# Periodic points and measures for a class of skew products

Maria Carvalho  Maria Carvalho
Centro de Matemática da Universidade do Porto
Rua do Campo Alegre 687
4169-007 Porto
Portugal<EMAIL_ADDRESS>and  Sebastián A. Pérez  Sebastián A. Pérez Opazo
Centro de Matemática da Universidade do Porto
Rua do Campo Alegre 687
4169-007 Porto
Portugal<EMAIL_ADDRESS>
###### Abstract.

We consider the open set constructed by M. Shub in [42] of partially
hyperbolic skew products on the space $\mathbb{T}^{2}\times\mathbb{T}^{2}$
whose non-wandering set is not stable. We show that there exists an open set
$\mathcal{U}$ of such diffeomorphisms such that if $F_{S}\in\mathcal{U}$ then
its measure of maximal entropy is unique, hyperbolic and, generically,
describes the distribution of periodic points. Moreover, the non-wandering set
of such an $F_{S}\in\mathcal{U}$ contains closed invariant subsets carrying
entropy arbitrarily close to the topological entropy of $F_{S}$ and within
which the dynamics is conjugate to a subshift of finite type. Under an
additional assumption on the base dynamics, we verify that $F_{S}$ preserves a
unique SRB measure, which is physical, whose basin has full Lebesgue measure
and coincides with the measure of maximal entropy. We also prove that there
exists a residual subset $\mathfrak{R}$ of $\mathcal{U}$ such that if
$F_{S}\in\mathfrak{R}$ then the topological and periodic entropies of $F_{S}$
are equal, $F_{S}$ is asymptotic per-expansive, has a sub-exponential growth
rate of the periodic orbits and admits a principal strongly faithful symbolic
extension with embedding.

###### Key words and phrases:

Skew product; Partial hyperbolicity; Entropy-expansive; Asymptotically per-
expansive; Symbolic extension; Physical measure; SRB measure.

###### 2010 Mathematics Subject Classification:

Primary 37D35, 37A35, 37D30. Secondary 37A05, 37A30.

This research was partially supported by CMUP (UID/MAT/00144/2019) which is
funded by FCT with national (MCTES) and European structural funds through the
programs FEDER, under the partnership agreement PT2020. SP also acknowledges
financial support from the postdoctoral grant PTDC/CTM/BIO-4043-2014, under
the project UID/MAT/00144/2019.

##  1\. Introduction

Let $f:M\to M$ be a diffeomorphism of a manifold into itself and $\Omega(f)$
be its non-wandering set. When $\Omega(f)$ does not admit a hyperbolic
structure, it may be difficult to describe completely its orbit structure.
Motivated by this problem, R. Bowen suggested to look for invariant components
of $\Omega(f)$ with large entropy on which the dynamics of $f$ may be simpler
to characterize. The key idea is to find closed invariant subsets, say
topological horseshoes, within which the dynamics is conjugate to subshifts of
finite type that may be good approximations, in some sense, of the global
dynamics. For instance, this strategy might provide information on the
topological entropy of a complicated dynamics by taking the least upper bound
over its restrictions to those horseshoes. In this case, the system is said to
be a limit of horseshoes in the sense of the entropy. L.-S. Young studies in
[45] systems that are limits of this type, including piecewise monotonic maps
of the interval, the Poincaré map of the Lorenz attractor [23] and Abraham-
Smale’s examples [1], leaving unsolved the case of the partially hyperbolic,
robustly transitive, entropy-expansive and non-$\Omega$-stable skew products
introduced by Shub in [42]. In this work we consider precisely these skew
products, explore the dynamical properties of their measures of maximal
entropy and thereby show, on Section 5, that they are indeed limits of
horseshoes.

The second question we address here concerns the study of the distribution of
periodic points and measures of maximal entropy. Denote by $\mathcal{U}$ the
open set of Shub’s examples as constructed in [35] and by $F_{S}$ any of its
elements. It is known that $F_{S}$ has a unique measure of maximal entropy,
and so one expects this measure to have a strong tie with other dynamical
properties. In particular, it would be relevant to show that this measure
describes the distribution of the periodic points of $F_{S}$ (meaning that it
is the weak∗-limit of the sequence of Dirac measures supported on the sets of
periodic points) and that the asymptotic exponential growth rate of the number
of periodic orbits with the period (the so called _periodic entropy_) is equal
to the topological entropy of $F_{S}$. We will prove that these two
attributes, which are known to be valid within the uniformly hyperbolic
setting (cf. [7]), also hold on a residual subset $\mathfrak{R}$ of
$\mathcal{U}$. Both properties are a consequence of the existence of a semi-
conjugation between $F_{S}$ and a uniformly hyperbolic dynamics, besides a
careful analysis of the periodic fibers induced by the semi-conjugation.
Thereby, our study conveys a satisfactory description of the symbolic dynamics
of Shub’s examples. More precisely, we show that in $\mathfrak{R}$ any
diffeomorphism has a sub-exponential growth rate of the periodic orbits in
arbitrarily small scales (the so called _asymptotic per-expansiveness_); this
result enables us to build a symbolic extension, from whose properties we
conclude that, generically in $\mathcal{U}$, the set of Borel invariant
probability measures is homeomorphic to the space of Borel probability
measures invariant by a subshift of finite type. The proofs of these
assertions will be presented on Sections 6 and 7.

For Anosov diffeomorphisms and, more generally, $C^{2}$ Axiom A attractors,
the work of Bowen, Ruelle and Sinai (we refer the reader to [9] and references
therein) proved the existence of a unique invariant probability measure, the
so-called SRB measure, that is characterized by obeying Pesin’s formula [37].
From Ledrappier and L.-S. Young’s work [31], the property that defines an SRB
measure is known to be equivalent to the existence of a disintegration of the
measure in conditional measures on unstable manifolds which are absolutely
continuous with respect to Lebesgue measure. Moreover, the SRB measure is also
the unique physical measure (cf. [9, Theorem 4.12]; a thorough essay on the
existence and uniqueness of both SRB and physical measures within more general
settings may be read in [46]). For Shub’s examples, the existence of an SRB
measure was proved in [17]. Besides, under the additional assumption that the
base dynamics is the product of two linear hyperbolic automorphisms of the
$2$-torus, $F_{S}$ is mostly contracting with a minimal strong unstable
foliation, and so it has a unique SRB measure whose basin of attraction has
full Lebesgue measure (cf. [3]). Consequently, the SRB measure of $F_{S}$ is
also its unique physical measure and coincides with its measure of maximal
entropy, thus inheriting this property from the conservative base dynamics.
More detailed information will be given on Section 8.

##  2\. Main results

It is known that a diffeomorphism $f:M\to M$ on a compact Riemannian manifold
$M$ satisfying the Axiom A condition and without cycles is expansive and has
the specification property. So it preserves a unique invariant probability
measure with maximal entropy which describes the distribution of the periodic
points. Moreover, for these systems the topological and periodic entropies are
equal (cf. [8]). Summarizing:

  * •

_Uniqueness_ : $f$ preserves a unique probability measure $\mu$ satisfying
$h_{\mu}(f)=h_{\mathrm{top}}(f)$, where $h_{\mathrm{top}}(f)$ denotes the
topological entropy of $f$ (definition in [44]).

  * •

_Distribution of periodic points_ : $\mu$ is the limit in the weak∗ topology
of the sequence of equidistributed averages supported on the periodic points
of $f$.

  * •

_Equal topological and periodic entropies_ :
$\lim_{n\,\to\,+\infty}\,\frac{1}{n}\,\log\,\\#\,\mathrm{Per}_{n}(f)=h_{\mathrm{top}}(f)$,
where $\mathrm{Per}_{n}(f)$ stands for the set of $n$th periodic points of
$f$.

  * •

_Symbolic dynamics_ : There exists a principal strongly faithful symbolic
extension with embedding (reminding a similar property valid for Axiom A
systems – see [33, 12]).

Since specification and expansivity are not valid in general outside the
hyperbolic world, the previous properties are not expected within this setting
(cf. [4, 29]). Nonetheless, in a broad class of non-hyperbolic systems the
existence of at least one probability measure of maximal entropy is also
guaranteed. For instance, this is valid for entropy-expansive diffeomorphisms
(cf. [34]). And it was shown in [17] (see also [18, 19] for generalizations)
that, when the central bundle of $f$ is one-dimensional, then $f$ is entropy-
expansive. So Shub’s examples are endowed with a probability measure of
maximal entropy. However, even if we assume that the system is topologically
mixing, uniqueness of such a special measure is not certain (cf. [25]). For
Shub’s examples the uniqueness of the measure of maximal entropy was obtained
in [35] (a generalization for equilibrium states may be read in [16]).
Nevertheless, without additional assumptions this measure may not describe the
distribution of the periodic points and the topological entropy may be
different from the periodic one. Yet, as we will explain, Shub’s examples,
which are obtained through a homotopic deformation of a direct product of two
hyperbolic diffeomorphisms, may be constructed in such a way that, if we
restrict to a suitable residual subset of them, then we are able to keep
control on the periodic orbits even at arbitrarily small scales. This is our
first result.

Denote by $\mathcal{U}$ the open set of Shub’s examples, whose construction
will be recalled on Section 4, and by $F_{S}$ any of its elements.

######  Theorem A.

There exists a residual subset $\mathfrak{R}$ of the open set $\mathcal{U}$
such that, if $F_{S}$ belongs to $\mathfrak{R}$, then:

  1. (a)

$h_{\mathrm{top}}(F_{S})=\lim_{n\,\to\,+\infty}\,\frac{1}{n}\,\log\,\\#\,\mathrm{Per}_{n}(F_{S}).$

  2. (b)

The maximal entropy measure of $F_{S}$ describes the distribution of periodic
points.

As previously mentioned, Shub’s examples are entropy-expansive, and this
property is a sufficient condition for the existence of a principal symbolic
extension. However, if we restrict to $\mathfrak{R}$, the diffeomorphisms
satisfy a stronger property, namely the asymptotic per-expansiveness, and such
an extension may be constructed preserving the periodic points and inducing a
homeomorphism between the corresponding spaces of probability measures.

######  Theorem B.

Any diffeomorphism of the residual subset $\mathfrak{R}$ built in Theorem A
has a principal strongly faithful symbolic extension with embedding.

The construction of Shub’s examples ensures that, if both $\Phi$ and $L$ are
linear hyperbolic automorphisms of $\mathbb{T}^{2}$, then the map $F_{S}$ is
mostly contracting and so, according to [3], it has a finite number of SRB
measures whose basins cover Lebesgue almost everywhere. In addition, the
strong unstable foliation of $F_{S}$ is robustly minimal (cf. Proposition
5.2), so $F_{S}$ has in fact a unique SRB measure and its basin has full
Lebesgue measure. Under this additional assumption on $\Phi$, the map $F_{S}$
inherits from $\Phi\times L$ other properties.

######  Theorem C.

Suppose that both $\Phi$ and $L$ are linear hyperbolic automorphisms of
$\mathbb{T}^{2}$. Then:

  1. (a)

The image by $H_{*}$ of the SRB measure of $F_{S}\in\mathcal{U}$ is the SRB
measure of $\Phi\times L$.

  2. (b)

The SRB measure of $F_{S}\in\mathcal{U}$ is its unique measure of maximal
entropy and its unique physical measure.

##  3\. Glossary

We begin introducing the main definitions used in this work. Given a compact
metric space $(X,d)$ and a continuous map $f:X\to X$, denote by
$\mathscr{P}(X)$ the set of Borel probability measures on $X$ endowed with the
weak∗-topology, and by $\mathscr{P}(X,f)$ and $\mathscr{P}_{e}(X,f)$ its
subsets of $f$-invariant and $f$-invariant ergodic elements, respectively.

###  3.1. Maximal entropy measures

For each $\mu$ in $\mathscr{P}(X,f)$, consider the metric entropy $h_{\mu}(f)$
of $f$ with respect to $\mu$. The Variational Principle [44, Theorem 9.10]
states that the topological entropy $h_{\mathrm{top}}(f)$ of $f$ coincides
with the least upper bound of the operator $\mu\,\mapsto\,h_{\mu}(f)$
restricted to either $\mathscr{P}(X,f)$ or $\mathscr{P}_{e}(X,f)$. A measure
$\mu\in\mathscr{P}(X,f)$ such that $h_{\mu}(f)=h_{\mathrm{top}}(f)$ is called
a measure of maximal entropy of $f$.

###  3.2. Distribution of periodic points

Assume that the cardinality $\\#\,\mathrm{Per}_{n}(f)$ of the set of fixed
points of $f^{n}$ is finite for every $n\in\mathbb{N}$. We say that a
probability measure $\mu\in\mathscr{P}(X,f)$ describes the distribution of the
periodic points of $f$ if $\mu$ is the weak∗ limit of the sequence of
probability measures

$n\in\mathbb{N}\quad\mapsto\quad\frac{1}{\\#\,\mathrm{Per}_{n}(f)}\sum_{x\,\in\,\mathrm{Per}_{n}(f)}\delta_{x}$

where $\delta_{x}$ denotes the Dirac measure supported at $x$.

