# Open sets of partially hyperbolic skew products having a unique SRB measure

Davi Obata 111D.O. was partially supported by the projects ANR BEKAM :
ANR-15-CE40-0001 and ERC project 692925 NUHGD.

###### Abstract

In this paper we obtain $C^{2}$-open sets of dissipative, partially hyperbolic
skew products having a unique SRB measure with full support and full basin.
These partially hyperbolic systems have a two dimensional center bundle which
presents both expansion and contraction but does not admit any further
dominated splitting of the center. These systems are non conservative
perturbations of an example introduced by Berger-Carrasco.

To prove the existence of SRB measures for these perturbations, we obtain a
measure rigidity result for $u$-Gibbs measures for partially hyperbolic skew
products. This is an adaptation of a measure rigidity result by A. Brown and
F. Rodriguez Hertz for stationary measures of random product of surface
diffeomorphisms. In particular, we classify all the possible $u$-Gibbs
measures that may appear in a neighborhood of the example. Using this
classification, and ruling out some of the possibilities, we obtain open sets
of systems, in a neighborhood of the example, having a unique $u$-Gibbs
measure which is SRB.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Proof of Theorem A assuming Theorems B and C
  4. 4 Center Lyapunov exponents for $u$-Gibbs measures
  5. 5 Proof of Theorem C
  6. 6 Rigidity of $u$-Gibbs measures
  7. 7 The non invariance of stable directions by $u$-holonomies
  8. 8 Measures with atomic center disintegration and the proof of Theorem B
  9. 9 The proof of Theorem 8.3: the $s$-invariance of the center conditional measures
  10. 10 Appendix A: $C^{2}$-regularity of unstable holonomies

##  1 Introduction

In dynamics one usually tries to understand the asymptotic behavior of the
orbit of many points. In this direction, it is natural to try to understand
properties, and the existence, of certain invariant measures that capture the
statistical behavior of a set of points that is relevant for the Lebesgue
measure. Let us make this more precise.

Let $f$ be a diffeomorphism of a closed, compact, connected, orientable
manifold $M$. Given an invariant ergodic222Recall that $\mu$ is ergodic if and
only if any $f$-invariant measurable set $\Lambda$ has measure $0$ or $1$.
probability measure $\mu$, its basin is defined as

$B(\mu)=\left\\{p\in
M:\displaystyle\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f^{j}(p)}\xrightarrow{n\to+\infty}\mu\right\\},$

where $\delta_{p}$ is the dirac measure on $p$ and the convergence is for the
weak*-topology. The measure $\mu$ is physical if its basin has positive
Lebesgue measure. In other words, physical measures are the measures that
capture the asymptotic behavior of many points in the Lebesgue point of view.

In the 1970s, Sinai, Ruelle and Bowen [56, 53, 14] proved that $C^{1+\alpha}$
uniformly hyperbolic systems have finitely many physical measures that
describes the statistical behavior of Lebesgue almost every point. Nowadays,
the measures they constructed are called SRB measures (SRB for Sinai-Ruelle-
Bowen), see Definition 2.17. These measures have an important geometrical
property: they admit conditional measures along unstable manifolds which are
absolutely continuous with respect to the volume of the unstable manifolds.
After the work of Ledrappier in [40], there is a well developed ergodic theory
for these measures. The hyperbolic333See section 2.2 for the definition of
hyperbolic measure. SRB measures form an important class of physical measures.

We remark that in the hyperbolic setting there are uniform
expansion/contraction, and a dominated splitting444An invariant set $\Lambda$
admits a dominated splitting if it admits an invariant splitting
$T_{\Lambda}M=E\oplus F$, such that $\exists N\in\mathbb{N}$ that verifies
$\|Df^{N}(p)|_{E_{p}}\|\|Df^{-N}(f^{N}(p))|_{F_{f^{N}(p)}}\|<\frac{1}{2}$.
(which implies that the angle between the expanding/contracting directions is
uniformly bounded from below). These two points are important to carry the
constructions of such measures.

There are many works that study conditions that guarantee the existence of
hyperbolic SRB measures outside the uniformly hyperbolic setting, see for
instance [63, 13, 3, 23, 24, 7]. We also refer the reader to the recent survey
[25] for a discussion on the different methods of construction of such
measures (with a focus on the geometrical method). We now mention some of the
examples of systems admitting hyperbolic SRB measures.

  * •

Some derived from Anosov examples, in particular the ones introduced by
Bonatti-Viana in [13]: these examples have a dominated splitting, and
nonuniform expansion, or contraction (also known as mostly contracting, or
mostly expanding), see also [3, 58]. It gives open sets of systems having an
unique hyperbolic SRB measure.

  * •

Hénon maps: in [8] it is proved that for a set of positive Lebesgue measure of
parameters $(a,b)$ with $b>0$ small, the map

$h_{a,b}(x,y)=(x^{2}+y+a,-bx),$

admits a hyperbolic SRB measure. This example has non-uniform
expansion/contraction, it is dissipative (it does not preserve the Lebesgue
measure), and it does not admit a dominated splitting. However, it is not
guaranteed the existence of an SRB measure for an open set of parameters
$(a,b)$.

  * •

Some robustly non-uniformly hyperbolic volume preserving diffeomorphisms whose
Oseledec’s splitting is not dominated555See section 2 for the definition of
non-uniform hyperbolicity and Oseledec’s splitting.: by the absolute
continuity of the unstable partition after the work of Pesin (see [47]), in
the volume preserving scenario, the existence of a hyperbolic SRB measure is
equivalent to prove non-uniform hyperbolicity. Let us mention a few of such
examples. The Berger-Carrasco’s example in [10] (which we will study in more
detail in this work). We also mention Avila-Viana in [5], and Liang-Marin-Yang
in [43], where they obtain $C^{2}$-open sets of symplectomorphisms which are
non-uniformly hyperbolic. These examples are conservative, they have non-
uniform expansion/contraction, and the expanding/contracting directions are
not dominated.

  * •

Some “large local” perturbations of Axiom A systems, which appeared in [23]:
these examples also present non-uniform expansion/contraction, and no
dominated splitting. But the proof of the existence of a hyperbolic SRB
measure does not guarantee the robust existence of a hyperbolic SRB measure.

We remark that the list above is not a complete list of examples, but they
represent well the examples according to the presence of non-uniform
expansion/contraction, domination, and volume preserving or not.

Berger-Carrasco introduced in [10] an example of a partially hyperbolic
system666See section 2 for the definition., with two dimensional center, and
such that among the volume preserving systems it is robustly non-uniformly
hyperbolic with both expansion/contraction along the center and it does not
admit a decomposition of the center in dominated directions.

In [45], the author proves that the Berger-Carrasco’s example and any
$C^{2}$-small volume preserving perturbation of it is ergodic. In this work we
study dissipative perturbations of this example. In particular, we will find
an open set of systems having an unique hyperbolic SRB measure with full
basin, and each system in this open set has non-uniform expansion/contraction
whose angle between the expanding/contracting directions is not bounded away
from zero.

### The example and precise statement of the results

For $N\in\mathbb{R}$ we denote by $s_{N}(x,y)=(2x-y+N\sin(x),x)$ the standard
map on $\mathbb{T}^{2}:=\mathbb{R}^{2}/2\pi\mathbb{Z}^{2}$. For every $N$ the
map $s_{N}$ preserves the Lebesgue measure induced by the usual metric of
$\mathbb{T}^{2}$. This map is related to several physical problems, see for
instance [21], [36] and [54].

It is conjectured that for $N\neq 0$ the map $s_{N}$ has positive entropy for
the Lebesgue measure, see [57] page $144$. By Pesin’s entropy formula, see
[47] Theorem $5.1$, this is equivalent to the existence of a set of positive
Lebesgue measure, whose points have a positive Lyapunov exponent. The
existence of those sets is not known for any value of $N$. See [11, 22, 27,
32] for some results related to this conjecture.

In what follows we refer the reader to Section 2 for some basic definitions
regarding partially hyperbolic dynamics. Let $A\in SL(2,\mathbb{Z})$ be a
hyperbolic matrix that defines an Anosov diffeomorphism on $\mathbb{T}^{2}$,
let $P_{x}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ be the projection on the first
coordinate of $\mathbb{T}^{2}$, this projection is induced by the linear map
of $\mathbb{R}^{2}$, which we will also write $P_{x}$, given by
$P_{x}(a,b)=(a,0)$.

Consider the torus $\mathbb{T}^{4}=\mathbb{T}^{2}\times\mathbb{T}^{2}$ and
represent it using the coordinates $(x,y,z,w)$, where $x,y,z,w\in[0,2\pi)$. We
may naturally identify a point $(z,w)$ on the second torus with a point
$(x,y)$ on the first torus by taking $x=z$ and $y=w$. For each
$N\in\mathbb{N}$ define

$\begin{array}[]{rcccc}f_{N}&:&\mathbb{T}^{2}\times\mathbb{T}^{2}&\longrightarrow&\mathbb{T}^{2}\times\mathbb{T}^{2}\\\
&&(x,y,z,w)&\mapsto&(s_{N}(x,y)+P_{x}\circ
A^{N}(z,w),A^{2N}(z,w)).\end{array}$

This diffeomorphism preserves the Lebesgue measure. For $N$ large enough it is
a partially hyperbolic diffeomorphism, with two dimensional center direction
given by $E^{c}=\mathbb{R}^{2}\times\\{0\\}$. This type of system was
considered by Berger-Carrasco in [10], where they proved that for $N$ large
enough $f_{N}$ is $C^{2}$-robustly non-uniformly hyperbolic among the volume
preserving diffeomorphisms.

For $r\geq 1$ we consider $\mathrm{Diff}^{r}(\mathbb{T}^{4})$ to be the set of
$C^{r}$-diffeomorphisms of $\mathbb{T}^{4}$. Inside
$\mathrm{Diff}^{r}(\mathbb{T}^{4})$, we may consider the subspace
$\mathrm{Sk}^{r}(\mathbb{T}^{2}\times\mathbb{T}^{2})$ of skew products, which
is the set of $C^{r}$-diffeomorphisms $g$ of the form

$g(x,y,z,w)=(g_{1}(x,y,z,w),g_{2}(z,w)),$

where $g_{2}(.,.)$ is a $C^{r}$-diffeomorphism of $\mathbb{T}^{2}$, and for
each $(z,w)\in\mathbb{T}^{2}$, $g_{1}(.,.,z,w)$ is a $C^{r}$-diffeomorphism of
$\mathbb{T}^{2}$ as well. Observe that
$f_{N}\in\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$. We also remark
that for $N$ large enough, if $g$ is a skew product $C^{1}$-close enough to
$f_{N}$, then $g_{2}$ is an Anosov diffeomorphism, and $g$ is partially
hyperbolic.

We recall that for a map $g$, a $g$-invariant measure $\mu$ is Bernoulli if
the system $(g,\mu)$ is measurably conjugated to a Bernoulli shift.

Our main result is the following:

######  Theorem A.

Let $\alpha\in(0,1)$. For $N$ large enough, there exist $\mathcal{U}_{N}^{sk}$
a $C^{2}$-neighborhood of $f_{N}$ contained in
$\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$, and $\mathcal{V}$ a
$C^{2}$-open and $C^{2}$-dense subset of $\mathcal{U}_{N}^{sk}$ such that for
any $g\in\mathcal{V}$ having regularity $C^{2+\alpha}$, there exists a unique
$g$-invariant measure $\mu_{g}$ with the following properties:

  1. 1.

$\mu_{g}$ is a hyperbolic SRB measure and Bernoulli;

  2. 2.

$\mathrm{Leb}(B(\mu_{g}))=1$;

  3. 3.

$\mathrm{supp(\mu_{g})}=\mathbb{T}^{4}$.

The proof of Theorem A is based in the study of the so called $u$-Gibbs
measures, see Definition 2.21. These measures play a key role in the study of
ergodic properties of partially hyperbolic systems. Indeed, they capture the
asymptotic statistical behavior of Lebesgue almost every point, see Theorem
2.22. For a partially hyperbolic diffeomorphism $g$, we write
$\mathrm{Gibbs}^{u}(g)$ as the set of $u$-Gibbs measures for $g$.

To prove Theorem A we will first classify all the ergodic $u$-Gibbs measures
that may appear in a neighborhood of $f_{N}$. This is given in the following
theorem:

######  Theorem B.

Let $\alpha\in(0,1)$. For $N$ large enough, there exists
$\mathcal{U}_{N}^{sk}$ a $C^{2}$-neighborhood of $f_{N}$ contained in
$\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$, such that for
$g\in\mathcal{U}_{N}^{sk}$ having regularity $C^{2+\alpha}$, if
$\mu\in\mathrm{Gibbs}^{u}(g)$ is ergodic, then either:

  1. 1.

$\mu$ is a hyperbolic SRB measure, or

  2. 2.

there exists a finite number of $C^{1}$ two dimensional tori
$T^{1}_{\mu},\cdots,T^{l}_{\mu}\subset\mathbb{T}^{4}$ such that each of them
is tangent to $E^{ss}_{g}\oplus E^{uu}_{g}$, and
$\mathrm{supp}(\mu)=\cup_{j=1}^{l}\mathbb{T}^{j}_{\mu}$.

The proof of Theorem B uses an adaptation to the partially hyperbolic skew
product setting of a recent result by Brown-Rodriguez Hertz in [17]. In their
paper they classify all the ergodic, hyperbolic stationary measures for random
products of surface $C^{2}$-diffeomorphisms. Their proof is inspired in ideas
from Benoist-Quint [9] and Eskin-Mirzakhani [30]. In the partially hyperbolic
skew product setting, we can actually get a result more general than Theorem
B, see Theorem D below.

We remark that there are also some recent works that “push” the ideas from [9,
30, 29, 17] to different settings. There is the work of Cantat-Dujardin in
[19] which attempt to classify stationary measures of random products of
automorphisms of real and complex projective surfaces. There is also the work
of Katz, [38], which pushes the ideas of [30, 29] to prove rigidity of
“$u$-Gibbs measures” of Anosov flows under a technical hypothesis called QNI
(quantified non-integrability).

The uniqueness of the SRB measure, and some other properties that appear in
the statement of Theorem A, will be a consequence of the following theorem:

######  Theorem C.

For $N$ large enough, there exists $\mathcal{U}_{N}$ a $C^{2}$-neighborhood of
$f_{N}$ in $\mathrm{Diff}^{2}(\mathbb{T}^{4})$ such that if
$g\in\mathcal{U}_{N}$, then $g$ has at most one SRB measure. Moreover, if
$\mu_{g}$ is an SRB measure for $g$, then
$\mathrm{supp}(\mu_{g})=\mathbb{T}^{4}$, it is Bernoulli and hyperbolic.

######  Remark 1.1.

Theorems A and B hold for a neighborhood of $f_{N}$ inside the set of skew
product diffeomorphisms,
$\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$. Theorem C guarantees
that there exists at most one SRB measure in a neighborhood of $f_{N}$ inside
$\mathrm{Diff}^{2}(\mathbb{T}^{4})$. However, it does not guarantee the
existence of an SRB measure.

As we mentioned before, the proof of Theorem B uses the following theorem,
which holds for more general partially hyperbolic skew products and not only
perturbations of Berger-Carrasco’s example. Let $S$ be a compact surface. We
can define $\mathrm{Sk}^{r}(S\times\mathbb{T}^{2})$ as the set of
$C^{r}$-diffeomorphisms $g$ of the form

$\begin{array}[]{rcc}S\times\mathbb{T}^{2}&\to&S\times\mathbb{T}^{2}\\\
(p_{1},p_{2})&\mapsto&(g_{1}(p_{1},p_{2}),g_{2}(p_{2}))\end{array}$

such that $g_{2}(.)$ is a $C^{r}$-diffeomorphism of $\mathbb{T}^{2}$ and for
each $p_{2}\in\mathbb{T}^{2}$, $g_{1}(.,p_{2})$ is a $C^{r}$-diffeomorphism of
$S$. We say that $g$ is a partially hyperbolic skew product of
$S\times\mathbb{T}^{2}$ if $g$ is partially hyperbolic and $g_{2}$ is an
Anosov diffeomorphism of $\mathbb{T}^{2}$. Let $g$ be a partially hyperbolic
skew product of $S\times\mathbb{T}^{2}$. In what follows we write
$\|Dg|_{E^{ss}}\|:=\displaystyle\sup_{p\in
S\times\mathbb{T}^{2}}\|Dg(p)|_{E^{ss}}\|$ and
$m(Dg|_{E^{c}}):=\displaystyle\inf_{p\in
S\times\mathbb{T}^{2}}m(Dg(p)|_{E^{c}})$, where
$m(Dg(p)|_{E^{c}}):=\|\left(Dg(p)|_{E^{c}}\right)^{-1}\|^{-1}$ is the co-norm
of $Dg(p)|_{E^{c}}$.

######  Theorem D.

Let $S$ be a compact surface and let $\alpha,\theta\in(0,1)$ be two constants.
Let $g\in\mathrm{Sk}^{2+\alpha}(S\times\mathbb{T}^{2})$ be a partially
hyperbolic skew product such that:

  1. (a)

$g$ is $(2,\alpha)$-center bunched (see (8) for the definition) ;

  2. (b)

$E^{uu}$ is $\theta$-Hölder and $\|Dg|_{E^{ss}}\|^{\theta}<m(Dg|_{E^{c}})$.

If $\mu\in\mathrm{Gibbs}^{u}(g)$ is an ergodic measure having one positive and
one negative Lyapunov exponent along the center direction, then either:

  1. 1.

$\mu$ is an SRB measure;

  2. 2.

the Oseledets direction $E^{-}$ is invariant by linear unstable holonomies
(see item $2$ of Theorem 6.3 for a precise definition);

  3. 3.

there exist a finite number of two dimensional $su$-tori
$T^{1}_{\mu},\cdots,T^{l}_{\mu}$, such that
$\mathrm{supp}(\mu)=\cup_{j=1}^{l}T^{j}_{\mu}$.

This theorem will be a direct consequence of the combination of Theorems 6.5
and 8.1 below (see also Remark 8.2).

### Discussion on the techniques and strategy of the proofs

Theorem A is an easy consequence of Theorems B, C, and of some recent results
on accessibility classes for skew products with two dimensional fibers from
[35], given by Theorem 2.13 below.

Using the calculations to prove non-uniform hyperbolicity of $f_{N}$ from
[10], and the adaptations made in [45], we prove that in a neighborhood of
$f_{N}$ in $\mathrm{Diff}^{2}(\mathbb{T}^{4})$, every $u$-Gibbs measure is
hyperbolic with both a positive and a negative Lyapunov exponent along the
center.

The proof of Theorem C is based on the techniques developed by the author in
[45]. Using such techniques we can prove that any $u$-Gibbs measure has a set
of large measure, whose points have “large” stable and unstable manifolds.
Furthermore, we can obtain precise control on the “geometry” of these
invariant manifolds. This allows us to prove that any two $u$-Gibbs measures
are homoclinically related (see Definition 2.19 and Theorem 5.1). Hence, we
conclude that in a neighborhood of $f_{N}$ (inside
$\mathrm{Diff}^{2}(\mathbb{T}^{4})$) there exists at most one SRB measure. The
techniques will also allow us to conclude that such a measure is Bernoulli.
Using some arguments from the recent work [20] of the author with P. Carrasco,
we prove that if there exists an SRB measure then it has full support. A key
point in this proof is a quantified version of Pesin theory that appeared in
[26]. We remark that this type of strategy using this quantified Pesin theory
allowed the author to prove the uniqueness of the measure of maximal entropy
for the standard map itself (see [46]).

One of the key ingredients in the proof of Theorem B is an adaptation for the
partially hyperbolic skew product setting of the main results from [17]. There
are two parts in this adaptation, which are given by Theorems 6.3 and 8.1. To
prove Theorem 6.3, we show that for $g$ sufficiently close to $f_{N}$ and for
an ergodic $u$-Gibbs measure $\mu$, after a measurable change of coordinates
using the unstable holonomies, we are in the setting of Theorem $4.10$ from
[17]. To justify that the change of coordinates mentioned above take us to the
setting of Brown-Rodriguez Hertz’s rigidity result, we use the version of the
invariance principle by Tahzibi-Yang in [59]. We then obtain that there are
only three possibilities for an ergodic $u$-Gibbs measure: either it is an SRB
measure; or it has atomic disintegrations along the center foliation; or the
Oseledets direction for the negative center Lyapunov exponent is invariant by
the derivative of unstable holonomies. Using some estimates from [10], we
prove that the third case never happens (see proposition 7.1). We are left to
deal with the $u$-Gibbs measures having an atomic disintegration along the
center foliation. This is done with Theorem 8.1.

