically related to a hyperbolic ergodic measure
$\mu$ is called the _measured homoclinic class_ of $\mu$.

######  Remark 1.11.

The homoclinic relation is an equivalence relation, moreover, two atomic
measures supported on two periodic orbits are homoclinically related if and
only if the two periodic orbits are hyperbolic and homoclinically related.

We have the following Theorem.

######  Theorem C.

For any $f\in{\mathcal{U}}^{r}$, all the hyperbolic ergodic measures of index
$2$ are homoclinically related. Let
$\phi:{\mathbb{T}}^{4}\rightarrow\mathbb{R}$ be any Hölder potential function
with $\max_{x,y\in{\mathbb{T}}^{4}}\|\phi(x)-\phi(y)\|<\log\lambda_{B}$, where
$f\in\operatorname{PH}_{A,B}^{0}$, then $f$ admits at most one equilibrium
state for the potential $\phi$. In particular, every $f\in{\mathcal{U}}^{r}$
admits at most one measure of maximal entropy.

A direct consequence of [29, 6] is the following.

######  Corollary 1.12.

Every $f\in{\mathcal{U}}^{\infty}\cap\operatorname{PH}_{A,B}^{0}$ admits a
unique equilibrium state for every Hölder potential satisfying
$\max_{x,y\in{\mathbb{T}}^{4}}\|\phi(x)-\phi(y)\|<\log\lambda_{B}$. In
particular every $f\in{\mathcal{U}}^{\infty}$ admits a unique measure of
maximal entropy.

For Shub’s example, some similar results were obtained under some extra
assumptions. For instance, by Newhouse, Young [20] and Carvalho, Pérez [8]
with the extra assumption that within the center foliation there exists a one-
dimensional invariant sub-foliation, and by Álvarez [1] assuming that the
center bundle admits a further dominated splitting.

##  2\. Preliminaries

###  2.1. Stable and unstable holonomies between center leaves.

As we mentioned before, the condition 1 implies that the holonomies between
the center leafs are uniformly $C^{1}$. In fact there exists a continuity of
the holonomies in the $C^{1}$ topology. If $y\in{\mathcal{F}}^{u}_{f}(x)$ let
us denote by
$h_{f,x,y}^{u}:{\mathcal{F}}^{c}_{f}(x)\rightarrow{\mathcal{F}}^{c}_{f}(y)$
the unstable holonomy between the two center leaves. Since it is of class
$C^{1}$, the derivative $Dh_{f,x,y}^{u}$ induces a continuous map between the
unit tangent bundles $Dh_{f,x,y*}^{u}:T^{1}{\mathcal{F}}^{c}_{f}(x)\rightarrow
T^{1}{\mathcal{F}}^{c}_{f}(y)$.

######  Lemma 2.1.

$Dh_{f,x,y*}^{u}$ is continuous with respect to $f\in\mathcal{SH}^{2}_{b}$
(the $C^{2}$ topology) and $(x,y)\in{\mathbb{T}}^{4},\ \
y\in{\mathcal{F}}^{u}_{f}(x)$. The same holds for the stable holonomy.

######  Remark 2.2.

The continuity in Lemma 2.1 means that if $f_{n}$ converges to $f$ in the
$C^{2}$ topology, $x_{n}$ converges to $x$ in ${\mathbb{T}}^{2}$, and
$y_{n}\in{\mathcal{F}}^{u}_{f_{n},loc}(x_{n})$ converges to $y$, then
$Dh_{f_{n},x_{n},y_{n}*}^{u}$ converges uniformly to $Dh_{f,x,y*}^{u}$. The
proof requires only the weaker global condition 1.

######  Remark 2.3.

Since in our case the stable and unstable bundles are one-dimensional, one
could approach the continuity question using the classical ODE theory of the
regularity of solutions with respect to the initial conditions and parameters.
We prefer to present a different proof which constructs the projectivized
holonomies as unstable foliations of the projectivization of $f$ along the
center bundle.

###### Proof.

Let $T^{1}{\mathbb{T}}^{4}$ be the unit tangent bundle of ${\mathbb{T}}^{4}$
(which can be identified with ${\mathbb{T}}^{4}\times\mathbb{S}^{3}$) with
$Df_{*}$ being the $C^{1}$ diffeomorphism induced by $f$. We will consider the
central unit tangent bundle $S_{f}:=\cup_{x\in{\mathbb{T}}^{4}}S(f,x)$ where
$S(f,x)=T_{x}^{1}{\mathcal{F}}^{c}_{f}(x)$ is the unit circle in
$E^{c}_{f}(x)$. Then $S_{f}$ is a Hölder submanifold of
$T^{1}{\mathbb{T}}^{4}$ invariant under $Df_{*}$, which is also a Hölder
bundle over ${\mathbb{T}}^{4}$.

We claim that there exists a continuous unstable foliation on $S_{f}$ and that
$Dh_{f,x,y*}^{u}$ is exactly the unstable holonomy for this foliation between
the transversals $T^{1}{\mathcal{F}}^{c}_{f}(x)$ and
$T^{1}{\mathcal{F}}^{c}_{f}(y)$. We apply the standard construction of the
local unstable leaves as the invariant section of a bundle contraction map
(see [12] for example), with a minor difficulty arising from the lack of
smoothness.

For any $x,y\in{\mathbb{T}}^{4}$ we define the
$\pi_{f,y,x}:E^{c}_{f}(y)\rightarrow E^{c}_{f}(x)$ the projection parallel to
$E^{s}_{f}(x)\oplus E^{u}_{f}(x)$. The maps $\pi_{f,y,x}$ depend continuously
on $x,y\in{\mathbb{T}}^{4}$ and $f$ (in the $C^{1}$ topology). For $x$ close
to $y$ this is invertible and close to the identity, and its projectivization
$\pi_{f,x,y*}$ is bi-Lipschitz with Lipschitz constant close to 1.

For $\delta>0$ and $x\in{\mathbb{T}}^{4}$ let
$\alpha_{f,x}:[-\delta,\delta]\rightarrow{\mathcal{F}}^{u}_{f}(x)$ be the
length parametrization of the local unstable manifold of $f$ at $x$. Since the
unstable foliation is orientable and depends continuously in the $C^{1}$
topology with respect to $x$ and $f$, we have that $\alpha_{f,x}$ is
continuous in $x$ and $f$ (in the $C^{1}$ topology).

For any $\delta>0$ there exists $\epsilon_{\delta}>0$ such that for any $x,y$
such that $d(x,y)<\delta$ we have:

  * •

$\|\pi_{f,x,y}^{\pm 1}-Id\|<\epsilon_{\delta}$,

  * •

$\pi_{f,x,y*}^{\pm 1}$ is bi-Lipschitz with constant $(1+\epsilon_{\delta})$,

  * •

$(1+\epsilon_{\delta})^{-1}\lambda_{c}^{-}(x)<\lambda_{c}^{-}(y)\leq\lambda_{c}^{+}(y)<(1+\epsilon_{\delta})\lambda_{c}^{+}(x)$,

  * •

If furthermore $y\in{\mathcal{F}}^{u}_{f,\delta}(x)$ then
$d_{u}(f(x),f(y))\geq(1+\epsilon_{\delta})^{-1}\lambda_{u}(x)d_{u}(x,y)$,
where $d_{u}$ is the distance along the unstable leaves.

We can choose $\epsilon_{\delta}$ independent of $f$ in a $C^{1}$ neighborhood
and $\lim_{\delta\rightarrow 0}\epsilon_{\delta}=0$.

Now we will construct the bundle with the candidates for the local unstable
manifolds in $S_{f}$. Consider $\delta>0$ (small) to be specified later. Let

$B=\\{\sigma:[-\delta,\delta])\rightarrow\mathbb{R}:\ \sigma(0)=0,\
\left|\frac{\sigma(t)}{t}\right|<\infty\\}.$

the Banach space of functions $\sigma$ bounded for the norm

$\|\sigma\|=\sup_{t\in[-\delta,\delta]}\left|\frac{\sigma(t)}{t}\right|.$

Then

$V(f):=S_{f}\times B$

is a continuous (in fact Hölder) Banach bundle over $S_{f}$.

######  Remark 2.4.

The maps $\sigma$ are candidates for unstable manifolds in $S_{f}$ in the
following sense. For any $(x,v)\in S_{f}$ and $\sigma\in B$ we can define a
section $\tilde{\sigma}:{\mathcal{F}}^{u}_{f,\delta}(x)\rightarrow S_{f}$ in
the following way:

$\tilde{\sigma}(y):=\pi_{f,y,x*}^{-1}(v+\sigma(\alpha_{f,x}^{-1}(y)))\in
S(f,y).$

The graph of this section $\tilde{\sigma}$ is a natural candidate for the
local unstable manifold in $(x,v)\in S_{f}$. We construct it as a fixed point
of the natural graph transformation.