###  3.3. Expansiveness

Denote by $B_{\rho}(x)$ the open ball in the metric $d$ centered at $x$ with
radius $\rho$, and by $\overline{B_{\rho}(x)}$ its closure. Define, for each
$n\in\mathbb{N}$, the equivalent metric

$(x,y)\in X\times X\quad\mapsto\quad
d_{n}(x,y)\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\,\max_{0\,\leqslant\,j\,\leqslant\,n-1}\,d(f^{j}(x),\,f^{j}(y)).$

Given $\varepsilon>0$ and a compact subset $Y\subset X$, a subset $S$ of $Y$
is said to be $(n,\varepsilon)$-spanning if for every $y\in Y$ there is $a\in
S$ such that $d_{n}(y,a)\leqslant\varepsilon$. The minimum cardinality of the
$(n,\varepsilon)$-spanning subsets of $Y$ is denoted by
$r_{n}(Y,\varepsilon)$. Define

$\displaystyle\overline{r}_{n}(Y,\varepsilon)\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\,\limsup_{n\,\to\,+\infty}\,\frac{1}{n}\,\log\,r_{n}(Y,\varepsilon)\quad\quad\mbox{and}\quad\quad\overline{h}_{\mathrm{top}}(f,Y)\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\lim_{\varepsilon\,\to\,0^{+}}\,\overline{r}_{n}(Y,\varepsilon).$

Having fixed $\varepsilon>0$ and $x\in X$, consider the set of points in $X$
whose forward orbits by $f$ are $\varepsilon$-close to the orbit of $x$, that
is,

$B^{f}_{\infty,\,\varepsilon}(x)\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\,\,\bigcap_{i\,\in\,\mathbb{A}}\,f^{-i}\,\Big{(}\,\overline{B_{\varepsilon}(f^{i}(x))}\Big{)}=\Big{\\{}y\in
X:\,\,d(f^{i}(x),f^{i}(y))\leqslant\varepsilon,\quad\forall\,i\in\mathbb{A}\Big{\\}}$

with $\mathbb{A}=\mathbb{Z}$ if $f$ is invertible and $\mathbb{A}=\mathbb{N}$
otherwise. Consider

$h^{*}_{\mathrm{top}}(f,\varepsilon)\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\,\,\sup_{x\,\in\,X}\,\overline{h}_{\mathrm{top}}(f,B^{f}_{\infty,\,\varepsilon}(x))\qquad\mbox{and}\qquad
h^{*}_{\mathrm{top}}(f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\lim_{\varepsilon\,\to\,0^{+}}\,h^{*}_{\mathrm{top}}(f,\varepsilon).$

The map $f$ is said to be _entropy-expansive_ if there is $\varepsilon_{0}>0$
such that $h^{*}_{\mathrm{top}}(f,\varepsilon)=0$ for every
$0<\varepsilon<\varepsilon_{0}$, and asymptotically entropy-expansive if
$h^{*}_{\mathrm{top}}(f)=0$. Misiurewicz has shown in [34] that for
asymptotically entropy-expansive maps the entropy operator
$\mu\in\mathscr{P}(X,f)\to h_{\mu}(f)$ is upper-semicontinuous, guaranteeing
the existence of at least a measure of maximal entropy for $f$.

Given $\varepsilon>0$, define

$Per(f,\varepsilon)\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\,\limsup_{n\,\to\,+\infty}\,\frac{1}{n}\,\sup_{x\,\in\,X}\,\log\,\\#\,\Big{(}\mathrm{Per}_{n}(f)\cap
B^{f}_{\infty,\,\varepsilon}(x)\Big{)}\qquad\mbox{and}\qquad
Per^{*}(f)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\lim_{\varepsilon\,\to\,0^{+}}\,Per(f,\varepsilon).$

Following [12], the map $f$ is said to be asymptotically per-expansive if
$Per^{*}(f)=0$. For instance, expansive or aperiodic maps are asymptotically
per-expansive. An interesting connection between the entropy, the growth of
the cardinality of the periodic orbits with the period and the asymptotic per-
expansiveness is given in the next lemma.

######  Lemma 3.1.

[13, Lemma 2.2]
$\quad\limsup_{n\,\to\,+\infty}\,\frac{1}{n}\,\log\,\\#\,\mathrm{Per}_{n}(f)\leqslant
h_{\mathrm{top}}(f)+Per^{*}(f).$

Thus, if $f$ is asymptotically per-expansive then
$\limsup_{n\,\to\,+\infty}\,\frac{1}{n}\,\log\,\\#\,\mathrm{Per}_{n}(f)\leqslant
h_{\mathrm{top}}(f)$, an inequality that generalizes [44, Theorem 8.16].

###  3.4. Symbolic extensions

A map $f$ has a _symbolic extension_ if there exists $m\in\mathbb{N}$, a
closed shift-invariant subset $\Sigma$ of the full shift
$\\{0,1,\cdots,m\\}^{\mathbb{Z}}$, and a continuous surjective map
$\pi:\Sigma\to X$ such that $f\circ\pi=\pi\circ\sigma$, where $\sigma$ stands
for the shift map.

A symbolic extension $(\Sigma,\sigma,\pi)$ is said to be _principal_ if $\pi$
preserves the metric entropy, that is, $h_{\eta}(\sigma)=h_{\mu}(f)$ for every
$f$-invariant measure $\mu$ and every $\sigma$-invariant measure $\eta$ such
that $\mu=\pi_{*}(\eta)$. If, in addition, there is a Borel measurable map
$\tau:X\to\Sigma$ such that $\pi\circ\tau=\text{Identity}_{X}$,
$\,\,\sigma\circ\tau=\tau\circ f$ and $\Sigma=\overline{\tau(X)}$, then
$(\Sigma,\sigma,\pi,\tau)$ is called a _symbolic extension with embedding_.

A symbolic extension $(\Sigma,\sigma,\pi)$ is _strongly faithful_ if the
induced map $\pi_{*}:\mathscr{P}(\Sigma,\sigma)\,\to\,\mathscr{P}(X,f)$ is an
homeomorphism and $\pi$ preserves periodic points, that is, for any
$n\in\mathbb{N}$ we have
$\pi(\mathrm{Per}_{n}(\sigma_{|\Sigma}))=\mathrm{Per}_{n}(f)$.

The existence of symbolic extensions seems to depend on hyperbolic-type
properties of $f$ and its degree of differentiability. For instance, in the
setting of $C^{\infty}$ diffeomorphisms on a compact Rimannian manifold, J.
Buzzi established in [14] that principal symbolic extensions always exist. On
the other hand, D. Burguet proved in [10] that, for $C^{2}$ diffeomorphisms on
surfaces, symbolic extensions are sure to exist. On the contrary, T.
Downarowicz and S. Newhouse proved in [20] that a generic $C^{1}$ area-
preserving diffeomorphism of a compact surface either is Anosov or has no
symbolic extension.

M. Boyle, D. Fiebig and U. Fiebig showed in [6] that, if $f$ is entropy-
expansive, then it has a principal symbolic extension. In addition, W.
Cowieson and L.-S. Young proved in [17] that every partially hyperbolic
$C^{1}$ diffeomorphism with a one-dimensional center bundle is entropy-
expansive (see generalizations in [18, 19]). Therefore, if $f$ is partially
hyperbolic with a one-dimensional center bundle then a principal symbolic
extension exists. In particular, every Shub’s example in $\mathcal{U}$ has a
principal symbolic extension. In addition, if we restrict to $\mathfrak{R}$,
the diffeomorphisms are asymptotically per-expansive, and we may find a
strongly faithtful extension with embedding.

For further use, we register that, according to [12, Main Theorem], the
following four conditions together are enough to guarantee that $f$ has a
principal strongly faithful symbolic extension with embedding:

  1. (1)

$f$ is entropy-expansive.

  2. (2)

$f$ is asymptotically per-expansive.

  3. (3)

$\mathrm{Per}(f)$ is zero dimensional.

  4. (4)

There exists $K>0$ such that

    1. (i)

$h_{\mathrm{top}}(f)<\log K$;

    2. (ii)

$\frac{1}{n}\,\log\\#\,\mathrm{Per}_{n}(f)\leqslant\log K$ for every
$n\in\mathbb{N}$.

###  3.5. Partial hyperbolicity

Assume in the following subsections that $X$ is a compact, connected
Riemannian manifold. An $f$-invariant compact set $\Lambda\subset X$ is
_partially hyperbolic_ if the tangent bundle on $\Lambda$ admits a
$Df$-invariant splitting $E^{\mathrm{s}}(f)\oplus E^{\mathrm{c}}(f)\oplus
E^{\mathrm{u}}(f)$ such that $E^{\mathrm{s}}$ is uniformly contracted and
$E^{\mathrm{u}}$ is uniformly expanded, and the possible contraction and
expansion of $Df$ in $E^{c}(f)$ are weaker than those in the complementary
bundles. More precisely, there exist constants $N\in\mathbb{N}$ and
$\lambda>1$ such that, for every $x\in\Lambda$ and every unit vector
$v^{\ast}\in E^{\ast}(x,f)$, where $\ast=\mathrm{s},\mathrm{c},\mathrm{u}$, we
have

  * (a)

$\quad\lambda\,\|\,Df_{x}^{N}(v^{\mathrm{s}})\,\|<\|\,Df_{x}^{N}(v^{\mathrm{c}})\,\|<\lambda^{-1}\,\|\,Df_{x}^{N}(v^{\mathrm{u}})\,\|$

  * (b)

$\quad\|\,Df_{x}^{N}(v^{\mathrm{s}})\,\|<\lambda^{-1}<\lambda<\|\,Df_{x}^{N}(v^{\mathrm{u}})\,\|$.

In particular, an $f$-invariant compact set $\Lambda\subset X$ is said to be a
partially hyperbolic attracting set if there exists an open neighborhood $U$
of $\Lambda$ such that $\overline{f(U)}\subset U$ and

$\Lambda=\bigcap_{n\,\in\,\mathbb{N}}\,f^{n}(U)$

on which there exist a continuous $Df$-invariant splitting of the tangent
bundle into a strong unstable sub-bundle $E^{\,\mathrm{uu}}$ and a center sub-
bundle $E^{\,\mathrm{c}}$ dominated by $E^{\,\mathrm{uu}}$. More precisely,
$T_{\Lambda}X=E^{\,\mathrm{uu}}\oplus E^{\,\mathrm{c}}$ and

$\|\,(Df\,|\,_{E^{\,\mathrm{uu}}})^{-1}\,\|<1\quad\quad\mbox{and}\quad\quad\|\,Df\,|\,_{E^{\,\mathrm{c}}}\,\|\,\,\|\,(Df\,|\,_{E^{\,\mathrm{uu}}})^{-1}\,\|<1.$

Partial hyperbolicity is a robust property, and a partially hyperbolic
diffeomorphism $f$ admits stable and unstable foliations, say
$W^{\mathrm{s}}(f)$ and $W^{\mathrm{u}}(f)$, which are $f$-invariant and
tangent to $E^{\mathrm{s}}(f)$ and $E^{\mathrm{u}}(f)$, respectively [5].
However, the center bundle $E^{\mathrm{c}}(f)$ may not have a corresponding
tangent foliation (cf. [26]). For a comprehensive exposition on partial
hyperbolicity, we refer the reader to [5].

Suppose that $f$ has a partially hyperbolic attracting set. We say that $f$ is
_mostly contracting_ if, from the point of view of the natural volume within
the strong unstable leaves, the asymptotic forward behavior along the central
direction is contracting: given any $uu$-dimensional disk $D$ inside a strong
unstable leaf of $W^{\,\mathrm{uu}}$, there exists a positive volume measure
subset $A\subset D$ whose points satisfy

$\limsup_{n\,\to\,+\infty}\,\frac{1}{n}\,\log\|\,Df^{n}\,|\,_{E^{\,\mathrm{c}}(x)}\,\|<0\quad\quad\forall\,\,x\in
A.$

We note that, according to [2], the set of partially hyperbolic
diffeomorphisms whose central direction is mostly contracting is open in the
$C^{k}$ topology for any $k>1$.

###  3.6. Hyperbolic measures

Given $x\in X$ and $v\in T_{x}X$, define the upper Lyapunov exponent of $v$ at
$x$ by

$\lambda^{+}(x,v)\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\,\limsup_{n\,\to\,+\infty}\,\,\frac{1}{n}\,\log\|\,D_{x}f^{n}(v)\,\|.$

The function $\lambda^{+}:TX\,\to\,\mathbb{R}$ can only take a finite number
$r(x)$ of different values on each space $T_{x}X$, say
$\lambda_{1}(x)<\lambda_{2}(x)<\cdots<\lambda_{r(x)}(x)$, and associated to
these there exist a filtration $L_{1}(x)\subset L_{2}(x)\subset\cdots\subset
L_{r(x)}(x)=T_{x}X$ such that $\lambda^{+}(x,v)=\lambda_{i}(x)$ for every
$x\in X$ and all $v\in L_{i}(x)\setminus L_{i-1}(x)$. Besides, the maps
$\big{(}\lambda_{i}(x)\big{)}_{1\,\leqslant\,i\,\leqslant\,r(x)}$ are
measurable and $f$-invariant; their values are called the Lyapunov exponents
of $f$ at $x$. For each $1\,\leqslant\,i\,\leqslant\,r(x)$ and $x\in X$, the
number $k_{i}(x)=\dim(L_{i}(x))-\dim(L_{i-1}(x))$ is the multiplicity of the
$i$-th exponent at $x$. Moreover, there exists a subset $\mathscr{O}(f)\subset
X$ such that, if $x$ belongs to $\mathscr{O}(f)$, then the limit

$\lim_{n\,\to\,+\infty}\,\frac{1}{n}\,\log\,\|\,D_{x}f^{n}(v)\,\|$

exists for all $v\neq 0$. The elements in $\mathscr{O}(f)$ are called regular
points. Oseledets’ Theorem [36] ensures that the set of regular points
$\mathscr{O}(f)$ has full $\mu$ measure for any $\mu\in\mathscr{P}(X,f)$. If,
in addition, $\mu$ is ergodic, then the functions $x\,\to\,\lambda_{i}(x)$ and
$x\,\to\,r(x)$ are constant at $\mu$-almost everywhere. We denote these
constants by $\lambda_{1}(\mu)<\cdots<\lambda_{r}(\mu)$. An ergodic
probability measure $\mu$ is said to be hyperbolic if $\lambda_{i}(\mu)\neq 0$
for every $i=1,...,r$.