Theorem 8.1 corresponds to the adaptation of Theorem $4.8$ from [17]. The
proof of this theorem is done in Sections 8 and 9. If the $u$-Gibbs measure
has atomic disintegration along the center foliation and the stable Oseledets
direction is not invariant by the derivative of unstable holonomies, we prove
that the center disintegration is invariant by stable and unstable holonomies.
Since the system also verifies a condition called center bunching (see
Definition 2.1), using some results on accessibility classes (see Theorem
2.14), we may conclude the existence of the tori tangent to the strong stable
and unstable directions (see Theorem 8.1) which contain the support of the
measure.

Let us finish with a remark on item $(b)$ in the hypothesis of Theorem D. This
condition states that we need $E^{uu}$ to be “Hölder enough” to apply the
theorem. It is well known that the invariant directions of a partially
hyperbolic diffeomorphism are usually Hölder. Let $g$ be a partially
hyperbolic skew product. If $\theta\in(0,1)$ is a number such that

$\displaystyle\frac{\|Dg(p)|_{E^{c}}\|}{m(Dg(p)|_{E^{uu}})}<m(Dg(p)|_{E^{ss}})^{\theta},$

for every point $p\in S\times\mathbb{T}^{2}$, then $E^{uu}$ is $\theta$-Hölder
(see Section 4 from [50]). This condition gives an upper bound on $\theta$.
Indeed, we obtain that

$\theta<\inf_{p\in S\times\mathbb{T}^{2}}\left\\{\displaystyle\frac{\log
m(Dg(p)|_{E^{uu}})-\log\|Dg(p)|_{E^{c}}\|}{-\log
m(Dg(p)|_{E^{ss}})}\right\\}.$

On the other hand, to obtain condition $(b)$ in the hypothesis of Theorem D we
need that $\|Dg|_{E^{ss}}\|^{\theta}<m(Dg|_{E^{c}})$, which implies

$\theta>\displaystyle\frac{\log m(Dg|_{E^{c}})}{\log\|Dg|_{E^{ss}}\|}.$

Thus, a sufficient condition to obtain the hypothesis $(c)$ is that

$\displaystyle\frac{\log m(Dg|_{E^{c}})}{\log\|Dg|_{E^{ss}}\|}<\inf_{p\in
S\times\mathbb{T}^{2}}\left\\{\displaystyle\frac{\log
m(Dg(p)|_{E^{uu}})-\log\|Dg(p)|_{E^{c}}\|}{-\log
m(Dg(p)|_{E^{ss}})}\right\\}.$

### Further remarks and questions

The $\alpha$ that appears in the statements of Theorems A and B and D, only
appears because in the statement of the main result from [17], the surface
diffeomorphisms they consider have regularity $C^{2}$. If one obtains a
version of their result for $C^{1+\beta}$-diffeomorphisms, then one could
remove the $\alpha$ from the statement (see section 6).

Let us make a few remarks about the skew product hypothesis in the statement
Theorems A, B and D. This condition implies that the center foliation is
smooth. This is used to prove Proposition 2.28, which states that we may use
the invariance principle (see also Corollary 2.30). We also use the smoothness
of the center foliation to prove that an $u$-Gibbs measure projects to the
unique SRB measure for the $C^{2}$-Anosov diffeomorphism on the basis (see
Lemma 2.29). This is important in our proof because SRB measures for a
$C^{2}$-Anosov diffeomorphism have a property called local product structure
(see Section 8). This local product structure is a key property used to obtain
Lemma 8.4, which is used in the proof of the existence of the $su$-tori in
item $2$ of the statement of Theorem B (and item $3$ of Theorem D). It is an
interesting question to know if one can remove the skew product condition in
the hypothesis to work with more general partially hyperbolic systems with two
dimensional center.

An important notion in the study of dynamical properties of partially
hyperbolic systems is accessibility (see Section 2 for the definition). It is
not known if $f_{N}$ is accessible or not. If it were, we would obtain several
interesting consequences, such as:

  * •

$f_{N}$ would be $C^{1}$-stably ergodic (we refer the reader to [45] for the
definition and discussion on stable ergodicity);

  * •

for $N$ large and $\mathcal{U}_{N}^{sk}$ small enough such that Theorem B is
verified, for any
$g\in\mathcal{U}_{N}^{sk}\cap\mathrm{Diff}^{2+\alpha}(\mathbb{T}^{4})$, there
would be an unique $SRB$ measure $\mu_{g}$, which is Bernoulli, it has full
support and full basin. Furthermore, this measure would be the unique
$u$-Gibbs measure for $g$.

We emphasize the question made by Berger-Carrasco in [10]:

######  Question 1.2.

For every $L>0$, does it exist $N\in[L,+\infty)$ such that $f_{N}$ is
accessible?

An interesting strategy to prove the existence of an SRB measure in a
neighborhood of $f_{N}$ inside $\mathrm{Diff}^{2}(\mathbb{T}^{4})$ is to use
the results from [23]. In order to do that, one needs to prove that the
condition called effective hyperbolicity is verified (see Section $1.2$ in
[23]). This condition seems hard to prove, however it could give the existence
of SRB measures outside the fibered case.

######  Question 1.3.

For $N$ large enough, for any diffeomorphism $g$ which is sufficiently
$C^{2}$-close to $f_{N}$, does it hold that $g$ is effective hyperbolic?

In [60], Viana introduced two examples of systems (sometimes called Viana
maps), which exhibit non-uniformly hyperbolic attractors . The first one is an
endomorphism (see Theorem A in [60]), which is an skew product over an
expanding map of the circle (on the basis), and the dynamics on the fiber is
based on the quadratic family. For this example there were several works that
studied its ergodic properties, in particular the existence of SRB measure,
see for instance [1, 2, 4, 18].

The second example introduced by Viana is a diffeomorphism on a
$5$-dimensional manifold (see Theorem B in [60]). It is an skew product with a
solenoid on the basis, and the dynamics on the fiber is based on the Hénon
maps (which are dissipative). Viana proved that Lebesgue almost every point
has a positive Lyapunov exponent along the fiber. This example is not well
understood. In particular, nothing has been done regarding the existence of
SRB measures for this type of Viana maps.

######  Question 1.4.

Can the same strategy we use to study SRB measures be applied to study the
existence of SRB measure for the second type of Viana maps?

### Organization of the paper

In Section 2, we review several tools that we will use in this work. In
particular, results on partially hyperbolic systems and accessibility classes,
$u$-Gibbs and SRB measures, and the invariance principle. In section 3 we
prove Theorem A assuming Theorems B and C. Sections 4 and 5 are dedicated to
prove Theorem C. In these sections we show how the techniques from [45], and
[10], are used to obtain precise control on the center Lyapunov exponents of
$u$-Gibbs measures, and how to obtain the uniqueness of the SRB measure.

In Section 6 we state Theorem $4.10$ from [17], and we show how after a
measurable change of coordinates of our systems we are in the setting of their
result. In Section 7 we prove that in a neighborhood of Berger-Carrasco’s
example, the Oseledets direction for the negative center Lyapunov exponent is
not invariant by the derivative of unstable holonomies, for any $u$-Gibbs
measure.

In Sections 8 and 9, we deal with the case where a $u$-Gibbs measure has
atomic center disintegration. This is done by using the invariance principle
and adapting the proof of Theorem $4.8$ from [17]. In the appendix we prove
that with some stronger bunching condition the strong unstable holonomy
between center manifolds has regularity $C^{2}$, this is used in the proof of
Theorem B.

### Acknowledgments

The author thanks Sylvain Crovisier for careful reading of this work, and
useful conversations. The author also would like to thank Aaron Brown, Alex
Eskin, Todd Fisher, Mauricio Poletti, Rafael Potrie, Federico Rodriguez Hertz,
Ali Tahzibi and Amie Wilkinson for useful conversations and comments.

##  2 Preliminaries

###  2.1 Partial hyperbolicity, holonomies and accessibility classes

#### Partial hyperbolicity and foliations

A $C^{r}$-diffeomorphism $f$, with $r\geq 1$, is partially hyperbolic if the
tangent bundle has a decomposition $TM=E^{ss}\oplus E^{c}\oplus E^{uu}$, there
is a riemannian metric on $M$ and continuous functions
$\chi^{ss},\chi^{uu},\chi^{c}_{-},\chi^{c}_{+}:M\to\mathbb{R}$, such that for
any $m\in M$

$\chi^{ss}(m)<1<\chi^{uu}(m)\textrm{ and
}\chi^{ss}(m)<\chi^{c}_{-}(m)\leq\chi^{c}_{+}(m)<\chi^{uu}(m),$

it also holds

$\begin{array}[]{c}\chi^{c}_{-}(m)\leq
m(Df(m)|_{E^{c}_{m}})\leq\|Df(m)|_{E^{c}_{m}}\|\leq\chi^{c}_{+}(m);\\\
\|Df(m)|_{E^{ss}_{m}}\|\leq\chi^{ss}(m)\textrm{ and }\chi^{uu}(m)\leq
m(Df(m)|_{E^{uu}_{m}}).\end{array}$

If the functions in the definition of partial hyperbolicity can be taken
constant, we say that $f$ is absolutely partially hyperbolic.

It is well known that the distributions $E^{ss}$ and $E^{uu}$ are uniquely
integrable, that is, there are two unique foliations $\mathcal{F}^{ss}$ and
$\mathcal{F}^{uu}$, with $C^{r}$-leaves, that are tangent to $E^{ss}$ and
$E^{uu}$ respectively. For a point $p\in M$ we will denote by $W^{ss}(p)$ a
leaf of the foliation $\mathcal{F}^{ss}$, we will call such leaf the strong
stable manifold of $p$. Similarly we define the strong unstable manifold of
$p$ and denote it by $W^{uu}(p)$.

######  Definition 2.1.

A partially hyperbolic diffeomorphism is center bunched if

$\chi^{ss}(m)<\frac{\chi^{c}_{-}(m)}{\chi^{c}_{+}(m)}\textrm{ and
}\frac{\chi^{c}_{+}(m)}{\chi^{c}_{-}(m)}<\chi^{uu}(m),\textrm{ for every $m\in
M$}.$

We denote $E^{cs}=E^{ss}\oplus E^{c}$ and $E^{cu}=E^{c}\oplus E^{uu}$.

######  Definition 2.2.

A partially hyperbolic diffeomorphism $f$ is dynamically coherent if there are
two invariant foliations $\mathcal{F}^{cs}$ and $\mathcal{F}^{cu}$, with
$C^{1}$-leaves, tangent to $E^{cs}$ and $E^{cu}$ respectively. From those two
foliations one obtains another invariant foliation
$\mathcal{F}^{c}=\mathcal{F}^{cs}\cap\mathcal{F}^{cu}$ that is tangent to
$E^{c}$. We call those foliations the center-stable, center-unstable and
center foliation.

For any $R>0$ we write $W^{*}_{R}(p)$ to be the disc of size $R$ centered on
$p$, for the Riemannian metric induced by the metric on $M$, contained in the
leaf $W^{*}(p)$, for $*=ss,c,uu$.

The definition below allows one to obtain higher regularity of the leaves of
such foliations.

######  Definition 2.3.

We say that a partially hyperbolic diffeomorphism $f$ is $r$-normally
hyperbolic if for any $m\in M$

$\chi^{ss}(m)<(\chi^{c}_{-}(m))^{r}\textrm{ and
}(\chi^{c}_{+}(m))^{r}<\chi^{uu}(m).$

######  Definition 2.4.

Let $f$ and $g$ be partially hyperbolic diffeomorphisms of $M$ that are
dynamically coherent. Denote by $\mathcal{F}^{c}_{f}$ and
$\mathcal{F}^{c}_{g}$ the center foliations. We say that $f$ and $g$ are leaf
conjugated if there is a homeomorphism $h:M\to M$ that takes leaves of
$\mathcal{F}^{c}_{f}$ to leaves of $\mathcal{F}^{c}_{g}$ and such that for any
$L\in\mathcal{F}^{c}_{f}$ it is verified

$h(f(L))=g(h(L)).$

One may study the stability of partially hyperbolic systems up to leaf
conjugacy. Related to this there is a technical notion called plaque
expansivity which we will not define here, see chapter 7 of [34] for the
definition. The next theorem is important for the theory of stability of
partially hyperbolic systems.

######  Theorem 2.5 ([34], Theorem $7.4$).

Let $f:M\to M$ be a $C^{r}$-partially hyperbolic and dynamically coherent
diffeomorphism. If $f$ is $r$-normally hyperbolic and plaque expansive then
any $g:M\to M$ in a $C^{r}$-neighborhood of $f$ is partially hyperbolic and
dynamically coherent. Moreover, $g$ is leaf conjugated to $f$ and the center
leaves of $g$ are $C^{r}$-immersed manifolds.

######  Remark 2.6.

Fix $R>0$, and let $f$ be a diffeomorphism that satisfies the hypothesis of
the previous theorem. The proof of this theorem implies that for $g$
sufficiently $C^{r}$-close to $f$, for any $m\in M$ we have that
$W^{c}_{f,R}(m)$ is $C^{r}$-close to $W^{c}_{g,R}(m)$. In particular, if the
center foliation is uniformly compact then for every $g$ sufficiently
$C^{r}$-close to $f$, for any $m\in M$, $W^{c}_{f}(m)$ is $C^{r}$-close to
$W^{c}_{g}(m)$.

It might be hard to check the condition of plaque expansiviness, but this is
not the case when the center foliation of a dynamically coherent, partially
hyperbolic diffeomorphism is at least $C^{1}$, see Theorem $7.4$ of [34].
Usually the invariant foliations that appear in dynamics are only Hölder.

We can also obtain a better regularity for the center direction given by the
following theorem, see section $4$ of [50] for a discussion on this topic.

######  Theorem 2.7.

Let $f$ be a $C^{2}$-partially hyperbolic diffeomorphism and let $\theta>0$ be
a number such that for every $m\in M$ it is verified

$\chi^{ss}(m)<\chi^{c}_{-}(m)m(Df(m)|_{E^{ss}})^{\theta}\textrm{ and
}\chi^{c}_{+}(m)\|Df(m)|_{E^{uu}}\|^{\theta}<\chi^{uu}(m),$

then $E^{c}$ is $\theta$-Hölder.

#### Unstable holonomies

Let $f$ be a partially hyperbolic, dynamically coherent diffeomorphism. Each
leaf of the foliation $\mathcal{F}^{cs}$ is foliated by strong stable
manifolds. For a point $p\in M$ and $q\in W^{ss}_{1}(p)$, where
$W^{ss}_{1}(p)$ is the strong stable manifold of size $1$, we can define the
stable holonomy map restricted to the center-stable manifold, between center
manifolds. Let us be more precise. We can choose two small numbers
$R_{1},R_{2}>0$, with the property that for any $z\in W^{c}_{R_{1}}(p)$, there
is only one point in the intersection $W^{ss}_{2}(z)\cap W^{c}_{R_{2}}(q)$. We
define $H^{s}_{p,q}(z)=W^{ss}_{2}(z)\cap W^{c}_{R_{2}}(q)$. With this
construction we obtain a map $H^{s}_{p,q}:W^{c}_{R_{1}}(p)\to
W^{c}_{R_{2}}(q)$. By the compactness of $M$ we can take the numbers $R_{1}$
and $R_{2}$ to be constants, independent of $p$ and $q$.

We can define analogously the unstable holonomy map, for $p\in M$ and $q\in
W^{uu}_{1}(p)$, which we will denote by $H^{u}_{p,q}:W^{c}_{R_{1}}(p)\to
W^{c}_{R_{2}}(q)$.

In [48] and [49], the authors prove that the map $H^{s}_{p,q}$ is $C^{1}$ if
$f$ is a partially hyperbolic, center bunched and dynamically coherent
$C^{2}$-diffeomorphism. Indeed, the authors prove that the strong stable
foliation is $C^{1}$ when restricted to a center-stable leaf. Consider the
family of $C^{1}$-maps $\displaystyle\\{H^{s}_{p,q}\\}_{p\in M,q\in
W^{ss}_{1}(p)}$.

######  Theorem 2.8.

Let $f$ be an absolutely partially hyperbolic, dynamically coherent
diffeomorphism with regularity $C^{2}$. Suppose also that $f$ verifies:

  1. 1.

$\chi^{c}_{-}<1$ and $\chi^{c}_{+}>1$;

  2. 2.

there exists $\theta\in(0,1)$, such that

$(\chi^{ss})^{\theta}<\frac{\chi^{c}_{-}}{\chi^{c}_{+}}\textrm{ and
}\frac{\chi^{c}_{+}}{\chi^{c}_{-}}<(\chi^{uu})^{\theta};$ (1)

and also

$\chi^{ss}<\chi^{c}_{-}m(Df|_{E^{ss}})^{\theta}\textrm{ and
}\chi^{c}_{+}\|Df|_{E^{uu}}\|^{\theta}<\chi^{uu}.$ (2)

Then the family $\\{H^{s}_{p,q}\\}_{p\in M,q\in W^{ss}_{1}(p)}$ is a family of
$C^{1}$-maps depending continuously in the $C^{1}$-topology with the choices
of the points $p$ and $q$. Furthermore, there exists a constant $C>0$ such
that for any $p\in M$, $q\in W^{ss}_{1}(p)$, and any unit vector $v\in
E^{c}_{p}$, it is verified

$d\left(\frac{H^{s}_{p,q}(p)v}{\|H^{s}_{p,q}(p)v\|},v\right)<Cd(p,q)^{\theta}.$
(3)

Similar results holds for the family of unstable holonomies
$\\{H^{u}_{p,q}\\}_{p\in M,q\in W^{uu}_{1}(p)}$.

The proof of this theorem can be found in [44], which is an adaptation of the
arguments from [15] by Brown. In what follows, we give the main points of this
proof mostly to justify (3). For all the details, we refer the reader to [44].

###### Sketch of the proof.

By Theorem 2.7, condition (2) implies that the center bundle $E^{c}$ is
$\theta$-Hölder (see section $4$ in [50]). The condition (1) is sometimes
called the strong bunching condition.

We may fix a local approximation of the holonomy $H^{s}_{*}$, which we will
denote by $\pi^{s}_{*}$, that verifies the following: there exists a constant
$\tilde{C}>0$ such that for any $p\in M$ and $q\in W^{ss}_{1}(p)$, there
exists a $C^{1+\theta}$-map, which is a diffeomorphism onto its image,
$\pi^{s}_{p,q}:W^{c}_{R_{1}}(p)\to W^{c}(q)$ that verifies

  1. 1.

$d(\pi^{s}_{p,q}(p),q)\leq\tilde{C}d(p,q)$;

  2. 2.

$d(D\pi^{s}_{p,q}(p).v,v)\leq\tilde{C}d(p,q)^{\theta}$, where $v\in
SE^{c}_{p}$, and $SE^{c}_{p}$ is the unit sphere on $E^{c}_{p}$;

  3. 3.

if $p^{\prime}\in W^{c}_{loc}(p)$ and $q^{\prime}\in
W^{ss}_{1}(p^{\prime})\cap W^{c}_{loc}(q)$, then $\pi_{p,q}^{s}$ coincides
with $\pi^{s}_{p^{\prime},q^{\prime}}$ on $W^{c}_{loc}(p)\cap
W^{c}_{loc}(p^{\prime})$.

This can be done in the following way: Consider a smooth subbundle
$\widetilde{E}$ inside a cone which is close to the direction perpendicular to
the subbundle $E^{c}$, with dimension $\mathrm{dim}(M)-\mathrm{dim}(E^{c})$.
Since $E^{c}$ is $\theta$-Hölder, the center manifolds are $C^{1+\theta}$.
Hence, the restriction of $\widetilde{E}$ to any center manifold is a
$C^{1+\theta}$-bundle. For each point $q\in M$ and $\rho>0$, consider
$L_{q,\rho}:=\exp_{q}(\widetilde{E}(q,\rho))$ to be the projection of the ball
of radius $\rho$ by the exponential map over $q$. By the uniform
transversality and the compactness of $M$, there exists a constant $\rho_{0}$
such that for any center leaf $W^{c}_{R_{1}}(p)$, the set
$\\{L_{q,\rho}\\}_{q\in W^{c}_{R_{1}}(p)}$ forms an uniform foliated
neighborhood of $W^{c}_{R_{1}}(p)$ (or a tubular neighborhood). Let
$\pi^{s}_{p,q}$ be the holonomy defined by this local foliation, up to
rescaling of the metric we may assume that it is well defined for $p\in M$ and
$q\in W^{ss}_{1}(p)$. By the compactness of $M$ we obtain the constant
$\tilde{C}>0$ above. Observe also that since the center leaves vary
continuously in the $C^{1}$-topology, we obtain that the map $\pi^{s}_{p,q}$
varies continuously in the $C^{1}$-topology with the points $p$ and $q$.

For any $p,q\in M$ and each $n\in\mathbb{N}$, write $p_{n}=f^{n}(p)$ and
$q_{n}=f^{n}(q)$. We define

$H^{s}_{p,q,n}=f^{-n}\circ\pi^{s}_{p_{n},q_{n}}\circ f^{n}.$

If it is clear that we are talking about two points $p$ and $q\in
W^{ss}_{1}(p)$ we will only write $H^{s}_{n}=H^{s}_{p,q,n}$ and similarly
$\pi^{s}_{n}=\pi^{s}_{p_{n},q_{n}}$.

