# Horseshoe and Periodic orbits for quasi-periodic forced systems

Wen Huang  Department of Mathematics
University of Science and Technology of China
Hefei, Anhui, China<EMAIL_ADDRESS>and  Zeng Lian  School of Mathematical
Sciences
Sichuan University
Chengdu, Sichuan, 610016, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we investigate chaotic phenomena on quasi-periodic forced
systems. We introduce the notion of random periodic orbits and random
horseshoe for a quasi-periodic forced system $(M\times\Omega,\phi)$ by viewing
$\phi$ as a random dynamical system over the metric dynamical system
$(\Omega,\theta)$. When $\phi$ is transitive and Anosov on fibers, we prove
that $\phi$ has an abundance of random periodic points, Spectral Decomposition
Theorem and existence of random horseshoes.

Huang is partially supported by NNSF of China (11225105, 11431012, 11571335).
Lian is partially supported by NNSF of China (11671279, 11541003).

##  1\. Introduction

Let $f:M\rightarrow M$ be a diffeomorphism of a compact $C^{\infty}$
Riemannian manifold $M$. The diffeomorphism $f$ is said to be Anosov if $M$ is
hyperbolic [3]. The definition of the more general class of Axiom A
diffeomorphisms is due to S. Smale [36]. A point $x\in M$ is nonwandering if
$U\cap\bigcup_{n\geq 1}f^{-n}U\neq\emptyset$ for every neighborhood $U$ of
$x$. The set $\Omega(f)$ of all nonwandering points is closed and
$f$-invariant. The diffeomorphism $f$ satisfies Axiom A if $\Omega(f)$ is
hyperbolic and is also equal to the closure of the set of all periodic points
of $f$.

It has been well known since the works of Berg, Sinai, Adler-Weiss and Bowen
around 1970 that uniformly hyperbolic diffeomorphisms (i.e., Anosov
diffemorphism or Axiom A diffeomorphisms) of a compact manifold has Markov
partition or are ”almost isomorphic” to subshifts of finite type [5, 33, 34,
1, 2, 7, 8, 11, 12]. Markov partitions are also implicit in the invariant
Cantor sets of the diffeomorphisms of the sphere constructed by Smale [36],
the simplest one of which is the famous horseshoe map (see section 2.5, also
[36] and [31, page 23]). In [36], Smale also proved the spectral decomposition
theorem: If the diffeomorphism $f$ satisfies Axiom A, then there is a unique
way of writing $\Omega(f)$ as the finite union of disjoint, closed, invariant,
indecomposable subsets

$\Omega(f)=\Omega_{1}\cup\Omega_{2}\cup\cdots\cup\Omega_{k}$

in such a way that on each $\Omega_{i}$, $f$ is transitive, i.e., has a dense
orbit. Further, Bowen showed that

$\Omega_{i}=X_{1,i}\cup\cdots\cup X_{n_{i},i},$

with the $X_{j,i}$ s pairwise disjoint closed sets,
$f(X_{j,i})=X_{j+1,i}\,(X_{n_{i}+1,i}=X_{1,i})$ and $f^{n_{i}}|_{X_{j,i}}$
topologically mixing [9]. It is also shown by Bowen in [9] that if the
diffeomorphism $f$ satisfies Axiom A, then the topological entropy
$h(f)=\lim_{n\rightarrow+\infty}\frac{1}{n}\log N_{n}(f)$, where $N_{n}(f)$ is
the number of fixed points of $f_{n}$, i.e., periodic points of period $n$.

In the ergodic theory of uniformly hyperbolic diffeomorphisms developed by
Sinai [35], Ruelle [29] and Bowen [12, 10], it was shown that for a transitive
Axiom A attractor there is a unique invariant measure which can be
characterized by the metric entropy being equal to the sum of positive Lypunov
exponents. Such invariant measure is called SRB measure.

On nonuniform hyperbolic theory, in his remarkable paper [20] A. Katok proved,
roughly speaking, the following: Let $f$ be a $C^{2}$ diffeomorphism of a
compact Riemannian manifold, and let $\mu$ be an $f$-invariant ergodic Borel
probability measure. Assume that $(f,\mu)$ has nonzero Lyapunov exponents and
positive metric entropy. Then horseshoes are present, and that implies, among
other things, existence of an abundance of periodic points. More preciesly,
the diffeomorphism $f$ has a closed invariant set $\Gamma$ such that
$f|_{\Gamma}$ is topologically conjugate to a topological Markov shift and the
topological entropy $h(f|_{\Gamma})>0$, and also
$\limsup_{n\rightarrow+\infty}\frac{1}{n}\log N_{n}(f)\geq h_{\mu}(f)$, where
$h_{\mu}(f)$ is the Metric entropy of $(M,f,\mu)$ introduced by Kolmogorov and
Sinai [22, 32].

In [24], Lian and Young extended Katok’s results to $C^{2}$ differentiable
maps with a hyperbolic invariant measure supported by a compact invariant set
in a separable Hilbert space. In their second paper [25], Lian and Young went
much further and studied a $C^{2}$ semiflow in a Hilbert space and proved that
if it has a hyperbolic invariant measure supported by a compact invariant set,
then the positive entropy implies the existence of horseshoes. In this case,
the semiflow has one simple zero Lyapunov exponent, which implies that the
associated time-one map restricted to this invariant set is partially
hyperbolic with one-dimensional center direction. This result is new even for
flows generated by ordinary differential equations.

The proofs of the results obtained by Katok and Lian-Young rely on not only
the positive of entropy but also the hyperbolic geometric structures of
systems. The horseshoes are constructed by using stable and unstable
manifolds. Recently, Huang and Lu studied the complicated dynamics of infinite
dimensional random dynamical systems which include deterministic dynamical
systems as their special cases in a Polish space. In [19], without assuming
any hyperbolicity, they proved if a continuous random map has a positive
topological entropy, then it contains a weak horseshoe (see also [18] for
topological dynamical systems in compact metric spaces and [21] for
$C^{*}$-dynamics).

In this paper, we intend to understand chaotic phenomena on quasi-periodic
forced systems. A quasi-periodic forced system is the skew product of $f$ and
$\theta$ induces a map $\phi:M\times\mathbb{T}^{d}\to M\times\mathbb{T}^{d}$
such that for any $x\in M,\omega\in\mathbb{T}^{d}$

$\phi(x,\omega)=(f(\omega)x,\theta\omega),$

where $M$ is compact $C^{\infty}$ Riemannian manifold, $\theta$ is a minimal
rotation on the torus $\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$ and
$f:\mathbb{T}^{d}\rightarrow\text{diff}^{2}(M)$ is a continuous map.

A quasi-periodic forced systems $(M\times\mathbb{T}^{d},\phi)$ can be
considered as a specific kind of partially hyperbolic system, whose each
invariant Borel probability measure $\mu$ has some zero Lyapunov exponents.
Hence the Katok’s result ([20]) doesn’t apply to the quasi-periodic forced
systems $(M\times\mathbb{T}^{d},\phi)$; and Lian and Young’s results ([24],
[25]) for semiflows or flows, which allow only one zero Lyapunov exponent
associated to the flow direction, are not applicable to this case. In fact,
the quasi-periodic forced systems $(M\times\mathbb{T}^{d},\phi)$ has no
periodic points, and so has no standard horseshoe.

To describe the periodic structure and the chaotic behavior accurately for
quasi-periodic systems, we introduce the concepts of random periodic orbit and
random horseshoe for $\phi$ in Section 2 by viewing $\phi$ as a random
dynamical system over the metric dynamical system $(\mathbb{T}^{d},\theta)$.
Random periodic orbit can be viewed as a generalization of the concept of
stationary orbit. As of our awareness, the concept of random periodic
solutions was first introduces by Zhao and Zheng [37] for systems generated by
random differential equations; while in this paper, random periodic orbit
(Definition 2.1) can be defined equivalently, which can be viewed as graph of
functions from the space $L^{\infty}(\mathbb{T}^{d},M)$ endowed with the
$L^{\infty}$-norm. Similar to case of autonomous systems (Anosov
diffeomorphisms, Axiom A maps, or non-uniform hyperbolic systems), we first
show that the following Theorem holds:

Theorem A(Theorem 2.3): If $\phi$ is transitive and Anosov on fibers,
$C(\mathbb{T}^{d},M)$, the continuous maps from $\mathbb{T}^{d}$ to $M$,
belongs to the closure of functions corresponding to random periodic points in
$L^{\infty}(\mathbb{T}^{d},M)$.

The main difficulties appear in non-autonomous systems is that the pseudo
orbit derived from Poincaré Recurrence Theorem only provides returns in
$M\times\mathbb{T}^{d}$, thus does not have shadowing property since
hyperbolicity only happens on fibers which is identified by $M$. To overcome
this difficulty, instead of looking at one single pseudo orbit, we construct a
group of globally returned pseudo orbits for all $\omega\in\mathbb{T}^{d}$ in
the same time. Behind this, equi-continuity of the driven system (rotation
$\theta:\mathbb{T}^{d}\to\mathbb{T}^{d}$) and uniform hyperbolicity on fibers
play the key roles. Theorem A provides both a fundamental description of the
dynamics of the system $(M\times\mathbb{T}^{d},\phi)$ and a pathway to achieve
the main aim of this paper:

Theorem B( part of Theorem 2.5): If $\phi$ is transitive and Anosov on fibers,
then there exists a full random horseshoe.

Here, roughly speaking, ”full random horseshoe” (Definition 2.2) can be viewed
as an embedding of a two-sides symbolic dynamical system into
$L^{\infty}(\Omega,M)$, which is a standard defined horseshoe in
$L^{\infty}(\Omega,M)$. The proof of Theorem B mainly contains three steps,
which still inherits the ideas from Katok [20]:

  * Step

I) Existence of invariant measures with positive entropy;

  * Step

II) Capturing entropy;

  * Step

III) Constructing Horseshoe by following up proper pseudo orbits.

In Section 4.3.1, we first apply Kifer and Gundlach’s results [17] to derive
the existence and uniqueness of the SRB measure, and then use Bogenschütz and
Gundlach’s results [13] further to show that such a SRB measure has full
support. It is worth to mention that, in order to apply [17], one need the
mixing property of the system, which motivates the proof of Spectral
Decomposition Theorem (Theorem 2.4) in Section 4.2; and the proof of Theorem
2.4 is essentially depending on Theorem 2.3, which is similar as in the case
of Axiom A systems.

In Section 4.3.2, we first prove a result on capturing entropy (Lemma 4.7) by
modifying an abstract result from [25] (Lemma 17), which together with the
Shadowing Lemma play the key roles in constructing the random horseshoe.
Section 4.3.3 shows that random horseshoe induces a standard horseshoe in the
space of Borel probability measures on $M\times\mathbb{T}^{d}$. Section 4.3.4
contains the construction of a weak horseshoe with bounded hitting times,
whose definition is given by 2.8.

Section 5 gives a type of quasi-periodic systems which is Anosov and
transitive, thus all the main theorems derived in this paper are applicable
there. Another interesting observation based on this example is that the
function corresponding to random periodic points are NOT necessarily
continuous. Actually, we have shown that under certain conditions, such
systems have no periodic point which is graph of continuous functions from
sample space $\mathbb{T}^{d}$ ( for the example $d=1$) to phase space (for the
example $M=\mathbb{T}^{2}$). This tells that, in general, one can not expects
that random periodic points persist smoothness, even the system is linear or
smooth. Nevertheless, when the system is provided by small perturbations of
uniformly hyperbolic system, continuity of random periodic points could be
guaranteed. For example, in [26], Liu studied the structural stability of
Axiom A attractors under i.i.d. small random perturbations. The concepts of
full random periodic orbits and full random horseshoe can be adapted into
Liu’s setting, whose existence is due to Theorem 1.1 of [26]. Moreover, under
Liu’s setting, one can also obtain that the graphs corresponding to random
periodic orbits and points of random horseshoes are all continuous.

Finally, we mention some existing results which are closely relevant to the
results derived in this paper, which may shed a light on further research
directions, for example, chaotic behavior of Anosov flows driven by quasi-
periodic noises. Of course, this is far from being a complete picture. In
[16], Gundlach studied the random homoclinic points for smooth invertible
random dynamical systems, and derived a random version of Birkhoff-Smale
Theorem. This result is applicable to systems which are provided by small
random perturbations of systems with homoclinic points. In the recent papers
[27] and [28], Lu and Wang studied the chaos behavior of a type of
differential equations driven by a nonautononous force deduced from stochastic
processes which are the truncated and classical Brownian motions in [27] and
in [28] respectively. The authors extended the concept of topological
horseshoe and showed that such structure (thus chaotic behavior) exists almost
surely. Moreover, they further applied the results to randomly forced Duffing
equations and pendulum equations. One interesting observation is that unlike
the systems driven by a periodic or almost periodic forcing, the transversal
intersections of the stable and unstable manifolds are neither necessary nor
sufficient for chaotic dynamics to exist.

##  2\. Settings and Main Results

As discussed in the Introduction, we have in mind the following three
settings: quasi-periodic forced systems, transitivity and Anosov on fibers.

Setting. Let $M$ be a compact $C^{\infty}$ Riemannian manifold with a metric
$d_{M}$ and $(\Omega,d_{\Omega})$ be a compact metric space. Let
$\theta:\Omega\to\Omega$ be a homeomorphism. We say $\theta$ is equicontinuous
if there exist $c>1$ and $\delta>0$ such that for any
$\omega_{1},\omega_{2}\in\Omega$ with
$d_{\Omega}(\omega_{1},\omega_{2})<\delta$,

$\frac{1}{c}d_{\Omega}(\omega_{1},\omega_{2})\leq
d_{\Omega}(\theta^{n}\omega_{1},\theta^{n}\omega_{2})\leq
cd_{\Omega}(\omega_{1},\omega_{2}),\forall n\in\mathbb{Z}.$

When $\theta$ is equicontinuous, if we take

$\widetilde{d}_{\Omega}(\omega_{1},\omega_{2})=\sup_{n\in\mathbb{Z}}d_{\Omega}(\theta^{n}\omega_{1},\theta^{n}\omega_{2}),\,\forall\omega_{1},\omega_{2}\in\Omega,$

then $\widetilde{d}_{\Omega}$ is a metric on $\Omega$ satisfying that

  * •

$\widetilde{d}_{\Omega}$ and $d_{\Omega}$ are equivalent;

  * •

$\widetilde{d}_{\Omega}(\theta^{n}\omega_{1},\theta^{n}\omega_{2})=\widetilde{d}_{\Omega}(\omega_{1},\omega_{2})$
for any $\omega_{1},\omega_{2}\in\Omega$ and $n\in\mathbb{Z}$.

Therefore, without losing any generality, we assume that

(1)
$d_{\Omega}(\theta^{n}\omega_{1},\theta^{n}\omega_{2})=d_{\Omega}(\omega_{1},\omega_{2}),\
\forall\omega_{1},\omega_{2}\in\Omega\text{ and }n\in\mathbb{Z}.$

Let $\mathcal{H}=\text{diff}^{2}(M)$ be the space of $C^{2}$ diffeomorphisms
on $M$ with $C^{2}$-norm, and $f:\Omega\to\mathcal{H}$ be a continuous map.
Then the skew product of $f$ and $\theta$ induces a map $\phi:M\times\Omega\to
M\times\Omega$ such that for any $x\in M,\omega\in\Omega$

$\phi(x,\omega)=(f(\omega)x,\theta\omega)=(f_{\omega}x,\theta\omega),$

where we rewrite $f(\omega)$ as $f_{\omega}$ for sake of convenience. Then
$\phi$ induces a cocycle over $(\Omega,\theta)$, which can be inductively
define by

$\begin{cases}\phi^{0}(x,\omega)&=id\\\
\phi^{n}(x,\omega)&=(f_{\theta^{n-1}\omega}\phi^{n-1}(x,\omega),\theta^{n}\omega),\
n\geq 1\\\
\phi^{n}(x,\omega)&=(f^{-1}_{\theta^{n+1}\omega}\phi^{n+1}(x,\omega),\theta^{n}\omega),\
n\leq-1\end{cases}.$

Note that $M\times\Omega$ is a compact metric space when we introduce a metric
$d$ by defining that

$d((x_{1},\omega_{2}),(x_{2},\omega_{2}))=d_{M}(x_{1},x_{2})+d_{\Omega}(\omega_{1},\omega_{2}),\forall(x_{1},\omega_{1}),(x_{2},\omega_{2})\in
M\times\Omega.$

It is obvious that $\phi$ and $D_{M}\phi$ are continuous on $M\times\Omega$.

######  Remark 2.1.

In this paper, we will consider $(\Omega,\theta)$ to be an irrational rotation
on torus (with infinite or finite dimensions). Then $\Omega$ is path connected
and $(\Omega,\theta^{n})$ is uniquely ergodic for all $n\in\mathbb{N}$. Also
note that Theorem 2.3 still holds without assuming that $(\Omega,\theta)$ is
an irrational rotation on torus.

The conditions transitivity and Anosov on fibers are defined as following:
First $\phi$ is said to be transitive if there is a dense orbit (in
$M\times\Omega$). Secondly $\phi$ is called Anosov on fibers if the following
holds: for every $(x,\omega)\in M\times\Omega$ there is a splitting of the
tangential fiber of $M_{\omega}:=M\times\\{\omega\\}$ at $x$

$T_{x}M_{\omega}=E_{(x,\omega)}^{u}\oplus E_{(x,\omega)}^{s}$

which depends continuously on $(x,\omega)\in M\times\Omega$ with $\dim
E_{(x,\omega)}^{u}>0$ and satisfies that for every $(x,\omega)\in
M\times\Omega$

$Df_{\omega}(x)E_{(x,\omega)}^{u}=E_{\phi(x,\omega)}^{u},\ \ \
Df_{\omega}(x)E_{(x,\omega)}^{s}=E_{\phi(x,\omega)}^{s},$

where $Df_{\omega}(x)$ is the derivative of $f_{\omega}$ at point $x$, and

(2) $\left\\{\begin{array}[]{ll}|Df_{\omega}(x)\xi|\geq
e^{\lambda_{0}}|\xi|,&\forall\ \xi\in E_{(x,\omega)}^{u},\\\
|Df_{\omega}(x)\eta|\leq e^{-\lambda_{0}}|\eta|,&\forall\ \eta\in
E_{(x,\omega)}^{s},\end{array}\right.$

where $\lambda_{0}>0$ is a constant. Some quasi-periodic forced examples with
transitivity and Anosov on fibers are given in Section 5.