Let $T:V(f)\rightarrow V(f)$ be the bundle map which fibers over $Df_{*}$ on
$S_{f}$ and is given by

$\displaystyle T\sigma_{(x,v)}(t)$ $\displaystyle=$
$\displaystyle\left(\pi_{f,f(y(t)),f(x)}\circ
Df(y(t))\circ\pi_{f,y(t),x}^{-1}\right)_{*}(v+\sigma(\alpha_{f,x}^{-1}(y(t)))-Df(x)_{*}(v),$
$\displaystyle y(t)$ $\displaystyle=$ $\displaystyle
f^{-1}\circ\alpha_{f,f(x)}(t).$

One can check that in fact $T$ is defined in such a way so that we have
$\tilde{T\sigma}=Df_{*}\tilde{\sigma}$. Let us check that $T$ is a continuous
bundle map on $V(f)$, which is also a fiber contraction.

Claim 1: If $\sigma\in B$ then $T\sigma_{(x,v)}\in B$.

Remember that $y(t)=f^{-1}\circ\alpha_{f,f(x)}(t)$, and let us denote

$G(t):=\left(\pi_{f,f(y(t)),f(x)}\circ Df(y(t))\circ\pi_{f,x,y}^{-1}\right).$

Observe that $G(t)_{*}$ is Lispchitz with the Lipschitz constant

$Lip(G(t)_{*})=(1+\epsilon_{\delta})^{2}\frac{\lambda_{c}^{+}(y(t))}{\lambda_{c}^{-}(y(t))}\leq(1+\epsilon_{\delta})^{4}\frac{\lambda_{c}^{+}(x)}{\lambda_{c}^{-}(x)}.$

Also

$|\alpha_{f,x}^{-1}(y(t))|=d_{u}(x,y(t))\leq(1+\epsilon_{\delta})\lambda_{u}(x)^{-1}d_{u}(f(x),f(y(t))=(1+\epsilon_{\delta})\lambda_{u}(x)^{-1}|t|.$

Then

$\displaystyle\|T\sigma\|$ $\displaystyle=$
$\displaystyle\sup_{t\in[-\delta.\delta]}\left|\frac{G(t)_{*}(v+\sigma(\alpha_{f,x}^{-1}(y(t)))-Df(x)_{*}(v)}{t}\right|$
$\displaystyle\leq$
$\displaystyle\sup_{t\in[-\delta.\delta]}\left|\frac{G(t)_{*}(v+\sigma(\alpha_{f,x}^{-1}(y(t)))-G(t)_{*}(v)}{t}\right|+\sup_{t\in[-\delta.\delta]}\left|\frac{G(t)_{*}(v)-Df(x)_{*}(v)}{t}\right|$
$\displaystyle\leq$
$\displaystyle(1+\epsilon_{\delta})^{4}\frac{\lambda_{c}^{+}(x)}{\lambda_{c}^{-}(x)}\sup_{t\in[-\delta.\delta]}\left|\frac{\sigma(\alpha_{f,x}^{-1}(y(t))}{t}\right|+\frac{\pi}{2}\sup_{t\in[-\delta.\delta]}\frac{1}{t}\left\|\frac{G(t)(v)}{\|G(t)(v)\|}-\frac{Df(x)(v)}{\|Df(x)(v)\|}\right\|$
$\displaystyle\leq$
$\displaystyle(1+\epsilon_{\delta})^{5}\frac{\lambda_{c}^{+}(x)}{\lambda_{c}^{-}(x)\lambda_{u}(x)}\|\sigma\|+\frac{\pi}{\lambda_{c}^{+}(x)}\sup_{t\in[-\delta.\delta]}\left\|\frac{G(t)(v)-Df(x)(v)}{t}\right\|,$

where in the last line we used the inequality

$\left\|\frac{a}{\|a\|}-\frac{b}{\|b\|}\right\|\leq\left\|\frac{a}{\|a\|}-\frac{a}{\|b\|}\right\|+\left\|\frac{a}{\|b\|}-\frac{b}{\|b\|}\right\|\leq\frac{2}{\|b\|}\|a-b\|.$

Let us remark that if $v\in E^{c}_{f}(x)$ then $\pi_{f,f(y(t)),f(x)}\circ
Df(x)\circ\pi_{f,y,x}^{-1}(v)=Df(x)(v)$, because the partially hyperbolic
splitting is invariant under $Df$. Then

$\displaystyle\|G(t)(v)-Df(x)(v)\|$ $\displaystyle=$
$\displaystyle\|\pi_{f,f(y(t)),f(x)}\circ(Df(y(t))-Df(x))\circ\pi_{f,y,x}^{-1}(v)\|$
$\displaystyle\leq$ $\displaystyle(1+\epsilon_{\delta})^{2}Lip(Df)d(y(t),x)$
$\displaystyle\leq$ $\displaystyle
Lip(Df)\frac{(1+\epsilon_{\delta})^{3}}{\lambda_{u}(x)}|t|.$

Finally we obtain the desired bound:

$\|T\sigma\|\leq\frac{(1+\epsilon_{\delta})^{5}\lambda_{c}^{+}(x)}{\lambda_{c}^{-}(x)\lambda_{u}(x)}\|\sigma\|+\frac{Lip(Df)(1+\epsilon_{\delta})^{3}\pi}{\lambda_{c}^{+}(x)\lambda_{u}(x)}.$

Claim 2: $T$ is a fiber contraction.

$\displaystyle\|T\sigma_{1}-T\sigma_{2}\|$ $\displaystyle=$
$\displaystyle\sup_{t\in[-\delta,\delta]}\left|\frac{T\sigma_{1}(t)-T\sigma_{2}(t)}{t}\right|$
$\displaystyle=$
$\displaystyle\sup_{t\in[-\delta,\delta]}\left|\frac{G(t)_{*}(v+\sigma_{1}(\alpha_{f,x}^{-1}(y(t)))-G(t)_{*}(v+\sigma_{2}(\alpha_{f,x}^{-1}(y(t)))}{t}\right|$
$\displaystyle\leq$
$\displaystyle(1+\epsilon_{\delta})^{4}\frac{\lambda_{c}^{+}(x)}{\lambda_{c}^{-}(x)}\sup_{t\in[-\delta,\delta]}\left|\frac{\sigma_{1}(\alpha_{f,x}^{-1}(y(t))-\sigma_{2}(\alpha_{f,x}^{-1}(y(t))}{t}\right|$
$\displaystyle\leq$
$\displaystyle(1+\epsilon_{\delta})^{5}\frac{\lambda_{c}^{+}(x)}{\lambda_{c}^{-}(x)\lambda_{u}(x)}\|\sigma_{1}-\sigma_{2}\|.$

Now all we have to do is to choose $\delta$ small enough so that
$\epsilon_{\delta}$ is close enough to zero so that we have

$(1+\epsilon_{\delta})^{5}\frac{\lambda_{c}^{+}(x)}{\lambda_{c}^{-}(x)\lambda_{u}(x)}<1.$

Claims 1 and 2 show that we are in the conditions of the Invariant Section
Theorem from [12], so there exists a unique bounded continuous invariant
section.

From [22] we know that the unstable holonomy along center leaves is uniformly
differentiable. The projectivisation of the derivative of this local holonomy
will then correspond to a bounded invariant section for the transfer operator
$T$, so it has to coincide with the unique continuous invariant section
constructed above. This concludes the proof of the continuity of
$Dh^{u}_{f,x,y*}$ with respect to the points $x,y$ (we proved it for
$d(x,y)<\delta$, but this can be easily extended to larger distances).

If $f_{n}$ converges to $f$ then $S_{f_{n}}$ converges to $S_{f}$ (this can be
made explicit by projecting $E_{f_{n}}^{c}$ to $E^{c}_{f}$ parallel to
$E_{f}^{s}\oplus E_{f}^{u}$ for example). One can check that the corresponding
transfer operators $T_{f_{n}}$ also converge to $T_{f}$. Since the invariant
section is continuous with respect to the bundle map, we obtain the continuity
of $Dh^{u}_{f,x,y*}$ with respect to $f$.

∎

######  Remark 2.5.

We gave the proof for our special setting, but the proof can be adapted to
general partially hyperbolic deffeomorphisms in higher dimensions. We used
that $Df$ is Lipschitz in order to show that the transfer operator $T$
verifies the conditions of the Invariant Section Theorem. The proof can be
adapted for $f$ of class $C^{1+\alpha}$ and the stronger bunching condition
$\lambda_{s}(x)^{\alpha}<\frac{\lambda_{c}^{-}(x)}{\lambda_{c}^{+}(x)}\leq\frac{\lambda_{c}^{+}(x)}{\lambda_{c}^{-}(x)}<\lambda_{u}(x)^{\alpha}$,
using the norm
$\|\sigma\|=\sup_{t\in[-\delta,\delta]}\left|\frac{\sigma(t)}{t^{\alpha}}\right|$.
Once one obtains the bounded invariant section for the projectivization
$Df_{*}$ on $S_{f}$, using the boundness of the Jacobian, one could try to
obtain the differentiability of the stable/unstable holonomy along center
leaves.