Since we are assuming that $f$ is absolutely partially hyperbolic, only for
this proof, we write its partially hyperbolic constants as
$\chi_{s}=\chi^{ss}$, $\chi_{c}=\chi^{c}_{-}$ and
$\widehat{\chi}_{c}=(\chi^{c}_{+})^{-1}$. Also, for a diffeomorphism
$g:N_{1}\to N_{2}$, between manifolds $N_{1}$ and $N_{2}$, we will write
$g_{*}:SN_{1}\to SN_{2}$, the action induced by the derivative on the unitary
bundles of $N_{1}$ and $N_{2}$.

The proof of Theorem 2.8 follows the steps in [15]. The first step is to prove
that $(H^{s}_{n})_{n\in\mathbb{N}}$ is uniformly Cauchy in the
$C^{0}$-topology. The second step is to prove that the sequence
$\left((H^{s}_{n})_{*}\right)_{n\in\mathbb{N}}$ is uniformly Cauchy. The third
step is to prove that for any vector $v\in E^{c}_{p}$, the sequence
$\left(\|DH^{s}_{n}(p)v\|\right)_{n\in\mathbb{N}}$ is also uniformly Cauchy.
In all these three steps it is obtained that the rate of convergence of these
sequence does not depend on the choices of the points $p$ and $q$. The uniform
convergence in the $C^{1}$-topology of the sequence
$(H^{s}_{n})_{n\in\mathbb{N}}$ then follows from these three steps. In this
paper, we only describe in more details step two, for the details of the other
two steps we refer the reader to [44].

Observe that the Lipschitz norm of $f^{-1}_{*}$ restricted to a fiber
$S_{x}E^{c}$ is $(\chi_{c}\widehat{\chi}_{c})^{-1}$. Since $f$ is a
$C^{2}$-diffeomorphism, then $f^{-1}_{*}$ is a $C^{1}$-diffeomorphism of $SM$,
let $C_{1}>0$ be the $C^{1}$-norm of $f^{-1}$ on $M$ and $C_{2}$ to be the
$C^{1}$-norm of $f^{-1}_{*}$ on $SM$. For $\xi=(x,v)\in S_{x}M$, write
$\xi_{k}=f^{k}_{*}(x,v)=(x_{k},v_{k})$, with $k\in\mathbb{Z}$.

In [15], the author uses the strong bunching condition (1) above, but he also
uses another type of bunching (see Theorem $4.1$ in [15]). In the proof, this
different type of bunching is only used to obtain a version of lemma 2.9
below. In our setting, instead of asking for this other type of bunching, we
ask that $\chi_{c}<1$ and $\hat{\chi}_{c}<1$. We obtain the following lemma.

######  Lemma 2.9.

There are constants $\delta,\alpha\in(0,1)$, that verify the following: if
$\xi=(x,v)$, $\zeta=(y,u)\in SW^{c}(p)$, $K>0$ and $n\geq 0$ verify
$d(x_{n},y_{n})<K\chi_{s}^{n}$, $d(\xi_{n},\zeta_{n})\leq
K\chi_{s}^{n\theta}$, and for every $0\leq k\leq n$,

$d(x_{k},y_{k})\leq\delta.$

Then, for all $0\leq k\leq n$,

$d(x_{k},y_{k})\leq K\chi_{s}^{n}.\chi_{c}^{-(n-k)}\textrm{ and
}d(\xi_{k},\zeta_{k})\leq
K\chi_{s}^{n\theta}.(\chi_{c}\widehat{\chi}_{c})^{-(n-k)(1+\alpha)}.$

In particular,

$d(\xi,\zeta)\leq
K\chi_{s}^{n\theta}.(\chi_{c}\widehat{\chi}_{c})^{-n(1+\alpha)}.$

Furthermore, $\alpha$ can be chosen such that

$\chi_{s}^{\theta}.(\widehat{\chi}_{c}\chi_{c})^{-(1+\alpha)}<1.$

###### Proof.

The proof is by backward induction in $k$. We will first denote by $\alpha$
and $\delta$ quantities that will be fixed later. Since $x_{k}$ and $y_{k}$
belongs to the same center manifold, we obtain

$d(x_{k-1},y_{k-1})\leq\chi_{c}^{-1}d(x_{k},y_{k})\leq
K\chi_{s}^{n}.\chi_{c}^{-n+k+1}.$

For any $\beta\in(0,1)$, and since $d(x_{k},y_{k})\leq\delta$, we have

$\begin{array}[]{rcl}d(f^{-1}_{*}(x_{k},v_{k}),f^{-1}_{*}(y_{k},u_{k}))&\leq&d(f^{-1}_{*}(x_{k},v_{k}),f^{-1}_{*}(x_{k},u_{k}))+d(f^{-1}_{*}(x_{k},u_{k}),f^{-1}_{*}(y_{k},u_{k}))\\\
&\leq&(\chi_{c}\widehat{\chi}_{c})^{-1}d(v_{k},u_{k})+C_{2}d(x_{k},y_{k}).\\\
&\leq&(\chi_{c}\widehat{\chi}_{c})^{-1}[1+C_{2}.(\chi_{c}\widehat{\chi}_{c})d(x_{k},y_{k})^{1-\beta}].\max\\{d(x_{k},y_{k})^{\beta},d(v_{k},u_{k})\\}\\\
&\leq&(\chi_{c}\widehat{\chi}_{c})^{-1}[1+C_{2}.(\chi_{c}\widehat{\chi}_{c})\delta^{1-\beta}]\\\
&&.K\max\\{\chi_{s}^{n\beta}.\chi_{c}^{-(n-k)\beta},\chi_{s}^{n\theta}.(\chi_{c}\widehat{\chi}_{c})^{-(n-k)(1+\alpha)}\\}.\end{array}$

We claim that we can choose $\alpha$ and $\beta$ such that for any
$n\in\mathbb{N}$ and $0\leq k\leq n$ it holds

$\chi_{s}^{n\beta}.\chi_{c}^{-(n-k)\beta}\leq\chi_{s}^{n\theta}.(\chi_{c}\widehat{\chi}_{c})^{-(n-k)(1+\alpha)}.$

This inequality is equivalent to

$1\leq\chi_{s}^{n(\theta-\beta)}.(\chi^{(\beta-1-\alpha)}_{c}\widehat{\chi}^{-(1+\alpha)}_{c})^{(n-k)}.$
(4)

Since $\widehat{\chi}^{-1}_{c}>1$, we can fix $\beta>\theta$ close enough to
$1$ such that $1<\chi^{(\beta-1-\alpha)}_{c}\widehat{\chi}^{-(1+\alpha)}_{c}$.
Let us explain. Observe that $(\chi_{c})^{-\alpha}>1$, for any $\alpha>0$.
Hence,

$\chi_{c}^{\beta-1}(\hat{\chi}_{c}\chi_{c})^{-\alpha}\hat{\chi}_{c}^{-1}>\chi_{c}^{\beta-1}\hat{\chi}_{c}^{-1}.$

From this, one can see that if $\beta$ is sufficiently close to $1$, we have
that $1<\chi^{(\beta-1-\alpha)}_{c}\widehat{\chi}^{-(1+\alpha)}_{c}$. Since
$\beta>\theta$, and hence $\theta-\beta$ is negative, we conclude (4).

We also need that

$\chi_{s}^{\theta}.(\widehat{\chi}_{c}\chi_{c})^{-(1+\alpha)}<1.$ (5)

By the strong center bunching condition (1), the inequality above holds if
$\alpha$ is sufficiently close to $0$. Fix $\alpha>0$ that verifies (5).

Now fix $\delta>0$ small enough such that

$[1+C_{2}.(\chi_{c}\widehat{\chi}_{c})\delta^{1-\beta}]\leq(\chi_{c}\widehat{\chi}_{c})^{-\alpha}.$

We conclude,

$\begin{array}[]{rcl}d(f^{-1}_{*}(\xi^{k}),f^{-1}_{*}(\zeta^{k}))&\leq&(\chi_{c}\widehat{\chi}_{c})^{-(1+\alpha)}.K\chi_{s}^{n\theta}.(\chi_{c}\widehat{\chi}_{c})^{-(n-k)(1+\alpha)}\\\
&=&K\chi_{s}^{n\theta}.(\chi_{c}\widehat{\chi}_{c})^{-(n-k-1)(1+\alpha)}\end{array}$

∎

Fix $\xi=(z,l)\in SW_{R_{1}}^{c}(p)$. Write $\zeta^{n}:=(H^{s}_{n})_{*}(\xi)$
and $\zeta^{n}_{j}:=f^{j}_{*}(\zeta^{n})$, for any $j\in\mathbb{Z}$. We also
write $w=H^{s}_{p,q}(z)$, $\zeta^{n}=(H^{s}_{n})_{*}(\xi)=(x,v)$ and
$\zeta^{n+1}=(H^{s}_{n+1})_{*}(\xi)=(y,u)$. Observe that
$\zeta^{n}_{n}=(\pi^{s}_{n})_{*}(\xi_{n})$ and
$\zeta^{n+1}_{n}=f^{-1}_{*}((\pi^{s}_{n+1})_{*}(\xi_{n+1}))$. First we have

$\begin{array}[]{rcl}d(\pi^{s}_{n}(z_{n}),f^{-1}(\pi^{s}_{n+1}(z_{n+1})))&\leq&d(z_{n},\pi^{s}_{n}(z_{n}))+d(f^{-1}(z_{n+1}),f^{-1}(\pi^{s}_{n}(z_{n+1}))\\\
&\leq&\tilde{C}\chi_{s}^{n}d(z,w)+C_{1}\tilde{C}\chi_{s}^{n+1}d(z,w)\\\
&\leq&2\tilde{C}C_{1}\chi_{s}^{n}d(z,w).\end{array}$

The previous estimate shows that $d(x_{n},y_{n})\leq
2\tilde{C}C_{1}d(z,w)\chi_{s}^{n}$. Also it is verified for any $0\leq k\leq
n$

$d(x_{k},y_{k})\leq 2\tilde{C}C_{1}d(z,w)\chi_{s}^{n}\chi_{c}^{-(n-k)}.$ (6)

Let $\delta$ be the constant given by lemma 2.9. By domination, if $n$ is
large enough, we conclude that $d(x_{k},y_{k})<\delta$. This $n$ can be taken
uniform, independently of $p$ and $q$.

Also, using that $f^{-1}_{*}(\xi_{n+1})=\xi_{n}$, we obtain

$\begin{array}[]{rcl}d(\zeta^{n}_{n},\zeta^{n+1}_{n})&=&d((\pi^{s}_{n})_{*}(\xi_{n}),f^{-1}_{*}(\pi^{s}_{n+1})_{*}(\xi_{n+1}))\\\
&\leq&d(\xi_{n},(\pi^{s}_{n})_{*}(\xi_{n}))+d(f^{-1}_{*}(\xi_{n+1}),f^{-1}_{*}(\pi^{s}_{n+1})_{*}(\xi_{n+1})).\end{array}$

By property $2$ of $\pi^{s}_{*}$, we have
$d(\xi_{n},(\pi^{s}_{n})_{*}(\xi_{n}))\leq\tilde{C}d(z,w)^{\theta}\chi_{s}^{n\theta}$.
For the second term in the inequality we have

$\begin{array}[]{lcl}d(f^{-1}_{*}(\xi_{n+1}),f^{-1}_{*}(\pi^{s}_{n+1})_{*}(\xi_{n+1}))&=&d(f^{-1}_{*}(z_{n+1},l_{n+1}),f^{-1}_{*}(y_{n+1},u_{n+1}))\\\
&\leq&d(f^{-1}_{*}(z_{n+1},l_{n+1}),f^{-1}_{*}(z_{n+1},u_{n+1}))\\\
&&+d(f^{-1}_{*}(z_{n+1},u_{n+1}),f^{-1}_{*}(y_{n+1},u_{n+1}))\\\
&\leq&C_{2}d(l_{n+1},u_{n+1})+C_{2}d(z_{n+1},y_{n+1})\\\
&\leq&\tilde{C}C_{2}d(z,w)^{\theta}\chi_{s}^{(n+1)\theta}+\tilde{C}C_{2}d(z,w)\chi^{n+1}_{s}\\\
&\leq&(\tilde{C}C_{2}+\tilde{C}C_{2}d(z,w)^{1-\theta}\chi_{s}^{(n+1)(1-\theta)})d(z,w)^{\theta}\chi^{(n+1)\theta}_{s}\\\
&\leq&(\tilde{C}C_{2}+\tilde{C}C_{2}d(z,w)^{1-\theta})d(z,w)^{\theta}\chi^{(n+1)\theta}_{s}.\end{array}$

Thus,

$d(\zeta^{n}_{n},\zeta^{n+1}_{n})\leq[\tilde{C}+(\tilde{C}C_{2}+\tilde{C}C_{2}d(z,w)^{1-\theta})]d(z,w)^{\theta}\chi^{n\theta}_{s}.$

By compactness, $d(z,w)$ is bounded from above independently of $p$ and $q$.
Hence, take a constant $C_{3}$ such that $d(\zeta^{n}_{n},\zeta^{n+1}_{n})\leq
C_{3}d(z,w)^{\theta}\chi^{n\theta}_{s}$. Fix
$K_{1}=\max\\{2\tilde{C}C_{1},C_{3}\\}$, and observe that we are in the
setting of lemma 2.9, for $K=K(z,w):=K_{1}d(z,w)^{\theta}$. Let $\alpha$ be
the constant given by the same lemma. We conclude that

$d(\zeta^{n},\zeta^{n+1})\leq
K\chi_{s}^{n\theta}.(\chi_{c}\widehat{\chi}_{c})^{-n(1+\alpha)}=K_{1}\chi_{s}^{n\theta}.(\chi_{c}\widehat{\chi}_{c})^{-n(1+\alpha)}d(z,w)^{\theta},\textrm{
for $n$ large enough.}$

In particular, the sequence $(\zeta^{n})_{n\in\mathbb{N}}$ is Cauchy. Since
this holds uniformly for any $\xi$, we obtain that
$\left((H^{s}_{n})_{*}\right)_{n\in\mathbb{N}}$ is a Cauchy sequence whose
speed of convergence does not depend on the choices of the the points $p$ and
$q$.

If $d(p,q)\leq\delta$ then for any $n\geq 0$ it holds that

$d((H^{s}_{n})_{*},(H^{s}_{n+1})_{*})\leq
K_{1}\chi_{s}^{n\theta}.(\chi_{c}\widehat{\chi}_{c})^{-n(1+\alpha)}d(p,q)^{\theta}.$

Write $(H^{s}_{p,q})_{*}=\displaystyle\lim_{n\to+\infty}(H^{s}_{n})_{*}$.
Hence, there exists a constant $K_{2}>0$ such that for $p,q\in M$ with
$d(p,q)<\delta$, we have

$d(Id_{*},(H^{s}_{p,q})_{*})\leq
d(Id_{*},(\pi^{s})_{*})+\displaystyle\sum_{j=0}^{+\infty}d((H^{s}_{j})_{*},(H^{s}_{j+1})_{*})\leq
K_{2}d(p,q)^{\theta}.$

Since $\delta>0$ is a constant, there is a maximum number
$T=[\frac{1}{\delta}]$ such that there are at most $T+1$ points,
$\\{x_{1},\cdots,x_{T+1}\\}\subset W^{s}_{1}(p)$ verifying $x_{1}=p$,
$x_{T+1}=q$ and $d(x_{i},x_{i+1})<\delta$. Since
$H^{s}_{p,q}(.)=H^{s}_{x_{T},x_{T+1}}\circ\cdots\circ H^{s}_{x_{1},x_{2}}(.)$,
we conclude that there exists a constant $C>0$ such that

$d(Id_{*},(H^{s}_{p,q})_{*})\leq Cd(p,q)^{\theta}.$ (7)

This concludes the proof of the second step that we mentioned above. In
particular, it also proves the conclusion (3) in the statement of this
theorem. ∎

Suppose that $f$ is a partially hyperbolic, center bunched skew product on
$\mathbb{T}^{4}=\mathbb{T}^{2}\times\mathbb{T}^{2}$, with the Anosov map on
the base $f_{2}:\mathbb{T}^{2}\to\mathbb{T}^{2}$. Observe that for any
$p\in\mathbb{T}^{4}$, its unstable manifold $W^{uu}(p)$ projects to the
unstable manifold of $\pi_{2}(p)$ of $f_{2}$. In particular, for each
$p\in\mathbb{T}^{4}$ and $q\in W^{uu}(p)$ and since the center leaves are
uniformly compact (indeed they are just the fibers), the unstable holonomy map
can be defined on the entire center leaf $H^{u}_{p,q}:W^{c}(p)\to W^{c}(q)$.
By Theorem 2.5, this property is $C^{1}$-open.

Using the $f$-invariance of the center and strong unstable foliations, it is
easy to see that for any $n\in\mathbb{Z}$, for each $p,q$ as above, we have

$H^{u}_{f^{n}(p),f^{n}(q)}\circ f^{n}=f^{n}\circ H^{u}_{p,q}.$

We remark that in the skew product case, we may also use the notation
$H^{u}_{p_{2},q_{2}}$ to denote the unstable holonomy between
$\pi_{2}^{-1}(p_{2})$ and $\pi^{-1}_{2}(q_{2})$, for $p_{2}$ and $q_{2}$
belonging to the same unstable manifold of $f_{2}$. Sometimes we will use this
notation.

### Higher regularity of unstable holonomies

Let $f$ be a $C^{2+\alpha}$ absolutely partially hyperbolic skew product of
$\mathbb{T}^{4}=\mathbb{T}^{2}\times\mathbb{T}^{2}$ and let
$\chi^{ss},\chi^{c}_{-},\chi^{c}_{+},\chi^{uu}$ be the partially hyperbolic
constants of $f$. We say that $f$ verifies the $(2,\alpha)$-center unstable
bunching condition if

$\displaystyle\left(\frac{\chi^{c}_{+}}{\chi^{c}_{-}}\right)^{2}<\chi^{uu}\textrm{
and }\frac{\chi^{c}_{+}}{(\chi^{c}_{-})^{2}}<(\chi^{uu})^{\alpha}.$ (8)

Similarly, $f$ verifies the $(2,\alpha)$-center stable bunching condition if

$\displaystyle\chi^{ss}<\left(\frac{\chi^{c}_{-}}{\chi^{c}_{+}}\right)^{2}\textrm{
and }(\chi^{ss})^{\alpha}<\frac{\chi^{c}_{-}}{(\chi^{c}_{+})^{2}}.$ (9)

If $f$ verifies condition (8) and (9) then we say that $f$ is
$(2,\alpha)$-center bunched.

We use the $(2,\alpha)$-center bunching condition to obtain $C^{2}$-regularity
of the unstable holonomy inside a center unstable leaf. This is given in the
following theorem.

######  Theorem 2.10.

Let $f$ be a $C^{2+\alpha}$ absolutely partially hyperbolic skew product of
$\mathbb{T}^{4}$, and fix $R>0$. If $f$ is $(2,\alpha)$-center unstable
bunched, then $\\{H^{u}_{p,q}\\}_{p\in\mathbb{T}^{4},q\in W_{R}^{uu}(p)}$ is a
family of $C^{2}$-diffeomorphisms of $\mathbb{T}^{2}$ whose $C^{2}$-norm
varies continuously with the choices of $p$ and $q$.

This theorem is proved in the appendix (see section 10).

####  Accessibility classes

For a partially hyperbolic diffeomorphism $f$, an $su$-path is a curve which
is the concatenation of finitely many curves, each of them being contained in
a stable or unstable leaf. Given a point $m\in M$, its accessibility class is
defined as

$AC(m)=\\{p\in M:\textrm{ there exists an $su$-path connecting $m$ and
$p$.}\\}$

We say that $f$ is accessible if for any $m\in M$, $AC(m)=M$. Suppose that $f$
is dynamically coherent, we say that an accessibility class $AC(m)$ is trivial
if $AC(m)\cap W^{c}(m)$ is totally disconnected. We say that $f$ has the
global product structure if there is a covering $\pi:\tilde{M}\to M$ and a
lift $\tilde{f}:\tilde{M}\to\tilde{M}$ for any
$\tilde{x},\tilde{y}\in\tilde{M}$ we have

$\\#\\{\tilde{\mathcal{F}}^{cs}(\tilde{x})\cap\tilde{\mathcal{F}}^{uu}(\tilde{y})\\}=1\textrm{
and
}\\#\\{\tilde{\mathcal{F}}^{cu}(\tilde{x})\cap\tilde{\mathcal{F}}^{ss}(\tilde{y})\\}=1,$

where $\tilde{\mathcal{F}}^{*}$ if the lift of the foliation
$\mathcal{F}^{*}$, for $*=ss,cs,cu,uu$. We now describe some results from
Horita-Sambarino in [35]. In what follows we will restrict ourselves to the
case that $M=\mathbb{T}^{4}$.