###  3.7. SRB measures

Let $x\in X$ be a regular point and consider the sum (with multiplicity) of
all the positive Lyapunov exponents at $x$

$\chi^{u}(x)\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\sum\limits_{\\{i\,:\,\lambda_{i}(x)\,>\,0\\}}\,k_{i}(x)\,\lambda_{i}(x).$

Margulis-Ruelle inequality [41] states that the metric entropy (definition in
[44, §4]) of every $\mu\in\mathscr{P}(X,f)$ is bounded above by the space
average of $\chi^{u}$, that is,

$h_{\mu}(f)\leqslant\int\,\chi^{u}\,\,d\mu.$

On the other hand, by Oseledets’ Theorem one knows that, if $E^{u}(x)$ stands
for the subspace of $T_{x}X$ corresponding to the positive Lyapunov exponents
at the regular point $x\in X$ and we denote by $J^{u}(x)$ the Jacobian of $Df$
restricted to the subspace $E^{u}(x)$, then

$\chi^{u}(x)=\lim_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum\limits^{n-1}_{i=0}\,\log|\,J^{u}\big{(}f^{i}(x)\big{)}\,|.$

Thus, for every Borel $f$-invariant probability measure $\mu$ one has

$h_{\mu}(f)\leqslant\int\,\log|\,J^{u}\,|\,d\mu.$ (3.1)

A probability measure $\mu$ attaining the equality in (3.1) is called an SRB
measure. Pesin proved in [37] that if $\mu$ is equivalent to Lebesgue measure
(the Riemannian volume) then $\mu$ is an SRB measure. Afterwards, Ledrappier
and L.-S. Young identified all the measures satisfying Pesin’s entropy
formula, establishing in [31] that the equality (3.1) holds if and only if the
conditional measures of $\mu$ on unstable manifolds are absolutely continuous
with respect to Lebesgue measure.

###  3.8. Physical measures

Let $\mu$ be a Borel $f$-invariant probability measure on $X$. A point $x\in
X$ is called _$\mu$ -generic_ if

$\lim_{n\,\to\,+\infty}\,\frac{1}{n}\,\sum^{n-1}_{i=0}\,\varphi(f^{i}(x))=\int\,\varphi\,d\mu\qquad\forall\,\,\varphi\in
C^{0}(X,\mathbb{R})$

where $C^{0}(X,\mathbb{R})$ stands for the space of continuous maps
$\varphi:X\,\to\,\mathbb{R}$ with the uniform norm. We will denote by
$\mathfrak{B}(\mu)$ the set of $\mu$-generic points, also called the basin of
attraction of $\mu$. The measure $\mu$ is called physical if
$\mathfrak{B}(\mu)$ has positive Lebesgue measure. Note that, if the basin of
$\mu$ has full Lebesgue measure, then $\mu$ is the unique physical measure of
$f$.

For Anosov diffeomorphisms and, more generally, $C^{2}$ Axiom A attractors,
there exists a unique invariant probability measure $\mu$ which is
characterized by each of the following properties, equivalent to one another
(cf. [9]):

  1. (1)

Equality (3.1) holds (that is, $\mu$ is SRB).

  2. (2)

The conditional measures on unstable manifolds of $\mu$ are absolutely
continuous with respect to Lebesgue measure.

  3. (3)

Lebesgue almost every point in a neighborhood of the attractor is generic with
respect to $\mu$ (that is, $\mu$ is physical).

##  4\. The setting

We now review the construction of Shub’s examples [42] with the additional
constraints imposed in [35]. Let $\Phi\colon\mathbb{T}^{2}\to\mathbb{T}^{2}$
be an Anosov diffeomorphism and
$\mathrm{T}\,\mathbb{T}^{2}=E^{\mathrm{ss}}\oplus E^{\mathrm{uu}}$ be its
hyperbolic splitting satisfying, for some uniform constant $0<\gamma<1$,

$\max\,\left\\{\|\,D\Phi|_{E^{\mathrm{ss}}}\,\|,\quad\|\,D\Phi^{-1}|_{E^{\mathrm{uu}}}\,\|\right\\}<\gamma.$

Assume that $\Phi$ has two fixed points $p$ and $q$. Note that they are
homoclinically related (that is, both intersections
$W^{\mathrm{s}}(p)\pitchfork W^{\mathrm{u}}(q)$ and
$W^{\mathrm{u}}(p)\pitchfork W^{\mathrm{s}}(q)$ are transversal and non-
empty). Afterwards, take a smooth family of torus $C^{1}$-diffeomorphisms
$f_{x}\colon\,\mathbb{T}^{2}\,\to\,\mathbb{T}^{2}$ indexed by
$x\in\mathbb{T}^{2}$ with the following properties:

  * (P1)

At each $x\in\mathbb{T}^{2}$, the tangent space at $x$ admits a splitting
$\mathrm{T}\,\mathbb{T}^{2}_{x}=E^{\mathrm{c}}(f_{x})\oplus
E^{\mathrm{u}}(f_{x})$ invariant under $Df_{x}$ and for which there exist
constants $0<\gamma_{1}<\gamma^{-1}_{2}<1$ such that

$\|\,D{f_{x}^{-1}}|_{E^{\mathrm{u}}(f_{x})}\,\|\leqslant\gamma_{1}\quad\quad\text{and}\quad\quad\gamma_{1}\leqslant\|\,Df_{x}|_{E^{\mathrm{c}}(f_{x})}\,\|\leqslant\,\gamma_{2}.$

We may assume, taking a power of $\Phi$ if necessary, that
$\gamma<\gamma_{1}$.

  * (P2)

For every $x\in\mathbb{T}^{2}$, the diffeomorphism $f_{x}$ preserves cone
fields $\mathcal{C}^{\mathrm{cs}}$ and $\mathcal{C}^{\mathrm{u}}$.

  * (P3)

The map $f_{p}$ is Anosov, while $f_{q}$ is a Derived from Anosov.

  * (P4)

There is $\theta_{0}\in\mathbb{T}^{2}$ such that
$f_{x}(\theta_{0})=\theta_{0}$ for every $x$, and $\theta_{0}$ is a saddle of
$f_{p}$ and a source for $f_{q}$.

Shub’s examples are precisely the skew products induced by the action of the
diffeomorphisms $(f_{x})_{x\,\in\,\mathbb{T}^{2}}$, namely
$F_{S}\colon\mathbb{T}^{2}\times\mathbb{T}^{2}\to\mathbb{T}^{2}\times\mathbb{T}^{2}$
defined by

$\displaystyle F_{S}(x,y)=\big{(}\Phi(x),\,f_{x}(y)\big{)}.$ (4.1)

It is not difficult to check that
$\Omega(F_{S})=\mathbb{T}^{2}\times\mathbb{T}^{2}$ and that $F_{S}$ is
partially hyperbolic with a one-dimensional center bundle and a splitting

$\mathrm{T}_{(x,y)}\big{(}\mathbb{T}^{2}\times\mathbb{T}^{2}\big{)}=E^{\mathrm{ss}}(x,y)\oplus
E^{\mathrm{c}}(x,y)\oplus E^{\mathrm{u}}(x,y)\oplus E^{\mathrm{uu}}(x,y).$

In what follows we denote by $\mathcal{W}^{*}$ the invariant foliation tangent
to $E^{*}$, with $*=$ ss, c, u, uu.

###  4.1. Additional assumptions

The selection of the family $x\in\mathbb{T}^{2}\to
f_{x}\in\mathrm{Diff}^{1}(\mathbb{T}^{2})$ in the construction of $F_{S}$ is
not unique. In this work we will add the conditions (A1)-(A3) below, as done
in [35]:

  * (A1)

The map $x\in\mathbb{T}^{2}\to f_{x}\in\mathrm{Diff}^{1}(\mathbb{T}^{2})$ is
continuous.

  * (A2)

$F_{S}$ is homotopic to $\Phi\times L$ as a bundle map (that is, the homotopic
path is made of skew-products with fixed base $\Phi$), where
$L:\mathbb{T}^{2}\to\mathbb{T}^{2}$ is a hyperbolic toral automorphism.

  * (A3)

Each $f_{x}$ preserves the stable foliation $\mathcal{F}$ of $L$ which is
tangent to the central direction $E^{\mathrm{c}}$. We also require that this
foliation is normally expanded, meaning that

$\inf_{(x,y)\,\in\,\mathbb{T}^{2}\times\mathbb{T}^{2}}\,\|\,D_{y}f_{x}\,|\,_{E^{\mathrm{u}}(x,y)}\,\|\,>\,\max\,\Big{\\{}1,\sup_{(x,y)\,\in\,\mathbb{T}^{2}\times\mathbb{T}^{2}}\,\|\,D_{y}f_{x}\,|\,_{E^{\mathrm{c}}(x,y)}\,\|\Big{\\}}$

###  4.2. Construction of Shub’s examples

Although Shub’s examples can be constructed quite generally, the natural way
to obtain them is through a small $C^{0}$-perturbation supported on a small
neighborhood of a fixed point of $\Phi\times L$. More precisely, suppose that
$0<\lambda_{s}<1$ and $\lambda_{u}=\lambda_{s}^{-1}>1$ are the eigenvalues
associated to the unstable and stable eigenvectors $\textbf{v}^{u}$ and
$\textbf{v}^{s}$ of the matrix $L$. Let $\theta_{0}\in\mathbb{T}^{2}$ be a
fixed point of the induced map by $L$ (which we still denote by $L$ if no
confusion arises) corresponding to 0 in $\mathbb{R}^{2}$. In a relatively
small neighborhood $W\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,W_{1}\times W_{2}$ of $(q,\theta_{0})$ we use coordinates
$u_{1}\textbf{v}^{u}+u_{2}\textbf{v}^{s}$ in each fiber $\\{w\\}\times W_{2}$,
where $w\in W_{1}$. Let $\varrho>0$ be small enough so that the ball
$B_{\varrho}(q,\theta_{0})=B_{\varrho}(q)\times B_{\varrho}(\theta_{0})$ of
radius $\varrho$ centered at $(q,\theta_{0})$ is contained in $W$. Take a bump
function $\delta:\mathbb{T}^{2}\times\mathbb{T}^{2}\to\mathbb{R}$ defined by
$\delta(x,y)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}b(x)\,b(y)$, where $b:\mathbb{T}^{2}\to\mathbb{R}$ is a bump
function satisfying $0\leqslant b(x)\leqslant 1$ for every
$x\in\mathbb{T}^{2}$, $\,b(x)=1$ if $|x|<\varrho/2$ and $b(x)=0$ if
$|x|>\varrho$. Afterwards consider the system of differential equations in
$\mathbb{T}^{2}\times\mathbb{T}^{2}$ given by

$\left\\{\begin{array}[]{lr}\dot{w}=0\quad\mbox{in}\quad\mathbb{T}^{2}&\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
(\dot{u}_{1},\,\dot{u}_{2})=\big{(}0,\,\,u_{2}\,\delta\,(|w-q|,\,|(u_{1},u_{2})|)\big{)}\quad\mbox{in}\quad\mathbb{T}^{2}&\end{array}\right.$
(4.2)

Let $\varphi^{t}$ be the flow of the differential equation (4.2), that is,

$\varphi^{t}(w,\,(u_{1},u_{2}))=\big{(}w,\psi^{t}_{w}(u_{1},u_{2})\big{)}\qquad\mbox{where}\qquad\psi^{t}_{w}(u_{1},u_{2})=\big{(}u_{1},\psi^{t}_{w,\,2}(u_{1},u_{2})\big{)}.$

Then the support of $\varphi^{t}-id$ is contained in $W$. Moreover, the
derivative of the flow at $(w,\theta_{0})$ in terms of the
$(w,u_{1},u_{2})$-coordinates is given by

$D_{(w,\,\theta_{0})}\,\varphi^{t}=\begin{pmatrix}\textbf{1}&\textbf{0}\\\
\textbf{0}&D_{\theta_{0}}\,\psi^{t}_{w}\end{pmatrix}\qquad\mbox{where}\qquad
D_{\theta_{0}}\,\psi^{t}_{w}=\begin{pmatrix}1&0\\\
0&e^{t\,b(|w-q|)}\end{pmatrix}$

where the bold numbers 0 and 1 stand for the null $2\times 2$ matrix and the
$2\times 2$ identity matrix, respectively.