Results. Under conditions transitivity and Anosov on fibers, it will be shown
that a quasi-periodic forced system $\phi$ has an abundance of Random periodic
points (Theorems 2.3), Spectral Decomposition Theorem (Theorem 2.4) and Random
horseshoes (Theorem 2.5).

By viewing $\phi$ as a random dynamical system over the metric dynamical
system $(\Omega,\theta)$, we now define random periodic orbits and random
horseshoe for $\phi$.

######  Definition 2.1.

If there exist $n$ Borel measurable maps $\\{g_{i}:\Omega\to M\\}_{1\leq i\leq
n}$ such that for all $1\leq i\leq n$

$\phi(graph(g_{i}))=graph(g_{i+1\mod n})\ \forall 1\leq i\leq n,$

then we call $\\{graph(g_{i})\\}_{1\leq i\leq n}$ a full random periodic orbit
of $\phi$ with period $n$, and each $graph(g_{i})$ a full random periodic
point. Moreover if in additional, each $g_{i}$ is continuous map then we call
each $graph(g_{i})$ a periodic continuous graph.

Let $\mathcal{S}_{k}=\\{1,\ldots,k\\}^{\mathbb{Z}}$ be the space of two sides
sequences of $k$ symbols, and $\sigma$ be the (left-)shift map. Denote
$L^{\infty}(\Omega)$ the space of Borel measurable maps from $\Omega$ to $M$
endowed with the following metric

$d_{L^{\infty}(\Omega)}(g_{1},g_{2})=\sup_{\omega\in\Omega}d_{M}(g_{1}(\omega),g_{2}(\omega)).$

######  Definition 2.2.

If there exists a continuous embedding $\Psi:\mathcal{S}_{k}\to
L^{\infty}(\Omega)$ such that the following hold

  1. i)

There exists $L,C>1$ such that for all
$\hat{a}_{1},\hat{a}_{2}\in\mathcal{S}_{k}\text{ with
}\hat{a}_{1}\neq\hat{a}_{2}$

$\inf_{\omega\in\Omega}\\{d_{M}(\Psi({\hat{a}_{1}})(\omega),\Psi({\hat{a}_{2}})(\omega))\\}>C^{-1}L^{-l},$

where $l=\inf\left\\{|n|\big{|}\ \hat{a}_{1}(n)\neq\hat{a}_{2}(n)\right\\}$;

  2. ii)

For all $\hat{a}\in\mathcal{S}_{k}$

$\phi(graph(\Psi({\hat{a}})))=graph(\Psi({\sigma\hat{a}}));$

then we call $\\{graph(\Psi({\hat{a}}))\\}_{\hat{a}\in\mathcal{S}_{k}}$ a
$k$-symbol full random horseshoe of $\phi$.

Denote $C(\Omega,M)$ the space of continuous maps from $\Omega$ to $M$ endowed
with the sup norm. Now we state the main results of this paper:

######  Theorem 2.3.

Assume that $\phi$ is transitive and Anosov on fibers, and $\theta$ is
equicontinuous. Then for any $g\in C(\Omega,M)$ and $\epsilon>0$, there exists
a full random periodic point $\\{graph(\tilde{g})\\}$ such that

(3) $\sup_{\omega\in\Omega}\\{\
d_{M}(g(\omega),\tilde{g}(\omega))\\}\leq\epsilon,$

We remark that there exist quasi-periodic forced systems which is transitivity
and Anosov on fibers but has no periodic continuous graph (see Lemma 5.2 in
Section 5). Note that since $M$ is a compact manifold, $M$ has finitely many
disjoint connected components, for which we write $\\{M_{1},\cdots,M_{k}\\}$.
These sets are both open and closed, and any two of them depart for a positive
distance. We say $\phi:\Omega\times M\to\Omega\times M$ topological mixing on
fibers if for any nonempty open sets $U,V\subset M$, there exists $N>0$ such
that for any $n\geq N$ and $\omega\in\Omega$

$\phi^{n}(\\{\omega\\}\times U)\bigcap\\{\theta^{n}\omega\\}\times
V\neq\emptyset.$

The following theorem is called Spectral Decomposition Theorem.

######  Theorem 2.4.

Assume that $\phi$ is transitive and Anosov on fibers, and $(\Omega,\theta)$
is an irrational rotation on torus. Then there exists a unique permutation, of
$\\{1,\cdots,k\\}$, $\tau$, such that

$\phi(M_{\tau(i)}\times\Omega)=M_{\tau(i+1\mod k)}\times\Omega\text{ when
}i\in\\{1,\cdots,k\\}\text{ and }\tau(1)=1.$

Moreover, for any $i\in\\{1,\cdots,k\\}$, $\phi^{k}|_{M_{i}\times\Omega}$ is
topological mixing on fibers.

Under the assumption of Theorem 2.4, as $h:\Omega\to M$ being measurable,
$\Omega_{i}=\\{\omega|h(\omega)\in M_{i}\\}$, $1\leq i\leq k$, are measurable
sets which are $\theta^{k}$-invariant by Theorem 2.4. On the other hand, since
$\theta^{n}$ is uniquely ergodic for all $n\in\mathbb{N}$, we have that
$\mathbb{P}(\Omega_{i})=1$ for some $i$. We summarize the results into the
following corollary: Under the assumption of Theorem 2.4, if $\\{graph(h)\\}$
be a full random periodic point with period $N$ then $N=nk$ for some
$k\in\mathbb{N}$ and $h(\Omega^{\prime})\subset M_{i}$ for a Borel subset
$\Omega^{\prime}$ of $\Omega$ and some $i\in\\{1,\cdots,k\\}$, where
$\mathbb{P}(\Omega^{\prime})=1$.

######  Theorem 2.5.

Assume that $\phi$ is transitive and Anosov on fibers, and $(\Omega,\theta)$
is an irrational rotation on torus, then there exists a $\phi$-invariant
ergodic Borel probability measure (SRB measure) $\mu$ such that
$h_{\mu}(\phi)>0$ and $supp\mu=M\times\Omega$. Moreover, for any $\gamma>0$,
there exists $n,k\in\mathbb{N}$ such that the following are satisfied

  1. i)

$\frac{1}{n}\ln k\geq h_{\mu}(\phi)-\gamma$;

  2. ii)

$\phi^{n}$ has a $k$-symbol full random horseshoe.

Another way to view the random horseshoe is to look at the dynamics in
$\mathcal{M}(M\times\Omega)$ induced by the original system
$(M\times\Omega,\phi)$, where $\mathcal{M}(M\times\Omega)$ is the space of
Borel probability measures on $M\times\Omega$ endowed with the weak* topology.
Recall that $\lim_{n\to\infty}\mu_{n}=\mu$ in $\mathcal{M}(M\times\Omega)$ if
the following is true

$\lim_{n\to\infty}\int_{M\times\Omega}hd\mu_{n}=\int_{M\times\Omega}hd\mu\text{
for all }h\in C(M\times\Omega),$

where $C(M\times\Omega)$ is the collection of continuous real functions
defined on $M\times\Omega$. Since $\phi$ is a homeomorphism,
$\phi^{*}:\mathcal{M}(M\times\Omega)\to\mathcal{M}(M\times\Omega)$ is defined
by the following: for any $\mu\in\mathcal{M}(M\times\Omega)$

(4) $\phi^{*}(\mu)(A)=\mu(\phi^{-1}(A))\text{ for all Borel }A\subset
M\times\Omega.$

A quick consequence of Theorem 2.5 is the following corollary.

######  Corollary 2.6.

Assume that $\phi$ is transitive and Anosov on fibers, and $(\Omega,\theta)$
is an irrational rotation on torus. Let $\mu$ be a $\phi$-invariant ergodic
probability measure (SRB measure) $\mu$ with $h_{\mu}(\phi)>0$ and
$supp\mu=M\times\Omega$, then for any $\gamma>0$, there exists
$n,k\in\mathbb{N}$ such that the following are satisfied

  1. i)

$\frac{1}{n}\ln k\geq h_{\mu}(\phi)-\gamma$;

  2. ii)

$(\phi^{*})^{n}$ has a bi-invariant horseshoe with $k$ symbols.

By the statement ”$(\phi^{*})^{n}$ has a bi-invariant horseshoe with $k$
symbols”, we mean that there is a continuous embedding
$\Psi:\mathcal{S}_{k}\to\mathcal{M}(M\times\Omega)$ such that if
$\Sigma=\Psi(\mathcal{S}_{k})$, then $(\phi^{*})^{n}|_{\Sigma}$ is one-to-one
and is conjugate to $\sigma$. This is the standard definition for horseshoe
with $k$-symbols, except that one can not talk about hyperbolicity in
$\mathcal{M}(M\times\Omega)$. For general definitions of horseshoes, we refer
the reader to [24].

Following from the recent result by Huang and Lu [19](see also [18, 21]), we
know that the existence of an SRB measure also yields the existence of a full
weak horseshoe since the entropy is positive. We state the definition of weak
horseshoe in the following for sake of completeness.

######  Definition 2.7.

By a full weak horseshoe of two symbols we mean that there exist subsets
$X_{1},X_{2}\subset M\times\Omega$ such that the following properties hold

  1. (1)

$X_{1}$ and $X_{2}$ are non-empty closed subsets of $M\times\Omega$ and

$\inf\\{d((x_{1},\omega_{1}),(x_{2},\omega_{2})|\ (x_{1},\omega_{1})\in
X_{1},\ (x_{2},\omega_{2})\in X_{2}\\}>0.$

  2. (2)

there exists a constant $b>0$ and $J\subset\mathbb{N}$ such that the limit

$\lim_{m\rightarrow+\infty}\frac{1}{m}|J\cap\\{0,1,2,\cdots,m-1\\}|$

exists and is larger than or equal to $b$ (positive density), and for any
$s\in\\{1,2\\}^{J}$, there exists $(x_{s},\omega_{s})\in M\times\Omega$ with
$\phi^{j}(x_{s},\omega_{s})\in X_{s(j)}$ for any $j\in J$.

In fact, under the setting of this paper, as a special case, one can expect
the weak horseshoe to have more properties. For sake of convenience, we
introduce a new definition of weak horseshoe for random case in the following.

######  Definition 2.8.

By a random full weak horseshoe of two symbols with bounded hitting time we
mean that there exist subsets $U_{1},U_{2}\subset M$ such that the following
properties hold

  1. (1)

$U_{1}$ and $U_{2}$ are non-empty closed subsets of $M$ and

$\inf\\{d_{M}(x_{1},x_{2})|\ x\in U_{1},x_{2}\in U_{2}\\}>0.$

  2. (2)

There exists a constant $K>0$ and Borel functions $n_{i}:\Omega\to\mathbb{N}$
for $i\in\mathbb{N}$ such that

$0<n_{i+1}(\omega)-n_{i}(\omega)<K,\text{ for }\mathbb{P}-a.e.\
\omega\in\Omega\text{ and }i\in\mathbb{N},$

and for any $s\in\mathcal{S}^{+}_{2}:=\\{1,2\\}^{\mathbb{N}}$, there exists a
Borel function $x_{s}:\Omega\to M$ with
$\pi_{M}\phi^{n_{j}(\omega)}(x_{s}(\omega),\omega)\in U_{s(j)}$ for any
$j\in\mathbb{N}$.

######  Theorem 2.9.

Assume that $\phi$ is transitive and Anosov on fibers, and $(\Omega,\theta)$
is an irrational rotation on torus, then there exists a random full weak
horseshoe of two symbols with bounded hitting time.

##  3\. Preliminary

Throughout this section, we assume that $\phi$ is transitive and Anosov on
fibers.

###  3.1. Invariant Manifolds

We define the local stable manifolds and unstable manifolds as the following:

$\displaystyle W^{s}_{\epsilon}(x,\omega)=\\{y\in M|\
d_{M}(\pi_{M}\phi^{n}(y,\omega),\pi_{M}\phi^{n}(x,\omega))\leq\epsilon\text{
for all }n\geq 0\\}$ $\displaystyle W^{u}_{\epsilon}(x,\omega)=\\{y\in M|\
d_{M}(\pi_{M}\phi^{n}(y,\omega),\pi_{M}\phi^{n}(x,\omega))\leq\epsilon\text{
for all }n\leq 0\\},$

where $\pi_{M}:M\times\Omega\to M$ is the nature coordinate projection. The
following lemma can mainly be viewed as a special version of Theorem 3.1 from
[17]. The only difference is that these local manifolds depends on $\omega$
measurably in [17] while depends on $\omega$ continuously in this paper. The
reason is that the invariant splitting $E^{u}_{(x,\omega)}\oplus
E^{s}_{(x,\omega)}$ varies continuously on both $x$ and $\omega$, and $\phi$
and $D_{M}\phi$ are also continuous on $\omega$. The proof of iii) employs
standard ideas for which we refer to the proof of Lemma 3.1 in [23]. So the
proof of Lemma 3.1 is omitted.

######  Lemma 3.1.

For any $\lambda\in(0,\lambda_{0})$, there exists $\epsilon_{0}>0$ such that
for any $\epsilon\in(0,\epsilon_{0}]$, the following hold:

  * i)

$W^{s}_{\epsilon}(x,\omega),W^{u}_{\epsilon}(x,\omega)$ are $C^{2}$ embedded
discs for all $(x,\omega)\in M\times\Omega$ with
$T_{x}W^{\tau}(x,\omega)=E^{\tau}_{(x,\omega)}$, $\tau=u,s$;

  * ii)

$d_{M}(\pi_{M}\phi^{n}(x,\omega),\pi_{M}\phi^{n}(y,\omega))\leq
e^{-n\lambda}d_{M}(x,y)$ for $y\in W^{s}_{\epsilon}(x,\omega)$, $n\geq 0$, and
$d_{M}(\pi_{M}\phi^{-n}(x,\omega),\pi_{M}\phi^{-n}(y,\omega))\leq
e^{-n\lambda}d_{M}(x,y)$ for $y\in W^{u}_{\epsilon}(x,\omega)$, $n\geq 0$;

  * iii)

$W^{s}_{\epsilon}(x,\omega),W^{u}_{\epsilon}(x,\omega)$ vary continuously on
$(x,\omega)$ (in $C^{1}$ topology).

The local stable and unstable manifolds can be used to construct the global
stable and unstable manifolds respectively,

$\displaystyle W^{s}(x,\omega)=\\{y\in M_{\omega}|\
|d_{M}(\pi_{M}\phi^{n}(y,\omega),\pi_{M}\phi^{n}(x,\omega))\to 0\text{ as
}n\to\infty\\}$ $\displaystyle W^{u}(x,\omega)=\\{y\in M_{\omega}|\
|d_{M}(\pi_{M}\phi^{n}(y,\omega),\pi_{M}\phi^{n}(x,\omega))\to 0\text{ as
}n\to-\infty\\},$

as

$\displaystyle
W^{s}(x,\omega)=\bigcup_{n=0}^{\infty}\phi^{-n}(W^{s}_{\epsilon}(\phi^{n}(x,\omega))),$
$\displaystyle
W^{u}(x,\omega)=\bigcup_{n=0}^{\infty}\phi^{n}(W^{u}_{\epsilon}(\phi^{-n}(x,\omega))),$

where $\epsilon$ is an arbitrarily fixed small positive number as in 3.1. The
following lemma provides local canonical coordinates on $M_{\omega}$.

######  Lemma 3.2.

For any $\epsilon\in(0,\epsilon_{0})$ there is a $\delta\in(0,\epsilon)$ such
that for any $x,y\in M$ with $d_{M}(x,y)<\delta$
$W^{s}_{\epsilon}(x,\omega)\bigcap W^{u}_{\epsilon}(y,\omega)$ consists of a
single point, which is denoted by $[x,y]_{\omega}$. Furthermore

$[\cdot,\cdot]_{\cdot}:\\{(x,y,\omega)\in M\times M\times\Omega|\
d_{M}(x,y)<\delta\\}\to M\text{ is continuous}.$

We say the system $(M\times\Omega,\phi)$ has local product structure with size
$\delta$.

###### Proof.

This simply follows iii) of Lemma 3.1, the continuous (thus uniformly
continuous) dependence of
$W^{s}_{\epsilon}(x,\omega),W^{u}_{\epsilon}(x,\omega)$ on $(x,\omega)$, and
the uniform continuity of the invariant splitting $E^{u}_{(x,\omega)}\oplus
E^{s}_{(x,\omega)}$. For details, we refer to [12]. ∎

######  Lemma 3.3.

There is an $\epsilon>0$ such that for any $\omega\in\Omega$ and $x,y\in
M_{\omega}$ with $y\neq x$, the following holds:

$d_{M}(\pi_{M}\phi^{k}(x,\omega),\pi_{M}\phi^{k}(y,\omega))>\epsilon\text{ for
some }k\in\mathbb{Z}.$

###### Proof.

Take $\epsilon$ smaller than $\epsilon_{0}$ as in Lemma 3.1. Otherwise,
$(y,\omega)\in W^{s}_{\epsilon}(x,\omega)\cap W^{u}_{\epsilon}(x,\omega)$
therefore $y=x$. ∎

###  3.2. Shadowing Lemma

We first introduce some concepts needed here. For any $\alpha>0$, an orbit
$\\{(x_{i},\theta^{i}\omega)\\}_{i\in\mathbb{Z}}\subset M\times\Omega$ is
called $(\omega,\alpha)$-pseudo orbit if for any $i\in\mathbb{Z}$

$d_{M}(f_{\theta^{i}\omega}(x_{i}),x_{i+1})<\alpha.$

######  Lemma 3.4.

For any $\beta>0$, there exists $\alpha>0$ for any $(\omega,\alpha)$-pseudo
orbit $\\{(x_{i},\theta^{i}\omega)\\}_{i\in\mathbb{Z}}$, there is a true orbit
$\\{(y_{i},\theta^{i}\omega)\\}_{i\in\mathbb{Z}}$ of $\phi$ such that

$\sup_{i\in\mathbb{Z}}d_{M}(y_{i},x_{i})<\beta.$

$\\{(y_{i},\theta^{i}\omega)\\}_{i\in\mathbb{Z}}$ is called
$(\omega,\beta)$-shadowing orbit of
$\\{(x_{i},\theta^{i}\omega)\\}_{i\in\mathbb{Z}}$.
Further more, there is a $\beta_{0}>0$ such that for any $0<\beta<\beta_{0}$,
the above true orbit is unique.