We define $\mathcal{E}=\mathcal{E}^{2}(\mathbb{T}^{4})$ to be the set of
$C^{2}$-partially hyperbolic diffeomorphisms $f$ such that

  * •

$f$ is dynamically coherent, $2$-normally hyperbolic and plaque expansive;

  * •

$f$ is center bunched;

  * •

$f$ has the global product structure;

  * •

the set of compact center leaves that are $f$-periodic is dense in $M$.

The set $\mathcal{E}$ is $C^{1}$-open in $\mathrm{Diff}^{2}(M)$.

Inside $\mathcal{E}$ let us define the set of skew-products over a fixed
Anosov diffeomorphism. Let $g:\mathbb{T}^{2}\to\mathbb{T}^{2}$ be a
$C^{2}$-Anosov diffeomorphism, let
$\mathcal{V}_{g}\subset\mathrm{Diff}^{2}(\mathbb{T}^{2})$ be the open set such
that if $h\in\mathcal{V}_{g}$ then $h\times g$ is partially hyperbolic, center
bunched and $2$-normally hyperbolic. Let $f:\mathbb{T}^{2}\to\mathcal{V}_{g}$
be a continuous map and denote by $f(.,y)$ the diffeomorphism
$f(y):\mathbb{T}^{2}\to\mathbb{T}^{2}$. We define the skew product given by
$f$ over $g$ as

$f_{g}(x,y)=(f(x,y),g(y)).$

Observe that $f_{g}\in\mathcal{E}$. Let $\mathcal{E}^{sp}_{g}$ be the set of
partially hyperbolic skew products over $g$ which take value on
$\mathcal{V}_{g}$. That is, $f_{g}\in\mathcal{E}^{sp}_{g}$ if and only if
$f_{g}(x,y)=(f(x,y),g(y))$, where $f(.,y)\in\mathcal{V}_{g}$ for every
$y\in\mathbb{T}^{2}$. Observe that $\mathcal{E}^{sp}_{g}$ can be identified
with $\mathcal{G}_{g}:=\\{f:\mathbb{T}^{2}\to\mathcal{V}_{g},\textrm{ s.t. $f$
is continuous}\\}$, since $g$ is fixed. We say that
$f,\tilde{f}\in\mathcal{G}_{g}$ are $C^{2}$-close if for each
$y\in\mathbb{T}^{2}$, the diffeomorphisms $f(.,y)$ and $\tilde{f}(.,y)$ are
$C^{2}$-close. Of course, $\mathcal{E}_{g}^{sp}\subset\mathcal{E}$. We state
the following theorem of Horita-Sambarino in our scenario, but we remark that
their theorem is more general than the statement we give.

######  Theorem 2.11 ([35], Theorem 2).

Let $g:\mathbb{T}^{2}\to\mathbb{T}^{2}$ be a $C^{2}$-Anosov diffeomorphism,
then the set $\mathcal{R}_{0}$ of diffemorphisms in $\mathcal{E}^{sp}_{g}$
whose accessibility classes are all non trivial is $C^{1}$-open and
$C^{2}$-dense.

Another important result from [35] is the following:

######  Proposition 2.12 ([35], Corollary $4.3$).

If $f\in\mathcal{E}$ has all its accessibility classes non trivial, then there
exists a $C^{1}$-open set $\mathcal{V}(f)$ in $\mathcal{E}$ of partially
hyperbolic diffeomorphisms whose accessibility classes are all non trivial.

Recall that $\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$ is the set
of $C^{2}$-diffeomorphisms $h$ that are skew products, that is,
$h(x,y)=(h_{1}(x,y),h_{2}(y))$ where $x,y\in\mathbb{T}^{2}$ and $h(.,y)$ is a
$C^{2}$-diffeomorphism of $\mathbb{T}^{2}$ that changes continuously with the
choice of $y$.

Recall that in the introduction we defined, for each $N\in\mathbb{N}$, the
diffeomorphism $f_{N}(x,y,z,w)=(s_{N}(x,y)+P_{x}\circ
A^{N}(z,w),A^{2N}(z,w))$, such that $s_{N}$ is the standard map, $P_{x}$ is
the projection on the horizontal direction, and $A$ is a linear Anosov
diffeomorphism on $\mathbb{T}^{2}$. Observe that $f_{N}$ belongs to
$\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$. Furthermore, for $N$
large enough we have that $f_{N}$ is $2$-normally hyperbolic and center
bunched, in particular, it belongs to $\mathcal{E}$. Using Theorem 2.11 and
Proposition 2.12, we obtain the following theorem.

######  Theorem 2.13.

For $N$ large enough, for each sufficiently small $C^{1}$-neighborhood
$\mathcal{W}$ of $f_{N}$ in
$\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$, there exists a set
$\mathcal{V}\subset\mathcal{W}$, which is $C^{1}$-open and $C^{2}$-dense in
$\mathcal{W}$ such that for any $g\in\mathcal{V}$ all its accessibility
classes are non trivial.

###### Proof.

If $\mathcal{W}$ is sufficiently $C^{1}$-small, then for any $g\in\mathcal{W}$
the basis dynamics $g_{2}$ is a $C^{2}$-Anosov diffeomorphism which is
$C^{1}$-close to $A^{2N}$.

Let $\mathcal{N}$ be a small $C^{1}$-neighborhood of $A^{2N}$ in
$\mathrm{Diff}^{2}(\mathbb{T}^{2})$. For each $g_{2}\in\mathcal{N}$ we
consider $\mathcal{W}_{g_{2}}=\mathcal{W}\cap\mathcal{E}^{sp}_{g_{2}}$ and
observe that this set is $C^{1}$-open in $\mathcal{G}_{g_{2}}$.

By Theorem 2.11, there exists a $C^{1}$-open and $C^{2}$-dense subset
$\tilde{\mathcal{V}}_{g_{2}}$ of $\mathcal{W}_{g_{2}}$ such that for each skew
product $g\in\tilde{\mathcal{V}}_{g_{2}}$ all its accessibility classes are
non trivial. By Proposition 2.12, for each $g\in\tilde{\mathcal{V}}_{g_{2}}$
there exists a $C^{1}$-open subset of $\mathcal{E}$, which we denote it by
$\mathcal{V}(g)$, of diffeomorphisms whose accessibility classes are all non
trivial. Now define

$\mathcal{V}:=\displaystyle\bigcup_{g_{2}\in\mathcal{N}}\bigcup_{g\in\tilde{\mathcal{W}}_{g_{2}}}\mathcal{V}(g).$

It is easy to see that $\mathcal{V}$ is $C^{1}$-open and $C^{2}$-dense in
$\mathcal{W}$. Moreover, for each $g\in\mathcal{V}$ all its accessibility
classes are non trivial. ∎

In our work we will also need the following result that describes the
structure of accessibility classes.

######  Theorem 2.14 ([51], Theorem B).

Let $f$ be a dynamically coherent $C^{2}$-partially hyperbolic diffeomorphism
with two dimensional center, and which is center bunched. Then every
accessibility class is an immersed $C^{1}$-submanifold.

###  2.2 Pesin’s theory and SRB measures

Let $f$ be a $C^{1}$-diffeomorphism. A number $\lambda\in\mathbb{R}$ is a
Lyapunov exponent if there exists a point $p\in M$ and a non zero vector $v\in
T_{p}M$ such that $\lim_{n\to\pm\infty}\frac{1}{n}\log\|Df^{n}(p)v\|=\lambda$.
We write $\lambda(p,v):=\lim_{n\to\pm\infty}\frac{1}{n}\log\|Df^{n}(p)v\|$.

We say that a set $R$ has full probability if for any $f$-invariant
probability measure $\nu$ it is verified that $\nu(R)=1$. The following
theorem is known as the Oseledets theorem.

######  Theorem 2.15 ([6], Theorems $2.1.1$ and $2.1.2$).

For any $C^{1}$-diffeomorphism $f$, there is a set $\mathcal{R}$ of full
probability, such that for every $\varepsilon>0$ it exists a measurable
function $C_{\varepsilon}:\mathcal{R}\to(1,+\infty)$ with the following
properties:

  1. 1.

for any $p\in\mathcal{R}$ there are numbers $s(p)\in\mathbb{N}$,
$\lambda_{1}(p)<\cdots<\lambda_{s(p)}(p)$ and a decomposition
$T_{p}M=E^{1}_{p}\oplus\cdots\oplus E^{s(p)}_{p}$;

  2. 2.

$s(f(p))=s(p)$, $\lambda_{i}(f(p))=\lambda_{i}(p)$ and
$Df(p).E^{i}_{p}=E^{i}_{f(p)}$, for every $i=1,\cdots,s(p)$;

  3. 3.

for every $v\in E^{i}_{p}-\\{0\\}$, $\lambda(p,v)=\lambda^{i}(p)$.

We call the set $\mathcal{R}$ the set of regular points. A point
$p\in\mathcal{R}$ has $k$ negative Lyapunov exponents if

$\displaystyle\sum_{i:\lambda_{i}(p)<0}dim(E^{i}_{p})=k.$

Similarly for positive or zero Lyapunov exponents. From now on, we assume that
$\nu$ is a $f$-invariant measure, not necessarily ergodic, and there are
numbers $k$ and $l$ such that $\nu$-almost every point $p\in\mathcal{R}$ has
$k$ negative and $l$ positive Lyapunov exponents.

For a regular point we write

$E^{s}_{p}=\displaystyle\bigoplus_{i:\lambda_{i}(p)<0}E^{i}_{p}\textrm{ and
}E^{u}_{p}=\bigoplus_{i:\lambda_{i}(p)>0}E^{i}_{p}.$ (10)

It is well known that for a $C^{2}$-diffeomorphism $f$ and an invariant
measure $\nu$, then for $\nu$-almost every $p$, the set defined by

$W^{s}(p)=\\{q\in M:\displaystyle\limsup_{n\to+\infty}\frac{1}{n}\log
d(f^{n}(p),f^{n}(q))<0\\}$

is an immersed submanifold such that $T_{p}W^{s}(p)=E^{s}_{p}$ (see section
$4$ of [47]). We call $W^{s}(p)$ the stable Pesin manifold of the point $p$.
Similarly, the set defined by

$W^{u}(p)=\\{q\in M:\displaystyle\limsup_{n\to+\infty}\frac{1}{n}\log
d(f^{-n}(p),f^{-n}(q))<0\\}$

is an immersed submanifold such that $T_{p}W^{u}(p)=E^{u}_{p}$. We call
$W^{u}(p)$ the unstable Pesin manifold of the point $p$. Since these manifolds
exist for $\nu$-almost every point, the unstable manifolds
$\\{W^{u}(p)\\}_{p\in\mathcal{R}}$ form a partition of a $\nu$-full measure
subset of $M$.

######  Remark 2.16.

If $f$ is also partially hyperbolic, with $TM=E^{ss}\oplus E^{c}\oplus E^{uu}$
then the Oseledets splitting refines the partially hyperbolic splitting. This
means that for a regular point $p\in\mathcal{R}$, there are numbers $1\leq
l_{1}<l_{2}<s(p)$ such that

$E^{ss}_{p}=\displaystyle\bigoplus_{i=1}^{l_{1}}E^{i}_{p},\textrm{
}E^{c}_{p}=\bigoplus_{i=l_{1}+1}^{l_{2}}E^{i}_{p}\textrm{ and
}E^{uu}_{p}=\bigoplus_{i=l_{2}+1}^{s(p)}E^{i}_{p}.$

This follows from a standard argument similar to the proof of the uniqueness
of dominated splittings, see section $B.1.2$ from [12]. It also holds that for
any regular point $p$, $E^{ss}_{p}\subset E^{s}_{p}$ and $E^{uu}_{p}\subset
E^{u}_{p}$.

A partition $\xi$ of $M$ is measurable with respect to a probability measure
$\nu$, if up to a set of $\nu$-zero measure, the quotient $M/\xi$ is separated
by a countable number of measurable sets. Denote by $\hat{\nu}$ the quotient
measure in $M/\xi$.

By Rokhlin’s disintegration theorem [52], for a measurable partition $\xi$,
there is set of conditional measures $\\{\nu_{D}^{\xi}:D\in\xi\\}$ such that
for $\hat{\nu}$-almost every $D\in\xi$ the measure $\nu_{D}^{\xi}$ is a
probability measure supported on $D$, for each measurable set $B\subset M$ the
application $D\mapsto\nu^{\xi}_{D}(B)$ is measurable and

$\nu(B)=\displaystyle\int_{M/\xi}\nu_{D}^{\xi}(B)d\hat{\nu}(D).$ (11)

From now on we suppose that $f$ is a $C^{2}$-diffeomorphism and $\nu$ has no
zero Lyapunov exponents. We call such a measure hyperbolic. We remark that
usually the unstable partition $\\{W^{u}(p)\\}_{p\in\mathcal{R}}$ is not a
measurable partition. We say that a $\nu$-measurable partition $\xi^{u}$ is
$u$-subordinated if for for $\nu$-almost every $p$, the following conditions
are verified:

  * •

$\xi^{u}(p)\subset W^{u}(p)$;

  * •

$\xi^{u}(p)$ contains an open neighborhood of $p$ inside $W^{u}(p)$.

######  Definition 2.17 (SRB measure).

A measure $\nu$ is SRB if for any $u$-subordinated measurable partition
$\xi^{u}$, for $\nu$-almost every $p$, the conditional measure
$\nu^{u}_{\xi^{u}(p)}$ is absolutely continuous with respect to the riemannian
volume of $W^{u}(p)$.

There is a well developed ergodic theory for hyperbolic SRB measures. We now
state some results obtained by Ledrappier in [40].

######  Theorem 2.18 ([40], Corollary $4.10$ and Theorem $5.10$.).

Let $f$ be a $C^{2}$-diffeomorphism and $\nu$ a hyperbolic SRB measure. Then
there are at most countably many ergodic components of $\nu$, that is,

$\nu=\displaystyle\sum_{i\in\mathbb{N}}c_{i}\nu_{i},$

where $c_{i}\geq 0$, $\displaystyle\sum_{i\in\mathbb{N}}c_{i}=1$, each
$\nu_{i}$ is an $f$-invariant ergodic SRB measure such that if $i\neq j$, and
$c_{i},c_{j}>0$ then $\nu_{i}\neq\nu_{j}$. Moreover, for each $i\in\mathbb{N}$
such that $c_{i}>0$, there exists $k_{i}\in\mathbb{N}$ such that

$\nu_{i}=\frac{1}{k_{i}}\displaystyle\sum_{j=1}^{k_{i}}\nu_{i,j},$

where each $\nu_{i,j}$ is an $f^{k_{i}}$-invariant probability measure, the
system $(f^{k_{i}},\nu_{i,j})$ is Bernoulli and $\nu_{i,j}\neq\nu_{i,l}$ if
$j\neq l$. Furthermore, $f$ permutes the measures $\nu_{i,j}$, that is,
$f_{*}(\nu_{i,j})=\nu_{i,j+1}$ for $j=1,\cdots,k_{i}-1$ and
$f_{*}(\nu_{i,k_{i}})=\nu_{i,1}$, where $f_{*}(\nu)$ denotes the pushforward
of a measure $\nu$ by $f$.

Now given two hyperbolic ergodic measure, $\mu$ and $\nu$, we say that stable
manifolds of $\mu$ intersects transversely unstable manifolds of $\nu$ if the
following holds: there exist a set $\Lambda^{s}$ with positive $\mu$-measure
and a set $\Lambda^{u}$ with positive $\nu$-measure, such that for each
$p\in\Lambda^{s}$ and $q\in\Lambda^{u}$, there exists
$n_{1},n_{2}\in\mathbb{Z}$ with

$W^{s}(f^{n_{1}}(p))\pitchfork W^{u}(f^{n_{2}}(q))\neq\emptyset.$

In this case we write $\mu\pitchfork_{su}\nu$.

######  Definition 2.19.

For $\mu$ and $\nu$ hyperbolic ergodic measures, we say that $\mu$ is
homoclinically related with $\nu$, if $\mu\pitchfork_{su}\nu$ and
$\nu\pitchfork_{su}\mu$. We write $\mu\sim_{hom}\nu$.

In the case that $\mu$ and $\nu$ are ergodic SRB measures, homoclinic relation
actually implies that they are the same.

######  Theorem 2.20.

Let $\mu$ and $\nu$ be two hyperbolic, ergodic SRB measures. If
$\mu\sim_{hom}\nu$ then $\mu=\nu$.

The proof of Theorem 2.20 is a consequence of Hopf’s argument adapted to the
non-uniformly hyperbolic scenario. This type of argument has been done in many
places, see for instance Lemma $3.2$ in [33].

We remark that all the results stated in this section were stated for
$C^{2}$-diffeomorphisms, but they hold for $C^{1+\alpha}$-diffeomorphisms.

###  2.3 $u$-Gibbs measures and the invariance principle

####  $u$-Gibbs measures

Let $f$ be a $C^{2}$-partially hyperbolic diffeomorphism and let $\mu$ be an
$f$-invariant measure. We say that a $\mu$-measurable partition $\xi^{uu}$ is
subordinated to the foliation $\mathcal{F}^{uu}$, if for $\mu$-almost every
$p$, $\xi^{uu}(p)\subset W^{uu}(p)$ and $\xi^{uu}(p)$ contains an open
neighborhood of $p$ inside $W^{uu}(p)$. For simplicity, we will write the
conditional measure $\mu^{uu}_{\xi^{uu}(p)}$ by $\mu^{uu}_{p}$.

######  Definition 2.21 ($u$-Gibbs).

An $f$-invariant measure $\mu$ is $u$-Gibbs if for any $\mu$-measurable
partition $\xi^{uu}$ subordinated to $\mathcal{F}^{uu}$, for $\mu$-almost
every point $p$, the conditional measure $\mu^{uu}_{p}$ is absolutely
continuous with respect to the Lebesgue measure of $W^{uu}(p)$. We denote the
set of $u$-Gibbs measures of $f$ by $Gibbs^{u}(f)$.

These measures have an important role in the study of ergodic theory of
partially hyperbolic systems. The next lemma states that they capture all
possible statistical behavior of Lebesgue almost every point. Recall that for
any $p\in M$ and $n\in\mathbb{N}$, we defined

$\mu_{n}(p)=\displaystyle\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f^{j}(p)}.$

######  Theorem 2.22 ([12], Theorem $11.16$).

Let $f$ be a $C^{2}$-partially hyperbolic diffeomorphism, then for Lebesgue
almost every point $p\in M$, every accumulation point of the sequence of
probability measures $(\mu_{n}(p))_{n\in\mathbb{N}}$ belongs to
$\mathrm{Gibbs}^{u}(f)$.

Let us consider the strong unstable foliation $\mathcal{F}^{uu}$ and $\mu$ an
$f$-invariant measure. We say that a $\mu$-measurable partition $\xi^{uu}$
subordinated to $\mathcal{F}^{uu}$ is increasing if for $\mu$-almost every
$p$, we have $\xi^{uu}(f(p))\subset f(\xi^{uu}(p))$. We define the
$\mu$-partial entropy along $\mathcal{F}^{uu}$ by

$\displaystyle
h_{\mu}(f,\mathcal{F}^{uu})=H_{\mu}(f^{-1}\xi^{uu}|\xi^{uu}):=-\int_{M}\log\mu^{uu}_{p}(f^{-1}\xi^{uu}(p))d\mu(p),$
(12)

where $f^{-1}\xi^{uu}(p)$ is the element of the partition $f^{-1}\xi^{uu}$
containing $p$. The definition above does not depend on the choice of the
$\mu$-measurable partition $\xi^{uu}$. The notion of partial entropy along
expanding foliations has been introduced in [61] and [62] (see also [41]).

Let $Jac^{uu}(p)=|\det(Df(p)|_{E^{uu}})|$. In the case that $E^{uu}$ has
dimension one, for any ergodic $f$-invariant measure, we write
$\lambda^{uu}_{\mu}$ to be the Lyapunov exponent of the strong unstable
direction. The following result can be found in [62] and [40].

######  Proposition 2.23 ([62], Proposition $5.2$, and [40], Theorem $3.4$).

Let $\mu$ be an $u$-Gibbs measure. Then

$h_{\mu}(f,\mathcal{F}^{uu})=\displaystyle\int_{M}\log Jac^{uu}(p)d\mu(p).$

In particular, if $E^{uu}$ is one dimensional and $\mu$ is ergodic then
$h_{\mu}(f,\mathcal{F}^{uu})=\lambda^{uu}_{\mu}$.

#### The invariance principle

An important tool in this work is the invariance principle which was first
developed by Furstenberg in [31] and by Ledrappier in [39]. We also mention
the work of Avila-Viana in [5]. In this work we use the version of the
invariance principle given by Tahzibi-Yang in [59], which we describe in this
section. This relates entropy along strong unstable foliations with the so
called $u$-invariance of certain measures. Their results hold for large
classes of partially hyperbolic skew products, however, we will state them for
skew products on $\mathbb{T}^{2}\times\mathbb{T}^{2}$.

