2.1. The setup of random dynamical systems

In this subsection, we recall the settings and related notions of random dynamical systems investigated in [Reference Arnold1, Reference Crauel5, Reference Kifer15].

Let $(\Omega,\mathscr{F},\mathbb{P}, \theta)$ be a measure-preserving system, where $(\Omega, \mathscr{F}, \mathbb{P})$ is countably generated probability space and θ is an invertible measure-preserving transformation. We always assume that $\mathscr{F}$ is complete, countably generated, and separates points. Hence $(\Omega,\mathscr{F},\mathbb{P})$ is a Lebesgue space. Let X be a compact metric space endowed with the Borel σ-algebra $\mathscr{B}_{X}$. This endows $\Omega \times X$ with the product σ-algebra $\mathscr{F}\otimes \mathscr{B}_X$. For a measurable subset ${\mathcal{E}}\subset \Omega\times X$, the fibers ${\mathcal{E}}_{\omega}=\left\lbrace x\in X: (\omega,x)\in {\mathcal{E}}\right\rbrace $ with $\omega\in \Omega$ are non-empty compact subsets of X. A continuous (or homeomorphic) bundle random dynamical system (RDS for short) over $(\Omega, \mathcal{F}, \mathbb{P}, \theta)$ is generated by mappings $T_{\omega}: \mathcal{E}_{\omega}\rightarrow \mathcal{E}_{\theta \omega}$ with iterates

\begin{eqnarray*} T_{\omega}^{n}=\left\{\begin{array}{rcl} & T_{\theta^{n-1}\omega}\circ\cdots \circ T_{\theta \omega}\circ T_{\omega},&\text{if}~ n \gt 0\\ &id, & \text{if}~ n=0 \end{array}\right. \end{eqnarray*}

such that $(\omega, x)\mapsto T_{\omega}x$ is measurable and $x\mapsto T_{\omega}x$ is continuous (or homeomorphic, respectively) for $\mathbb{P}$-almost all ω. The map $\Theta:\mathcal{E}\rightarrow \mathcal{E}$ defined by $\Theta(\omega,x)=(\theta\omega, T_{\omega}x)$ is called the skew product transformation.

A finite family $\mathcal{U}=\left\lbrace U_{i}\right\rbrace_{i=1}^{k}$ of measurable subsets of $\Omega \times X$ is said to be a cover if $\Omega\times X=\bigcup_{i=1}^{k}U_{i}$, and for each $i\in \left\lbrace 1,\cdots, k \right\rbrace $ the ω-section

\begin{equation*}U_{i}(\omega):=\left\lbrace x\in X: (\omega, x)\in U_{i} \right\rbrace\subseteq X\end{equation*}

is a Borel set of X. This implies that $\mathcal{U}(\omega)=\left\lbrace U_{i}(\omega) \right\rbrace_{i=1}^{k}$ is a Borel cover of X. A partition of $\Omega\times X$ is a cover of $\Omega\times X$ whose elements are mutually disjoint. An open cover of $\Omega \times X$ is a cover of $\Omega\times X$ whose ω-sections are open sets. Denoted by $\mathcal{P}_{\Omega\times X}$, $C_{\Omega\times X}$ and $C_{\Omega\times X}^{0}$ the set of all finite partitions, finite covers and finite open covers of $\Omega\times X$, respectively. Specially, by $C_{\Omega\times X}^{0'}$ we denote the set of ${\mathcal{U}}\in C_{\Omega\times X}^{0}$ formed by ${\mathcal{U}}=\left\lbrace \Omega\times U_{i} \right\rbrace $ with the finite open cover $\left\lbrace U_{i}\right\rbrace $ of X. The notions ${\mathcal{P}}_{{\mathcal{E}}}$, $C_{{\mathcal{E}}}$, $C_{{\mathcal{E}}}^{0}$ and $C_{{\mathcal{E}}}^{0'}$ denote the restriction of ${\mathcal{P}}_{\Omega\times X}$, $C_{\Omega\times X}$, $C_{\Omega\times X}^{0}$ and $C_{\Omega\times X}^{0'}$ on ${\mathcal{E}}$, respectively. Given the covers $\xi\in {C}_{\Omega}$ and $\mathcal{W}\in {C}_{X}$, we sometimes write $(\Omega\times \mathcal{W})_{\mathcal{E}}=\{(\Omega \times W)\cap \mathcal{E}: W \in\mathcal{W} \}$ and $(\xi\times X)_{\mathcal{E}}=\{(A \times X)\cap \mathcal{E}: A\in\xi \}$. Given two covers $\mathcal{U}$, $\mathcal{V}\in C_{\Omega\times X}$, $\mathcal{U}$ is said to be finer than $\mathcal{V}$ (denote as $\mathcal{U}\succeq \mathcal{V}$) if each element of $\mathcal{U}$ is contained in some element of $\mathcal{V}$. The join of $\mathcal{U}$ and $\mathcal{V}$ is defined by $\mathcal{U}\vee\mathcal{V}=\left\lbrace U\cap V: U\in \mathcal{U}, V\in \mathcal{V}\right\rbrace $. For $a, b\in \mathbb{N}$ with $a\leq b$ and $\mathcal{U}\in C_{\Omega\times X}$, we define $\mathcal{U}_{a}^{b}=\bigvee\limits_{n=a}^{b}\Theta^{-n}\mathcal{U}.$