###  2.2. Homoclinic holonomies

Let $f\in\mathcal{SH}^{2}_{b}$ and $p_{f}$ the fixed point such that
$f\mid_{{\mathcal{F}}^{c}_{f}(p_{f})}$ is Anosov. We will drop the index $f$
when it is not necessary to specify the dependence on the map $f$. From [12]
and the bunching conditions we know that
${\mathcal{F}}^{c}(p),{\mathcal{F}}^{cu}(p)$ and ${\mathcal{F}}^{cs}(p)$ are
$C^{2}$ submanifolds. Assume that $q$ is a homoclinic point of $W^{c}(p)$,
i.e. $q\in{\mathcal{F}}^{cu}(p)\cap{\mathcal{F}}^{cs}(p)$, then $W^{c}(q)$ is
also $C^{2}$ as a connected component of the intersection of the transverse
$C^{2}$ submanifolds ${\mathcal{F}}^{cu}(p)$ and ${\mathcal{F}}^{cs}(p)$. We
can define the stable holonomy
$h_{p,q}^{s}:{\mathcal{F}}^{c}(p)\rightarrow{\mathcal{F}}^{c}(q)$ and the
unstable holonomy
$h_{q,p}^{u}:{\mathcal{F}}^{c}(q)\rightarrow{\mathcal{F}}^{c}(p)$, and they
are both of class $C^{1}$. Then $\tilde{h}_{q}:=h_{q,p}^{u}\circ
h_{p,q}^{s}:{\mathcal{F}}^{c}(p)\rightarrow{\mathcal{F}}^{c}(p)$ is a $C^{1}$
diffeomorphism, so it induces a $C^{0}$ map on the unit tangent bundle
$T^{1}{\mathcal{F}}^{c}(p)$ which we denote by $D\tilde{h}_{q*}$.

Let $\tilde{v}^{s}(x)$ be the unit vector tangent in
$x\in{\mathcal{F}}^{c}(p)$ to the stable bundle of
$f\mid_{{\mathcal{F}}^{c}(p)}$ (we fix an orientation). Since
$f\mid_{{\mathcal{F}}^{c}(p)}$ is a $C^{2}$ Anosov map on a $C^{2}$ surface,
we have that $\tilde{v}^{s}:{\mathcal{F}}^{c}(p)\rightarrow
T^{1}{\mathcal{F}}^{c}(p)$ is $C^{1}$. We define the map
$\tilde{g}_{q}:{\mathcal{F}}^{c}(p)\rightarrow T^{1}{\mathcal{F}}^{c}(p)$,

(6)
$\tilde{g}_{q}(x):=D\tilde{h}_{q*}(\tilde{v}^{s}(h_{q}^{-1}(x)))=\frac{D\tilde{h}_{q}(h_{q}^{-1}(x))\tilde{v}^{s}(h_{q}^{-1}(x))}{\|D\tilde{h}_{q}(h_{q}^{-1}(x))\tilde{v}^{s}(h_{q}^{-1}(x))\|}.$

######  Remark 2.6.

In fact we consider the stable foliation of $f\mid_{{\mathcal{F}}^{c}(p)}$
inside the leaf ${\mathcal{F}}^{c}(p)$, we first push it forward using the
stable holonomy $h_{p,q}^{s}$ to the leaf ${\mathcal{F}}^{c}(q)$, and then we
push it again using the unstable holonomy $h_{q,p}^{u}$ back to the leaf
${\mathcal{F}}^{c}(p)$. Then $\tilde{g}_{q}(x)$ is in fact the unit tangent
vector in $x$ to this new foliation.

If furthermore $f\in\mathcal{SH}^{3}_{b}$ then the stable and unstable
holonomies along the fixed center-stable leaf $W^{cs}(p)$ and respectively the
fixed center-unstable leaf $W^{cu}(p)$ are $C^{2}$, so in this case
$D\tilde{h}_{q*}$ and $\tilde{g}_{q}$ are in fact $C^{1}$.

If the map $f^{\prime}$ is $C^{2}$ close to $f$, then the fixed Anosov center
leaf ${\mathcal{F}}^{c}(p)$ and its homoclinic center leaf
${\mathcal{F}}^{c}(q)$ will have continuations
${\mathcal{F}}^{c}_{f^{\prime}}(p(f^{\prime}))$ and
${\mathcal{F}}^{c}_{f^{\prime}}(q(f^{\prime}))$. Then we obtain the
continuations of the stable holonomy
$h_{p(f^{\prime}),q(f^{\prime}),f^{\prime}}^{s}:{\mathcal{F}}^{c}_{f^{\prime}}(p(f^{\prime}))\rightarrow{\mathcal{F}}^{c}_{f^{\prime}}(q(f^{\prime}))$
and the unstable holonomy
$h_{q(f^{\prime}),p(f^{\prime}),f^{\prime}}^{u}:{\mathcal{F}}^{c}_{f^{\prime}}(q(f^{\prime}))\rightarrow{\mathcal{F}}^{c}_{f^{\prime}}(p(f^{\prime}))$;
they are $C^{1}$ maps and depend continuously in the $C^{1}$ topology with
respect to $f^{\prime}$ (in the $C^{2}$ topology). We also have a continuation
of the homoclinic holonomy
$\tilde{h}_{q(f^{\prime}),f^{\prime}}:{\mathcal{F}}^{c}_{f^{\prime}}(p(f^{\prime}))\rightarrow{\mathcal{F}}^{c}_{f^{\prime}}(p(f^{\prime}))$
and also the continuation
$\tilde{g}_{q(f^{\prime}),f^{\prime}}:{\mathcal{F}}^{c}_{f^{\prime}}(p(f^{\prime}))\rightarrow
T^{1}{\mathcal{F}}^{c}_{f^{\prime}}(p(f^{\prime}))$, which is continuous both
with respect to $x\in{\mathcal{F}}^{c}_{f^{\prime}}(p(f^{\prime}))$ and with
respect to $f^{\prime}\in\mathcal{SH}^{2}_{b}$ (in the $C^{2}$ topology).

###  2.3. Hyperbolic measures

Let $\mu$ be an ergodic measure of a diffeomorphism $f$, then by the Theorem
of Oseledets, for $\mu$-almost every point $x\in M$, there exist
$k(\mu)\in\mathbb{N}$, real numbers $\lambda_{1}(\mu)>\cdots\lambda_{k}(\mu)$
and an invariant splitting $T_{x}M=E^{1}(x)\oplus\cdots\oplus E^{k}(x)$ of the
tangent bundle at $x$, depending measurably on the point, such that
$\lim_{n\to\pm\infty}\frac{1}{n}\log\|Df^{n}_{x}(v)\|=\lambda_{j}(\mu)$ for
all $0\neq v\in E^{j}(x)$. The real numbers $\lambda_{j}(\mu)$ are the
_Lyapunov exponents_ of $\mu$. We say that the ergodic measure $\mu$ is
_hyperbolic_ if all the Lyapunov exponents of $\mu$ are non-zero.

######  Theorem 2.7.

[Katok’s Horseshoe Theorem [13]:] For any $f\in{\rm Diff}^{r}(M)$, $r>1$ and
any hyperbolic ergodic measure $\mu$, there exists a hyperbolic periodic point
$p$, such that $\mu\overset{h}{\sim}\delta_{{\rm Orb}(p)}$, where
$\delta_{{\rm Orb}(p)}$ is the ergodic measure supported on the orbit ${\rm
Orb}(p)$.

If a diffeomorphism $f$ admits a dominated splitting, then the Oseledet’s
splitting must be subordinated to the dominated splitting. In particular,
since every $f\in\mathcal{SH}$ is partially hyperbolic on $\mathbb{T}^{4}$,
then for any ergodic measure $\mu$ of $f$, its biggest Lyapunov exponent is
positive ($\lambda^{u}>0$) and its associated Oseledet’s bundle is tangent to
the strong unstable bundle $E^{u}$ of $f$. A similar result holds for the
minimal Lyapunov exponent $\lambda^{s}<0$ with its associated Oseledet’s
bundle tangent to the strong stable bundle $E^{s}$. There are also two center
Lyapunov exponents (counted with multiplies)
$\lambda^{c}_{1}\geq\lambda^{c}_{2}$ whose associated Oseledet’s bundles are
tangent to the center bundle $E^{c}$ of $f$.

###  2.4. Criterion of uniqueness of equilibrium state

######  Definition 2.8.

Let $\mu$ be an ergodic hyperbolic measure of a diffeomorphism $f$. The
_stable index of $\mu$_ is the number of negative Lyapunov exponents, counted
with multiplicity.

######  Proposition 2.9.

Let $f:M\to M$ be a $C^{r}$ diffeomorphism $r>1$,
$\phi:M\rightarrow\mathbb{R}$ be a Hölder potential, and $p$ a hyperbolic
periodic point. Then there is at most one equilibrium state for $\phi$ which
is homoclinically related to $\delta_{Orb(p)}$, and its support coincides with
${\rm HC}(Orb(p))$.