###### Proof.

The proof follows the proof of Proposition 3.6 in [17]. For sake of
convenience, we always choose $\beta_{0}<\frac{1}{3}\delta$, where $\delta$ is
the one in Lemma 3.2 for $\epsilon=\frac{1}{2}\epsilon_{0}$. ∎

######  Lemma 3.5.

For a given $\beta\in(0,\beta_{0})$, let $\alpha$ be as in Lemma 3.4
corresponding to $\beta$. Then for any $\tau>0$, there exists
$N_{0}\in\mathbb{N}$ such that given any two $(\omega,\alpha)$-pseudo orbits
$\\{(x_{i},\theta^{i}\omega)\\}_{i\in\mathbb{Z}}$ and
$\\{(x_{i}^{\prime},\theta^{i}\omega)\\}_{i\in\mathbb{Z}}$ which are
$(\omega,\beta)$-shadowed by two uniquely defined true orbits
$\\{(y_{i},\theta^{i}\omega)\\}_{i\in\mathbb{Z}}$ and
$\\{(y_{i}^{\prime},\theta^{i}\omega)\\}_{i\in\mathbb{Z}}$ respectively, for
any $N>N_{0}$ if $x_{i}=x_{i}^{\prime},\ \forall i\in[-N,N]$, then
$d_{M}(y_{0},y_{0}^{\prime})<\tau$.

###### Proof.

Note that if $x_{i}=x_{i}^{\prime},\ \forall i\in[-N,N]$, then
$d_{M}(y_{i},y_{i}^{\prime})<2\beta,\ \forall i\in[-N,N]$. Let
$z=[y_{0},y^{\prime}_{0}]_{\omega}$, where $[\cdot,\cdot]_{\cdot}$ is defined
in 3.2 for a fixed $\delta$ corresponding to
$\epsilon=\frac{1}{2}\epsilon_{0}$. By the definition of local unstable
manifold and ii) of Lemma 3.1, we have that

$\displaystyle d_{M}(z,y^{\prime}_{0})$ $\displaystyle\leq
e^{-N\lambda}d_{M}(\pi_{M}\phi^{N}(z,\omega),y_{N}^{\prime})$
$\displaystyle\leq
e^{-N\lambda}(d_{M}(y_{N},y^{\prime}_{N})+d_{M}(\pi_{M}\phi^{N}(z,\omega),y_{N}))$
$\displaystyle\leq e^{-N\lambda}(2\beta+d_{M}(z,y_{0}))$ $\displaystyle\leq
e^{-N\lambda}(2\beta+\frac{1}{2}\epsilon_{0}).$

Same argument is applicable to $d_{M}(z,y_{0})$ if reverse the time. Therefore

$d_{M}(y_{0},y^{\prime}_{0})\leq
2e^{-N\lambda}(2\beta+\frac{1}{2}\epsilon_{0}).$

The proof is done. ∎

##  4\. Proof of main results

Throughout this section, we will assume that $\phi$ is transitive and Anosov
on fibers.

###  4.1. Proof of Theorem 2.3

In this subsection, we only assume that $\theta$ is equicontinuous.

Fix a given $g\in C(\Omega,M)$, and consider a finite Borel partition
$\xi=\\{\xi_{i}\\}_{1\leq i\leq p}$ of $\Omega$, of which each element
contains an nonempty open set and has a finite diameter. Such a partition
exists because of compactness of $\Omega$. For any $\omega\in\Omega$, define
$\xi(\omega)$ the element in $\xi$ which contains $\omega$, and also denote
that

$B_{g}(\xi_{i},\delta):=\\{(x,\omega)|\
\omega\in\xi_{i},d_{M}(x,g(\omega))<\delta\\}.$

Note that $\bigcup_{\xi_{i}\in\xi}B_{g}(\xi_{i},\delta)$ is a
$\delta$-neighborhood of $graph(g)$, which is denoted by $B_{g}(\delta)$.

######  Lemma 4.1.

For any $\delta>0$, there exists $m\in\mathbb{N}$ such that for any
$\xi_{i}\in\xi$, there exists $(x_{i},\omega_{i})\in B_{g}(\xi_{i},\delta)$
such that

$\phi^{m}(x_{i},\omega_{i})\in B_{g}(\xi_{i},\delta).$

###### Proof.

Since $\phi$ is transitive, there exists $(x_{0},\omega_{0})$ such that the
orbit of $(x_{0},\omega_{0})$,
$\\{\phi^{n}(x_{0},\omega_{0})\\}_{n\in\mathbb{Z}}$, is dense in
$M\times\Omega$. By the finiteness of the partition, we have that there exists
a point $(x,\omega)$ in the interior of $B_{g}(\xi_{1},\delta)$ and an
$m_{0}\in\mathbb{N}$ such that

$\phi^{n_{i}}(x,\omega)\in\text{interior of }B_{g}(\xi_{i},\delta)\text{ for
some }n_{i}\in[1,m_{0}],\ \forall i\in[2,p].$

By the continuity of $\phi$ and the finiteness of the partition, we have that
there exists an small open neighbourhood of $(x,\omega)$ in $M\times\Omega$,
which is denoted by $U$, such that

$U\subset B_{g}(\xi_{1},\delta)\text{ and
}\phi^{n_{i}}(U)\subset\text{interior of }B_{g}(\xi_{i},\delta),\ \forall
i\in[2,p].$

Then, by the transitivity of the system, there exists an $m\in\mathbb{N}$ and
$(x^{\prime},\omega^{\prime})\in U$ such that
$\phi^{m}(x^{\prime},\omega^{\prime})\in U$. Note that

$\phi^{m}(\phi^{n_{i}}(x^{\prime},\omega^{\prime}))=\phi^{n_{i}}(\phi^{m}(x^{\prime},\omega^{\prime}))\in\phi^{n_{i}}(U)\subset\text{interior
of }B_{g}(\xi_{i},\delta),\ \forall i\in[2,p].$

We complete the proof by letting
$(x_{1},\omega_{1})=(x^{\prime},\omega^{\prime})$ and
$(x_{i},\omega_{i})=\phi^{n_{i}}(x^{\prime},\omega^{\prime})$ for $i\in[2,p]$.
∎

The next, we will construct a $(\upsilon,\delta)$-pseudo orbit for
$\upsilon\in\Omega$, $g\in C(\Omega,M)$, and $\delta>0$. Given $\delta>0$, by
Lemma 4.1, there exists $m=m(\delta,g,\phi)>0$ and $(x_{i},\omega_{i})\in
B_{g}(\xi_{i},\delta)$ for $1\leq i\leq p$ such that

(5) $\phi^{m}(x_{i},\omega_{i})\in B_{g}(\xi_{i},\delta)\text{ for }1\leq
i\leq p.$

For $l\in\mathbb{Z}$, suppose that $\theta^{lm}\upsilon\in\xi_{i_{l}}$, then
we define

(6)
$(y_{j},\upsilon_{j})=\begin{cases}(g(\theta^{lm}\upsilon),\theta^{lm}\upsilon),&\text{
when }j=lm\\\
(\pi_{M}\phi^{j-lm}(x_{i_{l}},\omega_{i_{l}}),\theta^{j}\upsilon),&\text{ when
}j\in[lm+1,(l+1)m-1]\\\ \end{cases}.$

######  Lemma 4.2.

For any $\epsilon^{\prime}>0$, there exists
$\delta^{\prime}(\phi,g,\epsilon^{\prime})>0$ such that if $\delta$ and the
diameter of the partition $\xi$ are less than $\delta^{\prime}$, the pseudo
orbit defined by (6) starting from $(y,\upsilon)\in graph(g)$ is a
$(\upsilon,\epsilon^{\prime})$-pseudo orbit.

###### Proof.

Noting that $\Omega$ being compact, then $g$ is uniformly continuous.
Therefore, for any $\epsilon_{1}>0$, there exists $\delta_{1}>0$ such that for
any $\omega_{1},\omega_{2}\in\Omega$ if
$d_{\Omega}(\omega_{1},\omega_{2})<\delta_{1}$ then

(7) $d_{M}(g(\omega_{1}),g(\omega_{2}))<\epsilon_{1}.$

Also note that $\phi$ is uniformly continuous, so for any $\epsilon_{2}>0$,
there exists $\delta_{2}>0$ such that for any
$(x_{1},\omega_{1}),(x_{2},\omega_{2})\in M\times\Omega$ if
$d((x_{1},\omega_{1}),(x_{2},\omega_{2}))<\delta_{2}$ then

(8)
$\max\\{d(\phi(x_{1},\omega_{1}),\phi(x_{2},\omega_{2})),d(\phi^{-1}(x_{1},\omega_{1}),\phi^{-1}(x_{2},\omega_{2}))\\}<\epsilon_{2}.$

Note that $d_{\Omega}(\theta^{j}\upsilon,\theta^{j-lm}\omega_{i_{0}})$ is less
than the diameter of the partition $\xi$ (which is less than
$\delta^{\prime}(\phi,g,\epsilon^{\prime})$) for all $j\in[lm,(l+1)m]$ and
$l\in\mathbb{Z}$ by (1). For given $(y,\upsilon)\in graph(g)$ with
$\upsilon\in\xi_{i_{0}}$, we have the following:

  1. a)

When $j\in[lm+2,(l+1)m-1]$, taking $\delta^{\prime}<\delta_{2}$, then (8)
implies that

$\displaystyle d_{M}(y_{j},\pi_{M}\phi(y_{j-1},\upsilon_{j-1}))=$
$\displaystyle d_{M}(y_{j},\pi_{M}\phi(y_{j-1},\theta^{j-1}\upsilon))$
$\displaystyle\leq$ $\displaystyle
d(\phi(y_{j-1},\theta^{j-lm-1}\omega_{i_{l}}),\phi(y_{j-1},\theta^{j-lm-1}\upsilon_{lm}))$
$\displaystyle\leq$ $\displaystyle\epsilon_{2};$

  2. b)

When $j=lm+1$, taking $\delta^{\prime}<\frac{1}{2}\delta_{2}$ (thus
$\delta<\frac{1}{2}\delta_{2}$), then
$d((x_{i_{l}},\omega_{i_{l}}),(g(\theta^{lm}\upsilon),\theta^{lm}\upsilon))<\delta_{2}$,
therefore (8) implies that

$d_{M}(y_{j},\pi_{M}\phi(y_{j-1},\upsilon_{j-1}))=d_{M}\left(\pi_{M}\phi(x_{i_{l}},\omega_{i_{l}}),\pi_{M}\phi(g(\theta^{lm}\upsilon),\theta^{lm}\upsilon)\right)<\epsilon_{2};$

  3. c)

For $j=(l+1)m$, taking $\delta^{\prime}<\min\\{\delta_{1},\delta_{2}\\}$, then
(7), (5), and (8) implies that

$\displaystyle d_{M}(y_{j},\pi_{M}\phi(y_{j-1},\upsilon_{j-1}))$
$\displaystyle=$ $\displaystyle
d_{M}(g(\theta^{j}\upsilon),\pi_{M}\phi(y_{j-1},\upsilon_{j-1}))$
$\displaystyle\leq$ $\displaystyle
d_{M}(g(\theta^{m}\upsilon_{lm}),g(\theta^{m}\omega_{i_{l}}))+d_{M}(g(\theta^{m}\omega_{i_{l}}),\pi_{M}\phi^{m}(x_{i_{l}},\omega_{i_{l}}))$
$\displaystyle+d_{M}(\pi_{M}\phi^{m}(x_{i_{l}},\omega_{i_{l}}),\pi_{M}\phi(y_{j-1},\theta^{m-1}\upsilon_{lm}))$
$\displaystyle\leq$
$\displaystyle\epsilon_{1}+\delta+d_{M}(\pi_{M}\phi(y_{j-1},\theta^{m-1}\omega_{i_{l}}),\pi_{M}\phi(y_{j-1},\theta^{m-1}\upsilon_{lm}))$
$\displaystyle\leq$ $\displaystyle\epsilon_{1}+\delta+\epsilon_{2}.$

Therefore, the required estimate can be achieved if one choose
$,\epsilon_{1}+\delta+\epsilon_{2}<\epsilon^{\prime}$ and take a
$\delta^{\prime}>0$ which satisfies all the proposed conditions in a), b), and
c) above. So for such $\delta^{\prime}>0$ if $\delta$ and $diam(\xi)$ are less
than $\delta^{\prime}$, the pseudo-orbit defined by (6) is a
$(\upsilon,\epsilon^{\prime})$-pseudo orbit. ∎

The next lemma will be used later in Section 4.3.2.

######  Lemma 4.3.

For any $\epsilon^{\prime}>0$ and $L>0$, there exists
$\delta^{\prime}(\phi,L,\epsilon^{\prime})>0$ such that for any $g\in
C(\Omega,M)$ with $Lip\ g\leq L$ if $\delta$ and the diameter of the partition
$\xi$ are less than $\delta^{\prime}$, the pseudo orbit defined by (6)
starting from $(y,\upsilon)\in graph(g)$ is a
$(\upsilon,\epsilon^{\prime})$-pseudo orbit.

###### Proof.

In the proof of Lemma 4.2 above, the dependence of $\delta^{\prime}$ on $g$ is
given by (7) which appears in part c). So the exact property of “$g$” being
involved in this lemma is how uniformly continuous it is. Therefore, under the
setting of the current lemma, the exactly same argument can be applied here,
in which “$\delta^{\prime}$” depends on the Lipchitz constant “$L$” rather
than the choice of particular map “$g$”. The detailed proof is omitted. ∎

Now we are ready to prove Theorem 2.3. For a given small $\epsilon>0$ and
$g\in C(\Omega,M)$, choose $0<\beta<\min\\{\frac{1}{2}\epsilon,\beta_{0}\\}$.
By Lemma 3.4, there exists a corresponding $\alpha>0$ for $\beta$. Then, by
applying Lemma 4.2, for $\epsilon^{\prime}=\alpha$ there exists a
$\delta^{\prime}>0$ such that for any $\upsilon\in\Omega$ and
$\delta\in(0,\delta^{\prime})$, there exists $m\in\mathbb{N}$ which is
depending on $\delta,g,\phi$ only and a $(\upsilon,\alpha)$-pseudo orbit
defined by (6), which is denoted by
$\\{(y_{i},\theta^{n}\upsilon)\\}_{n\in\mathbb{Z}}$, such that

$(y_{km},\theta^{km}\upsilon)\in graph(g),\ \forall k\in\mathbb{Z}.$

By applying Lemma 3.4, there is a unique true orbit
$\\{(y^{\prime}_{i},\theta^{i}\upsilon)\\}_{i\in\mathbb{Z}}$ which is
$(\upsilon,\beta)$-shadowing
$\\{(y_{i},\theta^{i}\upsilon)\\}_{i\in\mathbb{Z}}$. Note that such
$y^{\prime}_{0}$ is depending on $\upsilon$ and this $\upsilon$ is arbitrarily
chosen, therefore, by repeating the previous procedure, one can define a map
$\tilde{g}:\Omega\to M$ by letting

$\tilde{g}(\upsilon)=y^{\prime}_{0},\ \forall\upsilon\in\Omega.$

Note that (3) holds automatically by (6) for $j=0$ and the choice of $\beta$.
It is also obvious that $\\{\phi^{i}(graph(\tilde{g}))\\}_{i=0,1,\cdots,m-1}$
is a full random periodic orbit with period $m$ as long as $\tilde{g}$ being
measurable: otherwise, there is an $\omega\in\Omega$ such that the
$(\omega,\alpha)$-pseudo orbit starting from $(g(\omega),\omega)$ is
$(\omega,\beta)$-shadowed by two distinct true orbits, which contradicts Lemma
3.4 as $\beta<\beta_{0}$.

It remains to show the measurability of $\tilde{g}$, which is also following
the uniqueness.

First, for the given $g\in C(\Omega,M)$, partition $\xi$ of $\Omega$, and the
return time $m$ above, we define maps $g_{j}:\Omega\to M$ for each
$j\in\mathbb{Z}$ as the following:

(9) $g_{j}(\omega)=\begin{cases}g(\omega),&\text{ when }j=lm,\\\
\pi_{M}\phi^{j-lm}(x_{i_{l}},\omega_{i_{l}}),&\text{ when
}j\in[lm+1,(l+1)m-1]\text{ and }\theta^{lm-j}\omega\in\xi_{i_{l}}\end{cases},$

where $l\in\mathbb{Z}$. It is easy to see that all the $g_{j}$s are
measurable, since they are either continuous functions ($=g$) or simple
functions.

Second, note that for any $\upsilon\in\Omega$, the orbit
$\\{\phi^{i}(\tilde{g}(\upsilon),\upsilon)\\}_{i\in\mathbb{Z}}$ is
$(\upsilon,\beta)$-shadowing the $(\upsilon,\alpha)$-pseudo orbit
$\\{(y_{i},\theta^{i}\upsilon)\\}_{i\in\mathbb{Z}}$ defined by (6). Then, by
(9), one has that for any $\upsilon\in\Omega$

$d_{M}\left(\pi_{M}\phi^{i}(\tilde{g}(\upsilon),\upsilon),g_{i}(\theta^{i}\upsilon)\right)<\beta.$

Therefore, we have that

$graph(\tilde{g})\subset\bigcap_{j\in\mathbb{Z}}\left(\phi^{-j}\left(B_{g_{j}}(\beta)\right)\right).$

By the choice of $\beta$ which implies the uniqueness of the
$(\cdot,\beta)$-shadowing true orbit, one can conclude that

$graph(\tilde{g})=\bigcap_{j\in\mathbb{Z}}\left(\phi^{-j}\left(B_{g_{j}}(\beta)\right)\right),$

since the orbit starting from each point
$(y,\upsilon)\in\bigcap_{j\in\mathbb{Z}}\left(\phi^{-j}\left(B_{g_{j}}(\beta)\right)\right)$
will be $(\upsilon,\beta)$-shadowing the $(\upsilon,\alpha)$-pseudo orbit
$\\{(y_{i},\theta^{i}\upsilon)\\}_{i\in\mathbb{Z}}$ defined by (6).

Finally, we define a multivalued function $\tilde{g}_{i}:\Omega\to 2^{M}$ for
any $i\in\mathbb{N}$ as the following:

$\tilde{g}_{i}(\upsilon):=\bigcap_{-i\leq j\leq
i}\pi_{M}\phi^{-j}\left\\{(x,\theta^{j}\upsilon)\Big{|}\
d_{M}(x,y_{j}(\upsilon))\leq\beta\right\\},\text{ for }\upsilon\in\Omega.$

It is easy to see that for all $\upsilon\in\Omega$, $\tilde{g}_{i}(\upsilon)$
is a non-empty closed subset of $M$ because of the continuity of
$\phi(x,\omega)$ on $x$.