We collect some examples of continuous bundle RDSs below.

Example 1. Among interesting examples of continuous bundle RDSs are random sub-shifts, which appeared in the literature [Reference Bogenschutz and Gundlach3, Reference Kifer16]. Let $(\Omega, \mathscr{F}, {\mathbb{P}})$ be a Lebesgue space and $\theta:(\Omega, \mathscr{F}, {\mathbb{P}})\rightarrow (\Omega,\mathscr{F}, {\mathbb{P}})$ an invertible measure-preserving transformation. Set $X=\left\lbrace (x_{i})_{i\in{\mathbb{Z}}}: x_{i}\in {\mathbb{N}}\cup \left\lbrace +\infty \right\rbrace, i\in {\mathbb{Z}} \right\rbrace $, a compact metric space equipped with the metric

\begin{align*} d((x_{i})_{i\in {\mathbb{Z}}}, (y_{i})_{i\in {\mathbb{Z}}})=\sum_{i\in {\mathbb{Z}}} \frac{1}{2^{|i|}}|x_{i}^{-1}-y_{i}^{-1}|, \end{align*}

and let $F: X\rightarrow X$ be the translation $(x_{i})_{i \in {\mathbb{Z}}}\rightarrow (x_{i+1})_{i\in {\mathbb{Z}}}$. Then, the integer group ${\mathbb{Z}}$ acts on $(\Omega\times X, \mathscr{F}\otimes \mathscr{B}_{X})$ measurably with $(\omega, x)\rightarrow (\theta^{i}\omega, F^{i}x)$ for each $i\in \mathbb{Z}$, where $\mathscr{B}_{X}$ denotes the Borel σ-algebra of the space X. Now let $\mathcal{E}\in \mathscr{F}\otimes \mathscr{B}_{X}$ be an invariant subset of $\Omega\times X$ such that $\mathcal{E}_{\omega}\subset X$ is compact for ${\mathbb{P}}$-a.e. $\omega\in \Omega$. This defines a continuous bundle RDSs, for ${\mathbb{P}}$-a.e. $\omega\in \Omega$, $F_{i, \omega}$ is just the restriction of F ⁱ over ${\mathcal{E}}_{\theta^{i}\omega}$ for $i\in {\mathbb{Z}}$.

A very special case is when the subset ${\mathcal{E}}$ is given as follows. Let k be a random ${\mathbb{N}}$-valued random variable satisfying

\begin{align*} 0 \lt \int_{\Omega} \log k(\omega) d {\mathbb{P}}(\omega) \lt \infty, \end{align*}

and, for $\omega\in {\mathbb{P}}$, let $M(\omega)$ be a random matrix $(m_{i,j}(\omega): i=1, \cdots, k(\omega), j=1, \cdots, k(\theta(\omega)))$ with entries 0 and 1. Then the random matrix M generates a random sub-shift of finite type, where

\begin{align*} {\mathcal{E}}=\left\lbrace (\omega, (x_{i})_{i\in {\mathbb{N}}}): \omega\in \Omega, 1\leq x_{i} \leq k(\theta^{i}\omega), m_{x_{i}, x_{i+1}}(\theta^{i}\omega)=1, i\in {\mathbb{Z}} \right\rbrace. \end{align*}

It is not hard to see that this is a continuous bundle RDS.

2.2. Metric mean dimension of RDSs

In this subsection, we recall the definitions of topological entropy [Reference Bogenschutz2, Reference Kifer15] and metric mean dimension introduced by Ma et al. [Reference Ma, Yang and Chen27] for continuous bundle random dynamical systems.