Let $f$ be a $C^{2}$-partially hyperbolic center bunched skew product and let
$f_{2}$ be the Anosov diffeomorphism on the base. We remark that on
$\mathbb{T}^{2}$, every Anosov diffeomorphism is transitive. Fix a
$f_{2}$-invariant measure $\nu$. Let $\mathcal{\xi}^{uu}_{2}$ be a
$\nu$-measurable partition of $\mathbb{T}^{2}$ which is subordinated to the
foliation $\mathcal{F}^{uu}_{2}$ (the unstable foliation of $f_{2}$ on
$\mathbb{T}^{2}$), and consider the $\mu$-measurable partition $\xi^{uu}$ of
$\mathbb{T}^{4}$ subordinated to $\mathcal{F}^{uu}$ which refines the
partition $\pi^{-1}_{2}(\xi^{uu}_{2})$ with the property that for $\mu$-almost
every $p$, $\pi_{2}(\xi^{uu}(p))=\xi^{uu}_{2}(\pi_{2}(p))$.

######  Definition 2.24.

We say that an $f$-invariant measure $\mu$ is an $u$-state projecting on
$\nu$, if $(\pi_{2})_{*}=\nu$ and for $\mu$-almost every $p$,

$(\pi_{2})_{*}\mu^{uu}_{p}=\nu^{uu}_{\pi_{2}(p)}.$ (13)

We denote the set of $u$-state measures projecting on $\nu$ by
$\mathrm{State}^{u}_{\nu}(f)$. We say that a measure $\mu$ projecting on $\nu$
is an $s$-state projecting on $\nu$, if
$\mu\in\mathrm{State}^{u}_{\nu}(f^{-1})$. We denote the set of $s$-state
measures by $\mathrm{State}^{s}_{\nu}(f)$.

######  Remark 2.25.

In [59], the authors call the measures from definition 2.24 $u$-Gibbs measures
projecting on $\nu$. Since we already use the name $u$-Gibbs for the measures
from definition 2.21, we changed the name in our paper. Even though later we
will see that in our setting both definitions coincide once the measure $\nu$
is an SRB-measure for the Anosov diffeomorphism on the basis (see proposition
2.28).

The following result is a characterization using entropy for a measure to
belong to $\mathrm{State}^{u}_{\nu}(f)$.

######  Theorem 2.26 ([59], Theorem A).

Let $f$ be a $C^{2}$-partially hyperbolic skew product as above and let $\nu$
be an $f_{2}$-invariant measure. Suppose that $\mu$ is an $f$-invariant
measure such that $(\pi_{2})_{*}\mu=\nu$. Then
$h_{\mu}(f,\mathcal{F}^{uu})\leq h_{\nu}(f_{2})$ and the equality holds if and
only if $\mu\in\mathrm{State}^{u}_{\nu}(f)$.

######  Proposition 2.27 ([59], Proposition $5.4$).

A measure $\mu$ is an $u$-state projecting on $\nu$ if and only if there
exists a set $X\subset\mathbb{T}^{2}$ of full $\nu$-measure such that for any
two points $p_{2},q_{2}\in X$ in the same unstable leaf, we have that

$\mu^{c}_{q_{2}}=(H^{u}_{p_{2},q_{2}})_{*}\mu^{c}_{p_{2}}.$ (14)

The property described by (14) is called $u$-invariance of the conditional
measures $\\{\mu^{c}_{p_{2}}\\}_{p_{2}\in\mathbb{T}^{2}}$.

Since $f_{2}$ is a transitive $C^{2}$-Anosov diffeomorphism, it is well known
that it admits an unique SRB measure $\nu$, see [14, 53, 55]. Consider now the
set $\mathrm{State}^{u}_{\nu}(f)$. In what follows we will show that
$\mathrm{Gibbs}^{u}(f)=\mathrm{State}^{u}_{\nu}(f)$.

######  Proposition 2.28.

For $f$ and $\nu$ as above,
$\mathrm{Gibbs}^{u}(f)=\mathrm{State}^{u}_{\nu}(f)$.

To prove this proposition, we will need the following lemma.

######  Lemma 2.29.

Let $\mu\in\mathrm{Gibbs}^{u}(f)$, then $(\pi_{2})_{*}\mu=\nu$.

###### Proof.

It is enough to prove that $\tilde{\nu}:=(\pi_{2})_{*}\mu$ is an SRB measure
for $f_{2}$. Since $f_{2}$ admits only one SRB measure, it follows that
$\tilde{\nu}=\nu$.

Let $\xi^{uu}_{2}$ be a $\tilde{\nu}$-measurable partition subordinated to
$\mathcal{F}^{uu}_{2}$. Observe that the partition
$\xi^{cu}=\pi_{2}^{-1}(\xi^{uu})$ is $\mu$-measurable and denote by
$\mu^{cu}_{p}$ the conditional measures of $\mu$ with respect to this
partition. The partition $\xi^{cu}$ is refined by the $\mu$-measurable
partition $\xi^{uu}$ which is subordinated to $\mathcal{F}^{uu}$ and such that
for $\mu$-almost every $p$, we have
$\pi_{2}(\xi^{uu}(p))=\xi^{uu}_{2}(\pi_{2}(p))$.

Take a $\tilde{\nu}$-generic point $p_{2}\in\mathbb{T}^{2}$ and let
$B\subset\xi^{uu}_{2}(p_{2})$ be a set of zero Lebesgue measure inside the
unstable manifold of $p_{2}$. Since the foliation by center fibers is smooth
(because we are in the skew product setting), and the strong unstable
manifolds of $f$ are uniformly transverse to the center direction inside the
$cu$-leaves, we have that for $\mu^{cu}_{p_{2}}$-almost every $q$ the set
$\xi^{uu}(q)\cap\pi^{-1}_{2}(B)$ has zero Lebesgue measure inside
$W^{uu}_{f}(q)$. In particular, the $u$-Gibbs property of $\mu$ implies that
$\mu^{uu}_{q}(\pi^{-1}_{2}(B))=0$. We conclude

$\tilde{\nu}^{uu}_{p_{2}}(B)=\displaystyle\int_{M}\mu^{uu}_{q}(\pi^{-1}_{2}(B))d\mu^{cu}(q)=0.$

This is true for any set $B$ of zero Lebesgue measure. This implies that
$\tilde{\nu}^{uu}_{p_{2}}$ is absolutely continuous with respect to the
Lebesgue measure of $W^{uu}_{f_{2}}(p_{2})$ and the measure $\tilde{\nu}$ is
SRB.

∎

###### Proof of Proposition 2.28.

From (13) and the fact that the foliation by horizontal fiber is smooth, it is
immediate that $\mathrm{State}^{u}_{\nu}(f)\subset\mathrm{Gibbs}^{u}(f)$.

Since the strong unstable direction is uniformly transverse to the center
fibers inside the $cu$-leaves and it projects to $E^{uu}_{f_{2}}$, and since
the center direction is orthogonal to the base, there exists a constant $C\geq
1$ such that for any $p\in\mathbb{T}^{4}$ and any $v^{uu}\in E^{uu}_{p}$ we
have

$\displaystyle\frac{1}{C}\|v^{uu}\|\leq\|D\pi_{2}(p)v^{uu}\|\leq\|v^{uu}\|.$

Suppose that $\mu\in\mathrm{Gibbs}^{u}(f)$ is an ergodic measure. Let $p$ be a
generic point for $\mu$ and let $v^{uu}\in E^{uu}_{p}$ be an unit vector.
Observe that for any $n\in\mathbb{N}$ we have

$\displaystyle\frac{1}{C}\|Df^{n}(p)v^{uu}\|\leq\|D\pi_{2}(f^{n}(p))Df^{n}(p)v^{uu}\|\leq\|Df^{n}(p)v^{uu}\|.$

Since $f$ is a skew product and $\pi_{2}\circ f=f\circ\pi_{2}$, we obtain that
$D\pi_{2}(f^{n}(p))Df^{n}(p)v^{uu}=Df^{n}_{2}(\pi_{2}(p))D\pi_{2}(p)v^{uu}$.
By lemma 2.29, we may assume that $\pi_{2}(p)$ is a generic point for $\nu$.
We conclude that

$\displaystyle\lambda^{uu}_{\mu}=\lim_{n\to+\infty}\frac{1}{n}\log\|Df^{n}(p)v^{uu}\|=\lim_{n\to+\infty}\frac{1}{n}\log\|Df^{n}_{2}(\pi_{2}(p))D\pi_{2}(p)v^{uu}\|=\lambda^{uu}_{\nu},$

where $\lambda^{uu}_{\mu}$ is the Lyapunov exponent of $f$ for $\mu$ along the
strong unstable direction and $\lambda^{uu}_{\nu}$ is the Lyapunov exponent of
$f_{2}$ for $\nu$ along the unstable direction.

It is well known that the measure $\nu$ verifies Pesin’s formula (since it is
also an SRB measure for $f_{2}$), see [40], and hence
$h_{\nu}(f_{2})=\lambda^{uu}_{\nu}$. By proposition 2.23, we have that
$h_{\mu}(f,\mathcal{F}^{uu})=\lambda^{uu}_{\mu}$. We conclude that
$h_{\mu}(f,\mathcal{F}^{uu})=h_{\nu}(f_{2})$. By Theorem 2.26 we obtain that
$\mu\in\mathrm{State}^{u}_{\nu}(f)$. ∎

The main conclusion of proposition 2.28 is the following corollary.

######  Corollary 2.30.

For $f$ as above, any $u$-Gibbs measure $\mu$ has $u$-invariant center
conditional measures.

##  3 Proof of Theorem A assuming Theorems B and C

Fix $N$ large enough and let $\mathcal{U}_{N}^{sk}$ be a $C^{2}$-neighborhood
of $f_{N}$ inside $\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$ small
enough such that it verifies the conclusions of Theorems B and C. By Theorem
C, any $g\in\mathcal{U}_{N}^{sk}$ has at most one SRB-measure. By Theorem B,
every $u$-Gibbs measure is either SRB or it is supported on a finite union of
two dimensional tori tangent to the strong stable and unstable directions. In
particular, in the second, the support of the $u$-Gibbs measure is contained
in the finite union of trivial accessibility classes.

By Theorem 2.13, there exists a subset
$\mathcal{V}\subset\mathcal{U}_{N}^{sk}$ which is $C^{2}$-open and
$C^{2}$-dense in $\mathcal{U}_{N}^{sk}$ such that any $g\in\mathcal{V}$ does
not have any trivial accessibility class. In particular, for such $g$, there
cannot be a two dimensional torus tangent to $E^{ss}_{g}\oplus E^{uu}_{g}$.
Therefore, as a consequence of Theorems B and C, we conclude that for any
$g\in\mathcal{V}$, there exists an unique $u$-Gibbs measure $\mu_{g}$. It is
hyperbolic SRB and Bernoulli. Moreover,
$\mathrm{supp}(\mu_{g})=\mathbb{T}^{4}$.

Fix $g\in\mathcal{V}$. By Theorem 2.22, for Lebesgue almost every point $p$,
any accumulation point of the sequence

$\mu_{n}(p)=\frac{1}{n}\displaystyle\sum_{j=0}^{n-1}\delta_{g^{j}(p)}$

is an $u$-Gibbs measure. Since there exists only one $u$-Gibbs measure
$\mu_{g}$, we conclude that for Lebesgue almost every point $p$,

$\displaystyle\lim_{n\to+\infty}\mu_{n}(p)=\mu_{g}.$

Therefore, $Leb(B(\mu_{g}))=1$ and we conclude the proof of Theorem A.

##  4 Center Lyapunov exponents for $u$-Gibbs measures

In this section we explain how the techniques developed by Berger-Carrasco in
[10], and the adaptations of their techniques made by the author in [45],
actually give estimates for the Lyapunov exponents for any $u$-Gibbs measure.
We prove the following theorem:

######  Theorem 4.1.

For every $\delta\in(0,1)$, there exists $N_{0}=N_{0}(\delta)$ such that for
every $N\geq N_{0}$, there exists $\mathcal{U}_{N}$ a $C^{2}$-neighborhood of
$f_{N}$ inside $\mathrm{Diff}^{2}(\mathbb{T}^{4})$ with the following
property. If $g\in\mathcal{U}_{N}$ and $\mu$ is an $u$-Gibbs measure, then
$\mu$-almost every point has a positive and a negative Lyapunov exponent along
the center whose absolute value is greater than $(1-\delta)\log N$.

######  Remark 4.2.

Even though the results from [10, 45] are in the volume preserving scenario,
several of the lemmas and propositions still valid for dissipative
perturbations. In what follows, we will use several results from these works.
The only point in this section that will need an adaptation for $u$-Gibbs
measures is given in proposition 4.11.

Let $A\in SL(2,\mathbb{Z})$ be the linear Anosov matrix considered in the
definition of the map $f_{N}$. Denote by
$0<\lambda<1<\tilde{\mu}=\lambda^{-1}$ the eigenvalues of $A$. Let $e^{s}$ and
$e^{u}$ be unit eigenvectors of $A$ for $\lambda$ and $\tilde{\mu}$,
respectively. Recall that we defined the linear map
$P_{x}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ given by $P_{x}(a,b)=(a,0)$.

######  Lemma 4.3 ([10], Proposition $1$).

There is a differentiable function $\alpha:\mathbb{T}^{4}\to\mathbb{R}^{2}$
such that the unstable direction of $f_{N}$ is generated by the vector field
$(\alpha(m),e^{u})$, where

$Df_{N}(m).(\alpha(m),e^{u})=\tilde{\mu}^{2N}(\alpha(f_{N}(m)),e^{u})\textrm{
and }\|\alpha(m)-\lambda^{N}P_{x}(e^{u})\|\leq\lambda^{2N}.$

Observe that $|\det Df_{N}|_{E^{c}_{f_{N}}}|=1$.

######  Lemma 4.4 ([45], Lemma 7.17).

For $\varepsilon_{1}>0$ and $\beta>0$ small, if $N$ is large and
$\mathcal{U}_{N}$ is small enough then for every $g\in\mathcal{U}_{N}$ and for
all unit vectors $v^{s}\in E^{ss}_{g}$, $v^{c}\in E^{c}_{g}$ and $v^{u}\in
E^{uu}_{g}$, the following holds:

  1. 1.

$e^{-\varepsilon_{1}}\tilde{\mu}^{2N}\leq\|Dg(v^{u})\|\leq
e^{\varepsilon_{1}}\tilde{\mu}^{2N}$;

  2. 2.

$\frac{1}{2N}\leq\|Dg(v^{c})\|\leq 2N;$

  3. 3.

$\|D^{2}g^{-1}\|\leq 2N$ and $\|D^{2}g\|\leq 2N$;

  4. 4.

$|\det Dg|_{E^{c}_{g}}|\in(e^{-\beta},e^{\beta})$;

  5. 5.

$E^{c}_{g}$ is $\frac{1}{2}$-Hölder.

A key element in Berger-Carrasco’s proof is to consider center vector fields
over certain pieces of strong unstable curve. Consider $g\in\mathcal{U}_{N}$.

######  Definition 4.5 ([10], Definition 7.18).

An $u$-curve for $g$ is a $C^{1}$-curve
$\gamma=(\gamma_{x},\gamma_{y},\gamma_{z},\gamma_{w}):[0,2\pi]\to M$ tangent
to $E^{uu}_{g}$ and such that $\left|\frac{d\gamma_{x}}{dt}(t)\right|=1$,
$\forall t\in[0,2\pi]$. For every $k\geq 0$ there exists an integer
$N_{k}=N_{k}(\gamma)\in\left[(e^{-\varepsilon_{1}}\tilde{\mu}^{2N})^{k},(e^{\varepsilon_{1}}\tilde{\mu}^{2N})^{k}\right]$
such that the curve $g^{k}\circ\gamma$ can be writen as

$g^{k}\circ\gamma=\gamma_{1}^{k}\ast\cdots\ast\gamma_{N_{k}}\ast\gamma_{N_{k}+1}^{k}$

where $\gamma_{j}^{k}$ for $j=1,\cdots,N_{k}$, are $u$-curves and
$\gamma_{N_{k}+1}^{k}$ is a segment of $u$-curve.

The following lemma controls the length of $u$-curves.

######  Lemma 4.6 ([10], Corollary $5$).

For $N$ is large and $\mathcal{U}_{N}$ small enough then for every
$g\in\mathcal{U}_{N}$ and any unit vector $v^{u}\in E^{uu}_{g,m}$, it holds
that

$|P_{x}(D\pi_{1}.v^{u})|\in[(\lambda^{N}(\|P_{x}(e^{u})\|-3\lambda^{N}),(\lambda^{N}(\|P_{x}(e^{u})\|+3\lambda^{N})].$

An easy consequence of this lemma is the following.

######  Corollary 4.7.

For any $\varepsilon_{2}>0$, if $N$ is large and $\mathcal{U}_{N}$ is small
enough, then any two $u$-curves $(\gamma,\gamma^{\prime})$ satisfy:

$e^{-\varepsilon_{2}}|\gamma|\leq|\gamma^{\prime}|\leq
e^{\varepsilon_{2}}l|\gamma|,$ (15)

where $|\gamma|$ denotes the length of the curve $\gamma$.

We define the unstable jacobian of $g^{k}$ as

$J^{uu}_{g^{k}}(m)=|\det Dg^{k}(m)|_{E^{uu}_{g}}|,\textrm{ $\forall
m\in\mathbb{T}^{4}$.}$ (16)

By item $2$ of lemma 4.4, for $g\in\mathcal{U}_{N}$ and for every
$m\in\mathbb{T}^{4}$

$e^{-\varepsilon_{1}}\lambda^{2N}\leq J^{uu}_{g^{-1}}(m)\leq
e^{\varepsilon_{1}}\lambda^{2N}.$ (17)

######  Lemma 4.8 ([45], Lemma 7.20).

For $\varepsilon_{3}>0$ small, if $N$ is large and $\mathcal{U}_{N}$ is small
enough, for every $g\in\mathcal{U}_{N}$ and any $u$-curve $\gamma$ for $g$,
for every $k\geq 0$, we have

$\forall m,m^{\prime}\in\gamma,\textrm{
}e^{-\varepsilon_{3}}\leq\frac{J^{uu}_{g^{-k}}(m)}{J^{uu}_{g^{-k}}(m^{\prime})}\leq
e^{\varepsilon_{3}}.$

This lemma implies that for $g\in\mathcal{U}_{N}$ and for any $u$-curve
$\gamma$ for $g$, if $A\subset\gamma$ is any measurable set, for every $k\geq
0$, it holds

$e^{-\varepsilon_{3}}\frac{Leb(A)}{Leb(\gamma)}\leq\frac{Leb(g^{-k}(A))}{Leb(g^{-k}(\gamma))}\leq
e^{\varepsilon_{3}}\frac{Leb(A)}{Leb(\gamma)}.$

######  Definition 4.9.

An adapted field $(\gamma,X)$ over an $u$-curve $\gamma$ is an unitary vector
field $X$ such that

  1. 1.

$X$ is tangent to the center direction;

  2. 2.

$X$ is $(C_{X},1/2)$-Hölder along $\gamma$, that is

$\|X_{m}-X_{m^{\prime}}\|\leq
C_{X}d_{\gamma}(m,m^{\prime})^{\frac{1}{2}},\textrm{ }\forall
m,m^{\prime}\in\gamma,$

where $C_{X}<30N^{2}\lambda^{N}$ and $d_{\gamma}$ is the distance measured
along $\gamma$.

######  Remark 4.10.

The estimate on the Hölder constant used in [10, 45] is $20N^{2}\lambda^{N}$,
instead of $30N^{2}\lambda^{N}$ as above. This is due to the fact that the
parametrization of the torus $\mathbb{T}^{4}$ is by intervals of length $2\pi$
instead of $1$ in the proof of lemma $2$ in [10]. However, this change on the
estimate of the Hölder constant does not affect the rest of the proof.

Let $(\gamma,X)$ be an adapted field, and define

$I_{n}^{\gamma,X}=\frac{1}{|\gamma|}\int_{\gamma}\log\|Dg^{n}.X\|d\gamma.$

######  Proposition 4.11.

Suppose that there exists $C>0$ with the following property: for every
$u$-curve $\gamma$ there exists an adapted vector field $(\gamma,X)$ for $g$
and for all $n>0$ large enough

$\frac{I_{n}^{\gamma,X}}{n}>C.$

Then any $u$-Gibbs measure $\mu$ for $g$ has a positive Lyapunov exponent
along the center direction greater than $e^{-2\varepsilon_{3}}C$.