For a given open set $U\subset M$ and any $i\in\mathbb{N}$, define

$\Gamma_{i}^{-}(U):=\\{\omega\in\Omega|\ \tilde{g}_{i}(\omega)\cap
U\neq\emptyset\\},$

which can be rewritten as the following

$\displaystyle\Gamma^{-}_{i}(U)$
$\displaystyle=\left\\{\omega\Big{|}U\bigcap\left(\bigcap_{-i\leq j\leq
i}\pi_{M}\phi^{-j}\left(\left\\{(x,\theta^{j}\omega)\Big{|}\
d_{M}(x,y_{j}(\omega))\leq\beta\right\\}\right)\right)\neq\emptyset\right\\}$
$\displaystyle=\bigcap_{-i\leq j\leq
i}\left\\{\omega\Big{|}U\bigcap\left(\pi_{M}\phi^{-j}\left(\left\\{(x,\theta^{j}\omega)\Big{|}\
d_{M}(x,y_{j}(\omega))\leq\beta\right\\}\right)\right)\neq\emptyset\right\\}$
$\displaystyle=\bigcap_{-i\leq j\leq
i}\pi_{\Omega}\left(\phi^{j}(U\times\Omega)\bigcap\overline{B_{g_{j}}(\beta)}\right),$

where
$\overline{B_{g_{j}}(\beta)}:=\bigcup_{\upsilon\in\Omega}\left\\{(x,\theta^{j}\upsilon)\Big{|}\
d_{M}(x,y_{j}(\upsilon))\leq\beta\right\\}$. Since $g_{j}$ is either a
continuous function (when $j=lm$) or a simple function (when $j\neq lm$),
$\overline{B_{g_{j}}(\beta)}$ is clearly a Borel subset in $M\times\Omega$;
while $\phi^{j}(U\times\Omega)$ is an open subset in $M\times\Omega$ because
$\phi$ is continuous. Therefore, one has that $\Gamma_{i}^{-}(U)\subset\Omega$
is Borel, which is true for arbitrary open set $U$ and $i\in\mathbb{N}$. Then,
by applying Theorem C.1 (Let $Y=M$, $X=\Omega$, and $\mathcal{L}$ be the Borel
$\sigma$-algebra of $\Omega$), one obtain that for any $i\in\mathbb{Z}$,
$\tilde{g}_{i}$ produces a measurable selection
$\tilde{g}_{i}^{\prime}:\Omega\to M$ satisfying that

$graph(\tilde{g}^{\prime}_{i})\subset graph(\tilde{g}_{i}).$

As $i\to\infty$, by Lemma 3.5, we have that $\tilde{g}_{i}^{\prime}$ converges
to $\tilde{g}$ pointwisely, thus $\tilde{g}$ is measurable.

The proof is completed.

###  4.2. Proof of Theorem 2.4

In this section, we investigate the structure of a quasi-periodic forced
system with transitivity and Anosov property on fibers to derive the so called
spectral decomposition. In the whole section, we will assume that $\Omega$ is
connected, for any $n\in\mathbb{N}$, $\theta^{n}$ is uniquely ergodic and
$\mathbb{P}$ is the ergodic probability measure.

By Theorem 2.3, we have that for any $g\in C(\Omega,M)$ and $\epsilon>0$,
there exists a full random periodic point $\\{graph(\tilde{g})\\}$ satisfying
(3). For this $\tilde{g}$, let

$X_{\tilde{g}}=\bigcup_{\omega\in\Omega}\\{(x,\omega)|x\in\overline{W^{u}(\tilde{g}(\omega),\omega)}\\},$

where $\overline{W^{u}(\tilde{g}(\omega),\omega)}$ is the closure of the
global unstable manifold of $(\tilde{g}(\omega),\omega)$ in $(M,\omega)$.
Since $\Omega$ is connected, one has that $g(\Omega)$ is connected thus
belongs to one of the $M_{i}$’s, where $\\{M_{1},\cdots,M_{k}\\}$ are the
connected components of $M$. At the beginning, we first propose the following
conditions on the main framework and some parameters:

C1: By taking $\epsilon$ small enough, one can make $\tilde{g}(\Omega)$ belong
to the same connected component. Additionally, we also require
$\epsilon<\frac{1}{2}\delta$ where $\delta$ is as in Lemma 3.2 corresponding
to a fixed small $\frac{1}{2}\epsilon_{0}>0$ with $\epsilon_{0}$ being smaller
than the minimum separation of $\\{M_{1},\cdots,M_{k}\\}$. In the rest of the
proof, we fix such $\epsilon$ and $\delta$, and without losing any generality,
we also suppose that $g(\Omega),\tilde{g}(\Omega)\subset M_{1}.$

First, we show that

(10) $X_{\tilde{g}}=M_{1}\times\Omega.$

The next lemma is the key result needed in the proof:

######  Lemma 4.4.

For any $(x,\omega)\in M_{1}\times\Omega$ and $g\in C(\Omega,M_{1})$, there
exists an nonempty open set $A\subset\Omega\setminus\\{\omega\\}$ and a
function $g^{\prime}\in C(\Omega,M_{1})$ such that

(11) $g^{\prime}(\omega)=x\text{ and }g^{\prime}|_{A}=g|_{A}.$

The proof of this lemma may be standard, we give it here for sake of
completeness.

###### Proof.

First, there exists a path $\gamma:[0,1]\to M_{1}$ such that
$\gamma(0)=g(\omega)$ and $\gamma(1)=x$. Note that there exists finitely many
numbers $0=s_{0}<s_{1}<s_{2}<\cdots<s_{n}=1$ such that $\gamma(s_{i})$ and
$\gamma(s_{i+1})$ are in a same chart of the atlas of $M$, say
$(U_{i},\psi_{i})$, for all $i=0,\cdots,n-1$. Consider an open ball centered
on $\omega$ in $\Omega$, $B(\omega,R_{1})$ with $R_{1}$ small enough so that
$\mathbb{P}(\overline{B(\omega,R_{1})})<1$ and $g(B(\omega,R_{1}))\subset
U_{1}$. Note that $\omega$ and $\Omega\setminus B(\omega,\frac{1}{2}R_{1})$
are disjoint closed sets. By Urysohn’s lemma, there exists a continuous
function $f_{1}:\Omega\to[0,1]$ such that

$f_{1}|_{\Omega\setminus B(\omega,\frac{1}{2}R_{1})}=0\text{ and
}f_{1}(\omega)=1.$

Then, we can define

$g_{1}(\upsilon)=\begin{cases}g_{0}(\upsilon)(\text{where }g_{0}:=g),&\text{
if }\upsilon\notin B(\omega,\frac{1}{2}R_{1})\\\
\psi_{1}^{-1}\left[f_{1}(\upsilon)\psi_{1}(\gamma(s_{1}))+(1-f_{1}(\upsilon))\psi_{1}(g_{0}(\upsilon))\right],&\text{
if }\upsilon\in B(\omega,\frac{1}{2}R_{1})\end{cases}.$

It is obvious that $g_{1}$ is continuous and satisfies that

$g_{1}(\omega)=\gamma(s_{1})\text{ and
}A_{1}:=\\{\omega|g_{1}(\omega)=g(\omega)\\}\text{ contains an open set}.$

Inductively, one can construct $g_{i+1}$ based on $g_{i}$ by using the exactly
same argument above. To guarantee the existence of $A$, it is sufficient to
take $R_{i+1}<R_{i}$. In this case, one can take
$A=\Omega\setminus\overline{B(\omega,\frac{1}{2}R_{1})}$. ∎

######  Remark 4.1.

Actually, it is easy to see that since $\mathbb{P}$ is not atomic,
$\mathbb{P}(A)$ can be taken arbitrarily close to $1$.

Now, we are ready to prove Theorem 2.4. For an arbitrary $(x,\omega)\in
M_{1}\times\Omega$, and $g$ chosen at the beginning of this subsection, by
Lemma 4.4, there exists $g^{\prime}\in C(\Omega,M)$ satisfying (11). For a
small $\epsilon^{\prime}\in(0,\frac{1}{2}\delta)$, where $\delta$ is as in
C1). We apply Theorem 2.3 to $g^{\prime}$, and then derive that there exists a
full random periodic point $\\{graph(\tilde{g}^{\prime})\\}$ such that

(12) $g^{\prime}(\Omega)\subset M_{1}\text{ and }sup_{\upsilon\in
A}\\{d_{M}(\tilde{g}^{\prime}(\upsilon),\tilde{g}(\upsilon))\\}\leq
2\epsilon^{\prime}.$

Let $m$ and $m^{\prime}$ be the periods of $\\{graph(\tilde{g})\\}$ and
$\\{graph(\tilde{g}^{\prime})\\}$ respectively. Note that
$\theta^{-mm^{\prime}}$ is $\mathbb{P}$-ergodic since $\theta$ is invertible
and $\theta^{mm^{\prime}}$ is $\mathbb{P}$-ergodic, and each $\omega\in\Omega$
is a $\theta^{mm^{\prime}}$-minimal point. That means there is an infinite
increasing sequence of positive integers, $0<n_{1}<n_{2}<n_{3}<\cdots$, such
that $\theta^{-n_{i}mm^{\prime}}\omega\in A\subset\Omega$ for $i\geq 1$ as
long as $A$ having nonempty interior.

For $i\geq 1$, let

$\displaystyle x^{\prime}_{i}$
$\displaystyle=[\pi_{M}\phi^{-n_{i}mm^{\prime}}(\tilde{g}^{\prime}(\omega),\omega),\tilde{g}(\theta^{-n_{i}mm^{\prime}}\omega)]_{\theta^{-n_{i}mm^{\prime}}\omega}$
$\displaystyle=W^{s}_{\frac{\epsilon_{0}}{2}}(\phi^{-n_{i}mm^{\prime}}(\tilde{g}^{\prime}(\omega),\omega))\bigcap
W^{u}_{\frac{\epsilon_{0}}{2}}(\tilde{g}(\theta^{-n_{i}mm^{\prime}}\omega),\theta^{-n_{i}mm^{\prime}}\omega).$

Note that, by Lemma 3.1, one has that

$\lim_{i\to\infty}\phi^{n_{i}mm^{\prime}}(x^{\prime}_{i},\theta^{-n_{i}mm^{\prime}}\omega)\to(\tilde{g}^{\prime}(\omega),\omega)\text{
and
}\phi^{n_{i}mm^{\prime}}(x^{\prime}_{i},\theta^{-n_{i}mm^{\prime}}\omega)\in
W^{u}(\tilde{g}(\omega),\omega),$

which implies that

$(\tilde{g}^{\prime}(\omega),\omega)\in\overline{W^{u}(\tilde{g}(\omega),\omega)}\subset
X_{\tilde{g}}.$

As $\epsilon^{\prime}$ can be taken arbitrarily small, we have that
$(x,\omega)\in X_{\tilde{g}}$, thus $M_{1}\times\Omega\subset X_{\tilde{g}}$
by the arbitrariness of $(x,\omega)$.

Next, we show that

$\phi(M_{1}\times\Omega)\bigcap M_{1}\times\Omega=\emptyset,\text{ if }k\geq
2.$

This is because $\phi^{-1}$ is continuous and $M_{1}\times\Omega$ is both open
and closed, which imply that $\phi(M_{1}\times\Omega)\bigcap
M_{1}\times\Omega$ is both open and closed thus is either $M_{1}\times\Omega$
or $\emptyset$. Since $\phi$ is transitive, $M_{1}\times\Omega$ should not be
an $\phi$-invariant set as long as $M$ has more than one connected components.
The same argument implies that $\Omega\times\bigcup_{i\in S}M_{i}$ is not
$\phi$-invariant, where $S\subset\\{1,\cdots,k\\}$ and
$S\neq\\{1,\cdots,k\\}$. So, there exists a unique permutation, $\tau$,
satisfies the required property.

Finally, we show that $\phi^{k}|_{M_{i}\times\Omega}$ is topological mixing on
fibers, for which we still work on $\phi^{k}|_{M_{1}\times\Omega}$. Roughly
speaking, the proof is based on the fact that the global stable manifold of an
full random periodic point is dense in $M_{1}\times\Omega$. The following
lemma is the key result needed to derive the mixing property.

######  Lemma 4.5.

For any $\epsilon_{1}>0$ and $g\in C(\Omega,M_{1})$, there exists
$\delta_{1}>0$ such that the following holds:
For any full random periodic point $graph(\tilde{g})\in B_{g}(\delta_{1})$
with period $l$, $\epsilon^{\prime}_{1}>0$ and $h\in C(\Omega,M_{1})$, there
exists $N\in\mathbb{N}$, and a measurable function $h_{1}:\Omega\to M_{1}$
satisfying that

$graph(h_{1})\subset B_{h}(\epsilon^{\prime}_{1})\text{ and
}\phi^{nl}(graph(h_{1}))\subset B_{g}(\epsilon_{1}),\forall n>N.$

###### Proof.

Here we only need to take $\delta_{1}\in(0,\frac{1}{6}\epsilon_{1})$ small
enough such that any $(\omega,2\delta_{1})$-pseudo orbit can be
$(\omega,\frac{1}{3}\epsilon_{1})$-shadowed by a unique true orbit. This can
be done by Lemma 3.4, and in the rest of this proof we fix such a
$\delta_{1}$.

It is easy to see that the existence of $\tilde{g}$ follows Theorem 2.3
directly. For an arbitrarily fixed $\omega\in\Omega$, by Lemma 4.4, there
exists an nonempty open set $A_{\omega}\subset\Omega$ and
$g^{\prime}_{\omega}\in C(\Omega,M)$ such that

$g^{\prime}_{\omega}(\omega)=h(\omega)\text{ and
}g^{\prime}_{\omega}|_{A_{\omega}}=g|_{A_{\omega}}.$

By applying Theorem 2.3, we have that there is a full random periodic point
$graph(\tilde{g}^{\prime}_{\omega})$ with period $l_{\omega}$ such that

$graph(\tilde{g}^{\prime}_{\omega})\in
B_{g^{\prime}_{\omega}}\left(\frac{1}{3}\epsilon^{\prime}_{1}\right),$

where, without losing any generality, we assume that
$\epsilon^{\prime}_{1}\in(0,\delta_{1})$.
Let $B_{\omega}\subset A_{\omega}$ be an open ball in $\Omega$ with radius
$r_{\omega}>0$, and $C_{\omega}$ be an open neighborhood of $\omega$ with
radius $\frac{1}{2}r_{\omega}$. Since $\theta^{l_{\omega}l}$ is an isometric
map on $\Omega$ and any nonempty open subset of $\Omega$ is not
$\mathbb{P}$-null set, there is an infinite set $I_{\omega}\subset\mathbb{N}$
such that the following holds

$\theta^{nl_{\omega}l}(C_{\omega})\subset B_{\omega},\ \forall n\in
I_{\omega}.$

For $\upsilon\in C_{\omega}$ and $n\in I_{\omega}$, define that

$(y_{i},\upsilon_{i})=\begin{cases}\phi^{i}(\tilde{g}^{\prime}_{\omega}(\upsilon),\upsilon)&\text{
when }i<nl_{\omega}l\\\
\phi^{i-nl_{\omega}l}(\tilde{g}(\theta^{nl_{\omega}l}\upsilon),\theta^{nl_{\omega}l}\upsilon)&\text{
when }i\geq nl_{\omega}l\end{cases}.$

It is easy to see that $\\{(y_{i},\upsilon_{i})\\}_{i\in\mathbb{Z}}$ is an
$(\upsilon,\frac{3}{2}\delta_{1})$-pseudo orbit. By the choice of $\delta_{1}$
at the beginning of the proof, we have that there exists a unique true orbit
$\\{\phi^{i}(z_{\omega}(\upsilon),\upsilon)\\}_{i\in\mathbb{Z}}$ which is
$(\upsilon,\frac{1}{3}\epsilon_{1})$-shadowing
$\\{(y_{i},\upsilon_{i})\\}_{i\in\mathbb{Z}}$. Note that
$\delta_{1}+\frac{3}{2}\delta_{1}+\frac{1}{3}\epsilon_{1}<\frac{3}{4}\epsilon_{1}$,
then it is obvious that

(13) $\phi^{ml}(z_{\omega}(\upsilon),\upsilon)\in
B_{g}\left(\frac{3}{4}\epsilon_{1}\right),\ \forall m\geq nl_{\omega}.$

By Lemma 3.5, there exist $N_{\omega}\in\mathbb{N}$ such that if one choose
$nl_{\omega}>N_{\omega}$, then
$d_{M}(z_{\omega}(\upsilon),\tilde{g}^{\prime}_{\omega}(\upsilon))<\frac{1}{2}\epsilon^{\prime}_{1}$.
Therefore, by

$|g^{\prime}_{\omega}(\upsilon)-h(\upsilon)|<\frac{1}{3}\epsilon^{\prime}_{1}\text{
and
}|\tilde{g}_{\omega}^{\prime}(\upsilon)-g_{\omega}^{\prime}(\upsilon)|<\frac{1}{3}\epsilon^{\prime}_{1},$

which implies that

(14) $d_{M}(z_{\omega}(\upsilon),h(\upsilon))<\epsilon^{\prime}_{1}.$

So far, we have defined a measurable function $z_{\omega}:C_{\omega}\to M_{1}$
with an integer $N_{\omega}\in\mathbb{N}$ satisfying (13) and (14) for all
$\upsilon\in C_{\omega}$(the measurability of $z_{\omega}$ can be proved in
the exactly same way as the argument in the proof of Theorem 2.3, thus is
omitted here).