###### Proof.

This is explained in [3][Theroem 2.4] and [7][Section 1.6]. See also [2] and
[7][Corollary 3.3]. ∎

###  2.5. Hyperbolicity of equilibrium states

If $f\in\operatorname{PH}_{A,B}^{0}$ then a standard results of Franks-Manning
[11, 19] imply that $f$ is semi-conjugate to $f_{A,B}$, i.e., there exists a
continuous surjection $h:\mathbb{T}^{4}\to\mathbb{T}^{4}$ homotopic to the
identity such that $f_{A,B}\circ h=h\circ f$. By Ledrappier-Walters
variational principle [16] we have

(7) $h_{top}(f)\geq h_{top}(f_{A,B})=\log\lambda_{A}+\log\lambda_{B}.$

For any invariant probability measure $\mu$ of $f$, we say that a measurable
partition $\xi$ is $\mu$ _adapted (sub-ordinated) to ${\mathcal{F}}^{u}$_ if
the following conditions are satisfied:

  * •

there is $r_{0}>0$ such that $\xi(x)\subset B^{{\mathcal{F}}^{u}}_{r_{0}}(x)$
for $\mu$ almost every $x$, where $B^{{\mathcal{F}}^{u}}_{r_{0}}(x)$ is a ball
of ${\mathcal{F}}^{u}(x)$ with radius $r_{0}$;

  * •

$\xi(x)$ contains an open neighborhood of $x$ inside ${\mathcal{F}}^{u}(x)$;

  * •

$\xi$ is increasing, that is, for $\mu$ almost every $x$, $\xi(x)\subset
f(\xi(f^{-1}(x)))$.

The existence of such a partition was provided by [15] (see also [14] and
[28]). The _partial entropy of $\mu$ along the expanding foliation
${\mathcal{F}}^{u}$_ is defined by

$h_{\mu}(f,{\mathcal{F}}^{u})=H_{\mu}(f^{-1}\xi\mid\xi).$

The definition of the partial entropy does not depend on the choice of the
partition.

The following two lemmas are important for our further discussion:

######  Lemma 2.10.

If $f\in\operatorname{PH}_{A,B}^{0}$, then
$h_{\mu}(f,{\mathcal{F}}^{u})\leq\log\lambda_{A}$.

###### Proof.

Denote by ${\mathcal{F}}^{c}_{f}$ the center foliation of $f$. By Proposition
1.3, the projection map $\pi^{c}_{f}$ along the center foliation induces a
topological Anosov homeomorphism $\overline{f}$ on the quotient space
$\overline{{\mathbb{T}}}^{2}_{f}={\mathbb{T}}^{4}/{\mathcal{F}}^{c}_{f}$,
which is topological conjugate to $A$, so we may in fact identify
$\overline{{\mathbb{T}}}^{2}_{f}$ with ${\mathbb{T}}^{2}$ and $\overline{f}$
with $A$.

Denote by ${\mathcal{F}}^{s}_{A}$ (resp. ${\mathcal{F}}^{u}_{A}$) the stable
(resp. unstable) foliation of $A$. The projection map $\pi^{c}_{f}$ maps each
center unstable leaf ${\mathcal{F}}^{cu}$ of $f$ to an unstable leaf
${\mathcal{F}}^{u}_{A}$ of $A$. In particular, $\pi^{c}_{f}$ maps every
unstable leaf ${\mathcal{F}}^{u}$ of $f$ to an unstable leaf
${\mathcal{F}}^{u}_{A}$ of $A$. Proposition 1.3 implies that all the
hypothesis of Tahzibi-Yang [26][Theorem A] are satisfied (see also [1][Section
2.5]), and this implies that $h_{\mu}(f,{\mathcal{F}}^{u})\leq
h_{top}(A)=\log\lambda_{A}$. ∎

The following lemma is a generalization of [1]:

######  Lemma 2.11.

Leb $f\in\operatorname{PH}_{A,B}^{0}$ be a $C^{r}$ diffeomorphism, $r>1$, and
$\mu$ an ergodic invariant measure of $f$ with $h_{\mu}(f)>\log\lambda_{A}$.
Then $\mu$ is an hyperbolic ergodic measure of $f$ with stable index $2$,
i.e., $\lambda^{c}_{1}>0>\lambda^{c}_{2}$.

###### Proof.

We will show that $\lambda^{c}_{1}>0$. To prove that $\lambda^{c}_{2}<0$, one
only needs to consider the diffeomorphism $f^{-1}$ instead of diffeomorphism
$f$.

Suppose by contradiction that $\lambda^{c}_{1}\leq 0$. The entropy formula of
Ledrappier-Young (see [15], [5]) implies that
$h_{\mu}(f)=h_{\mu}(f,{\mathcal{F}}^{u})$.

Combining with the previous lemma, we obtain that
$h_{\mu}(f)\leq\log\lambda_{A}$, which is a contradiction with the hypothesis
that $h_{\mu}(f)>\log\lambda_{A}$. The proof is complete. ∎

##  3\. Proof of theorem A

###  3.1. Definition of $\mathcal{U}$ and plan of the proof

######  Definition 3.1.

Let $\mathcal{U}_{s}^{r}\subset\mathcal{SH}^{r}_{b}$,

(8) $\mathcal{U}_{s}^{r}=\left\\{f\in\mathcal{SH}^{r}_{b}:\ \forall
x\in{\mathcal{F}}^{c}(p),\exists q\hbox{ homoclinic to ${\mathcal{F}}^{c}(p)$
such that }\tilde{g}_{q}(x)\neq\pm\tilde{v}^{s}(x)\right\\}.$

In a similar way we define $\mathcal{U}_{u}^{r}$. Let
$\mathcal{U}^{r}=\mathcal{U}_{s}^{r}\cap\mathcal{U}_{u}^{r}$.

The proof of Theorem A is divided in the following 3 propositions.

######  Proposition 3.2.

$\mathcal{U}^{r}$ is $C^{2}$ open.

###### Proof.

An immediate consequence of the compactness of ${\mathcal{F}}^{c}(p)$ and of
the fact that the stable and unstable holonomies depend continuously in the
$C^{1}$ topology with respect to the points (see Remark 1.7) is the following
lemma.

######  Lemma 3.3.

Let $f\in\mathcal{SH}^{r}_{b}$. Then $f\in\mathcal{U}_{s}^{r}$ if and only if
there exist $q_{1},q_{2},\dots q_{k}$ homoclinic points of
${\mathcal{F}}^{c}(p)$ such that the image of
$\tilde{g}_{q_{1}}\times\tilde{g}_{q_{2}}\times\dots\times\tilde{g}_{q_{k}}$
is disjoint from the image of $\pm\tilde{v}^{s^{k}}$.

On the other hand, the holonomies along the center leaves depend continuous in
the $C^{1}$ topology with respect to the map $f$ (in $C^{2}$ topology), so the
images of
$\tilde{g}_{q_{1}}\times\tilde{g}_{q_{2}}\times\dots\times\tilde{g}_{q_{k}}$
and $\pm\tilde{v}^{s^{k}}$ depend continuously on the map $f$. Since these
images are compact, this concludes the $C^{2}$ openness of
$\mathcal{U}_{s}^{r}$. The proof for $\mathcal{U}_{u}^{r}$ is similar, so
$\mathcal{U}^{r}=\mathcal{U}_{s}^{r}\cap\mathcal{U}_{u}^{r}$ is $C^{2}$ open.
∎

######  Proposition 3.4.

$\mathcal{U}^{r}$ is $C^{r}$ dense.

We will give the proof of the proposition in the subsection 3.2.

######  Proposition 3.5.

If $f\in\mathcal{U}^{r}$, then any two hyperbolic periodic points of $f$ of
index 2 are homoclinically related.

We will give the proof of the proposition in the subsection 3.3

###  3.2. Proof of $C^{r}$ density

We will show that $\mathcal{U}_{s}^{r}$ is $C^{r}$ dense in
$\mathcal{SH}^{r}_{b}$, the proof for $\mathcal{U}_{u}^{r}$ is similar. Then
$\mathcal{U}^{r}$ will be $C^{r}$ dense as the intersection of two $C^{r}$
open dense sets.

The main perturbation result which we will use is the following Lemma.

######  Lemma 3.6.

Let $f\in\mathcal{SH}^{3}_{b}$. Let $q$ be a homoclinic point of the fixed
Anosov leaf ${\mathcal{F}}^{c}(p)$. Then

  1. (1)

For any $C^{2}$ family of perturbations $(\phi_{T})_{T\in\mathbb{R}^{n}}$ of
the identity on $\mathbb{T}^{4}$ ($\phi_{0^{n}}=Id_{\mathbb{T}^{4}}$),
supported in a neighborhood of ${\mathcal{F}}^{c}(q)$ disjoint from all the
other iterates $f^{k}({\mathcal{F}}^{c}(q)),\ k\in\mathbb{Z}\setminus\\{0\\}$,
the map $(T,x)\mapsto\tilde{g}_{q(f\circ\phi_{T}),f\circ\phi_{T}}(x)$ is
$C^{1}$ on $[-\delta,\delta]^{n}\times{\mathcal{F}}^{c}(p)$, for some
$\delta>0$.