Finally, fix $T>0$ such that $1<\lambda_{s}\,e^{T}<\lambda_{u}$ and define
$F_{S}:\mathbb{T}^{2}\times\mathbb{T}^{2}\to\mathbb{T}^{2}\times\mathbb{T}^{2}$
by

$F_{S}\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\varphi^{T}\circ(\Phi\times L).$

This way the derivative of $F_{S}$ at $(q,\theta_{0})$ in the
$(w,u_{1},u_{2})$ coordinate system is precisely

$D_{(q,\,\theta_{0})}\,F_{S}=D_{(q,\,\theta_{0})}\,\varphi^{T}\,\,D_{(q,\,\theta_{0})}(\Phi\times
L)=\begin{pmatrix}\textbf{1}&\textbf{0}\\\
\textbf{0}&D_{\theta_{0}}\,\psi^{T}_{q}\end{pmatrix}\begin{pmatrix}D_{q}\Phi&\textbf{0}\\\
\textbf{0}&L\end{pmatrix}=\begin{pmatrix}D_{q}\Phi&\textbf{0}\\\
\textbf{0}&D_{\theta_{0}}\,\psi^{T}_{q}\,L\end{pmatrix}$

where

$D_{\theta_{0}}\,\psi^{T}_{q}\,L=\begin{pmatrix}\lambda_{u}&0\\\
0&\lambda_{s}\,e^{T}\end{pmatrix}.$

Therefore, $(q,\,\theta_{0})$ is a fixed point of unstable index $3$, while
$(p,\,\theta_{0})$ has unstable index $2$. Furthermore, the one-parameter
family $(f_{x})_{x\,\in\,\mathbb{T}^{2}}$ is given by

$f_{x}=\psi^{T}_{\Phi(x)}\circ L.$

\begin{overpic}[scale={.12}]{Sh.jpg} \put(26.0,68.0){\large{Anosov}}
\put(31.0,53.0){\large{$\theta$}} \put(34.0,20.0){\LARGE{$p$}}
\put(67.0,23.0){\LARGE{$q$}} \put(69.0,61.0){\large{$\theta$}}
\put(55.0,77.0){\large{Derived from Anosov}} \end{overpic} Figure 1. Homotopic
deformation from $\Phi\times L$ to $F_{S}$

######  Remark 4.1.

The previous construction provides an open set $\mathcal{U}$ of $C^{r}$
diffeomorphisms, $r>1$, with the properties (P1)-(P4) and (A1)-(A3) listed
above. Indeed, the conditions (P1)-(P4) are valid for
$\gamma_{1}:=\lambda_{s}$ and $\gamma_{2}:=e^{T}\,\lambda_{s}$; and (A1)-(A3)
are obtained by construction.

##  5\. Properties of $F_{S}\in\mathcal{U}$

The selection of the family $x\in\mathbb{T}^{2}\to
f_{x}\in\mathrm{Diff}(\mathbb{T}^{2})$ in the previous construction of $F_{S}$
induce several dynamical and ergodic features in $F_{S}$ we will now list.

###  5.1. Semi-conjugation with an Anosov diffeomorphism

Under the previous assumptions, it was shown in [35] the existence of a
continuous surjective map
$H:\mathbb{T}^{2}\times\mathbb{T}^{2}\,\to\mathbb{T}^{2}\times\mathbb{T}^{2}$
such that

$H\circ F_{S}=(\Phi\times L)\circ H.$ (5.1)

Moreover, the semi-conjugation $H$ is a skew product as well, that is,

$H(x,y)=(x,\,h_{x}(y))$

where each $h_{x}:\\{x\\}\times\mathbb{T}^{2}\to\\{x\\}\times\mathbb{T}^{2}$
is homotopic to the identity and satisfies

$h_{\Phi(x)}\circ f_{x}=L\circ h_{x},\quad\quad\forall\,x\in\mathbb{T}^{2}.$
(5.2)

The semi-conjugation $H$ can be seen as the result of a parameterized version
of a theorem due to Franks [21].

In [35], Newhouse and L.-S. Young also established the existence of a unique
probability measure $\mu_{\mathrm{max}}$ of maximal entropy for $F_{S}$, and
proved that $H_{*}(\mu_{\mathrm{max}})=\nu_{\mathrm{max}}$, where
$\nu_{\mathrm{max}}$ stands for the probability measure of maximal entropy of
$\Phi\times L$. Moreover, the pairs $(F_{S},\mu_{\mathrm{max}})$ and
$(\Phi\times L,\nu_{\mathrm{max}})$ are almost conjugate. More precisely,
there exists a set $B\times\mathbb{T}^{2}$ contained in the set of injectivity
points of $H$, say

$\mathcal{A}\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\Big{\\{}\,\,(x,y)\in\mathbb{T}^{2}\times\mathbb{T}^{2}\,\,\colon\,\,\\#\,H^{-1}(x,y)=1\,\,\Big{\\}}$

and such that

  * •

$\mu_{\mathrm{max}}(B\times\mathbb{T}^{2})=\nu_{\mathrm{max}}(B\times\mathbb{T}^{2})=1$;

  * •

$H:B\times\mathbb{T}^{2}\to B\times\mathbb{T}^{2}$ is a conjugation between
the restrictions $F_{S}|_{B\times\mathbb{T}^{2}}$ and $(\Phi\times
L)|_{B\times\mathbb{T}^{2}}$.

Actually, the set $B\times\mathbb{T}^{2}$ is contained in

$\mathcal{E}\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\Big{\\{}\,\,(x,y)\in\mathbb{T}^{2}\times\mathbb{T}^{2}\,\,\colon\,\,\lambda^{c}_{+}(F_{S})(x,y)<0\,\,\Big{\\}}\subset\mathcal{A}$

(cf. [35]), where $\lambda^{c}_{+}(F_{S})$ stands for the upper Lyapunov
exponent of $F_{S}$ along to the one-dimension central direction
$E^{\mathrm{c}}(F_{S})$.

The following proposition lists some useful consequences from the existence of
the semi-conjugation $H$ with the previous properties.

######  Proposition 5.1.

[35, Theorems 1 & 2]

  1. (a)

$h_{\mathrm{top}}(F_{S})=h_{\mathrm{top}}(\Phi\times
L)=h_{\mathrm{top}}(\Phi)+h_{\mathrm{top}}(L)$.

  2. (b)

$h_{\mathrm{top}}\big{(}H^{-1}(x,y)\big{)}=0,\quad\quad\forall\,\,(x,y)\,\in\,\mathbb{T}^{2}\times\mathbb{T}^{2}$.

  3. (c)

$\mu_{\mathrm{max}}\big{(}\,\mathcal{E}\,\big{)}=\nu_{\mathrm{max}}\big{(}\,\mathcal{E}\,\big{)}=1$.

We observe that $\nu_{\mathrm{max}}$ is a product measure and that, when
$\Phi$ is a linear hyperbolic automorphism, $\nu_{\mathrm{max}}$ is Lebesgue
measure on $\mathbb{T}^{2}\times\mathbb{T}^{2}$. In this case, Proposition 5.1
(c) indicates that $F_{S}$ is mostly contracting along the central direction
with respect to the splitting

$\mathbb{E}^{\,\mathrm{c}}=E^{\,\mathrm{ss}}\oplus
E^{\,\mathrm{c}}\quad\text{and}\quad\mathbb{E}^{\,\mathrm{u}}=E^{\,\mathrm{u}}\oplus
E^{\,\mathrm{uu}}$

since Lebesgue almost every point in $\mathbb{T}^{2}\times\mathbb{T}^{2}$ has
negative central Lyapunov exponent.

Taking into account that $\mathcal{E}\subset\mathcal{A}$, the previous
properties (b) and (c) of Proposition 5.1 allow us to apply [15, Theorem 1.5]
to $F_{S}$, and thereby show that $\mu_{\mathrm{max}}$ describes the
distribution of periodic classes of $F_{S}$. More precisely, consider the
equivalence relation on the set $\mathbb{T}^{2}\times\mathbb{T}^{2}$

$(x,y)\sim(x_{0},y_{0})\quad\quad\Leftrightarrow\quad\quad
H(x,y)=H(x_{0},y_{0})$

so the elements in the class $[(x,y)]$ are the ones in $H^{-1}(H(x,y))$. The
class $[(x,y)]$ is said to be $n$-periodic if $H(x,y)$ belongs to
$\mathrm{Per}_{n}(\Phi\times L)$. Denote by
$\widetilde{\mathrm{Per}_{n}}(F_{S})$ the set of periodic classes with period
$n$. Then $\mu_{\mathrm{max}}$ describes the distribution of periodic classes
of $F_{S}$ if $\mu_{\mathrm{max}}$ is the weak∗ limit of the sequence of
measures

$n\in\mathbb{N}\quad\mapsto\quad\zeta_{n}\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\frac{1}{\\#\,\widetilde{\mathrm{Per}}_{n}(F_{S})}\sum_{[(x,y)]\,\in\,\widetilde{\mathrm{Per}}_{n}(F_{S})}\,\delta_{[(x,y)]}$

where $\delta_{[(x,y)]}$ is any $F_{S}^{n}$-invariant probability measure
supported on the class $[(x,y)]$. We may wonder whether the elements of a
periodic class are $F_{S}$-periodic or if the class contains a periodic point
by $F_{S}$. We will answer to this question on Section 6, where we will also
establish that our assumptions about $F_{S}$ enable us to improve [15, Theorem
1.5], showing that the measure $\mu_{\mathrm{max}}$ describes the distribution
of the periodic points of $F_{S}$ (cf. Subsection 6.4).

###  5.2. Hyperbolicity of $\mu_{\mathrm{max}}$

A direct consequence of the construction of $F_{S}$ is the fact that the
measure $\mu_{\mathrm{max}}$ exhibits four Lyapunov exponents, namely

$\lambda^{ss}(\mu_{\mathrm{max}})<\lambda^{c}(\mu_{\mathrm{max}})<\lambda^{u}(\mu_{\mathrm{max}})<\lambda^{uu}(\mu_{\mathrm{max}})$

which are constant $\mu_{\mathrm{max}}$ almost everywhere (since
$\mu_{\mathrm{max}}$ is ergodic) and satisfy

$\lambda^{ss}(\mu_{\mathrm{max}})<0<\lambda^{u}(\mu_{\mathrm{max}})\quad\quad\text{and}\quad\quad\log(\gamma_{1})\leqslant\lambda^{c}(\mu_{\mathrm{max}})\leqslant\log(\gamma_{2}).$

Therefore:

######  Proposition 5.2.

  1. (a)

$\mu_{\mathrm{max}}$ is hyperbolic.

  2. (b)

The unstable foliation of $F_{S}$ tangent to the bundle $E^{\mathrm{u}}\oplus
E^{\mathrm{uu}}$ is robustly minimal.

  3. (c)

If $\Phi$ is a linear hyperbolic automorphism of the $2$-torus, the partially
hyperbolic attractor $\mathbb{T}^{2}\times\mathbb{T}^{2}$ of $F_{S}$ admits a
unique SRB measure, say $\mu_{\mathrm{SRB}}$, whose basin has full Lebesgue
measure. So $\mu_{\mathrm{SRB}}$ is the unique physical measure of $F_{S}$.

###### Proof.

To prove the hyperbolicity of $\mu_{\mathrm{max}}$ we need to verify that
$\lambda^{c}(\mu_{\mathrm{max}})\neq 0$. Let $\mathscr{O}(F_{S})$ be the set
of regular points of $F_{S}$ provided by Oseledets’s Theorem [36]. From
Proposition 5.1 (c), we know that the set
$\mathscr{O}(F_{S})\,\cap\,\mathcal{E}$ has full $\mu_{\mathrm{max}}$ measure.
Moreover, points $(x,y)$ in this intersection satisfy

$\lambda^{c}_{-}(F_{S})(x,y)=\lambda^{c}_{+}(F_{S})(x,y)=\lambda^{c}(\mu_{\mathrm{max}})<0$

where $\lambda^{c}_{-}(F_{S})(x,y)$ denotes the lower Lyapunov exponent of
$F_{S}$ at $E^{c}(x,y)$. This completes the proof of item (a).

The robust minimality, asserted in item (b), of the unstable foliation
$W^{\mathrm{u}}$ of $F_{S}$, tangent to $E^{\mathrm{u}}\oplus
E^{\mathrm{uu}}$, follows directly from [39, Theorem A].

Regarding item (c), we start noticing that, being a particular case to which
we may apply the results of [17], the skew product $F_{S}$ has at least one
SRB measure. As mentioned before, $\mathbb{T}^{2}\times\mathbb{T}^{2}$ is a
partially hyperbolic attractor for $F_{S}$ with a partially hyperbolic
splitting $\mathbb{E}^{c}=E^{ss}\oplus E^{c}$ and $\mathbb{E}^{u}=E^{u}\oplus
E^{uu}$. Under the additional assumption that $\Phi$ is a linear hyperbolic
automorphism of the $2$-torus, we know that $\nu_{\mathrm{max}}$ is the
Lebesgue measure on $\mathbb{T}^{2}\times\mathbb{T}^{2}$. Therefore, from
Proposition 5.1 (c) (which says that $F_{S}$ is mostly contracting) and the
previous item (b) (which states that the unstable foliation of $F_{S}$ is
robustly minimal), we conclude that $F_{S}$ satisfies the hypothesis of [3,
Theorem B]. Therefore, $F_{S}$ has a unique SRB measure $\mu_{\mathrm{SRB}}$
whose basin has full Lebesgue measure. Hence, $\mu_{\mathrm{SRB}}$ is the
unique physical measure of $F_{S}$ as well. ∎

From Proposition 5.1 (a) we know that $h_{\mu_{\mathrm{max}}}(F_{S})>0$. Then
we may apply the generalization of [28, Corollary 4.3] established in [22,
Theorem 1]. This result together with item (a) of Proposition 5.2 imply that
$F_{S}$ is a limit of horseshoes in the sense of the entropy.