###### Proof.

Suppose not, then there exist an $u$-Gibbs measure $\mu$ and a measurable set
$B$ with positive $\mu$-measure such that every point in $B$ has exponents in
the center direction strictly smaller than $e^{-2\varepsilon_{3}}C$. Since
$\mu$ has disintegration along unstable leaves equivalent to the Lebesgue
measure along the leaves, there is an unstable manifold $\gamma$ that
intersects $B$ on a set of positive measure for the Lebesgue measure of
$\gamma$. Let $b\in\gamma\cap B$ be a density point and take
$\gamma_{k}=g^{-k}\circ\beta_{k}$, where $\beta_{k}$ is a $u$-curve with
$\beta_{k}(0)=g^{k}(b)$. We have that $l(\gamma_{k})\to 0$ and by bounded
distortion (lemma 4.8)

$\frac{Leb(\gamma_{k}\cap B)}{Leb(\gamma_{k})}\longrightarrow 1.$

Take $k$ large enough such that

$\frac{Leb(\gamma_{k}\cap
B^{c})}{Leb(\gamma_{k})}<\frac{e^{-2\varepsilon_{3}}(e^{\varepsilon_{3}}-1)C}{2\log
2N}.$

Using bounded distortion again, for any $m^{k}\in g^{k}(\gamma_{k})$

$J^{uu}_{g^{-k}}(m^{k})\geq\frac{Leb(\gamma_{k})}{Leb(g^{k}(\gamma_{k}))}e^{-\varepsilon_{3}}.$

Define
$\chi_{k}(m)=\displaystyle\limsup_{n\to+\infty}\frac{1}{n}\log\|Dg^{n}(g^{k}(m)).X^{k}_{g^{k}(m)}\|$
for all $m\in\gamma_{k}$, where $X^{k}$ is the vector field such that
$(\beta_{k},X^{k})$ verifies the hypothesis of the proposition. Since for
$\mu$-almost every point the Lyapunov exponents exist, using the dominated
convergence theorem, we have

$\begin{array}[]{rcl}\displaystyle\int_{\gamma_{k}}\chi_{k}d\gamma_{k}&=&\displaystyle\int_{\beta_{k}}\chi_{k}\circ
g^{-k}J^{uu}_{g^{-k}}d\beta_{k}\\\ &\geq&\displaystyle
e^{-\varepsilon_{3}}\frac{Leb(\gamma_{k})}{Leb(\beta_{k})}\int_{\beta_{k}}\chi_{k}\circ
g^{-k}d\beta_{k}\\\ &=&\displaystyle
e^{-\varepsilon_{3}}\frac{Leb(\gamma_{k})}{Leb(\beta_{k})}\limsup_{n\to+\infty}\frac{I_{n}^{\beta_{k},X^{k}}}{n}.Leb(\beta_{k})\geq
e^{-\varepsilon_{3}}CLeb(\gamma_{k}).\end{array}$

On the other hand,

$\begin{array}[]{rcl}\displaystyle\int_{\gamma_{k}}\chi_{k}d\gamma_{k}&=&\displaystyle\int_{\gamma_{k}\cap
B}\chi_{k}d\gamma_{k}+\int_{\gamma_{k}\cap B^{c}}\chi_{k}d\gamma_{k}\\\
&\leq&\displaystyle e^{-2\varepsilon_{3}}CLeb(\gamma_{k})+\frac{\log
2Ne^{-2\varepsilon_{3}}(e^{\varepsilon_{3}}-1)CLeb(\gamma_{k})}{2\log 2N}\\\
&<&e^{-\varepsilon_{3}}CLeb(\gamma_{k})\end{array}$

which is a contradiction. ∎

Write

$E(\gamma,X)=\frac{1}{|\gamma|}\displaystyle\int_{\gamma}\log\|Dg(m).X_{m}\|d\gamma(m),$

where $(\gamma,X)$ is an adapted field. Let
$\pi_{1}:\mathbb{T}^{4}\to\mathbb{T}^{2}$ be the projection defined by
$\pi_{1}(x,y,z,w)=(x,y)$. For $X$ a vector field on $\gamma$ define

$\widetilde{X}_{m}=\frac{D\pi_{1}(X_{m})}{\|D\pi_{1}(X_{m})\|}.$

In what follows, we let $\tilde{\delta}>0$ to be a positive constant that we
will fix later.

######  Definition 4.12.

Consider the cone
$\Delta_{\tilde{\delta}}=\\{(u,v)\in\mathbb{R}^{2}:N^{\tilde{\delta}}|u|\geq|v|\\}$.
Let $(\gamma,X)$ be an adapted vector field. If for every $m\in\gamma$ we have
that $\tilde{X}(m)\in\Delta_{\tilde{\delta}}$ then we say that $(\gamma,X)$ is
a $\tilde{\delta}$-good adapted vector field. Otherwise we say that it is
$\tilde{\delta}$-bad.

Recall that for $k\geq 0$ and an $u$-curve $\gamma$ the number
$N_{k}=N_{k}(\gamma)$ denotes the maximum number of $u$ curves that subdivide
$g^{k}\circ\gamma$. For an adapted field $(\gamma,X)$ define the unit vector
field over $g^{k}(\gamma)$, $Y^{k}=\frac{g^{k}_{*}X}{\|g^{k}_{*}X\|}$, where
$g^{k}_{*}X_{m}=Dg^{k}(g^{-k}(m))X_{g^{-k}(m)}$.

######  Lemma 4.13 ([10], Lemma $9$).

For $N$ large and $\mathcal{U}_{N}$ small enough, let $g\in\mathcal{U}_{N}$
and $(\gamma,X)$ be an adapted field for $g$. For $k\geq 0$, every possible
pair $(\gamma_{j}^{k},Y^{k}|_{\gamma_{j}^{k}})$, with $1\leq j\leq
N_{k}(\gamma)$ is an adapted field.

The following formula is proved in section $6$ of [10].

######  Lemma 4.14.

For every adapted field $(\gamma,X)$ and any $n\in\mathbb{N}$

$I_{n}^{\gamma,X}=\displaystyle\sum_{k=0}^{n-1}\left(R_{k}+\sum_{j=0}^{N_{k}}\frac{1}{|\gamma|}\int_{\gamma_{j}^{k}}\log\|Dg(m).Y^{k}_{m}\|J^{uu}_{g^{-k}}d\gamma_{j}^{k}\right),$

where
$R_{k}=\frac{1}{|\gamma|}\int_{\gamma_{N_{k}+1}^{k}}\log\|Dg(m).Y^{k}_{m}\|J^{uu}_{g^{-k}}d\gamma_{N_{k}+1}^{k}$.

As a consequence of lemma 4.14, and using (15), we obtain

$\displaystyle
I_{n}^{\gamma,X}\geq\displaystyle\sum_{k=0}^{n-1}\left(R_{k}+e^{-\varepsilon_{2}}\sum_{j=0}^{N_{k}}(\min_{\gamma_{j}^{k}}J^{uu}_{g^{-k}})E(\gamma_{j}^{k},Y^{k})\right).$
(18)

Since $\gamma^{k}_{N_{k}+1}$ is a piece of an $u$-curve, then

$\displaystyle\frac{|\gamma^{k}_{N_{k}+1}|}{|\gamma|}<2.$

By (17), we have

$\begin{array}[]{rcl}\displaystyle|R_{k}|=\frac{1}{|\gamma|}\int_{\gamma_{N_{k}+1}^{k}}\log\|Dg(m).Y^{k}_{m}\|J^{uu}_{g^{-k}}d\gamma_{N_{k}+1}^{k}&<&\displaystyle\frac{|\gamma^{k}_{N_{k}+1}|}{|\gamma|}(e^{\varepsilon_{1}}\lambda)^{2Nk}\log
2N\\\ &<&\displaystyle 2(e^{\varepsilon_{1}}\lambda)^{2Nk}\log
2N\xrightarrow{k\rightarrow+\infty}0.\end{array}$

Hence,

$\displaystyle\frac{1}{n}\sum_{k=0}^{n-1}|R_{k}|\longrightarrow 0.$

The following is the key proposition that will give us the estimate that we
need.

######  Proposition 4.15 ([45], Proposition 7.29).

For $N$ large and $\mathcal{U}_{N}$ small enough, for every
$g\in\mathcal{U}_{N}$, any $\tilde{\delta}$-good adapted field $(\gamma,X)$
and every $k\geq 0$, we have

$e^{-\varepsilon_{2}}\displaystyle\sum_{j=0}^{N_{k}}(\min_{\gamma_{j}^{k}}J^{uu}_{g^{-k}})E(\gamma_{j}^{k},Y^{k})\geq(1-12\tilde{\delta})\log
N.$

######  Remark 4.16.

In [45], the term $e^{-\varepsilon_{2}}$ on the right hand side of the
equation (18) is missing. The same term is also missing in the statement of
proposition $7.29$ in [45]. Since we can fix $\varepsilon_{2}$ arbitrarily
close to $0$, this does not affect the rest of the proof in [45] to obtain the
estimate of the center Lyapunov exponents.

Now, we can proceed with the proof of Theorem 4.1.

###### Proof of Theorem 4.1.

Take $\tilde{\delta}=\frac{2\delta}{15}$. Let $N$ be large and let
$\mathcal{U}_{N}$ be small enough such that it verifies proposition 4.15. Fix
$g\in\mathcal{U}_{N}$ and let $\mu$ be an $u$-Gibbs measure for $g$. Consider
any $u$-curve $\gamma$ and any $\tilde{\delta}$-good vector field $X$ on
$\gamma$. By proposition 4.15, and using inequality (18), for $n$ large enough

$\frac{I^{\gamma,X}_{n}}{n}\geq(1-14\tilde{\delta})\log N.$ (19)

Since we could have chosen $\varepsilon_{3}>0$ small enough such that
$e^{-\varepsilon_{3}}(1-14\tilde{\delta})\geq(1-15\tilde{\delta})$ by
proposition 4.11, $\mu$-almost every point has a Lyapunov exponent for $g$ in
the center direction larger than

$(1-15\tilde{\delta})\log N=(1-2\delta)\log N.$

By condition ($4$) in lemma 4.4, we have that for $\mu$-almost every point $m$
the sum of the center Lyapunov exponents belongs to the interval
$(-\beta,\beta)$, that is, $-\beta<\lambda^{-}(m)+\lambda^{+}(m)<\beta$. By
taking $\beta>0$ small, after fixing $\delta$, we conclude that

$\lambda^{-}(m)<\beta-\lambda^{+}(m)<\beta-(1-2\delta)\log N<(1-\delta)\log
N.$

Therefore, we obtain that for $N$ large and $\mathcal{U}_{N}$ small enough,
for $g\in\mathcal{U}_{N}$, any $u$-Gibbs measure $\mu\in\mathrm{Gibbs}^{u}(g)$
verifies that $\mu$-almost every point $m$ has both a positive and a negative
Lyapunov exponent on the center with absolute value larger than
$(1-\delta)\log N$. ∎

##  5 Proof of Theorem C

Recall that in section 2 we defined the notion of homoclinically related
measures (see definition 2.19). The goal of this section is to prove Theorem
C. This is based in the techniques developed by the author in [45]. We
actually prove the following theorem, which is more general than Theorem C:

######  Theorem 5.1.

For $N$ large and $\mathcal{U}_{N}$ small enough, for any $k\in\mathbb{N}$ the
following holds: if $g\in\mathcal{U}_{N}$ and $\mu_{1},\mu_{2}$ are two
ergodic $u$-Gibbs measures for $g^{k}$, then $\mu_{1}$ is homoclinically
related to $\mu_{2}$.

For an SRB measure, we can also obtain the following proposition.

######  Proposition 5.2.

For $N$ large and $\mathcal{U}_{N}$ small enough, let $g\in\mathcal{U}_{N}$
and let $\mu$ be an SRB measure for $g$. Then
$\mathrm{supp}(\mu)=\mathbb{T}^{4}$.

###### Proof of Theorem C assuming Theorem 5.1 and Proposition 5.2.

Let $N$ be large and $\mathcal{U}_{N}$ be small enough such that Theorem 5.1
holds and fix $g\in\mathcal{U}_{N}$. If $\mu_{1}$ and $\mu_{2}$ are two
ergodic SRB measures for $g$, by Theorem 5.1, $\mu_{1}$ is homoclinically
related to $\mu_{2}$. By Theorem 2.20, $\mu_{1}=\mu_{2}$, and therefore $g$
has at most one SRB measure.

Suppose that $\mu$ is an SRB measure for $g$. By Theorem 2.18, there exist
$k\in\mathbb{N}$ and $k$ measures which are $g^{k}$-invariant and SRB,
$\mu_{1},\cdots,\mu_{k}$, such that $\mu_{i}\neq\mu_{j}$ for $j\neq i$ and

$\mu=\frac{1}{k}\displaystyle\sum_{j=1}^{k}\mu_{j}.$

Moreover, $g_{*}(\mu_{j})=\mu_{j+1}$, with the identification of $k+1=1$, and
$(g^{k},\mu^{k})$ is Bernoulli. Observe that if $k=1$, then $\mu$ is Bernoulli
for $g$.

Suppose $k>1$, by Theorem 5.1, we have that for any $i,j\in\\{1,\cdots,k\\}$
with $i\neq j$, the measures $\mu_{i}$ and $\mu_{j}$ are homoclinically
related. Since these measures are SRB, we obtain that $\mu_{i}=\mu_{j}$, which
is a contradiction with the fact that $\mu_{i}\neq\mu_{j}$. Hence, $k=1$ and
the measure $\mu$ is Bernoulli for $g$. Proposition 5.2 states that if $\mu$
is SRB then it has full support. ∎

The rest of this section is mostly dedicated to prove Theorem 5.1. As we will
see, the proof of this theorem is essentially contained in the proof of the
stable ergodicity for the map $f_{N}$ in [45]. We will refer the reader to
[45] for the proofs of several of the lemmas and propositions that we will use
in this section, and we remark that they are also valid outside the volume
preserving setting. At the end of the section we explain how to obtain
Proposition 5.2. The argument involved in the proof of Proposition 5.2 is a
combination of some estimates obtained to prove Theorem 5.1 and arguments from
[20].

###  5.1 Estimates for stable and unstable manifolds of $u$-Gibbs measures

For a vector $v\in T_{m}\mathbb{T}^{4}$, write $v_{1}=D\pi_{1}(m).v$. For a
direction $E\subset T_{m}\mathbb{T}^{4}$ we will write
$(E)_{1}=D\pi_{1}(m).E$. For this section we fix $0<\delta<<1$ small and we
are assuming that $N$ is large and $\mathcal{U}_{N}$ is small enough such that
Theorem 4.1 holds. For this subsection we fix two constants (depending on
$N$), $\theta_{1}:=N^{-\frac{2}{5}}$ and $\theta_{2}:=N^{-\frac{3}{5}}$.

Let $g\in\mathcal{U}_{N}$. For each ergodic measure $\mu$ for $g$ let
$\Lambda_{\mu}$ be the set of points $m\in\mathbb{T}^{4}$ such that

$\displaystyle\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f^{j}(m)}\xrightarrow[n\to+\infty]{}\mu\textrm{
and
}\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f^{-j}(m)}\xrightarrow[n\to+\infty]{}\mu\textrm{,
in the $weak^{*}$-topology.}$

Where $\delta_{p}$ is the dirac mass on the point $p$. Birkhoff’s theorem
implies that $\mu(\Lambda_{\mu})=1$. Recall that $\mathcal{R}_{g}$ is the set
of regular points for $g$. By Theorem 4.11, if $\mu$ is an $u$-Gibbs measure
for $g$, then for each $m\in\mathcal{R}_{g}\cap\Lambda_{\mu}$ there are two
directions $E^{-}_{g,m}$ and $E^{+}_{g,m}$ contained in $E^{c}_{g,m}$, which
are the Oseledets’ directions with respect to the negative and positive
Lyapunov exponent, respectively.

For each $\mu\in\mathrm{Gibbs}^{u}(g)$, we define the sets

$\begin{array}[]{rcl}Z_{\mu}^{-}&=&\left\\{m\in\mathcal{R}_{g}\cap\Lambda_{\mu}:\forall
n\geq 0\textrm{ it holds
}\displaystyle\left\|Dg^{n}(m)|_{E^{-}_{g,m}}\right\|<\left(N^{-\frac{4}{5}}\right)^{n}\right\\};\\\
Z_{\mu}^{+}&=&\left\\{m\in\mathcal{R}_{g}\cap\Lambda_{\mu}:\forall n\geq
0\textrm{ it holds
}\displaystyle\left\|Dg^{-n}(m)|_{E^{+}_{g,m}}\right\|<\left(N^{-\frac{4}{5}}\right)^{n}\right\\};\\\
Z_{\mu}&=&g(Z_{\mu}^{-})\cap g^{-1}(Z_{\mu}^{+});\\\
Z_{g}&=&\displaystyle\bigcup_{\mu\in\mathrm{Gibbs}^{u}(g)}Z_{\mu}.\end{array}$

The proof of the following lemma is the same as lemma $5.2$ in [45]. It is an
application of Pliss lemma.

######  Lemma 5.3 ([45], lemma $5.2$).

Let $g\in\mathcal{U}_{N}$. If $\mu$ is an ergodic $u$-Gibbs measure for $g$,
then $\mu(Z_{g})\geq\frac{1-7\delta}{1+7\delta}$.

Let $T=\left[\frac{1+7\delta}{28\delta}\right]$ and define

$X_{g}=\displaystyle\bigcap_{k=-T+1}^{T-1}g^{k}(Z_{g}).$ (20)

An easy consequence of the estimate in lemma 5.3 is given in the following
lemma.

######  Lemma 5.4 ([45], lemma $5.3$).

For $N$ large and $\mathcal{U}_{N}$ small enough, if $\mu$ is an $u$-Gibbs
measure for $g$ then

$\mu(X_{g})>0.$

For a vector $v\in\mathbb{R}^{2}$ we write $v=(v_{h},v_{v})$, where $v_{h}$
and $v_{v}$ are the coordinates of $v$ with respect to the basis $(1,0)$ and
$(0,1)$. For each $\theta>0$ we consider the horizontal and vertical cones

$\mathscr{C}^{hor}_{\theta}=\\{v\in\mathbb{R}^{2}:\theta\|v_{h}\|\geq\|v_{v}\|\\}\textrm{
and
}\mathscr{C}^{ver}_{\theta}=\\{v\in\mathbb{R}^{2}:\theta\|v_{v}\|\geq\|v_{h}\|\\}.$

One of the key ingredients in the proof of stable ergodicity of $f_{N}$ is
based in a version of the stable manifold theorem given by Crovisier-Pujals in
[26]. Using their construction we can obtain precise estimates on the sizes of
stable and unstable manifolds inside the center direction for $u$-Gibbs
measures. This is given in the following proposition. Fix
$\theta_{1}=N^{-\frac{2}{5}}$.

######  Proposition 5.5 ([45], Proposition $5.6$).

Let $N$ be large and $\mathcal{U}_{N}$ be small enough. For
$g\in\mathcal{U}_{N}$ and $m\in Z_{g}$, there are two $C^{1}$-curves,
$W^{*}_{g}(m)$, contained in $W^{c}_{g}(m)$, tangent to $E^{*}_{g,m}$ and with
length bounded from below by $r_{0}=N^{-7}$, for $*=-,+$. Those curves are
$C^{1}$-stable and unstable manifolds for $g$, respectively. Moreover,
$\left(T_{p}W^{+}_{g,r_{0}}(m)\right)_{1}\subset\mathscr{C}^{hor}_{\frac{4}{\theta_{1}}}(p)$
and
$\left(T_{q}W^{-}_{g,r_{0}}(m)\right)_{1}\subset\mathscr{C}^{ver}_{\frac{4}{\theta_{1}}}(q)$,
for every $p\in W^{+}_{g,r_{0}}(m)$ and $q\in W^{-}_{g,r_{0}}(m)$.

We remark that the proof of this proposition only uses the estimates for
points in the set $Z_{g}$ and estimates on the $C^{2}$-norm of $g$. The proof
is exactly the same as the proof of proposition $5.6$ in [45]

Let $\theta_{2}=N^{-\frac{3}{5}}$. Proposition 5.5 is one of the key
ingredients to prove the following lemma.

######  Lemma 5.6 ([45], Lemma $5.7$).

For $N$ large, $\mathcal{U}_{N}$ small and $n>15$, let $g\in\mathcal{U}_{N}$.
Then for every $m\in X_{g}$ there are two curves $\gamma_{g,-n}^{-}(m)\subset
g^{-n}(W^{-}_{g,r_{0}}(m))$ and $\gamma_{g,n}^{+}(m)\subset
g^{n}(W^{+}_{g,r_{0}}(m))$ with length greater than $4\pi$, such that
$\left(T\gamma^{-}_{g,-n}(m)\right)_{1}\subset\mathscr{C}^{ver}_{\theta_{2}}$
and
$\left(T\gamma^{+}_{g,n}(m)\right)_{1}\subset\mathscr{C}^{hor}_{\theta_{2}}$.