  2. (2)

For any $x_{0}\in{\mathcal{F}}^{c}(p)$ and any $r_{0}>0$ there exists a
$C^{\infty}$ family $(\phi_{t})_{t\in[-\delta,\delta]}$ of (volume-preserving)
perturbations of the identity on $\mathbb{T}^{4}$, supported in
$B(y_{0},r_{0})$ where $y_{0}:=h_{p,q}^{u}(x_{0})\in{\mathcal{F}}^{c}(q)$,
such that

(9) $\frac{\partial}{\partial
t}\tilde{g}_{q(f\circ\phi_{t}),f\circ\phi_{t}}(x)\mid_{(x,t)=(x_{0},0)}\neq
0.$

###### Proof.

Part (1). Since ${\mathcal{F}}^{c}(p)$ is compact, it is enough to prove that
$\tilde{g}_{q(f\circ\phi_{t}),f\circ\phi_{t}}(x)$ is $C^{1}$ in $(x,T)$ in a
small neighborhood of every point
$(x_{0},0^{n})\in{\mathcal{F}}^{c}(p)\times\mathbb{R}^{n}$.

Let $x_{0}\in{\mathcal{F}}^{c}(p)$ and denote
$y_{0}=h_{p,q}^{u}(x_{0})\in{\mathcal{F}}^{c}(q)$,
$y_{1}=f(y_{0})\in{\mathcal{F}}^{c}(f(q))$,
$z_{1}=h_{f(q),p}^{s}(y_{1})\in{\mathcal{F}}^{c}(p)$ and
$z_{0}=f^{-1}(z_{1})\in{\mathcal{F}}^{c}(p)$. The $f$-invariance of the stable
holonomy implies that $h_{q,p}^{s}(y_{0})=z_{0}$, or
$\tilde{h}_{q}(z_{0})=x_{0}$.

Let $\psi_{p}:\mathbb{T}^{2}\rightarrow{\mathcal{F}}^{c}(p)$ be a $C^{2}$
embedding, $a_{0}=\psi_{p}^{-1}(x_{0}),\ b_{0}=\psi_{p}^{-1}(z_{0})$. Let
$I_{\delta}=[-\delta,\delta]$. There exist $C^{2}$ foliations charts of
${\mathcal{F}}^{s}$ (respectively ${\mathcal{F}}^{u}$) on a small neighborhood
of $y_{1}$ (respectively $y_{0}$) inside ${\mathcal{F}}^{cs}(p)$ (respectively
${\mathcal{F}}^{cu}(p)$):

$\displaystyle\alpha^{s}:B^{c}_{y_{1}}\times I_{\delta}\rightarrow
B^{cs}_{y_{1}},$ $\displaystyle\
\alpha^{s}(\cdot,0)=Id_{{\mathcal{F}}^{c}_{loc}(y_{1})},\
\alpha^{s}(\\{y\\}\times I_{\delta})={\mathcal{F}}^{s}_{loc}(y),\ \forall y\in
B^{c}_{y_{1}};$ $\displaystyle\alpha^{u}:B^{c}_{y_{0}}\times
I_{\delta}\rightarrow B^{cu}_{y_{0}},$ $\displaystyle\
\alpha^{u}(\cdot,0)=Id_{{\mathcal{F}}^{c}_{loc}(y_{0})},\
\alpha^{u}(\\{y\\}\times I_{\delta})={\mathcal{F}}^{u}_{loc}(y),\ \forall y\in
B^{c}_{y_{0}},$

where $B_{x}^{*}$ denotes a small ball centered in $x$ inside
${\mathcal{F}}^{*}(x)$. Define the $C^{2}$ maps

$\displaystyle\beta^{s}:B_{b_{0}}\times I_{\delta}\rightarrow B^{cs}_{y_{1}},$
$\displaystyle\ \beta^{s}(b,s)=\alpha^{s}(h^{s}_{p,f(q)}(f(\psi_{p}(b))),s);$
$\displaystyle\beta^{u}:B_{a_{0}}\times I_{\delta}\rightarrow B^{cu}_{y_{0}},$
$\displaystyle\ \beta^{u}(a,r)=\alpha^{u}(h_{p,q}^{u}(\psi_{p}(a)),r),$

where $B_{x}$ is a small ball centered at $x$ in $\mathbb{T}^{2}$.

We know that the support of $\phi_{T}$ does not intersect
$f^{k}({\mathcal{F}}^{cs}_{loc}(y_{1})),\forall k\geq 0$ and
$f^{l}({\mathcal{F}}^{cu}_{loc}(y_{0})),\forall l<0$. This implies that
${\mathcal{F}}^{cs}_{loc}(y_{1})$ remains a local center-stable leaf for
$f_{T}:=f\circ\phi_{T}$ for all $T$, $\beta^{s}(a,\cdot)$ remain
parametrizations of the strong stable manifolds inside
${\mathcal{F}}^{cs}_{loc}(y_{i})$, ${\mathcal{F}}^{cs}_{loc}(y_{1})$ remains
inside ${\mathcal{F}}^{cs}(p)$ and the stable holonomy between
${\mathcal{F}}^{c}(p)$ and ${\mathcal{F}}^{cs}_{loc}(y_{1})$ is unchanged. A
similar statement holds for ${\mathcal{F}}^{cu}_{loc}(y_{0})$.

The maps $f_{T}$ do change the center leaves ${\mathcal{F}}^{c}(q)$, we have
that
${\mathcal{F}}^{c}(q(f_{T}),f_{T})=f^{-1}({\mathcal{F}}^{s}_{loc}({\mathcal{F}}^{c}(y_{1})))\cap{\mathcal{F}}^{u}_{loc}({\mathcal{F}}^{c}(y_{0}))$.
We can in fact compute implicitly the homoclinic stable-unstable holonomy
$\tilde{h}_{q(f_{T}),f_{T}}$ in a neighborhood of $z_{0}$ in the following
way:

$\displaystyle\psi_{p}(a)=\tilde{h}_{q(f_{T}),f_{T}}(\psi_{p}(b))$
$\displaystyle\iff$ $\displaystyle
h^{u}_{p,q(f_{T}),f_{T}}(\psi_{p}(a))=h^{s}_{p,q(f_{T}),f_{T}}(\psi_{p}(b))$
$\displaystyle\iff$ $\displaystyle
h^{u}_{p,q(f_{T}),f_{T}}(\psi_{p}(a))=f_{T}^{-1}(h^{s}_{p,q(f_{T}),f_{T}}(f(\psi_{p}(b)))$
$\displaystyle\iff$
$\displaystyle\beta^{u}(a,r)=f_{T}^{-1}(\beta^{s}(b,s))\hbox{ for some }r,s\in
I_{\delta}$ $\displaystyle\iff$
$\displaystyle\phi_{T}(\beta^{u}(a,r))=f^{-1}(\beta^{s}(b,s))\hbox{ for some
}r,s\in I_{\delta}.$

In conclusion, denoting
$h_{T}=\psi_{p}^{-1}\circ\tilde{h}_{q(f_{T}),f_{T}}\circ\psi_{p}$ (the map
$\tilde{h}_{q(f_{T}),f_{T}}$ in the chart $\psi_{p}$) we have

(10) $a=h_{T}(b)\iff\phi_{T}(\beta^{u}(a,r))=f^{-1}(\beta^{s}(b,s))\hbox{ for
some }r,s\in I_{\delta}.$

We choose a $C^{\infty}$ chart $\psi_{q}:B_{y_{0}}\rightarrow\mathbb{R}^{4}$
(can also be volume preserving) such that

  * •

$\psi_{q}(y_{0})=0^{4}$;

  * •

$D\psi_{q}(y_{0})(E^{c}(q_{0}))=sp\\{e_{1},e_{2}\\}$;

  * •

$D\psi_{q}(y_{0})(E^{u}(q_{0}))=sp\\{e_{3}\\}$;

  * •

$D\psi_{q}(y_{0})(E^{s}(q_{0}))=sp\\{e_{4}\\}$.

Let $E:B_{a_{0}}\times B_{b_{0}}\times
I_{\delta}^{n+2}\rightarrow\mathbb{R}^{4}$,

(11)
$E(a,b,r,s,T)=\psi_{q}(\phi_{T}(\beta^{u}(a,r)))-\psi_{q}(f^{-1}(\beta^{s}(b,s))).$

We have that $E$ is $C^{2}$ and
$E(a_{0},b_{0},0,0,0^{n})=\psi_{q}(y_{0})-\psi_{q}(f^{-1}(y_{1}))=0$.

Claim: $\frac{\partial E}{\partial(b,r,s)}(a_{0},b_{0},0,0,0^{N})$ is
invertible.
We observe that since $\alpha^{s}$ is a diffeomorphism such that
$\alpha^{s}(\\{y\\}\times I_{\delta})={\mathcal{F}}^{s}_{loc}(y)$, we have
that $D\alpha(y,0)\cdot\frac{\partial}{\partial s}$ is a nonzero vector in
$E^{s}(y)$. Since $Df$ preserves $E^{s}$ and $D\psi_{q}(y_{0})$ takes
$E^{s}(q_{0})$ to the line generated by $e_{4}$, we have that
$DE(a_{0},b_{0},0,0,0^{n})$ takes the line generated by
$\frac{\partial}{\partial s}$ isomorphically to the line generated by $e_{4}$.