######  Proposition 5.3.

For every $0<\varepsilon<h_{\mathrm{top}}(F_{S})$, there exists a compact
$F_{S}$-invariant set
$\Lambda_{\varepsilon}\subset\mathbb{T}^{2}\times\mathbb{T}^{2}$ such that
$F_{S}|_{\Lambda_{\varepsilon}}$ is conjugate to a subshift of finite type and

$h_{\mathrm{top}}(F_{S}|_{\Lambda_{\varepsilon}})\geqslant
h_{\mathrm{top}}(F_{S})-\varepsilon.$

###  5.3. SRB measures of $F_{S}$ and $\Phi\times L$

Given $(x,y)$ in $\mathbb{T}^{2}\times\mathbb{T}^{2}$, let
$J^{u}_{F_{S}}(x,y)$ be the Jacobian of $D_{(x,y)}F_{S}$ restricted to the
unstable bundle $E^{u}(x,y)\oplus E^{uu}(x,y)$ of $F_{S}$. Analogously, define
$J^{u}_{\Phi\times L}(x,y)$.

######  Proposition 5.4.

Let $\mu_{\mathrm{SRB}}$ be the SRB measure of $F_{S}$. Suppose that at
$\mu_{\mathrm{SRB}}$ almost every $(x,y)$ in
$\mathbb{T}^{2}\times\mathbb{T}^{2}$ we have

$|\,J^{u}_{\Phi\times L}\circ H(x,y)\,|\leqslant|\,J^{u}_{F_{S}}(x,y)\,|.$
(5.3)

Then $H_{*}(\mu_{\mathrm{SRB}})$ is the SRB measure of $\Phi\times L$.

###### Proof.

Set $\nu=H_{*}(\mu_{\mathrm{SRB}})$. After Margulis-Ruelle inequality (3.1),
we are left to verify that

$\int\,\log\,|\,J^{u}_{\Phi\times L}\,|\,d\nu\leqslant h_{\nu}(\Phi\times L).$

Firstly, we note that

$h_{\mu_{\mathrm{SRB}}}(F_{S})=h_{\nu}(\Phi\times L).$

Indeed, Proposition 5.1 (b) and Ledrappier-Walters’ formula [30, (1.2)] yield

$h_{\mu_{\mathrm{SRB}}}(F_{S})\leqslant h_{\nu}(\Phi\times L)$

which, together with the well-known fact [44, Theorem 4.11] that
$h_{\mu_{\mathrm{SRB}}}(F_{S})\geqslant h_{\nu}(\Phi\times L)$, imply the
equality. Thus, using (5.3) one gets

$\int\,\log\,|\,J^{u}_{\Phi\times L}\,|\,d\nu=\int\,\log\,|\,J^{u}_{\Phi\times
L}\circ
H\,|\,d\mu_{\mathrm{SRB}}\leqslant\int\,\log\,|\,J^{u}_{F_{S}}\,|\,d\mu_{\mathrm{SRB}}=h_{\mu_{\mathrm{SRB}}}(F_{S})=h_{\nu}(\Phi\times
L).$

∎

##  6\. Proof of Theorem A

As the argument on this section is rather long, for the reader’s convenience
we subdivide it according to the items of the theorem’s statement.

We consider the residual subset $\mathfrak{R}\subset\mathcal{U}$, defined by
$\mathfrak{R}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\mathcal{U}\cap\mathcal{K}\mathcal{S}$, where
$\mathcal{K}\mathcal{S}$ denotes the set of Kupka-Smale diffeomorphisms. Thus,
its elements have only hyperbolic periodic points, and so the whole set of the
periodic points is countable. In what follows, we will consider such generic
Shub’s examples in $\mathfrak{R}$.

###  6.1. Proof of Theorem A (a)

The key idea in the following argument is to assess how many periodic orbits
in $\mathrm{Per}_{n}(F_{S})$, for $n\in\mathbb{N}$, may occur in the
intersection $H^{-1}(P)\cap\mathrm{Per}_{n}(F_{S})$ for each periodic point
$P\in\mathrm{Per}_{n}(\Phi\times L)$.

###  6.2. Periodic classes

Consider a foliation $\mathcal{W}$ of a simply connected compact Riemannian
manifold $M$ and lift it to the universal cover $\widetilde{M}$, obtaining a
foliation we denote by $\widetilde{\mathcal{W}}$. For points $x,y$ on the same
leaf $\widetilde{W}$ of $\widetilde{\mathcal{W}}$, one can define a distance
$\mathcal{D}_{\widetilde{W}}(x,y)$ as the length of the shortest path inside
the leaf $\widetilde{W}$ linking $x$ and $y$. We say that the lifted foliation
$\widetilde{\mathcal{W}}$ of $\mathcal{W}$ is quasi-isometric if there is a
constant $C>1$ such that for any $x,y\in\widetilde{M}$ lying on the same leaf
of $\widetilde{\mathcal{W}}$ we have

$\mathcal{D}_{\widetilde{W}}(x,y)<C\,\mathcal{D}(x,y)+C$

where $\mathcal{D}$ denotes the metric on $\widetilde{M}$.

######  Claim 6.1.

Let $\mathcal{W}^{u}$ and $\mathcal{F}$ the foliations introduced on Sections
4 and 4.1, respectively. Then $\widetilde{\mathcal{W}}^{u}$ and
$\widetilde{\mathcal{F}}$ are quasi-isometric.

###### Proof.

Since we wish to estimate the intrinsic distance between two points of the
same leaf of either $\widetilde{\mathcal{W}}^{u}$ or
$\widetilde{\mathcal{F}}$, which is contained in some fiber
$\\{\tilde{x}\\}\times\mathbb{R}^{2}$ with $\tilde{x}\in\mathbb{R}^{2}$, it is
sufficient to consider the lift of $\mathcal{W}^{u}$ and $\mathcal{F}$, which
we still denote by $\widetilde{\mathcal{W}}^{u}$ and
$\widetilde{\mathcal{F}}$, to the universal cover
$\mathbb{T}^{2}\times\mathbb{R}^{2}$ of $\mathbb{T}^{2}\times\mathbb{T}^{2}$
with respect to the second factor.

Firstly we observe that from [38, Lemma 4.A.5] we know that, for each
$x\in\mathbb{T}^{2}$, the foliations $\widetilde{\mathcal{W}}^{u}(x,\cdot)$
and $\widetilde{\mathcal{F}}(x,\cdot)$ inside $\\{x\\}\times\mathbb{R}^{2}$
have a global product structure. Then the fact that
$\widetilde{\mathcal{W}}^{u}(x,\cdot)$ and $\widetilde{\mathcal{F}}(x,\cdot)$
are quasi-isometric follows from [38, Proposition 4.3.9] (see also the proof
of [38, Lemma 4.A.5] for more details). Indeed, this result informs that for
every $x\in\mathbb{T}^{2}$ there exist $C_{1,x},C_{2,x}>1$ such that, for
every $\tilde{y},\tilde{z}$ in $\mathbb{R}^{2}$ one has

$\mathcal{D}_{\widetilde{W}^{u}}\big{(}(x,\tilde{y}),(x,\tilde{z})\big{)}<C_{1,x}\,\|\,\tilde{y}-\tilde{z}\,\|+C_{1,x}\quad\mbox{and}\quad\mathcal{D}_{\widetilde{\mathcal{F}}}\big{(}(x,\tilde{y}),(x,\tilde{z})\big{)}<C_{2,x}\,\|\,\tilde{y}-\tilde{z}\,\|+C_{2,x}.$

Moreover, $C_{1,x}$ and $C_{2,x}$ can be taken independent of $x$. On the one
hand, note that $C_{1,x}\leqslant C_{1,q}$ for every $x\in\mathbb{T}^{2}$; on
the other hand, the foliation $\widetilde{\mathcal{F}}$ consists of the family
of lines obtained by the translation (in $\mathbb{R}^{2}$) of the stable
manifold of $L$, thus we can take $C_{2,x}=1$ for every $x$. This completes
the proof. ∎

The next result may be thought of as parameterized version of [43, Proposition
3.1].

######  Lemma 6.2.

For all $(x,y)\in\mathbb{T}^{2}\times\mathbb{T}^{2}$, the set $H^{-1}(x,y)$ is
a one-dimensional compact connected subset of a center manifold of $F_{S}$.

_Proof._ The equality (5.1) can be expressed in
$\mathbb{T}^{2}\times\mathbb{R}^{2}$ by lifting (5.2) to
$\\{x\\}\times\mathbb{R}^{2}$, which provides the equality
$\widetilde{H}\circ\widetilde{F_{S}}=(\Phi\times\widetilde{L})\circ\widetilde{H}$,
where
$\widetilde{H}(x,\tilde{y})=\big{(}x,\widetilde{h}_{x}(\tilde{y})\big{)}$ is a
proper map at a bounded distance from the identity map. The former property of
$\widetilde{H}$ implies that $\widetilde{h}_{x}^{-1}(\tilde{y})$ is a compact
subset of $\mathbb{R}^{2}$ for every
$(x,\tilde{y})\in\mathbb{T}^{2}\times\mathbb{R}^{2}$. The latter leads to the
following estimate: for every $x\in\mathbb{T}^{2}$ and
$\tilde{y},\,\tilde{z}\in\mathbb{R}^{2}$,

$\widetilde{h}_{x}(\tilde{y})=\widetilde{h}_{x}(\tilde{z})\qquad\Leftrightarrow\qquad\exists\,\,\
C>0:\,\,\|\,\widetilde{F_{S}}^{n}(x,\tilde{y})-\widetilde{F_{S}}^{n}(x,\tilde{z})\,\|<C\quad\,\forall
n\in\mathbb{Z}.$ (6.1)

Besides, if $\widetilde{W}^{s}_{\Phi\times L}$ stands for the lifts of the
weak stable foliation of $\Phi\times L$ to
$\mathbb{T}^{2}\times\mathbb{R}^{2}$, then (cf. [35, Lemma 2])

$\widetilde{h}_{x}\,\big{(}\widetilde{\mathcal{F}}(x,\tilde{y})\big{)}=\widetilde{W}^{s}_{\Phi\times
L}\,\big{(}\widetilde{H}(x,\tilde{y})\big{)}.$

We are left to verify that $\widetilde{h}_{x}^{-1}(\tilde{y})$ is a connected
set. To do it we show a parametrized version of [43, Lemma 3.2].

######  Claim 6.3.

If $\widetilde{h}_{x}(\tilde{y})=\widetilde{h}_{x}(\tilde{z})$, then
$(x,\tilde{z})\in\widetilde{\mathcal{F}}(x,\tilde{y})$.

###### Proof.

Suppose that $(x,\tilde{z})\notin\widetilde{\mathcal{F}}(x,\tilde{y})$. Let
$(x,\tilde{w})=\widetilde{W}^{u}(x,\tilde{z})\cap\widetilde{\mathcal{F}}(x,\tilde{y})$.
Note that such a point $(x,\tilde{w})$ exists and is unique (cf. [24,
Proposition 2.4]). Consider

$D_{c}=\mathcal{D}_{\widetilde{\mathcal{F}}}\big{(}(x,\tilde{y}),(x,\tilde{w})\big{)}\quad\mbox{and}\quad
D_{u}=\mathcal{D}_{\widetilde{\mathcal{W}}^{u}}\big{(}(x,\tilde{z}),(x,\tilde{w})\big{)}.$

Now, using the parameters $0<\gamma_{1}<\gamma_{2}^{-1}<1$ associated to the
partial hyperbolicity of $F_{S}$ (see Section 4), we can find constants
$0<\widetilde{\gamma}_{1}<\widetilde{\gamma}\,^{-1}_{2}<1$ such that

$\|\,\,\widetilde{F_{S}}^{n}(x,\tilde{y})-\widetilde{F_{S}}^{n}(x,\tilde{w})\,\,\|\leqslant\widetilde{\gamma}_{2}\,^{n}\,\,D_{c}\qquad\mbox{and}\qquad\mathcal{D}_{\widetilde{\mathcal{W}}^{u}}\big{(}\widetilde{F_{S}}^{n}(x,\tilde{z}),\widetilde{F_{S}}^{n}(x,\tilde{w})\big{)}\geqslant\widetilde{\gamma}\,^{-n}_{1}\,\,D_{u}.$

Since $\widetilde{W}^{u}$ is quasi-isometric (Claim 6.1), we also have

$\|\,\,\widetilde{F_{S}}^{n}(x,\tilde{z})-\widetilde{F_{S}}^{n}(x,\tilde{w})\,\,\|\geqslant\frac{1}{C}\big{(}\widetilde{\gamma}\,^{-n}_{1}\,\,D_{u}-C\big{)}.$

Therefore,

$\|\,\widetilde{F_{S}}^{n}(x,\tilde{y})-\widetilde{F_{S}}^{n}(x,\tilde{z})\,\|>\frac{1}{C}\big{(}\widetilde{\gamma}\,^{-n}_{1}\,\,D_{u}-C\big{)}-\widetilde{\gamma}\,^{n}_{2}\,\,D_{c}$

The last quantity goes to infinity as $n\to+\infty$, which implies, by (6.1),
that $\widetilde{h}_{x}(\tilde{y})\neq\widetilde{h}_{x}(\tilde{z})$. This
finishes the proof of the claim. ∎

######  Claim 6.4.

For every $x\in\mathbb{T}^{2}$ and $\tilde{y}\in\mathbb{R}^{2}$, the pre-image
$\widetilde{h}_{x}^{-1}(\tilde{y})$ is connected.