We remark that the statement of lemma $5.7$ from [45], which is the equivalent
of lemma 5.6 above, involves a measure $\nu_{g,i}$. However, the proof only
uses the estimates of the points in the set $Z_{g}$ and the definition of
$X_{g}$. Therefore, the proof of lemma 5.6 is exactly the same as the proof of
lemma $5.7$ from [45]

For $R>0$, let

$\displaystyle
W^{s}_{g,R,-n}(m)=\bigcup_{q\in\gamma_{g,-n}^{-}(m)}W_{g,R}^{ss}(q),$

where the curve $\gamma_{g,-n}^{-}(m)$ is the curve given by the previous
lemma. Define similarly the set $W^{u}_{g,R,n}(m)$, but using the strong
unstable manifolds.

Let $m\in X_{g}$ be a typical point for an $u$-Gibbs measure $\mu$. Recall
that the stable Pesin manifold is a $C^{1}$-immersed submanifold and it has a
topological characterization given by

$W^{s}(m)=\\{y\in\mathbb{T}^{4}:\displaystyle\limsup_{n\to+\infty}\frac{1}{n}\log
d(f^{n}(m),f^{n}(y))<0\\}.$

By the topological characterization of the stable Pesin manifold and by the
definition of $W^{s}_{g,R,-n}(m)$, it is easy to see that
$W^{s}_{g,R,-n}(m)\subset g^{-n}(W^{s}(m))$. Observe that the strong stable
manifolds subfoliate the Pesin stable manifold, in particular
$W^{s}_{g,R,-n}(m)$ is open inside the Pesin manifold $g^{-n}(W^{s}(m))$. We
conclude that $W^{s}_{g,R,-n}(m)$ is a $C^{1}$-submanifold. An analogous
conclusion holds for unstable manifolds.

The next lemma allows us to control the tangent space of these stable and
unstable manifolds inside the center direction.

######  Lemma 5.7 ([45], Lemma $5.8$).

Fix $\theta_{3}>0$ such that $\theta_{3}>\theta_{2}$ and satisfies
$\mathscr{C}^{hor}_{\theta_{3}}\cap\mathscr{C}_{\theta_{3}}^{ver}=\\{0\\}$.
For $g\in\mathcal{U}_{N}$, there exists $0<R<1$ such that if $n\geq 15$, $m\in
X_{g}$ and $m^{-}\in W^{s}_{g,R,-n}(m)\subset W^{s}_{g,2,-n}(m)$, then

$\left(T(W^{s}_{g,2,-n}(m)\cap
W_{g}^{c}(m^{-}))\right)_{1}\subset\mathscr{C}^{ver}_{\theta_{3}}.$

A similar result holds for $W^{u}_{g,R,n}(m)$.

The last ingredient for the proof we will need is the following proposition.

######  Proposition 5.8.

For $N$ large and $\mathcal{U}_{N}$ small enough, if $g\in\mathcal{U}_{N}$
then for any ergodic $u$-Gibbs measure $\mu$ for $g$ and for any
$k\in\mathbb{N}$, the following property holds: for $\mu$-almost every point
$m\in\mathbb{T}^{4}$, its center leaf $W^{c}_{g}(m)$ has dense $g^{k}$-orbit
in $\mathbb{T}^{4}$.

The proof of this proposition is essentially the same as the proof of
Proposition $5.9$ in [45]. For the sake of completeness we will include it
here.

###### Proof.

For $\mathcal{U}_{N}$ small enough, for every $g\in\mathcal{U}_{N}$ there is a
homeomorphism $h_{g}:\mathbb{T}^{4}\to\mathbb{T}^{4}$, that takes center
leaves of $f_{N}$ to center leaves of $g$, such that for every
$m\in\mathbb{T}^{4}$ it is verified

$g\circ h_{g}(W^{c}_{f}(m))=h_{g}\circ f(W^{c}_{f}(m))$

Consider the quotients $M_{f}=\mathbb{T}^{4}/\sim^{c}_{f}$ and
$M_{g}=\mathbb{T}^{4}/\sim^{c}_{g}$, where $p\sim^{c}_{*}q$ if and only if
$q\in W^{c}_{*}(p)$ for $*=f,g$. We denote $\pi_{f}:\mathbb{T}^{4}\to M_{f}$
and $\pi_{g}:\mathbb{T}^{4}\to M_{g}$ the respective projections. Observe that
$M_{f}=\mathbb{T}^{2}$ and that the induced dynamics $\tilde{f}:M_{f}\to
M_{f}$ of $f$ is given by $A^{2N}$. Endow $M_{g}$ with the distance $d_{g}$
given by the Hausdorff distance on the center leaves, that is,

$d_{g}(L,W)=d_{\textrm{Haus}}(\pi_{g}^{-1}(L),\pi_{g}^{-1}(W)).$

By the leaf conjugacy equation, the induced dynamics $\tilde{g}:M_{g}\to
M_{g}$ of $g$ is conjugated to the linear Anosov $A^{2N}$ on $\mathbb{T}^{2}$
by the homeomorphism induced by $h_{g}$, which we will denote by
$\tilde{h}_{g}$. Denote by $W^{uu}_{A^{2N}}(.)$ the stable manifold of
$A^{2N}$ on $\mathbb{T}^{2}$ and let

$W^{uu}_{\tilde{g}}(L)=\\{W\in
M_{g}:\displaystyle\lim_{n\to+\infty}d_{g}(\tilde{g}^{-n}(L),\tilde{g}^{-n}(W))=0\\},$

be the stable set of $L$.

######  Claim 5.9 (Claim 2 in the proof of Proposition $5.9$ from [45]).

For every $m\in\mathbb{T}^{4}$, for every $q\in W^{c}_{g}(m)$, it is verified
that

$\pi_{g}(W^{uu}_{g}(q))=W^{uu}_{\tilde{g}}(\pi_{g}(m))=\tilde{h}_{g}(W^{uu}_{A^{2N}}(\pi_{f}(h_{g}^{-1}(m)))),$

and $\pi_{g}$ is a bijection from $W^{uu}_{g}(q)$ to
$W^{uu}_{\tilde{g}}(\pi_{g}(m))$.

For the linear Anosov $A^{2N}$ the unstable foliation is minimal, that is,
every unstable manifold of $A^{2N}$ is dense in $\mathbb{T}^{2}$. Let $\mu$ be
an ergodic $u$-Gibbs measure for $g$ and fix $m$ a generic point for $\mu$.
Using the minimality of the unstable foliation of the linear Anosov and by the
leaf conjugacy $W^{uu}_{\tilde{g}}(\pi_{g}(m))$ is dense in $M_{g}$.

Take $U$ a small open set in $M_{g}$. Since the center foliation is uniformly
compact, $\hat{U}=\pi_{g}^{-1}(U)$ is a saturated open set such that any two
center leaves in $\hat{U}$ are close to each other. By the previous claim
$W^{uu}_{g}(m)\cap\hat{U}\neq\emptyset$.

Since $\mu$ is an $u$-Gibbs measure, we have that $W^{uu}_{g}(m)$ is contained
in the support of $\mu$. Hence, $\mathrm{supp}(\mu)\cap\hat{U}\neq\emptyset$.
In particular, $\mu(\hat{U})>0$. Recall that $m$ is a generic point for $\mu$,
therefore, its future orbit visits $\hat{U}$ infinitely many times. This is
true for any open set $U$ inside $M_{g}$, which concludes the proof of the
proposition for $k=1$.

For $k\in\mathbb{N}$, we remark that an unstable leaf for $A^{2N}$ is an
unstable leaf for $A^{2Nk}$, in particular, the unstable foliation of
$A^{2Nk}$ is minimal. The map $g^{k}$ is leaf conjugated to $A^{2Nk}$. The
same argument as above concludes the proof of the proposition. ∎

###### Proof of Theorem 5.1.

Let $N$ be large and $\mathcal{U}_{N}$ be small enough such that lemmas 5.6,
5.7 and proposition 5.8 hold. Fix $g\in\mathcal{U}_{N}$ and $\mu_{1},\mu_{2}$
be two ergodic $u$-Gibbs measures for $g$.

Recall that we defined the set $X_{g}$ in (20) and let $\Lambda_{\mu_{i}}$ be
the set of typical points that we defined before for the measures $\mu_{i}$,
for $i=1,2$. Since $\mu_{i}(X_{g})>0$ and $\mu_{i}(\Lambda_{\mu_{i}})=1$, the
set $X_{i}=X_{g}\cap\Lambda_{i}$ has positive $\mu_{i}$ measure as well, for
$i=1,2$.

For any two points $m_{1}\in X_{1}$ and $m_{2}\in X_{2}$, we will prove that
the stable manifold of $m_{1}$ has a transverse intersection with the unstable
manifold of $m_{2}$. Fix a center leaf $W^{c}_{g}(q)$, the center leaf of some
point $q\in\mathbb{T}^{4}$. By proposition 5.8 and remark
LABEL:rmk.pastiteratesdensity, the forward and past iterates of
$W^{c}_{g}(m_{i})$ are dense in $\mathbb{T}^{4}$, for $i=1,2$. Hence, there
are two sequences $n_{k}\to+\infty$ and $l_{j}\to+\infty$, such that

$g^{n_{k}}(W^{c}_{g}(m_{1}))\to W^{c}_{g}(q)\textrm{ and
}g^{-l_{j}}(W_{g}^{c}(m_{2}))\to W^{c}(q).$

By lemma 5.6, there are curves $\gamma_{g,n_{k}}^{+}(m_{1})$ and
$\gamma_{g,-l_{j}}^{-}(m_{2})$ with length bigger that $4\pi$ and contained in
the cone $\mathscr{C}^{hor}_{\theta_{2}}$ and
$\mathscr{C}^{ver}_{\theta_{2}}$, respectively. Take $R$ given by lemma 5.7
and consider the sets

$\displaystyle
L^{u}_{k}(m_{1})=\displaystyle\bigcup_{z\in\gamma^{+}_{g,n_{k}}(m_{1})}W^{uu}_{g,R}(z)\subset
W^{u}(g^{n_{k}}(m_{1}))$ $\displaystyle
L^{s}_{j}(m_{2})=\displaystyle\bigcup_{z\in\gamma^{-}_{g,-l_{j}}(m_{2})}W^{ss}_{g,R}(z)\subset
W^{s}(g^{-l_{j}}(m_{2})).$

For $k$ and $j$ large enough, $g^{n_{k}}(W^{c}_{g}(m_{1}))$ and
$g^{-l_{j}}(W^{c}_{g}(m_{2}))$ are very close to the leaf $W^{c}_{g}(q)$. Thus
by the control on the angles that we obtained in lemma 5.7, there is a
transverse intersection between $L^{u}_{k}(m_{1})$ and $L^{s}_{j}(m_{2})$. In
particular, $W^{u}_{g}(g^{n_{k}}(m_{1}))$ and $W^{s}_{g}(g^{-l_{j}}(m_{2}))$
intersect transversely. Since transverse intersections are invariant by
iterates, we conclude that $W^{u}_{g}(m_{1})$ and $W^{s}_{g}(m_{2})$ have a
transverse intersection.

Repeating this argument, exchanging the roles of $m_{1}$ and $m_{2}$, implies
that $W^{u}_{g}(m_{2})$ and $W^{s}_{g}(m_{1})$ have a transverse intersection.
Since the set $X_{i}$ has positive $\mu_{i}$ measure, for $i=1,2$, we conclude
that $\mu_{1}$ is homoclinically related to $\mu_{2}$. This finishes the proof
of Theorem 5.1 for $g$, in the case $k=1$.

Let $k\in\mathbb{N}$. Following the same steps as above, it is easy to prove
that any two ergodic $u$-Gibbs measures for $g^{k}$, $\mu_{1}$ and $\mu_{2}$,
are homoclinically related. ∎

###  5.2 Proof of proposition 5.2

We will need a few results from [20].

######  Lemma 5.10 ([20], Lemma $3.2$).

There exists a constant $R>0$ with the following property: for $N$
sufficiently large, there exists a $C^{1}$-neighborhood $\mathcal{U}$ of
$f_{N}$ such that for any $g\in\mathcal{U}_{N}$ and any two points
$p,q\in\mathbb{T}^{4}$ we have that for any $m_{p}\in W^{c}_{g}(p)$ there
exists $m_{q}\in W^{c}_{g}(q)$ such that $W^{uu}_{g,R}(m_{p})\cap
W^{ss}_{g,R}(m_{q})\neq\emptyset$.

Fix $\theta=N^{-\frac{3}{5}}$ and recall that in subsection 5.1, we defined
the vertical cone $\mathscr{C}^{ver}_{\theta}$.

######  Lemma 5.11 ([20], Proposition $3.3$).

If $N$ is sufficiently large there exists
$\mathcal{U}_{N}\subset\mathrm{Diff}^{2}(\mathbb{T}^{4})$ a
$C^{1}$-neighborhood of $f_{N}$ such that for any $g\in\mathcal{U}_{N}$ and
any open set $U\subset\mathbb{T}^{4}$, there exists $n_{s}\geq 0$ such that
for any $n\geq n_{s}$, there exists a $C^{1}$ curve $\gamma^{-}_{n}\subset
g^{-n}(U)$ satisfying:

  * •

$\gamma^{-}_{n}$ is contained in a center leaf.

  * •

$\pi_{1}(\gamma^{-}_{n})$ is tangent to $\mathscr{C}^{ver}_{\theta}$.

  * •

$\gamma^{-}_{n}$ has length greater than $4\pi$

  * •

$\displaystyle\bigcup_{q\in\gamma^{-}_{n}}W^{ss}_{g,R}(q)\subset g^{-n}(U).$

Consider the vertical foliation
$\mathcal{F}_{ver}=\\{\\{z\\}\times\mathbb{T}^{2}:z\in\mathbb{T}^{2}\\}$.
Observe that for any diffeomorphism $g$ sufficiently $C^{1}$-close to $f_{N}$,
we have that $W^{c}_{g}(m)$ intersects each vertical torus
$\\{z\\}\times\mathbb{T}^{2}$ in exactly one point, for any
$m\in\mathbb{T}^{4}$. Hence, for any two points
$m_{1},m_{2}\in\mathbb{T}^{4}$, the map from $W^{c}_{g}(m_{1})$ to
$W^{c}_{g}(m_{2})$ defined by
$h^{g}_{m_{1},m_{2}}(p)=W^{c}_{g}(m_{2})\cap\mathcal{F}_{ver}(p)$ is well
defined. Note that, after identifying all the horizontal tori with
$\mathbb{T}^{2}$, the map $h^{f_{N}}_{m_{1},m_{2}}$ is just the identity,
independently of the points $m_{1},m_{2}$.

######  Lemma 5.12 ([20], Lemma $3.4$).

For every $\varepsilon>0$, there exists $N_{0}:=N_{0}(\varepsilon)$ with the
following property: for $N\geq N_{0}$ there exists a $C^{1}$-neighborhood
$\mathcal{U}_{N}$ of $f_{N}$ such that if
$g\in\mathcal{U}_{N},p\in\mathbb{T}^{4}$ and $q\in W^{ss}_{g,R}(p)$ then
$d_{C^{0}}(h^{g}_{p,q},H^{s}_{p,q})<\varepsilon$. Analogous result holds for
the unstable holonomy.

###### Proof of proposition 5.2.

Let $N$ be large and $\mathcal{U}_{N}$ be small enough such that lemmas 5.6,
5.10, 5.11, and 5.12 hold. Let $g\in\mathcal{U}_{N}$ and suppose that $\mu$ is
an SRB measure for $g$. Fix $U\subset\mathbb{T}^{4}$ an open set, we must
prove that $\mathrm{supp}(\mu)\cap U\neq\emptyset$.

Since $\mu$ is SRB, its supports contains entire Pesin unstable manifolds. By
lemma 5.6, we can take a $\mu$-generic point $m_{u}$ with the property that
for $n_{u}$ large enough there exists $\gamma_{g,n_{u}}^{+}(m)\subset
g^{n_{u}}(W^{+}_{g,r_{0}}(m))$ a curve of length greater than $4\pi$ and whose
projection by $\pi_{1}$ is tangent to $\mathscr{C}^{hor}_{\theta}$.

For $n_{s}$ large enough, let $\gamma_{n_{s}}^{-}$ be the curve given by lemma
5.11 for $U$ and $g$. As a consequence of lemmas 5.10, and 5.12, we conclude
that

$\displaystyle\left(\bigcup_{q\in\gamma^{-}_{n_{s}}}W^{ss}_{g,R}(q)\right)\cap\left(\bigcup_{p\in\gamma^{+}_{n_{u}}}W^{uu}_{g,R}(p)\right)\neq\emptyset.$
(21)

We refer the reader to [20] for more details on this argument. By (21), we
obtain that $g^{n_{s}+n_{u}}(W^{u}(m_{u}))\cap U\neq\emptyset$, and since
$\mu$ is SRB we conclude that $\mathrm{supp}(\mu)\cap U\neq\emptyset$. ∎

##  6 Rigidity of $u$-Gibbs measures

The main tool to study the existence of SRB measures that we will use is a
recent result by Brown-Rodriguez Hertz on measure rigidity for random dynamics
of surface diffeomorphisms. The goal of this section is to explain the
statement of their result and how it can be applied to our scenario after a
measurable change of coordinates using the unstable holonomies (see Theorem
6.3).

###  6.1 Measure rigidity for general skew products

Let $(\Omega,\mathcal{B}_{\Omega},\nu)$ be a Polish probability space, that
is, $\Omega$ has the topology of a complete separable metric space,
$\mathcal{B}_{\Omega}$ is the Borel $\sigma$-algebra of $\Omega$ and $\nu$ is
a Borel probability measure on $\Omega$. Let
$\theta:(\Omega,\mathcal{B}_{\Omega},\nu)\to(\Omega,\mathcal{B}_{\Omega},\nu)$
be an invertible, measure-preserving and ergodic transformation. Let $S$ be a
compact smooth surface and $\mathrm{Diff}^{2}(S)$ be the set of
$C^{2}$-diffeomorphisms of $S$. We consider a measurable map that for each
point $\xi\in\Omega$ associates a diffeomorphism
$f_{\xi}\in\mathrm{Diff}^{2}(S)$. For each $n\in\mathbb{Z}$ we define

$\begin{array}[]{l}f^{0}_{\xi}:=Id,\\\
f^{n}_{\xi}:=f_{\theta^{n-1}(\xi)}\circ\cdots\circ f_{\xi}\textrm{ for
}n>0,\\\
f^{n}_{\xi}:=(f_{\theta^{n}(\xi)})^{-1}\circ\cdots\circ(f_{\theta^{-1}(\xi)})^{-1}\textrm{
for }n<0.\end{array}$

We consider the skew product over $\theta$ given by the map $\xi\mapsto
f_{\xi}$, which is defined by

$\begin{array}[]{rcl}F:S\times\Omega&\longrightarrow&S\times\Omega\\\
(x,\xi)&\mapsto&(f_{\xi}(x),\theta(\xi)).\end{array}$

With the notation above, we may write
$F^{n}(x,\xi)=(f^{n}_{\xi}(x),\theta^{n}(\xi))$. Write $X=S\times\Omega$ and
let $\pi_{2}:X\to\Omega$ be the natural projection on $\Omega$.

Let $\mu$ be an $F$-ergodic probability measure, such that
$(\pi_{2})_{*}\mu=\nu$. Observe that the partition by the fibers $S$ is
measurable. Therefore, we have a family of conditional measures defined in a
set $D$ of full $\nu$-measure $\\{\mu_{\xi}\\}_{\xi\in D}$ with respect to the
partition induced by $\pi_{2}$. For $\nu$-almost every $\xi$, the measure
$\mu_{\xi}$ is supported on $S_{\xi}:=S\times\\{\xi\\}$. There is a trivial
identification of $S_{\xi}$ with $S$, hence, by an abuse of notation we
consider the map $\xi\mapsto\mu_{\xi}$ to be a $\nu$-measurable map from
$\Omega$ to the space of Borel probability measures of $S$.

To talk about SRB measures in this setting, we need to first talk about
Lyapunov exponents and stable and unstable manifolds. Write
$TX:=TS\times\Omega$ and let $DF:TX\to TX$ to be the linear cocycle defined by

$DF((x,v),\xi)=((f_{\xi}(x),Df_{\xi}(x)v),\theta(\xi)).$

Suppose that the following integrability condition holds

$\displaystyle\int_{\Omega}\log^{+}(\|f_{\xi}\|_{C^{2}})+\log^{+}(\|f^{-1}_{\xi}\|_{C^{2}})d\nu(\xi)<\infty,$
(22)

where $\log^{+}(.)=\max\\{0,\log(.)\\}$ and $\|f_{\xi}\|_{C^{2}}$ is the
$C^{2}$-norm of $f_{\xi}$. Applying Oseledec’s theorem for the linear cocycle
$DF$, there is a $\mu$-measurable decomposition
$T_{(\xi,x)}X=\bigoplus_{j}E^{j}_{(x,\xi)}$ such that the space
$E^{j}_{(x,\xi)}$ is the space corresponding to the Lyapunov exponent
$\lambda^{j}_{\mu}$, where $\\{\lambda^{j}_{\mu}\\}_{j}$ are the Lyapunov
exponents of $DF$.