Let $\omega\in \Omega$, $n\in\mathbb{N}$ and ϵ > 0. For each $x,y\in {\mathcal{E}}_{\omega}$, the n-th Bowen metric $d_n^\omega$ on ${\mathcal{E}}_{\omega}$ is defined by

\begin{equation*} d_n^\omega(x,y)=\max\{d( T_\omega^ix,T_\omega^iy ): 0\leq i \lt n \}. \end{equation*}

Then the $(n,\epsilon, \omega)$-Bowen ball around x with radius ϵ in the metric $d_{n}^{\omega}$ is given by

\begin{equation*}B_{d_{n}^{\omega}}(x, \epsilon)=\left\lbrace y \in \mathcal{E}_{\omega}: d_{n}^{\omega}(x, y) \lt \epsilon\right\rbrace. \end{equation*}

Fix $\omega\in \Omega$ and let $\alpha=\left\lbrace A_{i}: 1\leq i \leq m \right\rbrace $ be a finite open cover of $\mathcal{E}_{\omega}$. We define the mesh of α with respect to the metric $d_{n}^{\omega}$ as follows

\begin{align*} {\rm diam}(\alpha, d_{n}^{\omega})=\max_{1\leq i \leq m } {\rm diam}(A_{i}, d_{n}^{\omega}), \end{align*}

where the diameter of a set A_i with respect to the metric $d_{n}^{\omega}$ is given by

\begin{align*} {\rm diam}(A_{i}, d_{n}^{\omega})=\sup \left\lbrace d_{n}^{\omega}(x,y): x, y\in A_{i} \right\rbrace. \end{align*}

Let $\#(\mathcal{E}, \omega, \epsilon, n)=\inf \left\lbrace |\alpha|: \alpha \in C_{\mathcal{E}_{\omega}}^{0}, {\rm diam}(\alpha, d_{n}^{\omega}) \lt \epsilon \right\rbrace. $ A set $E\subset{\mathcal{E}}_{\omega}$ is said to be an $(\omega,\epsilon,n)$-separated set if $x,y\in E$, x ≠ y implies that $d_n^\omega(x,y) \gt \epsilon$. The maximum cardinality of $(\omega,\epsilon,n)$-separated sets is denoted by ${{\rm sep}}({\mathcal{E}}, \omega,\epsilon,n)$. A subset F of ${\mathcal{E}}_{\omega}$ is said to be an $(\omega, \epsilon, n)$-spanning set if for any $x\in {\mathcal{E}}_{\omega}$, there exists $y\in F$ such that $d_{n}^{\omega}(x, y)\leq \epsilon$. The smallest cardinality of $(\omega, n, \epsilon)$-spanning sets is denoted by ${\rm span}({\mathcal{E}}, \omega, n,\epsilon)$. Let

\begin{equation*}S'({\mathcal{E}}, \omega, \epsilon)=\limsup\limits_{n\rightarrow \infty} \frac{1}{n}\log \#({\mathcal{E}}, \omega, \epsilon, n).\end{equation*}

Set

(2.2)\begin{align} S'({\mathcal{E}}, \epsilon)=\int S'({\mathcal{E}},\omega, \epsilon) d \mathbb{P}(\omega). \end{align}

The quantity (2.2) is non-decreasing as $\epsilon\rightarrow 0$. One can define a quantity to measure how fast $S'({\mathcal{E}}, \epsilon)$ increases as follows:

(2.3)\begin{align} &\mathbb{E}{{\rm\overline{mdim}_M}}(T,{\mathcal{E}}, d)=\limsup\limits_{\epsilon \rightarrow 0}\frac{S'({\mathcal{E}}, \epsilon)}{|\log \epsilon|}, \end{align}

\begin{align*} &\mathbb{E}{{\rm\underline{mdim}_M}}(T,{\mathcal{E}}, d)= \liminf\limits_{\epsilon \rightarrow 0}\frac{S'({\mathcal{E}}, \epsilon)}{|\log \epsilon|} . \end{align*}

We call (2.3) the upper and lower metric mean dimension of ${\mathcal{E}}$ for RDSs, respectively.

It is easy to show that

(2.4)\begin{align} \#({\mathcal{E}}, \omega, 2\epsilon, n) \leq {\rm sep}({\mathcal{E}}, \omega, \epsilon, n) \leq \#({\mathcal{E}}, \omega, \epsilon, n). \end{align}

Notice that ${\rm sep}({\mathcal{E}}, \omega,\epsilon,n)$ is measurable in ω [Reference Kifer16, lemma 2.1]. Then metric mean dimension can also defined by separated sets. Set

\begin{align*} S({\mathcal{E}}, \omega, \epsilon)=\limsup\limits_{n\rightarrow \infty} \frac{1}{n}\log {\rm sep}({\mathcal{E}}, \omega, \epsilon, n) \end{align*}

and

\begin{equation*} h_{top}^\textbf{r}(T,{\mathcal{E}}, d,\epsilon)=\int S({\mathcal{E}}, \omega, \epsilon) d \mathbb{P}(\omega).\end{equation*}