###### Proof.

We will see that given $\tilde{z}$ and $\tilde{w}$ in
$\widetilde{h}^{-1}_{x}(\tilde{y})$ then the arc in the center manifold
joining $\tilde{z}$ and $\tilde{w}$ is contained in
$\widetilde{h}^{-1}_{x}(\tilde{y})$. Let $\tilde{\vartheta}$ be a point in
this arc. From (6.1), we know that
$\|\,\widetilde{F_{S}}^{n}(x,\tilde{z})-\widetilde{F_{S}}^{n}(x,\tilde{w})\,\|<C$
for every $n\in\mathbb{Z}$. On the other hand, by Claim 6.1 we have, for every
$n$ in $\mathbb{Z}$,

$\begin{split}\|\,\widetilde{F_{S}}^{n}(x,\tilde{z})-\widetilde{F_{S}}^{n}(x,\tilde{\vartheta})\,\|&\leqslant\mathcal{D}_{\widetilde{\mathcal{F}}}\big{(}\widetilde{F_{S}}^{n}(x,\tilde{z}),\widetilde{F_{S}}^{n}(x,\tilde{\vartheta})\big{)}\\\
&\leqslant\mathcal{D}_{\widetilde{\mathcal{F}}}\big{(}\widetilde{F_{S}}^{n}(x,\tilde{z}),\widetilde{F_{S}}^{n}(x,\tilde{y})\big{)}\\\
&\leqslant C+1.\end{split}$

Therefore, $\tilde{\vartheta}$ belongs to $\widetilde{h}_{x}^{-1}(\tilde{y})$.
By projecting, the same property is valid for the map $h_{x}$. This ends the
proof of the claim and of Lemma 6.2. ∎

######  Corollary 6.5.

For every $n\in\mathbb{N}$ and $(x,y)\in\mathrm{Per}_{n}(\Phi\times L)$, the
interval $H^{-1}(x,y)$ intersect the set $\mathrm{Per}_{n}(F_{S})$ in at least
one point. Therefore,

$\\#\,\mathrm{Per}_{n}(\Phi\times
L)\,\leqslant\,\\#\,\mathrm{Per}_{n}(F_{S})\quad\,\,\forall\,n\in\mathbb{N}.$

###### Proof.

By Lemma 6.2, for every $(x,y)\in\mathrm{Per}_{n}(\sigma\times L)$ the map
$F_{S}^{n}:H^{-1}(x,y)\to H^{-1}(x,y)$ is a homeomorphism of a closed
(possibly degenerate) interval. Therefore, Brouwer’s Fixed Point Theorem
guarantees the existence of a fixed point for $F^{n}|_{H^{-1}(x,y)}$ for every
$(x,y)\in\mathrm{Per}_{n}(\sigma\times L)$. ∎

###  6.3. Cardinality of the periodic orbits

Recall from Section 4 that $F_{S}$ is a skew product defined by

$F_{S}(x,y)=\Big{(}\Phi(x),\,f_{x}(y)\Big{)},\quad\quad(x,y)\in\mathbb{T}^{2}\times\mathbb{T}^{2}$

so, for every $n\in\mathbb{N}$,

$F^{n}(x,y)=\big{(}\Phi^{n}(x),g^{n}_{x}(y)\big{)}$

where $g^{n}_{x}\colon\,\mathbb{T}^{2}\to\mathbb{T}^{2}$ is defined by

$g^{n}_{x}(y)\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,f_{\Phi^{n-1}(x)}\circ f_{\Phi^{n-2}(x)}\circ\cdots\circ
f_{x}(y),\quad\quad\quad y\in\mathbb{T}^{2}.$

######  Proposition 6.6.

Take $n\in\mathbb{N}$ and $x_{0}\in\mathrm{Per}_{n}(\Phi)$. Then either,
$g^{n}_{x_{0}}$ is a Anosov diffeomorphism (conjugated to $L^{n}$) or a
Derived from Anosov (obtained from $L^{n}$).

###### Proof.

Firstly, note that $g^{n}_{x_{0}}$ and $L^{n}$ are semi-conjugated. Indeed, as
$x_{0}\in\mathrm{Per}_{n}(\Phi)$ then $h_{\Phi^{n}(x_{0})}=h_{x_{0}}$ (see
(5.2)) and so we have for every $y\in\mathbb{T}^{2}$

$\begin{split}h_{x_{0}}\circ g^{n}_{x_{0}}(y)&=h_{\Phi^{n}(x_{0})}\circ
f_{\Phi^{n-1}(x_{0})}\circ g^{n-1}_{x_{0}}(y)=L\circ
h_{\Phi^{n-1}(x_{0})}\circ\circ g^{n-1}_{x_{0}}(y)=\,\ \cdots\,\,=L^{n}\circ
h_{x_{0}}(y).\end{split}$

Thus, if for every $y\in\mathbb{T}^{2}$, the interval
$H^{-1}(x_{0},y)=(x_{0},h_{x_{0}}^{-1}(y))$ is a point, then $y\to H(x_{0},y)$
is a conjugation between $g^{n}_{x_{0}}$ and $L^{n}$, and so $g^{n}_{x_{0}}$
is an Anosov diffeomorphism. The remaining case is dealt with on the next
lemma.

######  Lemma 6.7.

Take $n\in\mathbb{N}$ and $x_{0}\in\mathrm{Per}_{n}(\Phi)$. If for some
$y\in\mathbb{T}^{2}$ the interval $H^{-1}(x_{0},y)$ is non-degenerate, then
the diffeomorphism $g^{n}_{x_{0}}$ is a Derived from Anosov obtained from
$L^{n}$.

_Proof._ To check that $g^{n}_{x_{0}}$ satisfies the standard properties of a
Derived from Anosov we will follow the reference [40, Pag. 300].

######  Claim 6.8.

$\theta_{0}$ is a source of $g^{n}_{x_{0}}$.

###### Proof.

Since, by construction, when any expansion exists within $E^{c}$, the greatest
expansion is attained at $\theta_{0}$, we have that

$\|Dg^{n}_{x}|_{E^{c}(x,\theta_{0})}\|\,\geqslant\,\|Dg^{n}_{x}|_{E^{c}(x,y)}\|\,,\qquad\forall\,(x,y)\in\mathbb{T}^{2}\times\mathbb{T}^{2},\quad\forall\,n\in\mathbb{N}.$

On the other hand, if $H^{-1}(x_{0},y)$ is a non-degenerated interval then
$\lambda^{c}_{+}(x_{0},y)\geqslant 0$ (recall that
$\mathcal{E}\subset\mathcal{A}$). As $(x_{0},\theta_{0})$ is a fixed point of
$F_{S}^{n}$, the Lyapunov exponent $\lambda^{c}(F^{n}_{S})(x_{0},\theta_{0})$
is well defined and satisfies

$\lambda^{c}(F^{n}_{S})(x_{0},\theta_{0})=n\,\limsup_{k\,\to\,+\infty}\,\frac{1}{nk}\,\log\|Dg^{nk}_{x_{0}}|_{E^{c}(x_{0},\theta_{0})}\|=n\,\lambda^{c}_{+}(F_{S})(x_{0},\theta_{0})\geqslant
n\,\lambda^{c}_{+}(F_{S})(x_{0},y)\geqslant 0.$

Thus $\|Dg^{n}_{x_{0}}|_{E^{c}(x_{0},\theta_{0})}\|\,\geqslant\,1$. But, as
$F_{S}\in\mathfrak{R}$, one must have
$\|Dg^{n}_{x_{0}}|_{E^{c}(x_{0},\theta_{0})}\|>1$, and so $\theta_{0}$ is
indeed a source of $g^{n}_{x_{0}}$. ∎

######  Claim 6.9.

The map $g^{n}_{x_{0}}$ has three fixed points in $W^{s}(\theta_{0},L^{n})$,
namely $\theta_{0}$ and two new saddle points $\theta_{1}$ and $\theta_{2}$,
one in each connected component of
$W^{s}(\theta_{0},L^{n})\setminus\\{\theta_{0}\\}$.

###### Proof.

We recall the ball $B_{\varrho}(q)\subset\mathbb{T}^{2}$ and the subset
$W_{2}\subset\mathbb{T}^{2}$ introduced in Subsection 4.2. Since
$H^{-1}(x_{0},y)$ is non-degenerate interval, there exists $0\leqslant
i\leqslant n$ such that $\Phi^{i}(x_{0})\in B_{\varrho}(q)$. By construction,
outside the set $\\{\Phi^{i}(x_{0})\\}\times W_{2}$ introduced in Subsection
4.2, the slope of the graph of the restriction of the map

$g^{n}_{\Phi^{i}(x_{0})}\colon\,\Big{\\{}\Phi^{i}(x_{0})\Big{\\}}\times\mathbb{T}^{2}\quad\to\quad\Big{\\{}\Phi^{i}(x_{0})\Big{\\}}\times\mathbb{T}^{2}$

to $W^{s}(\theta_{0},L^{n})$ is smaller than one. Therefore, there must exist
two fixed points by the dynamics $g^{n}_{\Phi^{i}(x_{0})}$, say
$\theta^{i}_{1}$ and $\theta^{i}_{2}$, on each side of $\theta_{0}$ inside
$W^{s}(\theta_{0},L^{n})$. The points $\theta_{1}$ and $\theta_{2}$ we were
looking for are obtained intersecting the orbits of $\theta^{i}_{1}$ and
$\theta^{i}_{2}$ with the fibre $\\{x_{0}\\}\times\mathbb{T}^{2}$. ∎

Note that both $(x_{0},\theta^{i}_{1})$ and $(x_{0},\theta^{i}_{2})$ are
hyperbolic periodic points of $F_{S}$. Furthermore, the fixed points
$\theta^{i}_{1}$ and $\theta^{i}_{2}$ of $g^{n}_{\Phi^{i}(x_{0})}$ in
$\Big{\\{}\Phi^{i}(x_{0})\Big{\\}}\times\mathbb{T}^{2}$ are the unique saddles
inside this set fixed by $g^{n}_{\Phi^{i}(x_{0})}$. Indeed, denoting by
$[\theta_{0},w_{2}]\subset\Big{\\{}\Phi^{i}(x_{0})\Big{\\}}\times\mathbb{T}^{2}$
the closure of the connected component of
$\big{(}W^{s}(\theta_{0},L^{n})\setminus\\{\theta_{0}\\}\big{)}\cap W_{2}$
containing the saddle $\theta^{i}_{1}$ (the corresponding notation for
$\theta^{i}_{2}$ is $[-w_{2},\theta_{0}]$) and identifying all the fibers
$\Big{\\{}\Phi^{j}(x_{0})\Big{\\}}\times\mathbb{T}^{2}$ with $\mathbb{T}^{2}$,
we deduce that each one-dimensional maps

$f_{\Phi^{i}(x_{0})}\colon\,[\theta_{0},w_{2}]\quad\to\quad[\theta_{0},w_{2}]$

for $i=0,1,\cdots,n-1$ is a preserving orientation concave function
(including, possibly, affine components, as happens when
$\Phi^{j}(x_{0})\notin B_{\varrho}(q)$) such that

  * •

$f_{\Phi^{i}(x_{0})}(\theta_{0})=\theta_{0}$;

  * •

$f_{x_{0}}(w_{2})=f_{\Phi^{j}(x_{0})}(w_{2})$, for every
$j\in\\{0,\cdots,n-1\\}$;

  * •

there is $i\in\\{0,1,\cdots,n-1\\}$ such that the restriction
$f_{\Phi^{i}(x_{0})}|_{(\theta_{0},w_{2})}$ has a unique (saddle) fixed point
(different from $\theta_{0}$).

Similarly, for every $i=0,1,\cdots,n-1$, the map

$f_{\Phi^{i}(x_{0})}\colon\,[-w_{2},\theta_{0}]\quad\to\quad[-w_{2},\theta_{0}]$

preserves orientation and is concave, which ensures the existence of a unique
saddle $\theta^{i}_{2}$ inside $(-w_{2},\theta_{0})$ which is fixed by
$g^{n}_{\Phi^{i}(x_{0})}$. Consequently, apart from $\theta_{0}$, the points
$\theta^{i}_{1}$ and $\theta^{i}_{2}$ are the unique fixed points of
$g^{n}_{\Phi^{i}(x_{0})}$ in
$\Big{\\{}\Phi^{i}(x_{0})\Big{\\}}\times\mathbb{T}^{2}$.

######  Claim 6.10.

The non-wandering set of $g^{n}_{x_{0}}$ is given by
$\Omega(g^{n}_{x_{0}})=\\{\theta_{0}\\}\cup\Lambda^{n}_{x_{0}}$, where
$\Lambda^{n}_{x_{0}}$ is a hyperbolic attractor of topological dimension one.

_Proof._ Note that, regarding the splitting $E^{u}(L)\oplus E^{s}(L)$ of the
tangent space $T\,\mathbb{T}^{2}$, the derivative of each
$f_{\Phi^{i}(x_{0})}$ is determined by a matrix
$Df_{\Phi^{i}(x_{0})}=(a_{ij})$, which is lower triangular since
$a_{11}=\lambda_{u}$ and $a_{12}=0$ for the whole family
$(f_{x})_{x\,\in\,\mathbb{T}^{2}}$. Thus,

$Dg^{n}_{x_{0}}(y)=\begin{pmatrix}(\lambda_{u})^{n}&0\\\
b_{21}(y)&b_{22}(y)\end{pmatrix}$ (6.2)

with $0<b_{22}<1$ at the saddle fixed point $\theta_{1}$ and $\theta_{2}$.
Moreover, we can assume $b_{22}(\theta_{1})$, $b_{22}(\theta_{2})$
$\leq\lambda^{n}_{s}$. Let $V\subset\mathbb{T}^{2}$ be a neighborhood of
$\theta_{0}$ not containing $\theta_{1}$ and $\theta_{2}$, and such that

  1. i)

$b_{22}>1$ for $w\in V\quad$ (that is, $g^{n}_{x_{0}}$ is an expansion along
$E^{c}$ in $V$);

  2. ii)

$0<b_{22}<1$ for $w\notin g^{n}_{x_{0}}(V)\quad$ (that is, $g^{n}_{x_{0}}$ is
a contraction along $E^{c}$ outside $g^{n}_{x_{0}}(V)$);

  3. iii)

$g^{n}_{x_{0}}(V)\supset V$.