From now on, let us suppose that the measure $\mu$ is hyperbolic on the
fibers, meaning, all the Lyapunov exponents are non zero. The integrability
condition (22) is used to have Pesin’s theory for fibered systems. In
particular, for $\mu$-almost every point there exists stable and unstable
manifolds, which may possibly be just points in the case that all the
exponents are negative or positive. We refer the reader to section 6 in [17]
for more details.

Suppose that $\mu$ has at least one positive Lyapunov exponent. The family of
unstable manifolds $\\{W^{u}(x,\xi)\\}_{(x,\xi)\in X}$ forms a partition of a
$\mu$-full measure subset of $X$. Usually this partition is not measurable. In
this context, we say that a measurable partition $\mathcal{P}$ is
$u$-subordinated if for $\mu$-almost every $(x,\xi)$, there exists a positive
number $r>0$ such that $W^{u}_{r}(x,\xi)\subset\mathcal{P}(x,\xi)\subset
W^{u}(x,\xi)$.

######  Definition 6.1 (Fiber-wise SRB).

An $F$-invariant probability measure $\mu$ is fiber-wise SRB if for any
$u$-subordinated measurable partition $\mathcal{P}$, for $\mu$-almost every
$(x,\xi)$, the conditional measure $\mu_{(x,\xi)}^{\mathcal{P}}$ is absolutely
continuous with respect to the riemannian volume on $W^{u}(x,\xi)$.

Let $\mathcal{P}_{\Omega}$ be a measurable partition of $\Omega$. We say that
$\mathcal{P}_{\Omega}$ is increasing if for $\nu$-almost every point $\xi$ we
have

$\mathcal{P}_{\Omega}(\theta(\xi))\subset\theta(\mathcal{P}_{\Omega}(\xi)).$

Let $\hat{\mathcal{F}}(\mathcal{P}_{\Omega})\subset\mathcal{B}_{\Omega}$ be
the sub-$\sigma$-algebra generated by $\mathcal{P}_{\Omega}$. We say that
$\hat{\mathcal{F}}(\mathcal{P}_{\Omega})$ is an increasing
sub-$\sigma$-algebra. We remark that in [17], the authors call these
partitions and sub-$\sigma$-algebra decreasing instead of increasing. We
changed it here to be in harmony with the notion of increasing that we defined
in section 2.

Let $\mathcal{F}(\mathcal{P}_{\Omega})$ be the $\mu$-completion of
$\mathcal{B}_{S}\otimes\hat{\mathcal{F}}(\mathcal{P}_{\Omega})$, where
$\mathcal{B}_{S}$ is the Borel $\sigma$-algebra on $S$. For a hyperbolic
measure $\mu$, we may also look at the Oseledec’s direction $E^{s}(x,\xi)$ as
a measurable map of $X$ that takes values on the projectivization of $TX$. We
are now ready to state the main theorem in [17].

######  Theorem 6.2 ([17], Theorem $4.10$).

Let $F:X\to X$ be as above verifying the integrability condition (22), let
$\mathcal{P}_{\Omega}$ be a measurable increasing partition of $\Omega$ and
let $\mu$ be a hyperbolic $F$-invariant measure such that
$(\pi_{2})_{*}\mu=\nu$. Suppose that the family of conditional measures on the
fibers $\\{\mu_{\xi}\\}$ are non-atomic almost surely. Furthermore, assume
that

  1. 1.

$\xi\mapsto f_{\xi}^{-1}$ is
$\hat{\mathcal{F}}(\mathcal{P}_{\Omega})$-measurable, and

  2. 2.

$\xi\mapsto\mu_{\xi}$ is $\hat{\mathcal{F}}(\mathcal{P}_{\Omega})$-measurable.

Then either $(x,\xi)\mapsto E^{s}(x,\xi)$ is
$\mathcal{F}(\mathcal{P}_{\Omega})$-measurable of $\mu$ is fiber-wise SRB.

###  6.2 Change of coordinates

Fix $\alpha\in(0,1)$. In this section, we show how to use Theorem 6.2 to
obtain the following theorem:

######  Theorem 6.3.

For $N$ large enough, there exists $\mathcal{U}_{N}^{sk}$ a
$C^{2}$-neighborhood of $f_{N}$ in
$\mathrm{Sk}^{2}(\mathbb{T}^{2}\times\mathbb{T}^{2})$ such that for
$g\in\mathcal{U}_{N}^{sk}\cap\mathrm{Diff}^{2+\alpha}(\mathbb{T}^{4})$, for
any ergodic $\mu\in\mathrm{Gibbs}^{u}(g)$ one of the following holds:

  1. 1.

$\mu$ is SRB;

  2. 2.

for $\mu$-almost every $p\in\mathbb{T}^{4}$, and for Lebesgue almost every
point $q$ in $W^{uu}_{loc}(p)$

$E^{-}_{g,q}=DH^{u}_{p,q}(p)E^{-}_{g,p};$

  3. 3.

for $\mu$-almost every $p\in\mathbb{T}^{4}$ the measure $\mu^{c}_{p}$ is
atomic.

To prove Theorem 6.3 we will define a measurable change of coordinates using
the strong unstable holonomies, so that after this change of coordinates we
are in the setting of Theorem 6.2.

Recall that $\lambda<1$ is the rate of contraction of the linear Anosov $A$.
Let $N$ be large enough such that

$\displaystyle(4N^{2})^{2}\left(\lambda^{2N}\right)^{\alpha}<1.$

In particular, if the $C^{2}$-neighborhood $\mathcal{U}_{N}^{sk}$ of $f_{N}$
is sufficiently small, then for every $g\in\mathcal{U}_{N}^{sk}$ we have

$\displaystyle\left(\frac{\|Dg|_{E_{g}^{c}}\|}{m(Dg|_{E_{g}^{c}})}\right)^{2}(m(Dg|_{E^{uu}_{g}}))^{-\alpha}<1.$
(23)

Fix $g\in\mathcal{U}_{N}^{sk}\cap\mathrm{Diff}^{2+\alpha}(\mathbb{T}^{4})$ and
some $R>1$. Condition (23) above is the $(2,\alpha)$-unstable center bunching
condition defined in (8). By Theorem 2.10, for any $p\in\mathbb{T}^{4}$, $q\in
W^{uu}_{g,R}(p)$ the unstable holonomy $H^{u}_{p,q}:W^{c}_{g}(p)\to
W^{c}_{g}(q)$ is a $C^{2}$-diffeomorphism, whose $C^{2}$-norm varies
continuously with the choices of $p$ and $q$ as above.

Since $g$ is a partially hyperbolic skew product, we have that
$g(p_{1},p_{2})=(g_{p_{2}}(p_{1}),g_{2}(p_{2}))$, where $g_{2}(p_{2})$ is a
$C^{2+\alpha}$-Anosov diffeomorphism of $\mathbb{T}^{2}$ which is
topologically conjugated to $A^{2N}$. It is well known that a transitive
$C^{1+\alpha}$-Anosov diffeomorphism has an unique ergodic $u$-Gibbs measure.
Let $\nu$ be such a measure for $g_{2}$ on $\mathbb{T}^{2}$.

Fix $\mathcal{R}=\\{R_{1},\cdots,R_{m}\\}$ a small Markov partition for $A$
and observe that $\mathcal{R}$ is also a Markov partition for $A^{2N}$ for
every $N\in\mathbb{N}$. By taking $N$ sufficiently large we may suppose that
the transition matrix $P_{2N}$ associated with $\mathcal{R}$ for $A^{2N}$
verifies $(P_{2N})_{i,j}=1$, for every $i,j=1,\cdots m$. Let $\mathcal{R}_{g}$
be the image of $\mathcal{R}$ by the conjugacy map between $A^{2N}$ and
$g_{2}$. It is easy to see that $\mathcal{R}_{g}$ is a small Markov partition
for $g_{2}$ and the conjugacy implies that it has the same transition matrix
$P_{2N}$. Define $\Sigma:=\\{1,\cdots,m\\}^{\mathbb{Z}}$ which is the shift
space associated with $\mathcal{R}_{g}$ for $g_{2}$, let
$\sigma:\Sigma\to\Sigma$ be the left shift map, and let
$\Theta:\Sigma\to\mathbb{T}^{2}$ be the continuous surjection that defines the
semi-conjugacy between $\sigma$ and $g_{2}$.

Let us set some notations. Write $\Sigma^{-}=\\{(\xi_{i})_{i\leq
0}:\xi_{i}\in\\{1,\cdots,m\\}\\}$ and
$\Sigma^{+}:=\\{(\xi_{i})_{i>0}:\xi_{i}\in\\{1,\cdots,m\\}\\}$. Let
$\pi^{-}:\Sigma\to\Sigma^{-}$ and $\pi^{+}:\Sigma\to\Sigma^{+}$ be the natural
projections. For a point $\xi\in\Sigma$ we write $\xi^{-}:=\pi^{-}(\xi)$ and
$\xi^{+}:=\pi^{+}(\xi)$ and we use the notation $\xi=(\xi^{-},\xi^{+})$. The
local unstable set of a point $\xi\in\Sigma$ is

$\Sigma^{u}_{loc}(\xi)=\\{\eta\in\Sigma:\eta^{-}=\xi^{-}\\}.$

Define $\nu_{\sigma}:=\Theta_{*}\nu$, and observe that this is an ergodic,
$\sigma$-invariant measure. The partition $\Sigma^{u}_{loc}$ on local unstable
sets forms a $\nu_{\sigma}$-measurable partition of $\Sigma$. Let
$\mathcal{P}^{u}$ be the $\nu$-measurable $u$-subordinated partition given by
the intersection of local unstable manifolds of $g_{2}$ with the rectangles
from the Markov partition $\mathcal{R}_{g}$. Notice that $\mathcal{P}^{u}$ is
equivalent (on a set of full $\nu$-measure) to the partition
$\Sigma^{u}_{loc}$ (on a set of full $\nu_{\sigma}$-measure).

It is easy to see that the partition $\Sigma^{u}_{loc}$ is an increasing
partition. Let $\mathcal{B}^{u}$ be the sub-$\sigma$-algebra generated by the
partition on local unstable sets. This is an increasing sub-$\sigma$-algebra.

It is well known that $\Theta$ is bijective in a set of full
$\nu_{\sigma}$-measure, which we will denote by $\hat{D}$. We may further
assume that $\hat{D}$ is $\sigma$-invariant. Let $D:=\Theta(\hat{D})$ this is
a $g_{2}$-invariant set of full $\nu$-measure. Define
$\Psi=Id\times\Theta^{-1}$, and notice that it is an isomorphism between
$\mathbb{T}^{2}\times D$ and $\mathbb{T}^{2}\times\hat{D}$. Let
$\pi^{\prime}_{2}:\mathbb{T}^{2}\times\Sigma\to\Sigma$ be the natural
projection on the second coordinate.

Let $\mu$ be an ergodic $u$-Gibbs measure for $g$. By lemma 2.29,
$\nu=(\pi_{2})_{*}\mu$. Consider the measure $\hat{\mu}:=\Psi_{*}\mu$, and
observe that it verifies $(\pi^{\prime}_{2})_{*}\hat{\mu}=\nu_{\sigma}$. We
define the skew product on $\mathbb{T}^{2}\times\hat{D}$ by
$\hat{g}(x,\xi)=(\hat{g}_{\xi}(x),\sigma(\xi))$, where
$\hat{g}_{\xi}:=g_{\Theta(\xi)}$. We may extend $\hat{g}$ to
$\mathbb{T}^{2}\times\Sigma$ by setting $\hat{g}_{\xi}=Id$, for
$\xi\notin\hat{D}$. Observe that $(g,\mu)$ is isomorphic (or measurably
conjugated) to $(\hat{g},\hat{\mu})$ by the isomorphism $\Psi$. Since $\Psi$
is just the identity in the first coordinate, it is immediate that the center
Lyapunov exponents of $\mu$ are the same as the fiber Lyapunov exponents of
$\hat{\mu}_{\sigma}$. Furthermore, $\mu$ is SRB if and only if $\hat{\mu}$ is
fiber-wise SRB.

We now introduce a change of coordinate in the fibers for the skew product
$\hat{g}$ in a way that the new skew product will verify the conditions to
apply Theorem 6.2.

Fix $\eta^{+}\in\Sigma^{+}$ and define the function $\phi:\Sigma\to\Sigma$ by
$\phi(\xi)=(\xi^{-},\eta^{+})$ for every $\xi\in\Sigma$. Observe that for each
$\xi\in\Sigma$, $\phi(\xi)\in\Sigma^{u}_{loc}(\xi)$. In particular $\phi$ is
$\mathcal{B}^{u}$-measurable.

For each $\xi\in\hat{D}$, since $\Theta(\xi)$ and $\Theta(\phi(\xi))$ belongs
to the same local unstable manifold for $g_{2}$, we define

$\begin{array}[]{rcl}\Phi_{\xi}:\mathbb{T}^{2}&\longrightarrow&\mathbb{T}^{2}\\\
x&\mapsto&H^{u}_{\Theta(\xi),\Theta(\phi(\xi))}(x).\end{array}$

To simplify our notation, we write
$H^{u}_{\xi,\phi(\xi)}:=H^{u}_{\Theta(\xi),\Theta(\phi(\xi))}$. We also define
$\Phi:\mathbb{T}^{2}\times D\to\mathbb{T}^{2}\times D$ by
$\Phi(x,\xi)=(\Phi_{\xi}(x),\xi)$. We can extend the definition of $\Phi$ to
$\mathbb{T}^{2}\times\Sigma$ by setting $\Phi_{\xi}=Id$ for
$\xi\notin\hat{D}$. We consider a skew product $\tilde{g}$ on
$\mathbb{T}^{2}\times\Sigma$ defined by

$\tilde{g}=\Phi\circ\hat{g}\circ\Phi^{-1}.$ (24)

Consider the ergodic $\tilde{g}$-invariant measure
$\tilde{\mu}=\Phi_{*}\hat{\mu}$ and observe that
$(\pi^{\prime}_{2})_{*}\tilde{\mu}=\nu_{\sigma}$. The partition on the fibers
$\mathbb{T}^{2}$ forms a measurable partition of $\mathbb{T}^{2}\times\Sigma$.
Let $\\{\tilde{\mu}_{\xi}\\}_{\xi\in\Sigma}$ be the family of conditional
measures with respect to the fibers. Figure 1 represents all these changes of
coordinates that are conjugacies on subsets of full measure.

${(\mathbb{T}^{2}\times\mathbb{T}^{2},\mu)}$${(\mathbb{T}^{2}\times\mathbb{T}^{2},\mu)}$${(\mathbb{T}^{2}\times\Sigma,\hat{\mu})}$${(\mathbb{T}^{2}\times\Sigma,\hat{\mu})}$${(\mathbb{T}^{2}\times\Sigma,\tilde{\mu})}$${(\mathbb{T}^{2}\times\Sigma,\tilde{\mu})}$$\scriptstyle{g}$$\scriptstyle{\Psi}$$\scriptstyle{\Psi}$$\scriptstyle{\hat{g}}$$\scriptstyle{\Phi}$$\scriptstyle{\Phi}$$\scriptstyle{\tilde{g}}$
Figure 1: Changes of coordinates

######  Lemma 6.4.

The maps $\xi\mapsto\tilde{g}_{\xi}^{-1}$ and $\xi\mapsto\tilde{\mu}_{\xi}$
are $\mathcal{B}^{u}$-measurable.

###### Proof.

Recall that $\tilde{g}_{\xi}^{-1}=(\tilde{g}_{\sigma^{-1}(\xi)})^{-1}$. Since
the unstable holonomy commutes with $g$, and by the definition of $\hat{g}$,
in what follows we will use that
$H^{u}_{\xi,\eta}\circ\hat{g}_{\sigma^{-1}(\xi)}=\hat{g}_{\sigma^{-1}(\eta)}\circ
H^{u}_{\sigma^{-1}(\xi),\sigma^{-1}(\eta)}$. By (24), we have

$\begin{array}[]{rcl}\tilde{g}_{\sigma^{-1}(\xi)}(x)&=&H^{u}_{\xi,\phi(\xi)}\circ\hat{g}_{\sigma^{-1}(\xi)}\circ
H^{u}_{\phi(\sigma^{-1}(\xi)),\sigma^{-1}(\xi)}(x)\\\
&=&H^{u}_{\xi,\phi(\xi)}\circ
H^{u}_{\sigma(\phi(\sigma^{-1}(\xi))),\xi}\circ\hat{g}_{\phi(\sigma^{-1}(\xi))}(x)=H^{u}_{\sigma(\phi(\sigma^{-1}(\xi))),\phi(\xi)}\circ\hat{g}_{\phi(\sigma^{-1}(\xi))}(x).\end{array}$

Notice that $\phi(\xi)$ and $\phi(\sigma^{-1}(\xi))$ depend only on $\xi^{-}$,
in particular $\tilde{g}_{\sigma^{-1}(\xi)}$ depends only on $\xi^{-}$. If
$\eta\in\Sigma^{u}_{loc}(\xi)$, which means that $\eta^{-}=\xi^{-}$, then
$\tilde{g}_{\sigma^{-1}(\xi)}=\tilde{g}_{\sigma^{-1}(\eta)}$ and hence the map
$\xi\mapsto\tilde{g}_{\xi}^{-1}$ is constant on local unstable sets and it is
$\mathcal{B}^{u}$-measurable.

Since $\mu$ is an $u$-Gibbs measure, and it projects to $\nu$, corollary 2.30
implies that for $\nu$-almost every $p_{2}$, and for Lebesgue almost every
$q_{2}\in W^{uu}_{g_{2}}(p_{2})$ (for the riemannian volume of
$W^{uu}_{g_{2}}(p_{2})$), we have

$\mu^{c}_{q_{2}}=(H^{u}_{p_{2},q_{2}})_{*}\mu^{c}_{p_{2}}.$ (25)

At first, the disintegration $\mu^{c}_{q_{2}}$ is defined for almost every
point inside the unstable manifold of $p_{2}$. However, using (25), for any
$q_{2}\in W^{uu}_{g_{2}}(p_{2})$, we may consider the measure
$\mu_{q_{2}}=(H^{u}_{p_{2},q_{2}})_{*}\mu_{p_{2}}$. This defines a new
disintegration that coincides with the original one in $\mu$-almost every
point with the advantage that for $\nu$-almost every point the disintegration
is defined along entire unstable manifolds.

Since $\Psi$ is the identity on the fibers and a conjugation with the shift on
the basis, for $\nu_{\sigma}$-almost every $\xi$ we obtain
$\mu_{\Theta(\xi)}=\hat{\mu}_{\xi}$. Let us see the equivalent of property
(25) for $\hat{\mu}$. Consider the disintegration of $\nu_{\sigma}$ on the
measurable partition $\Sigma^{u}_{loc}$. For $\nu_{\sigma}$-almost every
$\xi$, let $\nu_{\sigma}^{\xi}$ be the conditional measure on
$\Sigma^{u}_{loc}(\xi)$. Hence, for $\nu_{\sigma}^{\xi}$-almost every $\eta$,
we have that $\hat{\mu}_{\eta}=(H^{u}_{\xi,\eta})_{*}\hat{\mu}_{\xi}$.

In an analogous way as we did for $\mu$, we define the measure $\mu_{\eta}$
for every $\eta$ in the local unstable set of $\xi$ and this defines a new
disintegration that coincides with the original disintegration on a set of
full measure. By an abuse of notation we will use the notation
$\hat{\mu}_{\xi}$ for the conditional measure of this new disintegration. We
remark that this disintegration has the advantage of being defined along
entire local unstable sets.

By the definition of $\Phi$ we see that for $\nu_{\sigma}$-almost every $\xi$
and for any $\eta\in\Sigma^{u}_{loc}(\xi)$ the measure
$\tilde{\mu}_{\eta}=(H^{u}_{\eta,\phi(\xi)})_{*}\hat{\mu}_{\eta}=\hat{\mu}_{\phi(\xi)}$.
In particular, the map $\xi\mapsto\tilde{\mu}_{\xi}$ is constant on local
unstable sets and it is $\mathcal{B}^{u}$-measurable. ∎

###### Proof of Theorem 6.3.

First, let us explain how the skew product $\tilde{g}$ verifies the hypothesis
of Theorem 6.2. Since $\Sigma^{u}_{loc}$ is a decreasing partition, we have
that $\mathcal{B}^{u}$ is a decreasing sub-$\sigma$-algebra. Let
$\mathcal{B}^{*}$ be the $\tilde{\mu}$-completion of
$\mathcal{B}_{\mathbb{T}^{2}}\otimes\mathcal{B}^{u}$, where
$\mathcal{B}_{\mathbb{T}^{2}}$ is the Borel $\sigma$-algebra on
$\mathbb{T}^{2}$. Recall that

$\tilde{g}_{\xi}=H^{u}_{\sigma(\phi(\xi)),\phi(\sigma(\xi))}\circ\hat{g}_{\phi(\xi)}.$

We claim that there exists a constant $R>0$ such that for any $\xi\in\Sigma$,