A similar argument shows that $DE(a_{0},b_{0},0,0,0^{n})$ takes the line
generated by $\frac{\partial}{\partial r}$ isomorphically to the line
generated by $e_{3}$ (remember that $\phi_{0^{n}}=Id_{\mathbb{T}^{4}}$).

Now let us analyze the action of $DE(a_{0},b_{0},0,0,0^{n})$ on the two-
dimensional space $T_{b_{0}}B_{b_{0}}$. It is not hard to see that
$D\beta^{s}(b_{0},0)$ takes $T_{b_{0}}B_{b_{0}}$ isomorphically to
$E^{c}(y_{1})$. Since $Df$ preserves $E^{c}$ and $D\psi_{q}(y_{0})$ takes
$E^{c}(q_{0})$ to the plane generated by $e_{1}$ and $e_{2}$, we have that
$DE(a_{0},b_{0},0,0,0^{n})$ takes $T_{b_{0}}B_{b_{0}}$ isomorphically to the
plane generated by $e_{1}$ and $e_{2}$. This concludes the proof of the claim.

Now let us finish the proof of the first part of the lemma. The Implicit
Function Theorem gives us the existence of a $C^{2}$ function
$H:B_{a_{0}}\times I_{\delta}^{N}\rightarrow B_{b_{0}}\times I_{\delta}^{2}$,
$H(a,T)=(h(a,T),r(a,T),s(a,T))$ such that $E(a,h(a,T),r(a,T),s(a,T),T)=0$
(eventually by making smaller the balls and the intervals). Then the map
$h_{T}(a)=h(a,T)$ is $C^{2}$ in both variables, which means that
$\tilde{h}_{q(f_{T}),f_{T}}(x)$ is $C^{2}$ in both variables, and then
$\tilde{g}_{q(f\circ\phi_{T}),f\circ\phi_{T}}(x)$ is $C^{1}$ in both
variables. This finishes the proof of the first part.

Part (2). We will use the same notations from part (1). Let
$\rho:\mathbb{R}^{4}\rightarrow[0,\infty)$ be a smooth bump function supported
on a small ball centered at the origin, and constantly equal to one near the
origin. The family $\phi_{t}:\mathbb{T}^{4}\rightarrow\mathbb{T}^{4}$ is
defined as

$\phi_{t}:=\psi_{q}^{-1}\circ(R_{\rho t}\times
Id_{\mathbb{R}^{2}})\circ\psi_{q},$

where $R_{t}$ is the rotation of angle $t$ in $\mathbb{R}^{2}$. Assume that
the support of $\rho$ is small enough so that the support of $\phi_{t}$ is
inside $B(y_{0},r_{0})$ and disjoint of all the other iterates of $W^{c}(q)$.
From part (1) we have

(12) $E(a,b,r,s,t)=(R_{\rho t}\times
Id|{\mathbb{R}^{2}})(\psi_{q}(\beta^{u}(a,r)))-\psi_{q}(f^{-1}(\beta^{s}(b,s)))$

We will compute $DE(a_{0},b_{0},0,0,t)$. Observe that
$L_{b}:=DE(a_{0},b_{0},0,0,t)\mid_{T_{b_{0}}B_{b_{0}}}:T_{b_{0}}B_{B_{0}}\rightarrow
sp\\{e_{1},e_{2}\\}$ and
$L_{s}:=DE(a_{0},b_{0},0,0,t)\mid_{sp\\{\frac{\partial}{\partial
s}\\}}:sp\\{\frac{\partial}{\partial s}\\}\rightarrow sp\\{e_{4}\\}$ are
isomorphisms independent of $t$. Since $D(R_{\rho t}\times
Id_{\mathbb{R}^{2}})$ keeps $e_{3}$ invariant we have that also
$L_{r}:=DE(a_{0},b_{0},0,0,t)\mid_{sp\\{\frac{\partial}{\partial
r}\\}}:sp\\{\frac{\partial}{\partial r}\\}\rightarrow sp\\{e_{3}\\}$ is also
an isomorphism independent of $t$, and

$\frac{\partial E}{\partial(b,r,s)}(a_{0},b_{0},0,0,t)=L_{b}\times L_{r}\times
L_{s}.$

From 12 we can compute

$DE(a_{0},b_{0},0,0,t)\mid_{T_{a_{0}}B_{a_{0}}}=R_{t}\circ
L_{a}:T_{a_{0}}B_{a_{0}}\rightarrow sp\\{e_{1},e_{2}\\},$

where
$L_{a}:=DE(a_{0},b_{0},0,0,0)\mid_{T_{a_{0}}B_{a_{0}}}:T_{a_{0}}B_{a_{0}}\rightarrow
sp\\{e_{1},e_{2}\\}$ is an isomorphism. From the Implicit Function Theorem we
deduce that

(13) $Dh_{t}(a_{0})=L_{b}^{-1}\circ R_{t}\circ L_{a}.$

Define $g:B_{a_{0}}\times I_{\delta}\rightarrow\mathbb{T}^{1}$,

$g(a,t):=D\psi_{p}^{-1}(a)_{*}\tilde{g}_{q(f\circ\phi_{t}),f\circ\phi_{t}}(\psi(a))=\frac{Dh_{t}(a)(v^{s}(a))}{\|Dh_{t}(a)(v^{s}(a))\|}$

where $D\psi_{p}^{-1}(a)_{*}$ is the diffeomorphism induced by
$D\psi_{p}^{-1}(a)$ on the unit tangent bundles and
$v^{s}(a)=D\psi_{p}^{-1}(a)(\tilde{v}^{s}(\psi_{p}(a)))$. In other words,
$g(\cdot,t)$ is in fact the map $\tilde{g}_{q(f\circ\phi_{t}),f\circ\phi_{t}}$
seen in the chart $\psi_{p}$ which identifies $W^{c}(p)$ with $\mathbb{T}^{2}$
and the unit tangent spaces to $W^{c}(p)$ with $\mathbb{T}^{1}$. In order to
prove 9 it is enough to show that

$\frac{\partial}{\partial t}g(a,t)\mid_{(a,t)=(a_{0},0)}\neq 0,$

which in turns is equivalent to the fact that $Dh_{0}(a_{0})(v^{s}(a_{0}))$
and $\frac{\partial}{\partial t}Dh_{t}(a_{0})(v^{s}(a_{0}))\mid_{t=0}$ are not
collinear. Using 13 we obtain $Dh_{0}(a_{0})(v^{s}(a_{0}))=L_{b}^{-1}\circ
L_{a}(v^{s}(a_{0}))$ and $\frac{\partial}{\partial
t}Dh_{t}(a_{0})(v^{s}(a_{0}))\mid_{t=0}=L_{b}^{-1}\circ R_{\frac{\pi}{2}}\circ
L_{a}(v^{s}(a_{0}))$, which are clearly non-collinear since $L_{a}$ and
$L_{b}$ are isomorphisms while $v^{s}(a_{0})$ is non-zero. This finishes the
proof of part (2). ∎

Now let us prove Proposition 3.4.

###### Proof of Proposition 3.4.

Let $f\in\mathcal{SH}^{r}_{b}$. We need to find maps in $\mathcal{U}^{r}_{s}$
arbitrarily $C^{r}$ close to $f$. Since the $C^{\infty}$ maps are dense in the
$C^{r}$ maps in the $C^{r}$ topology (even inside the volume preserving
class), we can assume that $f$ is $C^{\infty}$.

Choose $q_{1},q_{2},q_{3}$ homoclinic points of ${\mathcal{F}}^{c}(p)$ such
that the orbits of the homoclinic leaves ${\mathcal{F}}^{c}(q_{i})$ are
mutually disjoint, $i\in\\{1,2,3\\}$.

For any $x\in{\mathcal{F}}^{c}(p)$ and any $1\leq i\leq 3$, there exists
$r_{x,i}>0$ such that if
$y_{i}:=h_{p,q_{i}}^{u}(x)\in{\mathcal{F}}^{c}(q_{i})$, then the ball
$B(y_{i},r_{x,i})$ is disjoint from ${\mathcal{F}}^{c}(p)$, from all the
iterates $f^{k}({\mathcal{F}}(q_{i})),\forall k\neq 0$, and from all the
iterates of ${\mathcal{F}}^{c}(q_{j})$, $j\neq i$. Applying Lemma 3.6 part (2)
we obtain the family of perturbations $\phi_{t,x,i}$ such that the derivative
of $\tilde{g}_{q_{i},f\circ\phi_{t,x,i}}$ with respect to $t$ in $(x,0)$ does
not vanish. By the continuity of the derivative there exists a neighborhood
$U_{x,i}$ of $x$ such that $\frac{\partial}{\partial
t}\tilde{g}_{q_{i},f\circ\phi_{t,x,i}}$ is nonzero on
$\overline{U}_{x,i}\times\\{0\\}$.

Let $U_{x}=\cap_{i=1}^{3}U_{x,i}$. By compactness of ${\mathcal{F}}^{c}(p)$
there exist finitely many $x^{1},x^{2},\dots x^{K}\in{\mathcal{F}}^{c}(p)$
such that ${\mathcal{F}}^{c}(p)=\cup_{j=1}^{K}U_{x^{j}}$.