We observe that such a neighborhood $V$ exists (cf. Exercise 7.36 of [40]) and
$V\subset W^{u}(\theta_{0},g^{n}_{x_{0}})$. So it is a local unstable manifold
of $\theta_{0}$ and
$W^{u}(\theta_{0},g^{n}_{x_{0}})=\bigcup_{i\,\geqslant\,1}\,g^{i\,n}_{x_{0}}(V).$
Let $N=\mathbb{T}^{2}\setminus V$. Then $N$ is a trapping region because
$g^{n}_{x_{0}}(V)\supset V$. Set

$\Lambda^{n}_{x_{0}}\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\bigcap_{i\,\geqslant\,1}\,g^{i\,n}_{x_{0}}(N).$

This is an attracting set and $\Lambda^{n}_{x_{0}}=\mathbb{T}^{2}\setminus
W^{u}(\theta_{0},g^{n}_{x_{0}}).$ Thus,
$\Omega(g^{n}_{x_{0}})=\\{\theta_{0}\\}\cup\Lambda^{n}_{x_{0}}$.

We are left to show that $\Lambda^{n}_{x_{0}}$ is hyperbolic. Due to (6.2),
$E^{s}(L)=E^{c}(F_{S})$ is an invariant bundle and every vector in this bundle
is contracted by $D_{z}\,g^{n}_{x_{0}}$ for $z\in\Lambda^{n}_{x_{0}}$. This is
precisely the stable bundle on $\Lambda^{n}_{x_{0}}$. Let $C>0$ be a global
upper bound of $|b_{21}|$. Consider
$\alpha=C\,[(\lambda_{u})^{n}-(\lambda_{s})^{n}]^{-1}$ and take the cones

$\mathcal{C}\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\big{\\{}(v_{1},v_{2})\in E^{u}(L)\oplus
E^{s}(L):\,\,|v_{2}|<\alpha\,|v_{1}|\big{\\}}$

Then it can be checked, using the lower triangular nature of the derivative of
$f_{x}$, that these cones are invariant and

$E^{u}(g^{n}_{x_{0}},z)=\bigcap^{\infty}_{i=1}\,D_{g^{-jn}_{x_{0}}(z)}\,g^{jn}_{x_{0}}\,\,\Big{(}\mathcal{C}\,\big{(}g^{-jn}_{x_{0}}(z)\big{)}\Big{)}$

is an invariant bundle on which the derivative is an expansion for every point
$z\in\Lambda^{n}_{x_{0}}$. This provides the unstable bundle on
$\Lambda^{n}_{x_{0}}$, hence completing the hyperbolic splitting at the points
of this set. This ends the proofs of the last claim, of Lemma 6.7 and of
Proposition 6.6. ∎

######  Corollary 6.11.

For every $n\in\mathbb{N}$ and every $(x,y)\in\mathrm{Per}_{n}(\Phi\times L)$,
we have

$1\,\leqslant\,\\#\big{(}H^{-1}(x,y)\cap\,\,\mathrm{Per}_{n}(F_{S})\big{)}\,\leqslant\,3.$

In particular,

$\\#\mathrm{Per}_{n}(\Phi\times
L)\,\leqslant\\#\mathrm{Per}_{n}(F_{S})\,\leqslant\,3\,\,\\#\mathrm{Per}_{n}(\Phi\times
L)\qquad\forall\,n\in\mathbb{N}$

thus

$\lim_{n\to+\infty}\frac{1}{n}\,\log\\#\,\mathrm{Per}_{n}(F_{S})=h_{\mathrm{top}}(F_{S}).$

###### Proof.

From Proposition 6.6, given $x\in\mathrm{Per}_{n}(\Phi)$, either $g^{n}_{x}$
is Anosov or a Derived from Anosov. In the former case, the interval
$H^{-1}(x,y)$ is a point. In the latter, the interval $H^{-1}(x,\theta_{0})$
associated to the fixed point $(x,\theta_{0})$ has exactly three fixed points
by $g^{n}_{x}$. We also must to estimate the cardinality of
$H^{-1}(x,y)\cap\mathrm{Per}_{n}(F_{S})$ when $(x,y)$ is different from of the
fixed point $(x,\theta_{0})$. The last equality is due to Proposition 5.1.

######  Claim 6.12.

Let $(x,y)\in\mathrm{Per}_{n}(\Phi\times L)$ and suppose that $g^{n}_{x}$ is a
Derived from Anosov. If $y\neq\theta_{0}$, then $H^{-1}(x,y)$ is a point.

###### Proof.

Suppose, on the contrary, that $H^{-1}(x,y)$ is a non-degenerated interval.
Then

$F_{S}^{n}:\,H^{-1}(x,y)\quad\to\quad H^{-1}(x,y)$

is a Morse-Smale diffeomorphism of this interval (recall that
$F_{S}\in\mathfrak{R}$). Since $F_{S}$ is a preserving orientation map, the
boundary points of the interval $H^{-1}(x,y)$, say $(x,a_{1})$ and
$(x,a_{2})$, are necessarily fixed by $F_{S}^{n}$. This implies, using the
fact that $H^{-1}(x,\theta_{0})\cap H^{-1}(x,y)=\emptyset$, that

$\Big{\\{}(x,a_{1}),\,(x,a_{2})\Big{\\}}\subset\\{x\\}\times\Omega(g^{n}_{x})\setminus\\{(x,\theta_{0})\\}=\\{x\\}\times\Lambda^{n}_{x}$

and therefore $(x,a_{1})$ and $(x,a_{2})$ are two sinks of
$F_{S}^{n}|_{H^{-1}(x,y)}$. This forces the existence of a third point

$(x,a_{3})\,\,\in\,\,H^{-1}(x,y)\setminus\Big{\\{}(x,a_{1}),\,(x,a_{2})\Big{\\}}$

such that $F_{S}^{n}(x,a_{3})=(x,a_{3})$ and $(x,a_{3})$ is a source of
$F_{S}^{n}|_{H^{-1}(x,y)}$. But $(x,a_{3})$ also belongs to
$\\{x\\}\times\Omega(g^{n}_{x})\setminus\\{(x,\theta_{0})\\}=\\{x\\}\times\Lambda^{n}_{x}$,
so this conclusion contradicts Claim 6.10. ∎

Finally, we observe that, for every $n\in\mathbb{N}$,

$\mathrm{Per}_{n}(F_{S})\,=\,H^{-1}\big{(}\mathrm{Per}_{n}(\Phi\times
L)\big{)}\cap\mathrm{Per}_{n}(F_{S})\,=\,\bigcup_{(x,y)\,\in\,\mathrm{Per}_{n}(\Phi\times
L)}\,\,H^{-1}(x,y)\cap\mathrm{Per}_{n}(F_{S}).$ (6.3)

Thus,
$\,\,\\#\,\mathrm{Per}_{n}(F_{S})\,\leqslant\\#\,\mathrm{Per}_{n}(F_{S})\leqslant\,3\,\,\\#\mathrm{Per}_{n}(\Phi\times
L)$ for every $n\in\mathbb{N}$, as claimed. ∎

The proof of Theorem A is complete.

######  Remark 6.13.

Every $F_{S}$ belonging to the residual $\mathfrak{R}$ satisfies the
conditions:

  1. (1)

All the periodic points of $f$ are hyperbolic (in particular, the set
$\mathrm{Per}_{n}(f)$ is finite for every positive integer $n$).

  2. (2)

There exists $K>0$ such that
$\,\,\frac{1}{n}\,\log\\#\,\mathrm{Per}_{n}(f)\leqslant\log K\,\,$ for every
$n\in\mathbb{N}$.

So, the class of skew product we consider provides a local residual subset
where both properties hold. We note that according to [29] the set of $C^{r}$
diffeomorphisms for which the properties (1) and (2) are valid is dense in the
space of $C^{r}$ diffeomorphisms, $r\geqslant 1$.

###  6.4. Proof of Theorem A (b)

We now prove that the measure $\mu_{\mathrm{max}}$ is the weak∗ limit of the
sequence of probability measures on $\mathbb{T}^{2}\times\mathbb{T}^{2}$

$n\in\mathbb{N}\quad\mapsto\quad\mu_{n}\,\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\,\frac{1}{\\#\,\mathrm{Per}_{n}(F_{S})}\,\sum_{(x,y)\,\in\,\mathrm{Per}_{n}(F_{S})}\,\delta_{(x,y)}.$

To do it, we will show that $\mu_{\mathrm{max}}$ is the unique weak∗ limit
point of the sequence $(\mu_{n})_{n}$.

Consider the sequence of probabilities $(\nu_{n})_{n\,\in\,\mathbb{N}}$ on
$\mathbb{T}^{2}\times\mathbb{T}^{2}$ defined by

$n\in\mathbb{N}\quad\mapsto\quad\nu_{n}\,\,\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\,\,\frac{1}{\\#\,\mathrm{Per}_{n}(\Phi\times
L)}\,\sum_{(x,y)\,\in\,\mathrm{Per}_{n}(\Phi\times L)}\,\delta_{(x,y)}.$

We know that this sequence of measures converges in the weak∗ topology to the
measure of maximal entropy $\nu_{\mathrm{max}}$ of $\Phi\times L$.

######  Proposition 6.14.

The sequence $\big{(}\,H_{*}(\mu_{n})\,\big{)}_{n\,\in\,\mathbb{N}}$ converges
to $\nu_{\mathrm{max}}$ in the weak∗ topology.

To prove this assertion it is sufficient to show that the weak∗ limit of any
convergent sub-sequence of $\big{(}\,H_{*}(\nu_{n})\,\big{)}_{n}$ is equal to
$\nu_{\mathrm{max}}$. This will be a consequence of the following two
statements.

######  Lemma 6.15.

Let $f:X\to X$ be a continuous map defined on a compact metric space $(X,d)$.
Consider two sequences of $f$-invariant Borel probability measures
$(\eta_{k})_{k\,\in\,\mathbb{N}}$ and $(\zeta_{k})_{k\,\in\,\mathbb{N}}$ on
$X$ satisfying

$\exists\,C>1:\quad\quad
C^{-1}\,\zeta_{k}\,\,\leqslant\,\,\eta_{k}\,\,\leqslant\,\,C\,\zeta_{k}\qquad\forall\,k\in\mathbb{N}.$
(6.4)

Assume that $(\zeta_{k})_{k\,\in\,\mathbb{N}}$ and
$(\eta_{k})_{k\,\in\,\mathbb{N}}$ converge in the weak∗ topology to
probability measures $\zeta$ and $\eta$ respectively. Then
$C^{-1}\,\zeta\,\,\leqslant\,\,\eta\,\,\leqslant\,\,C\,\zeta$. In particular,
$\zeta$ and $\eta$ are equivalent.

######  Lemma 6.16.

If $\eta$ and $\zeta$ are $f$-invariant probability measures on $X$ such that
$\eta$ is ergodic and $\zeta$ is absolutely continuous with respect to $\eta$,
then $\zeta=\eta$.

Let us postpone for the moment the proofs of these lemmas to complete the
proof of Proposition 6.14.

###### Proof of Proposition 6.14.

Using the fact that for every $(x,y)\in\mathbb{T}^{2}\times\mathbb{T}^{2}$ we
have $H_{*}\delta_{(x,y)}=\delta_{H(x,y)}$, we deduce from (6.3) that the
$(\Phi\times L)$-invariant probability measure $H_{*}(\mu_{n})$ satisfies

$\begin{split}H_{*}(\mu_{n})&=\frac{1}{\\#\,\mathrm{Per}_{n}(F_{S})}\sum_{(x,y)\,\in\,\mathrm{Per}_{n}(\Phi\times
L)}\,\\#\,\big{(}H^{-1}(x,y)\cap\mathrm{Per}_{n}(F_{S})\big{)}\,\,\delta_{(x,y)}\\\
&=\left(\frac{\\#\,\mathrm{Per}_{n}(\Phi\times
L)}{\\#\,\mathrm{Per}_{n}(F_{S})}\right)\,\,\frac{1}{\\#\,\mathrm{Per}_{n}(\Phi\times
L)}\,\sum_{(x,y)\,\in\,\mathrm{Per}_{n}(\Phi\times
L)}\,\\#\,\big{(}H^{-1}(x,y)\cap\mathrm{Per}_{n}(F_{S})\big{)}\,\,\delta_{(x,y)}.\end{split}$

Besides, after Corollary 6.11 we know that

$\forall\,n\in\mathbb{N},\quad\quad
1\leqslant\,\\#\,\big{(}H^{-1}(x,y)\cap\mathrm{Per}_{n}(F_{S})\big{)}\,\leqslant
3\quad\quad\mbox{and}\quad\quad\frac{1}{3}\leqslant\frac{\\#\,\mathrm{Per}_{n}(\Phi\times
L)}{\\#\,\mathrm{Per}_{n}(F_{S})}\leqslant 1.$

Thus,

$\forall\,n\in\mathbb{N},\quad\forall\,\text{ Borel set
$A\subset\mathbb{T}^{2}\times\mathbb{T}^{2}$}\quad\Rightarrow\quad\frac{1}{3}\,\nu_{n}(A)\leqslant
H_{*}(\mu_{n})(A)\leqslant 3\,\nu_{n}(A).$

Let $\eta_{k}:=H_{*}(\mu_{n_{k}})$ be a subsequence converging to a
probability measure $\nu_{0}$ in the weak∗ topology. Since
$\zeta_{k}:=\nu_{n_{k}}$ converges to $\nu_{\mathrm{max}}$, it follows from
Lemma 6.15 that $\nu_{0}$ and $\nu_{\mathrm{max}}$ are equivalent measures. On
the other hand, as $\nu_{\mathrm{max}}$ is ergodic, Lemma 6.16 implies that
$\nu_{0}=\nu_{\mathrm{max}}$. ∎

We now return to the proof of the two pending lemmas.