Note that $\\{C_{\omega}\\}_{\omega\in\Omega}$ forms an open cover of the
compact set $\Omega$, thus there is an finite open sub-cover which is denoted
by $\\{C_{\omega_{1}},\cdots C_{\omega_{s}}\\}$. The proof is completed if we
take $N=\max\\{N_{\omega_{1}},\cdots,N_{\omega_{s}}\\}$ and define

$h_{1}(\omega)=\begin{cases}z_{\omega_{1}}(\omega)&\text{ when }\omega\in
C_{\omega_{1}}\\\ z_{\omega_{i}}(\omega)&\text{ when }\omega\in
C_{\omega_{i}}\setminus\bigcup_{1\leq j\leq i-1}C_{\omega_{j}},\ \forall 2\leq
i\leq s\end{cases}.$

∎

Let $U$ and $V$ be open sets in $M_{1}$ with an $x\in U$ and a $y\in V$ fixed.
Then there exists $\epsilon_{2}>0$ such that

$B_{M}(x,\epsilon_{2})\subset U\text{ and }B_{M}(y,\epsilon_{2})\subset V.$

Applying Lemma 4.5 by taking $\epsilon_{1}=\epsilon_{2}$ and $g=h_{y}$ where
$h_{y}(\omega)=y$ for all $\omega\in\Omega$, one can derive a $\delta_{2}>0$
correspondingly. By Theorem 2.3, there exists a full random periodic point
$graph(\tilde{h}_{y})\in B_{h_{y}}(\delta_{2})$ with period $pk$. Since $\phi$
is homeomorphism and $\phi^{k}(M_{1}\times\Omega)=M_{1}\times\Omega$, for any
$i\in\mathbb{N}$, $\phi^{ik}(U\times\Omega)$ is open in $M_{1}\times\Omega$
while $\phi^{ik}(graph(h_{x}))$ is compact in $\phi^{ik}(U\times\Omega)$.
Therefore, there exists $\epsilon_{3}\in(0,\epsilon_{2})$ such that

$B_{g_{i}}(\epsilon_{3})\subset\phi^{ik}(U\times\Omega),\ \forall 0\leq i\leq
p-1,$

where $g_{i}\in C(\Omega,M_{1})$ satisfies
$graph(g_{i})=\phi^{ik}(graph(h_{x}))$. Applying Lemma 4.5 $p$ times, we have
that there exist $\\{N_{0},\cdots,N_{p-1}\\}\subset\mathbb{N}$ and measurable
functions $\\{g^{\prime}_{i}:\Omega\to M_{1}\\}_{0\leq i\leq p-1}$ such that
for all $0\leq i\leq p-1$ the following hold

(15) $\displaystyle\begin{split}&graph(g^{\prime}_{i})\subset
B_{g_{i}}(\epsilon_{3})\subset\phi^{ik}(U\times\Omega)\\\
&\quad\quad\quad\quad\quad\quad\quad\text{and}\\\
&\phi^{npk}(graph(g^{\prime}_{i}))\subset B_{h_{y}}(\epsilon_{2})\subset
V\times\Omega,\forall n>N_{i}.\end{split}$

Obviously, (LABEL:E:3) holds if replace $N_{i}$ by
$N:=\max\\{N_{0},\cdots,N_{p-1}\\}$. Therefore, for any $m\in\mathbb{N}$, if
$mk=npk+jk$ where $n>N$ and $j\in[0,p)$, then

$\phi^{npk}(graph(g^{\prime}_{i}))\subset B_{g}(\epsilon_{2})\bigcap\phi^{mk-
jk}(\phi^{jk}(U\times\Omega))\subset
V\times\Omega\bigcap\phi^{mk}(U\times\Omega),$

which implies that for any $\omega\in\Omega$,

$\phi^{mk}(U\times\\{\omega\\})\cap
V\times\\{\theta^{mk}\omega\\}\neq\emptyset.$

The proof of Theorem 2.4 is completed.

###  4.3. Proof of Theorem 2.5, Corollary 2.6 and Theorem 2.9

####  4.3.1. Existence of ergodic measure with positive entropy and full
support.

In this section, we prove the following proposition:

######  Proposition 4.6.

Assume that $\phi$ is transitive and Anosov on fibers, and $(\Omega,\theta)$
is the irrational rotation on Torus. Then there exists a unique
$\phi$-invariant measure (SRB measure) $\mu$ with full support. Moreover

(16) $h_{\mu}(\phi)=\int\sum\lambda_{i}^{+}d\mu>0,$

where $\lambda_{i}$ are the Lyapunov exponents corresponding to $\mu$.

###### Proof.

This proposition can be mainly viewed as a corollary of Theorem B.7 and Remark
B.1. By Theorem 2.4, we have known that $\phi^{k}|_{M_{i}\times\Omega}$ is
topological mixing on fibers, where $k$ is the cardinality of $M_{i}$’s which
are connected components of $M$. Therefore, by Lemma B.3, we have that for
each $i\in\\{1,\cdots,k\\}$, $\phi^{k}|_{M_{i}\times\Omega}$ is random
topological transitive (see Definition B.2). By taking $F=\phi^{k}$ and
$\Lambda=M_{i}\times\Omega$, it is obviously that
$\phi^{k}|_{M_{i}\times\Omega}$ satisfies the conditions of Theorem B.7.
Therefore, for any $i\in\\{1,\cdots,k\\}$ there is a unique
$\phi^{k}$-invairant SRB measure $\mu_{i}$ supported in $M_{i}\times\Omega$.
By Theorem 2.4 and the invertibility of $\phi$, we have that for any $1\leq
i<j\leq k$, $\phi^{k}|_{M_{i}\times\Omega}$ is conjugate to
$\phi^{k}|_{M_{j}\times\Omega}$, and the conjugacy mapping is given by
$\phi^{n}$ for some $n\in\\{1,\cdots,k-1\\}$. Thus
$\mu_{i}=(\phi^{n})^{*}\mu_{j}$ since $\phi^{n}$ is $C^{2}$-map. So the unique
$\phi$-invariant SRB measure $\mu$ satisfies that

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

(16) follows i) of Theorem B.7 and the hyperbolicity of $\phi$.

It remains to show that $supp\ \mu=M\times\Omega$. By iv) of Theorem B.7, each
$\mu_{i}$ is the pullback (by ”$\psi$” in (iv) of Theorem B.7) of a
equilibrium state for an invertible symbolic dynamical system (a two-sided
shift $\sigma$ on $\Sigma_{A}$). By i) of Remark B.1, there is a
$\mathbb{P}$-full measure set, say $\Omega^{\prime}$, such that for any
$\omega\in\Omega^{\prime}$, $\psi|_{\Sigma_{A}(\omega)}$ is a homeomorphism
between $\Sigma_{A}(\omega)$ and an open dense set of $M$, which is denoted by
$M^{\prime}(\omega)$. Since $supp\ \mathbb{P}=\Omega$, any $\mathbb{P}$-full
measure set is dense in $\Omega$. Therefore, the random open set
$U:=\cup_{\omega\in\Omega^{\prime}}M^{\prime}(\omega)\times\\{\omega\\}$ is
dense in $M\times\Omega$. By iii) of Remark B.1, we have that for any
$\omega\in\Omega^{\prime}$ and any open set $V\subset M^{\prime}(\omega)$,
$\mu_{\omega}(V)>0$, where $\mu_{\omega}$ is the disintegration of $\mu$ on
fibers. This means that for $\mathbb{P}$-a.s. $\omega\in\Omega$, one has that
$supp\ \mu_{\omega}=M$, which implies that $supp\ \mu=M\times\Omega$. The
proof is done. ∎

######  Remark 4.2.

Of course, the properties stated in Theorem B.7 are hold by $\mu$ in
Proposition 4.6. However, to focus on the proof of 2.5 and for sake of
simplicity, only the properties of full support and positive entropy are
proposed here.

####  4.3.2. Existence of horseshoe

Let $\mu$ be a $\phi$-invariant ergodic probability measure with
$h_{\mu}(\phi)>0$ and $supp\mu=M\times\Omega$. Let $\mathbb{P}$ be the induced
measure of $\mu$ on $\Omega$, which is a $\theta$-invariant ergodic
probability measure. Note that $supp\ \mathbb{P}=\Omega$.

Consider a finite partition $\xi=\\{\xi_{i}\\}_{1\leq i\leq p}$ of $\Omega$
such that $\xi_{i}$ has nonempty interior and $\mathbb{P}(\partial\xi_{i})=0$
for each $1\leq i\leq p$, where $\partial\xi_{i}$ is the boundary of
$\xi_{i}$. Denote $diam(\xi)$ the maximum of diameters of elements of $\xi$.
In the following, for a $g\in C(\Omega,M)$, we borrow the notations
$B_{g}(\xi_{i},\delta)$ and $B_{g}(\delta)$ as in Section 4.1. Additionally,
we have the following property

$B_{g}(\delta)=\\{(x,\omega)|\omega\in\Omega,\
d_{M}(x,g(\omega))<\delta\\}=\text{ disjoint union of
}B_{g}(\xi_{i},\delta)\text{'s}.$

Note that each $B_{g}(\delta)$ is an open set in $M\times\Omega$, so there are
finite $B_{g}(\delta)$’s which can cover $M\times\Omega$.

Note that the following lemma is under the assumptions that $\mu$ is a
$\phi$-ergodic probability measure with $h_{\mu}(\phi)>0$ and
$supp\mu=M\times\Omega$.

######  Lemma 4.7.

For any $\gamma>0$, there exists $\alpha_{0}>0$ such that for all
$\alpha\in(0,\alpha_{0})$ the following hold: For $\delta>0$ and a finite
Borel partition $\xi=\\{\xi_{i}\\}_{1\leq i\leq p}$ of $\Omega$ described
above with $p$ elements, given any $n_{0}\in\mathbb{N}$, there exist $g\in
C(\Omega,M)$, $\\{E_{i}\\}_{1\leq i\leq p}$, $n>n_{0}$, and $k\in\mathbb{N}$
such that

  * i)

For all $1\leq i\leq p$, $E_{i},\phi^{n}(E_{i})\subset B_{g}(\xi_{i},\delta)$;

  * ii)

For all $1\leq i\leq p$, $E_{i}$ is an $(n,\alpha)$-separated set with respect
to the metric $d(\cdot,\cdot)$ on space $M\times\Omega$;

  * iii)

$|E_{i}|=k$ for all $1\leq i\leq p$, where $|E_{i}|$ is the cardinality of set
$E_{i}$;

  * iv)

$\frac{1}{n}\ln k\geq h_{\mu}(\phi)-\gamma$.

###### Proof.

Consider constant functions in $C(\Omega,M)$. Recall that $h_{x}$ is the
constant function who takes the value $x\in M$. Since $supp\mu=M\times\Omega$,
for any $x_{0}\in M$ and $r_{0}\in(0,1)$ one has that that
$\mu({B_{h_{x_{0}}}(r_{0})})>0$. Note that $h_{\mu}(\phi)>0$, then we apply
Lemma A.1 by taking a small $\gamma>0$,
$\beta=\frac{1}{4}\mu({B_{h_{x_{0}}}(r_{0})})$ to derive an $\alpha_{0}>0$.
Given $\delta>0$, a finite Borel partition $\xi=\\{\xi_{i}\\}_{1\leq i\leq p}$
and $n_{0}\in\mathbb{N}$ as in the statement of Lemma 4.7, we fix an
$\alpha\in(0,\alpha_{0})$ in the rest of the proof.

Next, we construct a Borel subset $S\subset{B_{h_{x_{0}}}(r_{0})}$ with
$\mu(S)\geq 2\beta$ and a finite Borel partition $\xi^{\prime}$ of $S$ which
is based on $\xi$:

Since $\overline{B_{h_{x_{0}}}(r_{0})}$ is compact, there is a finite open
cover of $\overline{B_{h_{x_{0}}}(r_{0})}$ with the form of open straps with
$\delta$ width, which is denoted by
$\\{B_{h_{x_{i}}}(\frac{\delta}{2})\\}_{1\leq i\leq q}$.

Let

$r_{1}=\min_{1\leq i\leq q,1\leq j\leq p}\\{\text{ maximum of the radius of
open balls in }B_{h_{x_{i}}}(\xi_{j},\frac{\delta}{2})\\}.$

It is obvious that $r_{1}>0$, and for each
$B_{h_{x_{i}}}(\xi_{j},\frac{\delta}{2})$, there exists an open ball
$V_{x_{i},\xi_{j}}\subset B_{h_{x_{i}}}(\xi_{j},\frac{\delta}{2})$ with radius
$\frac{1}{3}r_{1}$ such that for any $(x,\omega)\in V_{x_{i},\xi_{j}}$ and
$(y,\upsilon)\in M\times\Omega\setminus
B_{h_{x_{i}}}(\xi_{j},\frac{\delta}{2})$, one has
$d((x,\omega),(y,\upsilon))\geq\frac{1}{2}r_{1}$.

Let

$\Lambda_{N}=\left\\{(x,\omega)\in M\times\Omega\big{|}\ \forall 1\leq i\leq
q,\ 1\leq j\leq p,\exists n\in[1,N]\ s.t.\ \phi^{n}(x,\omega)\in
V_{x_{i},\xi_{j}}\right\\}.$

Note that, since $supp\mu=M\times\Omega$, one has that
$\mu(V_{x_{i},\xi_{j}})>0$ for all $1\leq i\leq q$ and $1\leq j\leq p$.
Therefore, by the Birkhoff ergodic theorem, we have that

$\lim_{N\to\infty}\mu(\Lambda_{N})=1.$

Now, by taking $N$ large enough so that $\mu(\Lambda_{N})>1-\beta$, then one
has that $\mu(\Lambda_{N}\bigcap{B_{h_{x_{0}}}(r_{0})})\geq 3\beta$. Let $V$
be an neighborhood of $M\times\left(\bigcup_{1\leq i\leq
p}\partial\xi_{i}\right)$ and $r_{2}\in(0,\frac{1}{4}r_{1})$ such that

(17) $d((x,\omega),(y,\upsilon))\geq r_{2},\text{ for any }(x,\omega)\in
M\times\left(\bigcup_{1\leq i\leq p}\partial\xi_{i}\right)\text{ and
}(y,\upsilon)\in M\times\Omega\setminus V.$

Note that one can take $r_{2}$ small enough to make $\mu(V)<\beta$ since
$\mathbb{P}\left(\bigcup_{1\leq i\leq p}\partial\xi_{i}\right)=0$.

Now we take

(18) $S=\Lambda_{N}\bigcap\left(B_{h_{x_{0}}}(r_{0})\setminus V\right),$

which clearly implies that $\mu(S)\geq 2\beta$.

By the definition of $\Lambda_{N}$, we have that for any
$l\in\\{1,\cdots,p\\}$ and any $(x,\omega)\in S$ (without losing any
generality, we suppose that $(x,\omega)\in
B_{h_{x_{i}}}(\xi_{j},\frac{\delta}{2})$), there exists
$n_{l}(x,\omega)\in[1,N]$ such that $\phi^{n_{l}(x,\omega)}(x,\omega)\in
V_{x_{i},\xi_{l}}$. Noting that each $\phi^{n_{l}}$ is uniformly continuous
and there are at most $N$ different $n_{l}(x,\omega)$’s, so there exists a
$r_{3}>0$ such that for any $(x,\omega)\in S$, the $r_{3}$-ball in
$M\times\Omega$ centered on $(x,\omega)$, $U^{r_{3}}(x,\omega)$, satisfies the
following property:

$\phi^{n_{l}(x,\omega)}(U^{r_{3}}(x,\omega))\subset\text{interior of
}B_{h_{x_{i}}}(\xi_{l},\delta),\ \forall l\in[1,p]$

It is clearly that, by (17), one has that $U^{r_{3}}(x,\omega)\subset
B_{h_{x_{i}}}(\xi_{j},\delta)$ as long as
$r_{3}\in(0,r_{2})\subset(0,\frac{\delta}{4})$. Here, we also require $r_{3}$
to be small enough such that

(19) $\text{ any two points from }U^{r_{3}}(x,\omega)\text{ are {\bf NOT}
}(N+n_{0}+1,\alpha)-\text{separated}.$

Also note that for any $(y,\upsilon)\in U^{r_{3}}(x,\omega)$, if
$\phi^{m}(y,\upsilon)\in U^{r_{3}}(x,\omega)$, then for any $l\in[1,p]$

(20)
$\phi^{m}(\phi^{n_{l}(x,\omega)}(y,\upsilon))=\phi^{n_{l}(x,\omega)}(\phi^{m}(y,\upsilon))\in\phi^{n_{l}(x,\omega)}(U^{r_{3}}(x,\omega))\subset\text{interior
of }B_{h_{x_{i}}}(\xi_{l},\delta).$

Since each $U^{r_{3}}(x,\omega)$ has the fixed radius $r_{3}$, we have that
$\\{U^{r_{3}}(x,\omega)\\}_{(x,\omega)\in S}$ forms an open cover of
$\overline{S}$, thus there is a finite open sub-cover of $S$, which is denoted
by $\\{U^{r_{3}}(x^{\prime}_{i},\omega_{i})\\}_{1\leq i\leq p^{\prime}}$. Now
we take $\xi^{\prime}$ to be the partitions generated by
$\\{U^{r_{3}}(x^{\prime}_{i},\omega_{i})\bigcap S\\}_{1\leq i\leq
p^{\prime}}$.

Finally, for the fixed $\alpha\in(0,\alpha_{0})$ and a given
$n_{0}\in\mathbb{N}$, $S$ defined in (18) and $\eta=\xi^{\prime}$, then by
applying Lemma A.1, we have that there exist $\hat{C}\in\xi^{\prime}$,
$n>n_{0}+N+1$ and an $(n,\alpha)$-separated set $E$ such that the following
are satisfied:

  * (a)

$E,\phi^{n}(E)\subset\hat{C}$,

  * (b)

$\frac{1}{n}\ln|E|\geq h_{\mu}(\phi)-\gamma$.

Without losing any generality, we suppose $E\subset
U^{r_{3}}(x_{1}^{\prime},\omega_{1})\subset B_{h_{x_{1}}(\xi_{1},\delta)}$.
Then, we define

$g=h_{x_{1}},\ E_{1}=E\text{ and
}E_{i}=\phi^{n_{i}(x_{1}^{\prime},\omega_{1})}(E)\ \forall i\in[2,p],$

where $n_{i}(x_{1}^{\prime},\omega)\in[1,N]$ satisfies that
$\phi^{n_{i}(x_{1}^{\prime},\omega)}(x_{1}^{\prime},\omega_{1})\in
V_{x_{1},\xi_{i}}$. It is obviously that i) follows (20) and (a) above, ii)
follows (19) and the separating property of $E$, and iii) and iv) follow (b)
above. The proof is completed. ∎

######  Remark 4.3.

Note that, from the proof, one can expect additional properties of $g$ and
$E_{i}$ to be satisfied: 1) one can chose the continuous function $g$ to be a
constant function; 2) one can request $E_{i}$ not to be a
$(n_{0},\alpha)$-separated set for $1\leq i\leq p$.