Let us fix some notations. Denote

$T=(t_{i}^{j})_{1\leq i\leq 3,\\\ 1\leq j\leq K}=(T_{i})_{1\leq i\leq
3}=(T^{j})_{1\leq j\leq K}\in I^{3K}:=[-\delta,\delta]^{3K},$

with $T_{i}=(t_{i}^{j})_{1\leq j\leq K}\in I^{K}$, $i\in\\{1,2,3\\}$ and
$T^{j}=(t_{i}^{j})_{1\leq i\leq 3}\in I^{3}$, $j\in\\{1,2,\dots K\\}$.

For every $1\leq i\leq 3$ we let $\phi_{i}:\mathbb{T}^{4}\times
I^{K}\rightarrow\mathbb{T}^{4}$ given by

(14)
$\phi_{i}(\cdot,T_{i})=\phi_{t_{i}^{1},x^{1},i}\circ\phi_{t_{i}^{2},x^{2},i}\circ\dots\circ\phi_{t_{i}^{K},x^{K},i},\
\ \forall T_{i}\in I^{K}.$

We define $\phi,\ F:\mathbb{T}^{4}\times I^{3K}\rightarrow\mathbb{T}^{4}$,

$\displaystyle\phi(\cdot,T)=\phi_{T}(\cdot)$ $\displaystyle:=$
$\displaystyle\phi_{1}(\cdot,T_{1})\circ\phi_{2}(\cdot,T_{2})\circ\phi_{3}(\cdot,T_{3}),$
$\displaystyle F(\cdot,T)=F_{T}(\cdot)$ $\displaystyle:=$ $\displaystyle
f\circ\phi_{T}.$

The maps $\phi_{i}$, $\phi$ and $F$ have the following properties:

  1. (1)

$\phi_{i}$, $\phi$ and $F$ are of class $C^{\infty}$ on $(x,T)$;

  2. (2)

$\phi_{i}$ is a small perturbation of the identity on a small neighborhood of
${\mathcal{F}}^{c}(q_{i})$, in particular it leaves the other homoclinic
orbits of ${\mathcal{F}}^{c}(q_{j})$ unchanged for $j\neq i$;

  3. (3)

$F_{T}$ is equal to $f$ on a neighborhood of ${\mathcal{F}}^{c}(p)$, so it
does not change ${\mathcal{F}}^{c}(p)$ and the function $\tilde{v}^{s}$;

Let $V_{j}=\psi_{p}^{-1}(U_{x^{j}})$, where
$\psi_{p}:\mathbb{T}^{2}\rightarrow{\mathcal{F}}^{c}(p)$ is the $C^{2}$
embedding. For every $1\leq i\leq 3$ define $g_{i}:\mathbb{T}^{2}\times
I^{3K}\rightarrow\mathbb{T}^{1}$,

$g_{i}(x,T)=D\psi_{p}^{-1}(x)_{*}\tilde{g}_{q_{i}(f_{T}),f_{T}}(\psi_{p}(x)).$

In other words, $g_{i}(\cdot,T)$ is again the map
$\tilde{g}_{q_{i}(f_{T}),f_{T}}$ seen in the chart $\psi_{p}$ which identifies
${\mathcal{F}}^{c}(p)$ with $\mathbb{T}^{2}$ and the unit tangent spaces
$T^{1}{\mathcal{F}}^{c}(p)$ with $\mathbb{T}^{1}$. Lemma 3.6 part (1) tells us
that $g_{i}$ is $C^{1}$ with respect to $(x,T)\in\mathbb{T}^{2}\times I^{3K}$
(maybe for a smaller interval $I$). Furthermore

(15) $\frac{\partial g_{i}}{\partial t_{i}^{j}}(x,T)\neq 0,\
\forall(x,T)\in\overline{V}_{j}\times\\{0\\}^{3K},\ \forall 1\leq j\leq K.$

On the other hand, because for $l\neq i$, the perturbation $\phi_{l}$ does not
touch the orbit of ${\mathcal{F}}^{c}(q_{i})$, we have

(16) $\frac{\partial g_{i}}{\partial t_{l}^{j}}(x,T)=0,\ \forall(x,T),\forall
l\neq i,\ \forall 1\leq j\leq K.$

Define $G:\mathbb{T}^{2}\times I^{3K}\rightarrow\mathbb{T}^{3}$

(17) $G(x,T)=(g_{1}(x,T),g_{2}(x,T),g_{3}(x,T)).$

Again $G$ is $C^{1}$ in $(x,T)\in\mathbb{T}^{2}\times I^{3K}$. The formulas 15
and 16 tell us that for every $1\leq j\leq K$ we have

$\det\left(\frac{\partial G}{\partial
T^{j}}(x,T)\right)=\det\left(\frac{\partial g_{i}}{\partial
t_{i}^{j}}(x,T)\right)=\prod_{i=1}^{3}\frac{\partial g_{i}}{\partial
t_{i}^{j}}(x,T)\neq 0,\ \forall(x,T)\in\overline{V}_{j}\times\\{0\\}^{3K}$

From the compactness of $\overline{V}_{j}$ and the $C^{1}$ continuity of $G$
with respect to $T$, there exists $0\in J\subset I$ such that, for all $1\leq
j\leq K$ we have

(18) $\det\left(\frac{\partial G}{\partial T^{j}}(x,T)\right)\neq 0,\
\forall(x,T)\in V_{j}\times J^{3K},$

and since every point from $\mathbb{T}^{2}$ is inside some $V_{j}$ we conclude
that $G$ has maximal rank at every point in $\mathbb{T}^{2}\times J^{3K}$.

Remember that $v^{s}:\mathbb{T}^{2}\rightarrow\mathbb{T}^{1}$ is the $C^{1}$
map given by $v^{s}(x)=D\psi_{p}^{-1}(x)_{*}\tilde{v}^{s}(\psi(x))$. Let
$A:=\\{(x,T)\in\mathbb{T}^{2}\times J^{3K}:\
G(x,T)\in\\{-v^{s}(x),v^{s}(x)\\}^{3}\\}$ and $B=\pi_{2}(A)$, where $\pi_{2}$
is the projection from $\mathbb{T}^{2}\times J^{3K}$ on the $T$ component in
$J^{3K}$.

A simple consequence of the above definitions is the fact that if $T\notin B$
then $f_{T}\in\mathcal{U}_{s}^{r}$. In order to finish the proof of the
density of $\mathcal{U}_{s}^{r}$ we have to find $T$ arbitrarily close to
$0^{NK}$ such that $T\notin B$. We will prove in fact that $B$ has Lebesgue
measure zero in $J^{3K}$.

It is enough to show this for $B_{1}=\pi_{2}(A_{1})$ where
$A_{1}=\\{(x,T)\in\mathbb{T}^{2}\times J^{3K}:\ G(x,T)=v^{s}(x)^{3}\\}$, the
other combinations of $\pm v^{s}$ work similarly. Let
$H(x,T)=G(x,T)-v^{s}(x)^{3}$, this is a $C^{1}$ map from $\mathbb{T}^{2}\times
J^{3K}$ to $\mathbb{T}^{3}$. Formula 18 tells us that $H$ has maximal rank
equal to 3 at every point ($v^{s}$ is independent of $T$), so $H^{-1}(0^{3})$
is a $C^{1}$ submanifold of codimension 3 (or dimension $3K-1$) inside
$\mathbb{T}^{2}\times J^{3K}$. Since
$\pi_{2}\mid_{H^{-1}(0^{3})}:H^{-1}(0^{3})\rightarrow J^{3K}$ is a $C^{1}$
map, Sard’s Theorem tells us that the image $B_{1}$ has Lebesgue measure zero.

This implies that we can find arbitrarily small $T\notin B$, which finishes
the proof of the $C^{r}$ density of $\mathcal{U}_{s}^{r}$.

∎

###  3.3. Proof of Proposition 3.5

###### Proof.

We first remark that, because of the transitivity of the homoclinic relation,
it is enough to show that every hyperbolic periodic point of index 2 of
$f\in\mathcal{U}^{r}$ is homoclinically related to the fixed point $p$ of the
hyperbolic fixed leaf ${\mathcal{F}}^{c}(p)$.

Let $x$ be a hyperbolic point of $f\in\mathcal{U}^{r}$ of index 2. Let
$\tilde{v}^{u}(x)$ be the unit tangent vector to the weak unstable direction
inside $T_{x}{\mathcal{F}}^{c}(x)$. The strong unstable manifold
${\mathcal{F}}^{u}(x)$ must accumulate on the fixed hyperbolic leaf
${\mathcal{F}}^{c}(p)$, so there exists a sequence of homoclinic points
$p_{n}\in{\mathcal{F}}^{u}(x)\cap{\mathcal{F}}^{s}_{loc}({\mathcal{F}}^{c}(p))$
such that $\lim_{n\rightarrow\infty}p_{n}=p_{0}\in{\mathcal{F}}^{c}(p)$. If
for some $p_{n}$ we have that $Dh^{u}_{x,p_{n}*}(\tilde{v}^{u}(x))\neq\pm
Dh^{s}_{p,p_{n}*}(\tilde{v}^{s}(h^{s^{-1}}_{p,p_{n}}(p_{n})))$ then the
2-dimensional unstable manifold of $x$, $W^{u}(x)$, intersects
${\mathcal{F}}^{c}(p_{n})$ in a $C^{1}$ curve locally transverse to the weak
stable foliation inside ${\mathcal{F}}^{c}(p_{n})$ (which is the pushed
forward by the stable holonomy of the weak stable foliation in
${\mathcal{F}}^{c}(p)$). Since the 2-dimensional global stable manifold of
$p$, $W^{s}(p)$, is dense inside the weak stable foliation of
${\mathcal{F}}^{c}(p_{n})$, we obtain a transverse homoclinic intersection
from $x$ to $p$.