###### Proof of Lemma 6.15.

By symmetry of the inequality (6.4) it is enough to check that for every open
set $U$ of $\mathbb{T}^{2}\times\mathbb{T}^{2}$ we have $\eta(U)\,\leqslant
C\,\zeta(U).$ Indeed, due the regularity of the measures $\zeta$ and $\eta$,
from the previous inequality we get, for every Borel set $A$ in
$\mathbb{T}^{2}\times\mathbb{T}^{2}$,

$\eta(A)=\inf\,\\{\eta(G):\,G\,\,\mbox{is open and}\,\,A\subset
G\\}\,\,\leqslant\,\,C\,\inf\,\\{\zeta(G):\,G\,\,\mbox{is open
and}\,\,A\subset G\\}=C\,\zeta(A).$

So, $\zeta(A)=0$ implies $\eta(A)=0$.

Now, consider the sequence of closed sets in
$\mathbb{T}^{2}\times\mathbb{T}^{2}$ defined by

$k\,\in\,\mathbb{N}\quad\mapsto\quad F_{k}=\Big{\\{}x\in X:\,d(x,X\setminus
U)\geqslant\frac{1}{k}\Big{\\}}.$

From Uryshon’s Lemma there exists a continuous function $g_{k}:X\to[0,1]$ such
that

$\mathbbm{1}_{F_{k}}\leqslant
g_{k}\leqslant\mathbbm{1}_{U}\,,\qquad\forall\,k\in\mathbb{N}.$

We may assume that $g_{k}$ converges to $\mathbbm{1}_{U}$ in a monotonic and
increasing way. Thus,

$\displaystyle\eta(U)$ $\displaystyle=$ $\displaystyle\sup_{k}\int
g_{k}\,d\eta\qquad\quad\text{(by the Monotone Convergence Theorem)}$
$\displaystyle=$ $\displaystyle\sup_{k}\,\lim_{n}\int
g_{k}\,d\eta_{n}\qquad\quad\text{(by the weak${}^{*}$ convergence of
$(\eta_{n})_{n\,\in\,\mathbb{N}}$)}$ $\displaystyle\leqslant$ $\displaystyle
C\,\sup_{k}\,\lim_{n}\int g_{k}\,d\zeta_{n}\qquad\quad\text{(by
equation~{}\eqref{e.noite})}$ $\displaystyle=$ $\displaystyle C\,\sup_{k}\int
g_{k}\,d\zeta\qquad\quad\text{(by the weak${}^{*}$ convergence of
$(\zeta_{n})_{n\,\in\,\mathbb{N}}$)}$ $\displaystyle=$ $\displaystyle
C\,\zeta(U)\qquad\quad\text{(by the Monotone Convergence Theorem)}.$

∎

###### Proof of Lemma 6.16.

Consider a Borel set $A\subset X$. By Birkhoff’s Ergodic Theorem we have

$\phi_{A}(x)\,:=\,\lim_{n\,\to\,+\infty}\,\frac{1}{n}\,\Big{\\{}0\leqslant
j\leqslant n-1:\,f^{j}(x)\in A\Big{\\}}=\mu(A)$

for $\mu$-almost every $x\in X$, and $\nu(A)=\int\phi_{A}(x)\,d\nu(x)$. Since
$\nu\ll\mu$, we also get $\phi_{A}(x)=\mu(A)$ for $\nu$-almost every $x$. So,
$\int\phi_{A}(x)\,d\nu(x)=\mu(A)$. Hence $\nu(A)=\mu(A)$. ∎

######  Corollary 6.17.

The sequence $(\mu_{n})_{n\,\in\,\mathbb{N}}$ converges to
$\mu_{\mathrm{max}}$ in the weak∗ topology.

###### Proof.

We will show that $\mu_{\mathrm{max}}$ is the unique weak∗ accumulation point
of the sequence $(\mu_{n})_{n\in\mathbb{N}}$. Suppose that the subsequence
$(\mu_{n_{k}})_{k}$ converges to a probability measure $\mu_{0}$. We will
verify that $h_{\mu_{0}}(F_{S})=h_{\mathrm{top}}(F_{S})$, and so, by the
uniqueness of the measure of maximal entropy of $F_{S}$, we deduce that
$\mu_{0}=\mu_{\mathrm{max}}$.

Using Proposition 5.1 (b) and Ledrappier-Walters’ formula, if follows that

$h_{\eta}(F_{S})=h_{H_{*}(\eta)}(\Phi\times
L),\quad\quad\forall\,\eta\,\in\,\mathscr{P}(\mathbb{T}^{2}\times\mathbb{T}^{2},\,F_{S}).$
(6.5)

Now, from Proposition 6.14 and the continuity of $\eta\to H_{*}(\eta)$, it
follows that $H_{*}(\mu)=\nu_{\mathrm{max}}$. Then, by (6.5) and Proposition
5.1 (a), we obtain

$h_{\mu}(F_{S})=h_{H_{*}(\mu)}(\Phi\times L)=h_{\nu_{\mathrm{max}}}(\Phi\times
L)=h_{\mathrm{top}}(\Phi\times L)=h_{\mathrm{top}}(F_{S}).$

∎

##  7\. Proof of Theorem B

We start observing that, as the periodic points of $F_{S}$ are hyperbolic, the
set of periodic points of $F_{S}$ is countable, and so zero dimensional.
Besides, $F_{S}$ has the small boundary property (cf. [12, Subsection 2.1]; it
was proved in [32] that on a finite dimensional manifold any dynamical system
whose set of periodic points is countable have this property). Moreover, as
already mentioned, the central direction of $F_{S}$ is one-dimensional, and so
$F_{S}$ is entropy-expansive. After summoning Remark 6.13 and Proposition 5.1
(a), to show the existence of a principal strongly faithful symbolic extension
with embedding for $F_{S}$ we are left to control of the growth rate of the
periodic points with the period at arbitrarily small scales.

######  Lemma 7.1.

If $F_{S}$ belongs to the residual $\mathfrak{R}$, then $F_{S}$ is
asymptotically per-expansive.

###### Proof.

Given $\varepsilon>0$ and
$(x_{0},y_{0})\in\mathbb{T}^{2}\times\mathbb{T}^{2}$, define

$B^{F_{S}}_{\infty,\,\varepsilon}(x_{0},y_{0}):=\Big{\\{}\,\,(x,y)\in\mathbb{T}^{2}\times\mathbb{T}^{2}\,:\,\,d(F_{S}^{i}(x,y),F_{S}^{i}(x_{0},y_{0}))\leqslant\varepsilon,\quad\forall\,i\in\mathbb{Z}\,\,\Big{\\}}.$

We claim that

$\forall\,\varepsilon>0\quad\forall\,n\in\mathbb{N}\quad\forall\,(x_{0},y_{0})\in\mathbb{T}^{2}\times\mathbb{T}^{2}\quad\quad\\#\,\left(\mathrm{Per}_{n}(F_{S})\cap
B^{F_{S}}_{\infty,\,\varepsilon}(x_{0},y_{0})\right)\leqslant 3.$

Firstly note that the central foliation of $F_{S}$ is plaque expansive (cf.
[27, Sections 7 & 8]): there exists $\varepsilon_{0}>0$ such that, for every
$0<\varepsilon<\varepsilon_{0}$, if $(x,y)$ belongs to
$B^{F_{S}}_{\infty,\,\varepsilon}(x_{0},y_{0})$, then both points
$(x_{0},y_{0})$ and $(x,y)$ lie on the same leaf of the central foliation,
which is sent by the semi-conjugation $H$ into a stable leaf. On the other
hand, if $\mathrm{Per}_{n}(F_{S})\cap
B^{F_{S}}_{\infty,\,\varepsilon}(x_{0},y_{0})\neq\emptyset$ then $x_{0}$ is
periodic and so, by Proposition 6.6, $g^{n}_{x_{0}}$ is an Anosov or a Derived
from Anosov. In the former case,
$B^{F_{S}}_{\infty,\,\varepsilon}(x_{0},y_{0})\subset
B^{F^{n}_{S}}_{\infty,\,\varepsilon}(x_{0},y_{0})=\\{(x_{0},y_{0})\\}$. In the
latter case, the intersection cannot have more than three periodic points:
otherwise, if we assume the existence of at least four elements in
$\mathrm{Per}_{n}(F_{S})$ in $B^{F_{S}}_{\infty,\,\varepsilon}(x_{0},y_{0})$,
then we may find two hyperbolic point $(x_{0},y_{1})$ and $(x_{0},y_{2})$ in
$\mathrm{Per}_{n}(F_{S})\cap B^{F_{S}}_{\infty,\,\varepsilon}(x_{0},y_{0})$
such that $H(x_{0},y_{1})\neq H(x_{0},y_{2})$ are in
$\mathrm{Per}_{n}(\Phi\times L)$ and belong to the same stable leaf of
$\Phi\times L$. This contradicts the known dynamics within stable leaves. ∎

To end the proof of Theorem B we just make a straightforward application of
the Main Theorem of [12].

##  8\. Proof of Theorem C

Suppose that $\Phi$ is a linear hyperbolic automorphism of $\mathbb{T}^{2}$
and let $\nu_{\mathrm{SRB}}$ be the SRB measure of $\Phi\times L$. Denote by
$\mu_{\mathrm{max}}$ and $\nu_{\mathrm{max}}$ the measures of maximal entropy
of $F_{S}$ and $\Phi\times L$, respectively. Similarly, let
$\mu_{\mathrm{SRB}}$ and $\nu_{\mathrm{SRB}}$ the SRB measures of $F_{S}$ and
$\Phi\times L$.

Consider the expanding eigenvalues $\beta_{1}>1$ and $\beta_{2}>1$ of $\Phi$
and $L$, respectively. By Pesin’s formula, the topological entropy of
$\Phi\times L$ is given by

$h_{\mathrm{top}}(\Phi\times L)=\log\,\beta_{1}+\log\,\beta_{2}.$

Note also that, on the corresponding regular sets, the positive Lyapunov
exponents $\lambda^{uu}>\lambda^{u}>0$ of $\mu_{\mathrm{max}}$ and
$\nu_{\mathrm{max}}$ are given by

$\lambda^{uu}\big{(}\mu_{\mathrm{max}}\big{)}=\lambda^{uu}\big{(}\nu_{\mathrm{max}}\big{)}=\log\,\beta_{1}\,\,\quad\text{and}\,\,\quad\lambda^{u}\big{(}\mu_{\mathrm{max}}\big{)}=\lambda^{u}\big{(}\nu_{\mathrm{max}}\big{)}=\log\,\beta_{2}.$

Moreover, under the assumption that both $\Phi$ and $L$ are linear
automorphisms of $\mathbb{T}^{2}$, the measure $\nu_{\mathrm{max}}$ coincides
with the Lebesgue measure in $\mathbb{T}^{2}\times\mathbb{T}^{2}$ (cf. [44,
Theorem 8.15]). Besides, $\nu_{\mathrm{max}}=\nu_{\mathrm{SRB}}$ since the
mapping $(x,y)\,\mapsto\,J^{u}_{\Phi\times L}(x,y)$ is constant and equal to
$\beta_{1}\,\beta_{2}$, and so

$h_{\nu_{\mathrm{SRB}}}(\Phi\times L)=\int\,\log\,J^{u}_{\Phi\times
L}\,\,d\nu_{\mathrm{SRB}}=\log\,\beta_{1}+\log\,\beta_{2}=h_{\mathrm{top}}(\Phi\times
L).$

To prove Theorem C (a), we note that, by construction, for every
$(x,y)\in\mathbb{T}^{2}\times\mathbb{T}^{2}$ we have

$J^{u}_{F_{S}}(x,y)\geqslant\beta_{1}\,\beta_{2}.$

So, $J^{u}_{F_{S}}$ and $J^{u}_{\Phi\times L}$ satisfy the assumption (5.3) of
Proposition 5.4. Therefore, one has

$H_{*}(\mu_{\mathrm{SRB}})=\nu_{\mathrm{SRB}}.$

To prove Theorem C (b), we use Proposition 5.1 (a) to deduce that

$h_{\mu_{\mathrm{SRB}}}(\Phi\times
L)=\int\,\log\,J^{u}_{F_{S}}\,\,d\mu_{\mathrm{SRB}}\geqslant\log\,\beta_{1}+\log\,\beta_{2}=h_{\mathrm{top}}(\Phi\times
L)=h_{\mathrm{top}}(F_{S})$

and thereby conclude that $h_{\mu_{\mathrm{SRB}}}(\Phi\times
L)=h_{\mathrm{top}}(F_{S})$, as claimed. The fact that $\mu_{\mathrm{SRB}}$ is
also the unique physical measure of $F_{S}$ has already been proved in
Proposition 5.2.

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