Construction of Horseshoe:

First, we fix a partition of $\Omega$, $\xi$, and make $\hat{\delta}_{1}$, the
diameter of $\xi$, small enough (the condition will be proposed later). Then,
we fix a $\gamma>0$, an $n_{0}=1$ and a $\hat{\delta}_{2}>0$ small enough
(again the condition will be given later). By applying Lemma 4.7 and also keep
in mind Remark 4.3, we have that for a fixed $\alpha\in(0,\alpha_{0})$, there
exist a constant function $h_{x_{0}}\in C(\Omega,M)$, $\\{E_{i}\\}_{1\leq
i\leq p}$, $n>1$, and $k\in\mathbb{N}$ such that

  * E1)

For all $1\leq i\leq p$, $E_{i},\phi^{n}(E_{i})\subset
B_{h_{x_{0}}}(\xi_{i},\hat{\delta}_{2})$;

  * E2)

For all $1\leq i\leq p$, $E_{i}$ is an $(n,\alpha)$-separated set with respect
to the metric $d(\cdot,\cdot)$ on space $M\times\Omega$;

  * E3)

$|E_{i}|=k$ for all $1\leq i\leq p$, where $|E_{i}|$ is the cardinality of set
$E_{i}$;

  * E4)

$\frac{1}{n}\ln k\geq h_{\mu}(\phi)-\gamma$.

Now we fix a label method of each $E_{i}$ using $\\{1,2,\cdots,k\\}$, and then
write $E_{i}=\\{(x_{i,1},\omega_{i,1}),\cdots,(x_{i,k},\omega_{i,k})\\}$.

Second, for a given point
$\hat{a}=(a_{i})_{i\in\mathbb{Z}}\in\mathcal{S}_{k}(=\\{1,\cdots,k\\}^{\mathbb{Z}})$,
we construct two-sides pseudo orbits starting from $graph(h_{x_{0}})$, which
are similarly defined as in (6) for $g=h_{x_{0}}$. For $\omega\in\Omega$ and
$l\in\mathbb{Z}$, suppose that $\theta^{ln}\omega\in\xi_{i_{l}}$, then define

(21)
$(y_{j}^{\prime}(\hat{a},\omega),\omega_{j})=\begin{cases}(x_{0},\theta^{ln}\omega),&\text{
when }j=ln\\\
(\pi_{M}\phi^{j-ln}(x_{i_{l},a_{l}},\omega_{i_{l},a_{l}}),\theta^{j}\omega),&\text{
when }j\in[ln+1,(l+1)n-1]\\\ \end{cases}.$

The following lemma follows the exactly same arguments used in the proof of
Lemma 4.2 and 4.3. The only difference between (9) and (21) is that the point
chosen from $B_{g}(\xi_{i_{l}},\delta)$ may vary in (21), i.e.
$(x_{i_{l},a_{l}},\omega_{i_{l},a_{l}})$ is depending on $a_{l}$, while keep
constant in (9), i.e. $(x_{i_{l}},\omega_{i_{l}})$ is fixed for given $l$.
However, in the proof of Lemma 4.2, the only property being used is (5) which
is satisfied by (21) because of E1). The proof of the following lemma is
omitted.

######  Lemma 4.8.

For any $\epsilon^{\prime}>0$, there exists $\delta^{\prime}>0$ depending on
$\epsilon^{\prime}$ only (in particular, $\delta^{\prime}$ is not depending on
$x_{0}$) such that if $\hat{\delta}_{1}(\text{the diameter of
}\xi),\hat{\delta}_{2}(\text{as in E1})<\delta^{\prime}$, then for any
$\hat{a}\in\mathcal{S}_{k}$ and $\omega\in\Omega$, the pseudo orbit defined by
(21) is an $(\omega,\epsilon^{\prime})$-pseudo orbit.

######  Remark 4.4.

Note that, $\hat{\delta}_{1}$ and $\hat{\delta}_{2}$ need to be fixed at the
very beginning of this construction, but Lemma 4.7 indicate that the constant
function $h_{x_{0}}\in C(\Omega,M)$, $\\{E_{i}\\}_{1\leq i\leq p}$, $n$ and
$k$ used here is depending on the partition $\xi$ (thus $\delta_{1}$) and
$\delta_{2}$. It seems to have a logic circle by proposing more conditions on
$\hat{\delta}_{1}$ and $\hat{\delta}_{2}$ afterwards, since that will change
$h_{x_{0}}$, $\\{E_{i}\\}_{1\leq i\leq p}$, $n$ and $k$ accordingly.
Nevertheless, as long as we are dealing with constant functions only, Lemma
4.8 (also see the argument for Lemma 4.3) tells the insurance of (21) forming
an $(\omega,\epsilon^{\prime})$-pseudo orbit is not depending on the choice of
particular $h_{x_{0}}$. That is why one need the $\delta^{\prime}$ in Lemma
4.8 to be depending on $\epsilon^{\prime}$ only.

By Lemma 3.4, for any $\beta\in(0,\beta_{0})$, if $\epsilon^{\prime}$ (as in
Lemma 4.8) is small enough, the true orbit which is $(\omega,\beta)$-shadowing
the pseudo orbit defined in (21) is unique, for which we denote
$\\{\phi^{i}(x_{0}^{\prime}(\hat{a},\omega),\omega)\\}_{i\in\mathbb{Z}}$. Now
we are ready to define the map $\Psi:\mathcal{S}_{k}\to L^{\infty}(\Omega)$ by
letting

(22) $\Psi(\hat{a})(\omega)=x_{0}^{\prime}(\hat{a},\omega),\
\forall\hat{a}\in\mathcal{S}_{k},\omega\in\Omega,$

where the measurability of $\Psi(\hat{a})$ follows the same argument as in the
end of Section 4.1, and the continuity of $\Psi$ follows Lemma 3.5.
By (21) and the uniqueness of the true orbit, we have that

$\phi^{n}(x_{0}^{\prime}(\hat{a},\omega),\omega)=(x_{0}^{\prime}(\sigma\hat{a},\theta^{n}\omega),\theta^{n}\omega),\
\forall\hat{a}\in\mathcal{S},\omega\in\Omega,$

which implies that (ii) in Definition 2.2 holds for $\phi^{n}$.
To ensure that (i) in Definition 2.2 holds, we need to propose more conditions
on $\hat{\delta}_{1},\hat{\delta}_{2}$:

  * R1)

By Lemma 4.8, taking $\hat{\delta}_{1},\hat{\delta}_{2}$ small enough, one can
make $\epsilon^{\prime}$ small enough so that
$\epsilon^{\prime}<\frac{1}{8}\alpha$ and the true orbit
$\\{\phi^{i}(x_{0}^{\prime}(\hat{a},\omega),\omega)\\}_{i\in\mathbb{Z}}$ is
$\frac{1}{8}\alpha$-shadowing the pseudo orbit defined by (21).

  * R2)

Another concern is that, the $(n,\alpha)$-separation of $E_{i}$ is from both
$M$ direction and $\Omega$ direction. Note that $M$-direction separation come
from the expansive property of the system, while the $\Omega$-direction
separation come from the initial position of the points which is invariant
under iterations. In order to make the $\Omega$-direction separation harmless
(not to potentially fail (i) Definition 2.2), one need to take
$\hat{\delta}_{1}$ small enough (e.g. taking
$\hat{\delta}_{1}<\frac{1}{8}\alpha$ is sufficient).

For any
$\hat{a}_{1}=(a_{1}(i))_{i\in\mathbb{Z}},\hat{a}_{2}=(a_{2}(i))_{i\in\mathbb{Z}}\in\mathcal{S}_{k}$
with $\hat{a}_{1}\neq\hat{a}_{2}$, let $s=\min\\{|i||\ a_{1}(i)\neq
a_{2}(i)\\}$. Then for any $\omega\in\Omega$, by E2) and R2) above, we have
that

$d\left(\phi^{r}(\Psi(\hat{a}_{1})(\omega),\omega),\phi^{r}(\Psi(\hat{a}_{2})(\omega),\omega)\right)>\frac{1}{2}\alpha,\text{
for some }r\in[-(s+1)n,(s+1)n].$

Note that there is a constant $L>1$ such that for any $\omega\in\Omega$ and
$x,y\in M$ with $d(x,y)>0$, one has

$d(\phi^{\pm 1}(x,\omega),\phi^{\pm 1}(y,\omega))\leq Ld_{M}(x,y).$

Therefore

$\displaystyle d_{M}(\Psi(\hat{a}_{1})(\omega),\Psi(\hat{a}_{2})(\omega))$
$\displaystyle=$ $\displaystyle
d((\Psi(\hat{a}_{1})(\omega),\omega),(\Psi(\hat{a}_{2})(\omega),\omega))$
$\displaystyle\geq$ $\displaystyle
L^{-|r|}d\left(\phi^{r}(\Psi(\hat{a}_{1})(\omega),\omega),\phi^{r}(\Psi(\hat{a}_{2})(\omega),\omega)\right)$
$\displaystyle\geq$ $\displaystyle\frac{1}{2}\alpha L^{-(s+1)n},$

which implies both (i) in Definition 2.2 and the continuity of $\Psi^{-1}$.
The proof is completed.

####  4.3.3. Proof of Corollary 2.6

For $g\in L^{\infty}(\Omega)$, one can define a Borel probability measure,
$\mu_{g}$, in $\mathcal{M}(M\times\Omega)$ by the following: for any Borel set
$A\subset M\times\Omega$

(23) $\mu_{g}(A)=\mathbb{P}\left(\left\\{\omega\in\Omega\big{|}\
(g(\omega),\omega)\in A\cap M\times\\{\omega\\}\right\\}\right).$

Then for any Borel function $h:M\times\Omega\to\mathbb{R}$

$\int_{M\times\Omega}hd\mu_{g}=\int_{\Omega}h(g(\omega))d\mathbb{P}.$

Note that $g_{n}\to g$ in $L^{\infty}(\Omega)$ means that $g_{n}(\omega)\to
g(\omega)$ uniformly on $\omega$, therefore for any continuous
$h:M\times\Omega\to\mathbb{R}$ (thus is uniformly continuous because of
compactness of $M\times\Omega$), the following is true:

$\displaystyle\left|\int_{M\times\Omega}hd\mu_{g_{i}}-\int_{M\times\Omega}hd\mu_{g}\right|=$
$\displaystyle\left|\int_{\Omega}h(g_{i}(\omega))d\mathbb{P}-\int_{\Omega}h(g(\omega))d\mathbb{P}\right|$
$\displaystyle\leq$
$\displaystyle\sup_{\omega\in\Omega}\left\\{|h(x,\omega)-h(y,\omega)|\big{|}\
d_{M}(x,y)\leq d_{L^{\infty}}(g_{i},g)\right\\}$ $\displaystyle\to$
$\displaystyle 0\quad\quad\text{ as }i\to\infty.$

So the map $\tilde{\mu}:L^{\infty}(\Omega)\to\mathcal{M}(M\times\Omega)$ with
$\tilde{\mu}(g):=\mu_{g}$ is continuous.

Let $\\{graph(\Psi(\hat{a}))\\}_{\hat{a}\in\mathcal{S}_{k}}$ be a full random
horseshoe of $\phi$ of $k$ symbols, where $\Psi:\mathcal{S}_{k}\to
L^{\infty}(\Omega)$ satisfies the conditions in Definition 2.2. Then $\Psi$
together with (23) induces a map
$\tilde{\Psi}:\mathcal{S}_{k}\to\mathcal{M}(M\times\Omega)$ by taking

$\tilde{\Psi}(\hat{a})=\mu_{\Psi(\hat{a})}\text{ for all
}\hat{a}\in\mathcal{S}_{k}.$

Since $\Psi$ is the composition of two continuous maps, thus $\Psi$ is
continuous.

Next we are to show $\tilde{\Psi}$ is injective. For any
$\hat{a}_{1},\hat{a}_{2}\in\mathcal{S}_{k}$ with $\hat{a}_{1}\neq\hat{a}_{2}$,
by i) of Definition 2.2 there exists $\epsilon\in(0,\frac{1}{6})$ such that

$\inf_{\omega\in\Omega}\\{d_{M}(\Psi({\hat{a}_{1}})(\omega),\Psi({\hat{a}_{2}})(\omega))\\}>\epsilon.$

By Lusin’s Theorem, there are two continuous maps $g_{1},g_{2}\in C(\Omega,M)$
and a compact subset $E$ of $\Omega$ such that $\mathbb{P}(\Omega\setminus
E)<\epsilon$ and $g_{i}(w)=\Psi({\hat{a}_{i}})(w)$ for $w\in E$ and $i=1,2$.
Then by Uryson’s Lemma, there exists $h:M\times\Omega\to[0,1]$ such that the
following holds

  * a)

$\left|\int_{M\times\Omega}hd\mu_{\Psi(\hat{a}_{i})}-\int_{M\times\Omega}hd\mu_{g_{i}}\right|<\epsilon$
for $i=1,2$;

  * b)

$\left|\int_{M\times\Omega}hd\mu_{g_{1}}-\int_{M\times\Omega}hd\mu_{g_{2}}\right|>1-4\epsilon$.

Thus

$\int_{M\times\Omega}hd\mu_{\Psi(\hat{a}_{1})}\neq\int_{M\times\Omega}hd\mu_{\Psi(\hat{a}_{2})}\text{
i.e. }\tilde{\Psi}(\hat{a}_{1})\neq\tilde{\Psi}(\hat{a}_{2}),$

which implies that $\tilde{\Psi}$ is injective. Since $\mathcal{S}_{k}$ is
compact and $\tilde{\Psi}$ is continuous, $\tilde{\Psi}(\mathcal{S}_{k})$ is
compact. So far, we have shown that
$\tilde{\Psi}:\mathcal{S}_{k}\to\tilde{\Psi}(\mathcal{S}_{k})$ is a one-to-one
continuous map from $\mathcal{S}_{k}$ (compact set) to
$\tilde{\Psi}(\mathcal{S}_{k})$ (compact set). Therefore,
$(\tilde{\Psi})^{-1}|_{\tilde{\Psi}(\mathcal{S}_{k})}$ is continuous
automatically. Thus, $\tilde{\Psi}$ is an embedding of $\mathcal{S}_{k}$ into
$\mathcal{M}(M\times\Omega)$.

Another quick observation based on (4) and (23) is the following

(24) $\phi^{*}(\mu_{g})=\mu_{g^{\prime}},\text{ where
}graph(g^{\prime})=\phi(graph(g)),$

which together with ii) of Definition 2.2 implies that

$\phi^{*}(\tilde{\Psi}(\hat{a}))=\phi^{*}(\mu_{\Psi(\hat{a})})=\mu_{\Psi(\sigma\hat{a})}=\tilde{\Psi}(\sigma\hat{a}),\
\forall\hat{a}\in\mathcal{S}_{k}.$

Hence we conclude that full random horseshoe of $\phi$ of $k$ symbols will
induce a bi-invariant horseshoe with $k$ symbols of $\phi^{*}$, thus Theorem
2.5 implies Corollary 2.6.

####  4.3.4. Proof of Theorem 2.9

Note that, unlike the deterministic case, Theorem 2.9 can not be viewed as a
corollary of Theorem 2.5. Nevertheless, one can construct the desired random
weak horseshoe (see Definition 2.8) by modifying the process of construction
of full random horseshoe in Section 4.3.2.

Suppose that we already have $\xi=\\{\xi_{i}\\}_{1\leq i\leq p}\ (\text{a
partition of }\Omega),\ \hat{\delta}_{1}(\text{diameter of }\xi),\ \gamma>0,\
n_{0}\geq 1,\ \hat{\delta}_{2}>0,\ \alpha\in(0,\alpha_{0}),\text{ constant
function }h_{x_{0}}\in C(\Omega,M),\ \\{E_{i}\\}_{1\leq i\leq p},\ n>1,\
k\in\mathbb{N}$ been settled as in Section 4.3.2 (right after Remark 4.3), and
all the conditions proposed for these parameters are satisfied (including
E1)-E4) and R1), R2)).

To construct the weak horseshoe, we first choose three distinct points,
denoted by $(z_{1,1},\omega_{1,1})$, $(z_{1,2},\omega_{1,2})$ and
$(z_{1},\omega_{1})$, from $E_{1}$, and for each $2\leq i\leq p$, one point,
denoted $(z_{i},\omega_{i})$, from $E_{i}$. Since
$(z_{1,1},\omega_{1,1}),(z_{1,2},\omega_{1,2})$ are $(n,\alpha)$-separated,
and since $n_{0}\geq 1$ and 1) of Remark 4.3, there exists $q\in(0,n-1]$ such
that

$d(\phi^{q}(z_{1,1},\omega_{1,1}),\phi^{q}(z_{1,2},\omega_{1,2}))>\alpha.$

By R2), we have that

$d_{M}(\pi_{M}\phi^{q}(z_{1,1},\omega_{1,1}),\pi_{M}\phi^{q}(z_{1,2},\omega_{1,2}))>\frac{7}{8}\alpha.$

Define the closed balls in $M$ as the following

(25)
$U_{i}=\overline{B\left(\pi_{M}\phi^{q}(z_{1,i},\omega_{1,i}),\frac{1}{8}\alpha\right)}\text{
for }i=1,2.$

It is easy to see that $U_{1},U_{2}$ satisfies (1) of Definition 2.8. Since
$\xi_{1}$ has nonempty interior, and $\theta^{n}$ is an minimal map on
$\Omega$, there exists an integer $K_{1}>0$ such that

$\bigcup_{j=1}^{K_{1}}\theta^{-jn}\xi_{1}=\Omega,$

which implies that for any $\omega\in\Omega$,
$n^{\prime}(\omega):=\min\\{j\in\mathbb{N}|\ \theta^{jn}\omega\in\xi_{1}\\}$
exists and is $\leq K_{1}$. It is easy to see that
$n^{\prime}:\Omega\to\mathbb{N}$ is measurable. Inductively, we define that

$\displaystyle n_{1}^{\prime\prime}(\omega)$
$\displaystyle=nn^{\prime}(\omega)$ $\displaystyle
n_{2}^{\prime\prime}(\omega)$
$\displaystyle=n(n^{\prime}(\theta^{nn_{1}^{\prime\prime}(\omega)}\omega)+n_{1}^{\prime\prime}(\omega))$
$\displaystyle\cdots$ $\displaystyle n_{i+1}^{\prime\prime}(\omega)$
$\displaystyle=n(n^{\prime}(\theta^{nn_{i}^{\prime\prime}(\omega)}\omega)+n_{i}^{\prime\prime}(\omega))$
$\displaystyle\cdots$

Note that, by definition of $n_{i}^{\prime\prime}$’s the following are
satisfied:

  * N1)

For each $i\in\mathbb{N}$, $n_{i}^{\prime\prime}:\Omega\to\mathbb{N}$ is
measurable.