Suppose that $W^{u}(x)\cap W^{s}(p)=\emptyset$. The above argument implies
that

$Dh^{u}_{x,p_{n}*}(\tilde{v}^{u})=\pm
Dh^{s}_{p,p_{n}*}(\tilde{v}^{s}(h^{s^{-1}}_{p,p_{n}}(p_{n})))\ \hbox{ for all
}\ n\in\mathbb{N}.$

Since $f\in\mathcal{U}$, there exists a homoclinic point $q$ of
${\mathcal{F}}^{c}(p)$ such that
$\tilde{g}_{q}(p_{0})\neq\pm\tilde{v}^{s}(p_{0})\in
T^{1}{\mathcal{F}}^{c}(p)$. Let $q_{0}:=h^{u}_{p,q}(p_{0})$, consider the
strong unstable holonomy
$h^{u}_{loc}:{\mathcal{F}}^{cs}_{loc}(p_{0})\rightarrow{\mathcal{F}}^{cs}_{loc}(q_{0})$
and let $q_{n}:=h^{u}_{loc}(p_{n})\in{\mathcal{F}}^{cs}_{loc}(q_{0})$. Then
$q_{n}\rightarrow q_{0}$. The lack of homoclinic relations between $x$ and $p$
implies that also

$Dh^{u}_{x,q_{n}*}(\tilde{v}^{u})=\pm
Dh^{s}_{p,q_{n}*}(\tilde{v}^{s}(h^{s^{-1}}_{p,q_{n}}(q_{n})))\ \hbox{ for all
}\ n\in\mathbb{N}.$

Since $h_{x,q_{n}}^{u}=h_{p_{n},q_{n}}^{u}\circ h_{x,p_{n}}^{u}$ we obtain
that

$Dh^{s}_{p,q_{n}*}(\tilde{v}^{s}(h^{s^{-1}}_{p,q_{n}}(q_{n})))=\pm
Dh^{u}_{p_{n},q_{n}*}\circ
Dh^{s}_{p,p_{n}*}(\tilde{v}^{s}(h^{s^{-1}}_{p,p_{n}}(p_{n}))).$

Using the continuity of $\tilde{v}^{s}$ and of $Dh^{s,u}$ we can pass to the
limit and obtain that

$Dh^{s}_{p,q*}(\tilde{v}^{s}(h^{s^{-1}}_{p,q}(q_{0})))=\pm
Dh^{u}_{p,q*}(\tilde{v}^{s}(p_{0})),$

or $\tilde{g}_{q}(p_{0})=\pm\tilde{v}^{s}(p_{0})$, which is a contradiction.

The proof of the intersection of the global stable manifold of $x$ with the
global stable manifold of $p$ is similar. This concludes the proof. ∎

##  4\. Proof of Corollary B

###### Proof.

We have to show that for any $f\in\mathcal{U}^{r}$, the transverse homoclinic
intersections of the invariant manifolds of the fixed hyperbolic point $p_{f}$
are dense in ${\mathbb{T}}^{4}$. The proof uses the same ideas from the proof
of Proposition 3.5

Let $f\in\mathcal{U}^{r}$, and $p$ the hyperbolic fixed point of $f$ (for
simplicity we will drop the index $f$ in the following arguments). Let $U$ be
an open set in ${\mathbb{T}}^{4}$. Since $W^{u}(p)\cap
W^{s}({\mathcal{F}}^{c}(p))$ is dense in ${\mathbb{T}}^{4}$, choose $x\in
W^{u}(p)\cap W^{s}({\mathcal{F}}^{c}(p))$ such that $B(x,\delta)\subset U$ for
some $\delta>0$. If $Dh^{s}_{p,x*}(\tilde{v}^{s}(h^{s}_{x,p}(x)))\notin
T_{x}W^{u}(p)$, then clearly there is a transverse homoclinic intersection
between $W^{s}(p)$ and $W^{u}(p)$ arbitrarily close to $x$.

Suppose that $v:=Dh^{s}_{p,x*}(\tilde{v}^{s}(h^{s}_{x,p}(x)))\in
T_{x}W^{u}(p)$. Then there exists a subsequence $n_{k}\rightarrow\infty$ and
$(p_{0},v_{0})\in T^{1}{\mathcal{F}}^{c}(p)$ such that
$Df^{n_{k}}_{*}(x,v)\rightarrow(p_{0},\tilde{v}^{s}(p_{0}))$. There exists a
homoclinic point $q$ of ${\mathcal{F}}^{c}(p)$ such that
$\tilde{g}_{q}(p_{0})\neq\pm\tilde{v}^{s}(p_{0})\in
T^{1}{\mathcal{F}}^{c}(p)$. We consider again the strong unstable holonomy
$h^{u}_{loc}:{\mathcal{F}}^{cs}_{loc}(p_{0})\rightarrow{\mathcal{F}}^{cs}_{loc}(q_{0})$
and let $q_{k}:=h^{u}_{loc}(f^{n_{k}}(x))\in{\mathcal{F}}^{cs}_{loc}(q_{0})$,
where $q_{0}:=h^{u}_{p,q}(p_{0})$. We have that

$Dh^{s}_{p,q*}(\tilde{v}^{s}(h^{s^{-1}}_{p,q}(q_{0})))\neq\pm
Dh^{u}_{p,q*}(\tilde{v}^{s}(p_{0})),$

and by continuity for all $k$ large enough we have

$Dh^{s}_{p,q_{k}*}(\tilde{v}^{s}(h^{s^{-1}}_{p,q_{k}}(q_{k})))\neq\pm
Dh^{u}_{f^{n_{k}}(x),q_{k}*}(\tilde{v}^{s}(f^{n_{k}}(x))).$

Iterating by $f^{-n_{k}}$ and denoting $f^{-n_{k}}(q_{k})=x_{k}\rightarrow x$
we obtain

$Dh^{s}_{p,x_{k}*}(\tilde{v}^{s}(h^{s^{-1}}_{p,x_{k}}(x_{k})))\neq\pm
Dh^{u}_{x,x_{k}*}(\tilde{v}^{s}(x))$

Since $Dh^{u}$ preserves $TW^{u}(p)$ we obtain that
$Dh^{s}_{p,x_{k}*}(\tilde{v}^{s}(h^{s}_{x_{k},p}(x_{k})))\notin
T_{x_{k}}W^{u}(p)$, with $x_{k}\in W^{u}(p)\cap W^{s}({\mathcal{F}}^{c}(p))$,
and this implies again that arbitrarily close to $x_{k}$ (and thus close to
$x$) there are transverse homoclinic intersection between $W^{s}(p)$ and
$W^{u}(p)$. This finishes the proof. ∎

##  5\. Proof of Theorem C

###### Proof.

For simplicity, we may assume

(19) $0\leq\inf\phi\leq\sup\phi\leq\log\lambda_{B}.$

First by the variation principle, there is a sequence of ergodic measures
$\mu_{n}$ of $f$ such that

$\limsup h_{\mu_{n}}=h_{top}(f)\geq\log\lambda_{A}+\log\lambda_{B},$

the last inequality comes from (7).

Again by the variation principle, pressure of the function $\phi$:

$P_{top}(\phi)\geq\limsup(h_{\mu_{n}}+\int\phi d\mu_{n})\geq\limsup
h_{\mu_{n}}\geq\log\lambda_{A}+\log\lambda_{B},$

where the last inequality comes from the assumption that $\phi>0$ ((19)).

Thus for any ergodic measure $\mu$ with pressure sufficient large, i.e.,

(20) $h_{\mu}+\int\phi d\mu>P_{top}(\phi)+(\int\phi d\mu-\log\lambda_{B}),$

we have

$h_{\mu}>P_{top}(\phi)-\log\lambda_{B}\geq\log\lambda_{A}.$

As a consequence of Lemma 2.11, $\mu$ is a hyperbolic measure with stable
index $2$. By Lemma 2.7, $\mu$ is homoclinically related to the atomic measure
supported on a hyperbolic periodic point $O$. Since $f\in{\mathcal{V}}$, by
Theorem A, all the hyperbolic periodic orbits with stable index $2$ are
homoclinically related, as a consequence of Remark 1.11, all the hyperbolic
ergodic measures with stable index $2$ are homoclinically related. In
particular, all the ergodic measures satisfies (20) are homoclinically
related.

Thus, all equilibrium states for the Hölder potential $\phi$ are
homoclinically related, if they do exist. By Proposition 2.9, there exists at
most one equilibrium state. The proof is complete. ∎

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