  * N2)

For all $\omega\in\Omega$,
$\theta^{n_{i}^{\prime\prime}(\omega)}\omega\in\xi_{1}$, and for any
$l\in\mathbb{N}$ with
$ln\in(n_{i}^{\prime\prime}(\omega),n_{i+1}^{\prime\prime}(\omega))$,
$\theta^{ln}\omega\notin\xi_{1}$.

  * N3)

For all $\omega\in\Omega$ and $i\in\mathbb{N}$,
$0<n_{i+1}^{\prime\prime}(\omega)-n_{i}^{\prime\prime}(\omega)<nK_{1}$.

Now, for a given point
$\hat{a}=(a_{i})_{i\in\mathbb{N}}\in\mathcal{S}_{2}^{+}\
(=\\{1,2\\}^{\mathbb{N}})$, we are ready to construct two-sides pseudo orbits
starting from $graph(h_{x_{0}})$, which are similarly defined as in (21). For
$\omega\in\Omega$ and $l\in\mathbb{N}$, suppose that
$\theta^{ln}\omega\in\xi_{i_{l}}$, then define

(26)
$(y_{j}^{\prime}(\hat{a},\omega),\omega_{j})=\begin{cases}\phi^{j}(x_{0},\omega),&\text{
when }j\leq 0\\\ (x_{0},\theta^{ln}\omega),&\text{ when }j=ln,\ l\geq 1\\\
(\pi_{M}\phi^{j-n_{i}^{\prime\prime}(\omega)}(z_{1,a_{i}},\omega_{1,a_{i}}),\theta^{j}\omega),&\text{
when }j\in[n_{i}^{\prime\prime}(\omega)+1,n_{i}^{\prime\prime}(\omega)+n-1]\\\
(\pi_{M}\phi^{j-ln(\omega)}(z_{i_{l}},\omega_{i_{l}}),\theta^{j}\omega),&\text{
otherwise, when }j\in[ln+1,(l+1)n-1]\\\ \end{cases}.$

By the exact same argument in Section 4.3.2 (thus details of the proof are
omitted), $\\{(y_{j}^{\prime}(\hat{a},\omega),\omega_{j})\\}_{i\in\mathbb{Z}}$
is $(\omega,\frac{1}{8}\alpha)$-shadowed by a true orbit
$\\{\phi^{j}(x_{0}^{\prime}(\hat{a},\omega),\omega)\\}_{j\in\mathbb{Z}}$.
Therefore, for any $\omega\in\Omega$ and $i\in\mathbb{N}$, let
$n_{i}(\omega)=n_{i}^{\prime\prime}(\omega)+q$, then one has that

$\displaystyle
d_{M}(\pi_{M}\phi^{n_{i}(\omega)}(x_{0}^{\prime}(\hat{a},\omega),\omega),\pi_{M}\phi^{n_{i}(\omega)-n_{i}^{\prime\prime}(\omega)}(z_{1,a_{i}}))$
$\displaystyle=$ $\displaystyle
d_{M}(\pi_{M}\phi^{n_{i}(\omega)}(x_{0}^{\prime}(\hat{a},\omega),\omega),\pi_{M}\phi^{q}(z_{1,a_{i}}))$
$\displaystyle\leq$ $\displaystyle\frac{1}{8}\alpha,$

which implies that
$\pi_{M}\phi^{n_{i}(\omega)}(x_{0}^{\prime}(\hat{a},\omega),\omega)\in
U_{a_{i}}$ (see defintion of $U_{i}$ in (25)). By N3), we have that
$0<n_{i+1}(\omega)-n_{i}(\omega)<K$ if one take $K=nK_{1}$, thus (2) of
Definition 2.8 is satisfied.

The proof is completed.

##  5\. Examples: Fiber Anosov maps on 2-d tori

Let
$\phi:\mathbb{T}^{2}\times\mathbb{T}\rightarrow\mathbb{T}^{2}\times\mathbb{T}$
with

$\phi\left(\binom{x}{y},\omega\right)=\left(A\binom{x}{y}+h(\omega),\omega+\alpha\right),$

where $\alpha\in\mathbb{R}\setminus\mathbb{Q}$,
$A=\left(\begin{array}[]{cc}1&1\\\ 2&1\end{array}\right)$ and
$h(\omega)=\binom{h_{1}(\omega)}{h_{2}(\omega)}$ is a continuous map from
$\mathbb{T}$ to $\mathbb{T}^{2}$. It is clear that $\phi$ is a quasi-periodic
forced system and $\phi$ induces a cocycle over $(\mathbb{T},\theta)$, where
$\theta(\omega)=\omega+\alpha$.

######  Lemma 5.1.

$\phi$ is transitive and Anosov on fibers.

###### Proof.

Let $f_{\omega}\binom{x}{y}=A\binom{x}{y}+h(\omega)$ for
$\omega\in\mathbb{T}$. Then
$\phi\left(\binom{x}{y},\omega\right)=\left(f_{\omega}\binom{x}{y},\omega+\alpha\right)$.
Note that $Df_{\omega}\binom{x}{y}=A$ for any
$(\binom{x}{y},\omega)\in\mathbb{T}^{2}\times\mathbb{T}$ and $A$ is a
hyperbolic matrix. Hence $(\mathbb{T}^{2}\times\mathbb{T},\phi)$ is Anosov on
fibers.

Let $\mu$ be the Lebesgue measure on $\mathbb{T}^{2}\times\mathbb{T}$. To show
that $T$ is transitive, it is sufficient to show that
$(\mathbb{T}^{2}\times\mathbb{T},\mathcal{B}_{\mathbb{T}^{2}\times\mathbb{T}},\phi,\mu)$
is an ergodic measure-preserving system. First, it is clear that
$(\mathbb{T}^{2}\times\mathbb{T},\mathcal{B}_{\mathbb{T}^{2}\times\mathbb{T}},\phi,\mu)$
is a measure-preserving system. In the following we show that
$(\mathbb{T}^{2}\times\mathbb{T},\mathcal{B}_{\mathbb{T}^{2}\times\mathbb{T}},\phi,\mu)$
is ergodic.

Let $f\in L^{2}(\mu)$ with $f\circ\phi=f$ $\mu$-a.e.. Let

$f\left(\binom{x}{y},\omega\right)=\sum_{(k,l,n)\in\mathbb{Z}^{3}}c_{k,l,n}e^{2\pi
i\left(k\omega+(l,n)\binom{x}{y}\right)}$

be the Fourier series of $f$ on $\mathbb{T}^{2}\times\mathbb{T}$. Then

$\sum_{(k,l,n)\in\mathbb{Z}^{3}}|c_{k,l,n}|^{2}=\|f\|_{L^{2}(\mu)}<+\infty.$

Fix $(n^{\prime},l^{\prime})\in\mathbb{Z}^{2}$. For any $g\in
L^{2}(m_{\mathbb{T}})$, where $m_{\mathbb{T}}$ is the Lebesgue measure on
$\mathbb{T}$, one has

$\displaystyle\hskip
14.22636pt\int\left(\sum_{k\in\mathbb{Z}}c_{k,l^{\prime},n^{\prime}}e^{2\pi
ik\omega}\right)g(\omega)dm_{\mathbb{T}}(\omega)$
$\displaystyle=\int\left(\sum_{(k,l,n)\in\mathbb{Z}^{3}}c_{k,l,n}e^{2\pi
i\left(k\omega+(l,n)\binom{x}{y}\right)}\right)g(\omega)e^{-2\pi
i(l^{\prime},n^{\prime})\binom{x}{y}}d\mu\left(\binom{x}{y},\omega\right)$
$\displaystyle=\int f\left(\binom{x}{y},\omega\right)g(\omega)e^{-2\pi
i(l^{\prime},n^{\prime})\binom{x}{y}}d\mu\left(\binom{x}{y},\omega\right)$
$\displaystyle=\int
f\left(\phi^{-1}\left(\binom{x}{y},\omega\right)\right)g(\omega)e^{-2\pi
i(l^{\prime},n^{\prime})\binom{x}{y}}d\mu\left(\binom{x}{y},\omega\right)$
$\displaystyle=\int f\left(\binom{x}{y},\omega\right)g(\omega+\alpha)e^{-2\pi
i(l^{\prime},n^{\prime})\left(A\binom{x}{y}+h(\omega)\right)}d\mu\left(\binom{x}{y},\omega\right)$
$\displaystyle=\int\left(\sum_{(k,l,n)\in\mathbb{Z}^{3}}c_{k,l,n}e^{2\pi
i\left(k\omega+(l,n)\binom{x}{y}\right)}\right)g(\omega+\alpha)e^{-2\pi
i(l^{\prime},n^{\prime})\left(A\binom{x}{y}+h(\omega)\right)}d\mu\left(\binom{x}{y},\omega\right)$
$\displaystyle=\int\left(\sum_{k\in\mathbb{Z}}c_{k,(l^{\prime},n^{\prime})A}e^{2\pi
ik\omega}\right)e^{-2\pi
i(l^{\prime},n^{\prime})h(\omega)}g(\omega+\alpha)dm_{\mathbb{T}}(\omega)$
$\displaystyle=\int\left(\sum_{k\in\mathbb{Z}}c_{k,(l^{\prime},n^{\prime})A}e^{2\pi
ik(\omega-\alpha)}\right)e^{-2\pi
i(l^{\prime},n^{\prime})h(\omega-\alpha)}g(\omega)dm_{\mathbb{T}}(\omega).$

Since the above equality holds for any $g\in L^{2}(m_{\mathbb{T}})$, one has

(27) $\sum_{k\in\mathbb{Z}}c_{k,l^{\prime},n^{\prime}}e^{2\pi
ik\omega}=\left(\sum_{k\in\mathbb{Z}}c_{k,(l^{\prime},n^{\prime})A}e^{2\pi
ik(\omega-\alpha)}\right)e^{-2\pi i(l^{\prime},n^{\prime})h(\omega-\alpha)}$

in $L^{2}(m_{\mathbb{T}})$. Thus

$\int\left|\sum_{k\in\mathbb{Z}}c_{k,l^{\prime},n^{\prime}}e^{2\pi
ik\omega}\right|^{2}dm_{\mathbb{T}}(\omega)=\int\left|\left(\sum_{k\in\mathbb{Z}}c_{k,(l^{\prime},n^{\prime})A}e^{2\pi
ik(\omega-\alpha)}\right)e^{-2\pi
i(l^{\prime},n^{\prime})h(\omega-\alpha)}\right|^{2}dm_{\mathbb{T}}(\omega).$

Hence

$\sum_{k\in\mathbb{Z}}|c_{k,l^{\prime},n^{\prime}}|^{2}=\sum_{k\in\mathbb{Z}}|c_{k,(l^{\prime},n^{\prime})A}|^{2}.$

When $(l^{\prime},n^{\prime})\neq(0,0)$,
$(l^{\prime},n^{\prime})A^{m}\neq(l^{\prime},n^{\prime})A^{k}$ for
$m,k\in\mathbb{N}$ with $m\neq k$. Hence if $(l^{\prime},n^{\prime})\neq(0,0)$
then for any $N\in\mathbb{N}$,

$\displaystyle\sum_{k\in\mathbb{Z}}|c_{k,l^{\prime},n^{\prime}}|^{2}=\frac{1}{N}\sum_{i=0}^{N-1}\sum_{k\in\mathbb{Z}}|c_{k,(l^{\prime},n^{\prime})A^{i}}|^{2}\leq\frac{1}{N}\sum_{(k,l,n)\in\mathbb{Z}^{3}}|c_{k,l,n}|^{2}\leq\frac{1}{N}\|f\|_{L^{2}(\mu)}.$

Let $N\nearrow+\infty$, we get
$\sum_{k\in\mathbb{Z}}|c_{k,l^{\prime},n^{\prime}}|^{2}=0$ for
$(l^{\prime},n^{\prime})\neq(0,0)$. When $(l^{\prime},n^{\prime})=(0,0)$, by
(27) one has

$\sum_{k\in\mathbb{Z}}c_{k,0,0}e^{2\pi
ik\omega}=\sum_{k\in\mathbb{Z}}c_{k,0,0}e^{-2\pi ik\alpha}e^{2\pi ik\omega}$

in $L^{2}(m_{\mathbb{T}})$. Thus $c_{k,0,0}=c_{k,0,0}e^{-2\pi ik\alpha}$ for
$k\in\mathbb{Z}$. When $k\neq 0$, one has $c_{k,0,0}=0$ as
$\alpha\in\mathbb{R}\setminus\mathbb{Q}$. Summing over we get $f=c_{0,0,0}$.
This shows that
$(\mathbb{T}^{2}\times\mathbb{T},\mathcal{B}_{\mathbb{T}^{2}\times\mathbb{T}},\phi,\mu)$
is ergodic. The proof is completed. ∎

For a continuous function $p$ from $\mathbb{T}$ to $\mathbb{T}$, we denote the
degree of $p$ by $deg(p)$.

######  Lemma 5.2.

Let $deg(h_{i})=n_{i}$ for $i=1,2$. If $n_{2}$ is odd number, then there are
no $g\in C(\mathbb{T},\mathbb{T}^{2})$ and positive integer $n$ such that
$\phi^{n}(graph(g))=graph(g)$, that is,
$(\mathbb{T}^{2}\times\mathbb{T},\phi)$ has no periodic continuous graph.

###### Proof.

If this is not true, then there are
$g(\omega)=\binom{g_{1}(\omega)}{g_{2}(\omega)}\in
C(\mathbb{T},\mathbb{T}^{2})$ and positive integer $n$ such that
$\phi^{n}(graph(g))=graph(g)$, i.e.

$\phi^{n}(g(\omega),\omega)=(g(\omega+n\alpha),\omega+n\alpha)$

for $\omega\in\mathbb{T}$. Let $deg(g_{i})=k_{i}$ for $i=1,2$. Note that

$\phi^{n}(g(\omega),\omega)=\left(A^{n}\binom{g_{1}(\omega)}{g_{2}(\omega)}+A^{n-1}\binom{h_{1}(\omega)}{h_{2}(\omega)}+\cdots+\binom{h_{1}(\omega+(n-1)\alpha)}{h_{2}(\omega+(n-1)\alpha)},\omega+n\alpha\right)$

Hence

$\binom{g_{1}(\omega+n\alpha)}{g_{2}(\omega+n\alpha)}=A^{n}\binom{g_{1}(\omega)}{g_{2}(\omega)}+A^{n-1}\binom{h_{1}(\omega)}{h_{2}(\omega)}+A^{n-2}\binom{h_{1}(\omega+\alpha)}{h_{2}(\omega+\alpha)}+\cdots+\binom{h_{1}(\omega+(n-1)\alpha)}{h_{2}(\omega+(n-1)\alpha)}.$

Compare the degree of continuous functions from $\mathbb{T}$ to $\mathbb{T}$
appearing in the above equation, one has

$\binom{k_{1}}{k_{2}}=A^{n}\binom{k_{1}}{k_{2}}+A^{n-1}\binom{n_{1}}{n_{2}}+A^{n-2}\binom{n_{1}}{n_{2}}+\cdots+\binom{n_{1}}{n_{2}}.$

That is

$(I-A^{n})\binom{k_{1}}{k_{2}}=(I+A+\cdots+A^{n-1})\binom{n_{1}}{n_{2}}.$

Since $\det(I-A^{n})\neq 0$ and $(I-A)(I+A+\cdots+A^{n-1})=I-A^{n}$, one has

$(I-A)\binom{k_{1}}{k_{2}}=\binom{n_{1}}{n_{2}}.$

This implies $k_{1}=-\frac{n_{2}}{2}\not\in\mathbb{Z}$, a contradiction. The
proof is completed. ∎

##  Appendix A Capturing Entropy

The results in this section are mainly borrowed from Section 6 in [25], which
are essentially taken from [20]. The notation and setting here is separated
from that in the rest of this paper.

Let $T:X\to X$ be a continuous map of a compact metric space with metric
$d(\cdot,\cdot)$, and let $\nu$ be an ergodic $T$ -invariant Borel probability
measure on $X$. For $n\in\mathbb{Z}^{+}$, recall that the $d^{T}_{n}$ -metric
on X is defined by

$d^{T}_{n}(x,y)=\max_{0\leq i\leq n-1}d(T^{i}(x),T^{i}(y)).$

Balls in this metric are denoted by $B_{d^{T}_{n}}(\cdot)$. For
$\alpha,\beta>0$, let $N(n,\alpha;\beta)$ denote the minimum number of
$\alpha-balls$ in the $d^{T}_{n}$-metric needed to cover a set of measure
$\geq\beta$ in $X$. Then for any $\beta\in(0,1)$,

$h_{\nu}(T)=\lim_{\alpha\to 0}\liminf_{n\to\infty}\frac{1}{n}\ln
N(n,\alpha;\beta).$

######  Lemma A.1.

Assume $h_{\nu}(T)>\in(0,\infty)$. Given $\gamma>0$ and
$\beta\in(0,\frac{1}{2})$, there exists $\alpha_{0}>0$ such that the following
holds for all $\alpha\in(0,\alpha_{0})$: Let $S\subset X$ be a Borel subset
with $\nu(S)\geq 2\beta$, and let $\eta$ be a finite measurable partition of
$S$. Then given any $n_{0}\in\mathbb{Z}^{+}$, there exist $\hat{C}\in\eta$,
$n>n_{0}$ and an $(n,\alpha)$-separated set $E$ such that

  * (a)

$E,T^{n}(E)\subset\hat{C}$,

  * (b)

$\frac{1}{n}\ln|E|\geq h_{\nu}(T)-\gamma$.

##  Appendix B Random Dynamical Systems

Let $(\Omega,\mathcal{B},\mathbb{P})$ be a probability space, $\mathbb{E}$ be
a locally compact Hausdorff second countable topological space. Let
$\mathcal{F},\mathcal{K},\mathcal{G}$ denote respectively the family of all
closed, compact and nonempty open sets of $\mathbb{E}$.

######  Definition B.1.

A map $X:\Omega\to\mathcal{F}$ is called a random closed set if, for every
compact set $K\subset\mathbb{E}$,

$\\{\omega\in\Omega|X(\omega)\cap K\neq\emptyset\\}\in\mathcal{B}.$

A map $Y:\Omega\to\mathcal{G}$ is called a random open set if its complement
$X=Y^{c}$ (by this, we mean $X(\omega)=E\setminus Y(\omega)$ for
$\mathbb{P}$-a.e. $\omega\in\Omega$) is a random closed set. Since
$\\{\omega\in\Omega|Y^{c}(\omega)\cap
F=\emptyset\\}=\\{\omega\in\Omega|F\subset Y(\omega)\\}$, $Y$ is random open
set if and only if $\\{\omega\in\Omega|F\subset Y(\omega)\\}$ is a measurable
event for every $F\in\mathcal{F}$.

Let $\theta:(\Omega,\mathcal{B},\mathbb{P})\to(\Omega,\mathcal{B},\mathbb{P})$
be an invertible ergodic metric dynamical system, and
$F:\mathbb{E}\times\Omega\to\mathbb{E}\times\Omega$ be a continuous random
dynamical system over this metric dynamical system.

######  Definition B.2.

$F$ is called random topological transitive if for any random open sets $U,V$
with $U(\omega),V(\omega)\neq\emptyset$ for all $\omega\in\Omega$, there
exists a random variable $n$ taking values in $\mathbb{Z}$ such that the
intersection
$F^{n(\omega)}(\theta^{-n(\omega)}\omega,U(\theta^{-n(\omega)}\omega))\cap
V(\omega)$ is non-empty $\mathbb{P}$-a.s..

The following is a technical lemma whose proof is included for sake of
completeness.

######  Lemma B.3.

$F$ is topological mixing on fibers (see Section 2 for definition) implies
that $F$ is random topological transitive.

###### Proof.

Let $U$ and $V$ be random open sets with $U(\omega),V(\omega)\neq\emptyset$
for $\mathbb{P}$-a.e. $\omega\in\Omega$. Let $\\{x_{i}\\}_{i\in\mathbb{N}}$ be
a countable dense set of $\mathbb{E}$. For $i,j\in\mathbb{N}$, define

$\Omega^{U}_{i,j}=\left\\{\omega|\
\overline{B\left(x_{i},\frac{1}{j}\right)}\subset U(\omega)\right\\}\text{ and
}\Omega^{V}_{i,j}=\left\\{\omega|\
\overline{B\left(x_{i},\frac{1}{j}\right)}\subset V(\omega)\right\\}.$

It is easy to see that each $\Omega^{U}_{i,j}$ or $\Omega^{V}_{i,j}$ is
measurable, and
$\mathbb{P}(\cup_{i,j}\Omega^{U}_{i,j})=\mathbb{P}(\cup_{i,j}\Omega^{V}_{i,j})=1$.
Without losing any generality, we suppose that
$\mathbb{P}(\Omega^{U}_{i_{0},j_{0}})>0$ for some $i_{0},j_{0}\in\mathbb{N}$.
Since $F$ is topological mixing on fibers, for any $i,j\in\mathbb{N}$, there
exists $N_{i,j}\in\mathbb{N}$ such that for any $k>N_{i,j}$

(28)
$F^{k}\left(B\left(x_{i_{0}},\frac{1}{j_{0}}\right)\times\\{\theta^{-k}\omega\\}\right)\cap
B\left(x_{i},\frac{1}{j}\right)\times\\{\omega\\}\neq\emptyset,\
\forall\omega\in\Omega.$

By ergodicity of $\theta^{-1}$ and Poincare Recurrence Theorem, we have that
for $\mathbb{P}$-a.s. $\omega\in\Omega$,
$\\{\theta^{-k}\omega\\}_{k\in\mathbb{N}}$ will visit
$\Omega^{U}_{i_{0},j_{0}}$ infinitely many times. Given $i,j\in\mathbb{N}$,
for $\omega\in\Omega^{V}_{i,j}$, define that

$n(\omega)=\min\\{k|\ k>N_{i,j},\
\theta^{-k}\omega\in\Omega^{U}_{i_{0},j_{0}}\\}.$

It is easy to see that $n:\Omega_{i,j}^{V}\to\mathbb{N}$ is measurable and
$n(\omega)<\infty$ for $\mathbb{P}$-a.s. $\omega$. Additionally, (28) implies
that for $\mathbb{P}$-a.s. $\omega\in\Omega^{V}_{i,j}$, the following holds

(29)
$\displaystyle\begin{split}&F\left(n(\omega),U(\omega)\times\left\\{\theta^{-n(\omega)}\omega\right\\}\right)\cap
V(\omega)\times\\{\omega\\}\\\
\supset&F\left(n(\omega),B\left(x_{i_{0}},\frac{1}{j_{0}}\right)\times\left\\{\theta^{-n(\omega)}\omega\right\\}\right)\cap
B\left(x_{i},\frac{1}{j}\right)\times\\{\omega\\}\\\
\neq&\emptyset.\end{split}$

Although we only give the definition of the integer valued function $n$ on
$\Omega_{i,j}^{V}$, it is easy to extend $n$ onto $\Omega$ for which
(LABEL:E:TTran) still holds for $\mathbb{P}$-a.e. $\omega\in\Omega$, since
there are only countable many $\Omega^{V}_{i,j}$s. Thus $F$ is random
topological transitive. ∎

The definitions and results collected in this section are mainly from [17] and
[13] whose proofs are omitted. Let $\Lambda$ be a random hyperbolic attractor
of $F$, where the $F(\omega)$ is uniformly hyperbolic on $\Lambda(\omega)$
$\mathbb{P}$-a.s.. For definition of random hyperbolic attractors, we refer
the reader to [17]; while for more details about random attractors, we refer
the reader to [4].

######  Definition B.4.

Let $\delta$ be a strictly positive random variable on
$(\Omega,\mathcal{B},\mathbb{P})$. Then for any $\omega\in\Omega$ a sequence
$\\{y_{n}\\}_{n\in\mathbb{Z}}$ in $\mathbb{E}$ is called an $(\omega,\delta)$
pseudo-orbit of $F$ if

$d(y_{n+1},F(y_{n},\theta^{n}\omega))\leq\delta(\theta^{n+1}\omega)\quad\text{
for all }n\in\mathbb{Z}.$

For a strictly positive random variable $\epsilon$ and any $\omega\in\Omega$ a
point $x\in\mathbb{E}$ is said to $(\omega,\epsilon)$-shadow the
$(\omega,\delta)$ pseudo-orbit $\\{y_{n}\\}_{n\in\mathbb{Z}}$ if

$d(F^{n}(x,\omega),y_{n})\leq\epsilon(\theta^{n}\omega)\quad\text{ for all
}n\in\mathbb{Z}.$

The following lemma is the Proposition 3.6 of [17], which is called Random
Shadowing Lemma.

######  Lemma B.5.

Let the random hyperbolic set $\Lambda$ have local product structure. Then for
every tempered random variable $\epsilon>0$ there exists a tempered random
variable $\beta>0$ such that $\mathbb{P}$-a.s. every $(\omega,\beta)$ pseudo-
orbit $\\{y_{n}\\}_{n\in\mathbb{Z}}$ with $y_{n}\in\Lambda(\theta^{n}\omega)$
can be $(\omega,\epsilon)$-shadowed by a point $x\in\Lambda(\omega)$. If
$2\epsilon$ is chosen as an expansivity characteristic, then the shadowing
point $x$ is unique. Moreover, if the $y_{n}$ are chosen to be random
variables such that for $\mathbb{P}$-almost all $\omega\in\Omega$ the sequence
$\\{y_{n}(\omega)\\}_{n\in\mathbb{Z}}$ is an $(\omega,\beta)$ pseudo -orbit,
then the starting point $x(\omega)$ of the corresponding
$(\omega,\epsilon)$-shadowing orbit depends measurably on $\omega$.

Let $V^{s},V^{u}$ be the local invariant manifolds of $F$ respectively,
moreover take them of random size $\eta$ being an expansivity characteristic.
Take a small enough strictly positive tempered random variable $\varpi$ the
smallness of which is only depending on $\beta$, $\eta$ and the hyperbolicity
of the system. Then one can find a random variable $k:\Omega\to\mathbb{N}$
such that $\Lambda(\omega)$ can be covered by $k(\omega)$ open balls of radius
less than $\varpi(\omega)$ and centres $p_{i}(\omega)$, $i=1,\cdots,k(\omega)$
with $p_{i}:\Omega\to\mathbb{E}$ measurable. Actually $\varpi$ can always be
chosen log-integrable, and $k$ can be chosen log-integrable if $F$ is of
tempered continuity.

Denote these balls by $B_{\varpi}(p_{i},\omega)$. Then $A$ is called a random
matrix if, for each $\omega\in\Omega$,
$A(\omega)\in\mathbb{R}^{k(\omega)\times k(\theta\omega)}$ such that

$A(\omega)_{i,j}=\begin{cases}1\quad&\text{ if }F(p_{i}(\omega),\omega)\in
B_{\varpi}(p_{j},\theta\omega)\\\ 0&\text{ otherwise.}\end{cases}$

######  Definition B.6.

Let $k:\Omega\to\mathbb{N}^{+}$ be a random variable, $A$ a corresponding
random transition matrix, and define for $\omega\in\Omega$

$\Sigma_{k}(\omega):=\prod_{i=-\infty}^{\infty}\\{1,\cdots,k(\theta^{i}\omega)\\},$
$\Sigma_{A}(\omega):=\\{\bar{x}=(x_{i})\in\Sigma_{k}(\omega):A_{x_{i},x_{i+1}}(\theta^{i}\omega)=1\text{
for all }i\in\mathbb{Z}\\}.$

Let $\sigma$ be the standard (left-) shift. The families
$\\{\sigma:\Sigma_{k}(\omega)\to\Sigma_{k}(\theta\omega)\\}$ and
$\\{\sigma:\Sigma_{A}(\omega)\to\Sigma_{A}(\theta\omega)\\}$ are called random
$k$-shift and random subshift of finite type, respectively. Moreover, define
$\Sigma_{A}:=\\{(\bar{x},\omega):\bar{x}\in\Sigma_{A}(\omega),\omega\in\Omega\\}$,
which is a measurable bundle over $\Omega$, and also denote the respective
skew-product transformation on $\Sigma_{A}$ by $\sigma$.

One can also define one-sided versions of random $k$-shifts and subshifts of
finite type on
$\Sigma_{k}^{+}:=\prod_{i=0}^{\infty}\\{1,\cdots,k(\theta^{i}\omega)\\}\text{
and corresponding }\Sigma_{A}^{+}(\omega),\omega\in\Omega$ respectively. Let
$C(\Sigma_{A}^{+}(\omega))$ denote the space of random continuous functions on
$\Sigma_{A}^{+}$ which are measurable on $\omega$ and continuous on
$x\in\mathbb{E}$ for fixed $\omega$. If a random continuous function is
$\mathbb{P}$-a.s. Hölder continuous with uniform exponent, such function is
called random Hölder continuous function. For a random continuous function
$\varphi$ and $\omega\in\Omega$ random transfer operators
$\mathcal{L}_{\varphi}(\omega):C(\Sigma_{A}^{+}(\omega))\to
C(\Sigma_{A}^{+}(\theta\omega))$ is defined by

$(\mathcal{L}_{\varphi}(\omega)h)(x)=\sum_{y\in\Sigma_{A}^{+}(\omega):\sigma
y=x}\exp(\varphi(\omega,y))h(y)$

for $h\in C(\Sigma_{A}^{+}(\omega)),x\in C(\Sigma_{A}^{+}(\theta\omega))$.
$\mathcal{L}_{\varphi}^{*}(\omega)$ denotes the random dual operator mapping
finite signed measures on $C(\Sigma_{A}^{+}(\theta\omega))$ to those on
$C(\Sigma_{A}^{+}(\omega))$ by

$\int hd\mathcal{L}_{\varphi}^{*}(\omega)m=\int\mathcal{L}_{\varphi}hdm\text{
for all }h\in C(\Sigma_{A}^{+}(\omega))$

for a finite signed measure $m$ on $C(\Sigma_{A}^{+}(\theta\omega))$.

The following theorem is the Theorem 4.3 of [17], which is the main result
about SRB measures needed in Section 4.3.1.

######  Theorem B.7.

Let $F$ be a $C^{1+\alpha}$ random dynamical system with a random
topologically transitive hyperbolic attractor
$\Lambda(\subset\mathbb{E}\times\Omega)$. Then there exists a unique
$F$-invariant measure (SRB measure) $\nu$ supported by $\Lambda$ and
characterized by each of the following:

  * (i)

$h_{\nu}(F)=\int\sum\lambda_{i}^{+}d\nu$ where $\lambda_{i}$ are the Lyapunov
exponents corresponding to $\nu$;

  * (ii)

$\mathbb{P}$-a.s. the conditional measures of $\nu_{\omega}$ on the unstable
manifolds are absolutely continuous with respect to the Riemannian volume on
these submanifolds;

  * (iii)

$h_{\nu}(F)+\int fd\nu=\sup_{F-\text{invariant measure }m}\\{h_{m}(F)+\int
fd\nu\\}$ and the latter is the topological pressure $\pi_{F}(f)$ of $f$ which
satisfies $\pi_{F}(f)=0$;

  * (iv)

$\nu=\psi\tilde{\mu}$ where $\tilde{\mu}$ is the equilibrium state for the
two-sided shift $\sigma$ on $\Sigma_{A}$ and the function $f\circ\psi$. The
measure $\tilde{\mu}$ can be obtained as a natural extension of the
probability measure $\mu$ which is invariant with respect to the one-sided
shift $\sigma$ on $\Sigma^{+}_{A}$ and such that
$\mathcal{L}^{*}_{\eta}(\omega)\mu_{\theta\omega}=\mu_{\omega}$
$\mathbb{P}$-a.s. where $\eta-f\circ\psi=h-h\circ(\theta\times\sigma)$ for
some random H older continuous function $h$;

  * (v)

$\nu$ can be obtained as a weak limit
$\nu_{\omega}=\lim_{n\to\infty}F(n,\theta^{-n}\omega)m_{\theta^{-n}\omega}$
$\mathbb{P}$-a.s. for any measure $m_{\omega}$ absolutely continuous with
respect to the Riemannian volume such that sup $m_{\omega}\subset U(\omega)$.

Here $f(x,\omega)=-\log\|\det D_{x}F(x,\omega)|_{E^{u}(x,\omega)}\|$, where
$E^{u}(x,\omega)$ is the invariant unstable Oseledets subspace of $D_{x}F$ on
$(x,\omega)$; and $\psi$ is the conjugation between $F$ on $\Lambda$ and
$\sigma$ on $\Sigma_{A}$ which is constructed based on Markov partitions, then
reduce $\sigma$ on $\Sigma_{A}$ to the image of an unstable manifold to obtain
$\sigma$ on $\Sigma_{A}^{+}$. The Hölder continuity follows the
$C^{1+\alpha}$ness of $F$, and the existence of Markov partitions are given by
Theorem 3.9 of [17].

######  Remark B.1.

For the purpose of Section 4.3.1, we need to notify the following facts:

  * i)

From the construction of Markov partitions in [17], one can obtain that for
$\mathbb{P}$-a.s. $\omega\in\Omega$, $\psi|_{\Sigma_{A}(\omega)}$ is a
continuous onto map from $\Sigma_{A}(\omega)$ to $\Lambda(\omega)$ and

$\\{x\in\Lambda(w):(\psi|_{\Sigma_{A}(\omega)})^{-1}(x)\text{ is not
singleton}\\}$

is a countable set.

  * ii)

For detailed discussion on one-sided random symbolic dynamical system
$(\Sigma_{A}^{+},\sigma)$ and the corresponding Ruelle’s transfer operator, we
refer the reader to [13]. Here, one property of the random equilibrium state
$\mu$ is useful (see Proposition 4.7 and Theorem 2.3 of [13]): there exists
positive constant $c_{1},c_{2}$ and strictly positive random variable
$\lambda$ with $\log\lambda\in\mathbb{L}^{1}(\Omega,\mathbb{P})$ such that
$\mathbb{P}$-a.s.

$c_{1}\leq\frac{\mu_{\omega}(y\in\Sigma_{A}^{+}(\omega):x_{i}=y_{i}\text{ for
}i=0,\cdots,m-1)}{\exp\left(\sum_{i=0}^{m-1}f\circ\psi(\sigma^{i}\bar{x},\theta^{i}\omega)-\log\lambda_{m}(\omega)\right)}\leq
c_{2},$

for all $m\in\mathbb{N}$ and all $\bar{x}\in\Sigma^{+}_{A}(\omega)$, where
$\lambda_{m}(\omega):=\lambda(\theta^{m-1}\omega)\cdot\ldots\cdot\lambda(\omega)$.

  * iii)

By the uniqueness of natural extension and the invariance of $\tilde{\mu}$,
one can obtain that $\mathbb{P}$-a.s.

$\displaystyle\tilde{\mu}_{\omega}(y\in\Sigma_{A}(\omega):x_{i}=y_{i}\text{
for }i=n,\cdots,n+m-1)$ $\displaystyle=$
$\displaystyle\tilde{\mu}_{\sigma^{n}\omega}(y\in\Sigma_{A}(\omega):\sigma
x_{i}=y_{i}\text{ for }i=0,\cdots,m-1)$ $\displaystyle=$
$\displaystyle\mu_{\sigma^{n}\omega}(y\in\Sigma_{A}^{+}(\omega):\sigma
x_{i}=y_{i}\text{ for }i=0,\cdots,m-1)$ $\displaystyle>$ $\displaystyle 0,$

for all $\bar{x}\in\Sigma_{A}(\omega)$ and all
$n\in\mathbb{Z},m\in\mathbb{N}$.

##  Appendix C Measurable Selection

In this section, we state a theorem about measurable selection, which is taken
from [14].

######  Theorem C.1.

Let $Y$ be a complete separable metric space, $(X,\mathcal{L})$ be a
measurable space, and $\Gamma$ be a multifunction from $X$ to $\mathcal{F}(Y)$
which is the collection of non-empty closed sets of $Y$. If for any open set
$U\subset Y$, $\Gamma^{-}(U)(:=\\{\omega|\ \Gamma(\omega)\cap
U\neq\emptyset)\in\mathcal{L}$, then $\Gamma$ admits a measurable selection.

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