By (2.4) and the fact that $\frac{|\log \epsilon|}{|\log 2\epsilon|}=1$, we have

\begin{align*} &\mathbb{E}{{\rm\overline{mdim}_M}}(T,{\mathcal{E}},d)=\limsup\limits_{\epsilon \rightarrow 0}\frac{h_{top}^\textbf{r}(T,{\mathcal{E}}, d,\epsilon)}{|\log \epsilon|} ,\\ &\mathbb{E}{{\rm\underline{mdim}_M}}(T,{\mathcal{E}},d)= \liminf\limits_{\epsilon \rightarrow 0}\frac{h_{top}^\textbf{r}(T,{\mathcal{E}}, d,\epsilon)}{|\log \epsilon|} . \end{align*}

Clearly, the metric mean dimension depends on the metrics on X and hence is not topologically invariant. Furthermore, Ma et al. [Reference Ma and Chen26] proved that any finite entropy systems have zero metric mean dimension in the setting of random dynamical systems. So metric mean dimension is a useful quantity to describe the topological complexity of infinite random entropy systems.

3.1. Variational principle I: Kolmogorov-Sinai ϵ-entropy

In this subsection, we first introduce the local variational principle for the topological entropy of a fixed finite open covers in terms of measure-theoretic entropy of a fixed finite open covers given in [Reference Dooley and Zhang6, Reference Ma and Chen26]. Subsequently, we prove the first main result Theorem 3.2 by using the local variational principle of RDSs.

By $\mathcal{P}_{\mathbb{P}}(\Omega\times X)$ we denote the space of probability measures on $\Omega\times X$ with the marginal $\mathbb{P}$ on Ω. Let $\mathcal{P}_{\mathbb{P}}(\mathcal{E})=\left\lbrace \mu \in \mathcal{P}_{\mathbb{P}}(\Omega\times X): \mu(\mathcal{E})=1 \right\rbrace $. It is well-known [Reference Crauel5, proposition 3.6] that $\mu\in \mathcal{P}_{\mathbb{P}}(\mathcal{E})$ on $\mathcal{E}$ can be disintegrated as $d \mu(\omega, x)=d \mu_{\omega}(x) d {\mathbb{P}}(\omega)$, where µ_ω is the regular conditional probabilities with respect to the σ-algebra $\mathscr{F}_{\mathcal{E}}$ formed by all sets $(A\times X)\cap \mathcal{E}$ with $A\in \mathscr{F}$. The set of Θ-invariant measures µ of $\mathcal{P}_{\mathbb{P}}(\mathcal{E})$ is denoted by $M_{\mathbb{P}}(\mathcal{E})$. By Bogenschutz [Reference Bogenschutz2], the measure $\mu\in M_{\mathbb{P}}(\mathcal{E})$ if and only if $T_{\omega} \mu_{\omega}=\mu_{\theta \omega}$ for ${\mathbb{P}}$-a.e. ω. And the set of ergodic elements in $M_{\mathbb{P}}(\mathcal{E})$ is denoted by $E_{\mathbb{P}}(\mathcal{E}) $. This means that µ_ω is a probability measure on $\mathcal{E}_{\omega}$ for $\mathbb{P}$-a.e.ω and for any measurable set $R\subset \mathcal{E}$, $\mu_{\omega}(R(\omega))=\mu(R|\mathscr{F}_{\mathcal{E}})$, where $\mu(R|\mathscr{F}_{\mathcal{E}})$ is the conditional expectation of the characterization function 1_R of R with respect to $\mathscr{F}_{\mathcal{E}}$, $R_{\omega}=\{x\in \mathcal{E}_{\omega}: (\omega, x)\in R_{i}\}$ and so $\mu(R)=\int \mu_{\omega}(R(\omega)) d \mathbb{P}(\omega)$. Let $\mathcal{R}=\left\lbrace {R}_{i}\right\rbrace $ be a finite measurable partition of $\mathcal{E}$ and ${R}_{i}(\omega)=\left\lbrace x \in \mathcal{E}_{\omega}: (\omega, x)\in {R}_{i} \right\rbrace $. Then $\mathcal{R}(\omega)=\left\lbrace {R}_{i}(\omega) \right\rbrace $ is a finite partition of $\mathcal{E}_{\omega}$. Set $\mathscr{F}_{\mathcal{E}}=\left\lbrace(A\times X)\cap \mathcal{E}: A\in \mathscr{F} \right\rbrace $.

The conditional entropy of $\mathcal{R}$ for the given σ-algebra $\mathscr{F}_{\mathcal{E}}$ is defined by

\begin{align*} H_{\mu}(\mathcal{R}|\mathscr{F}_{\mathcal{E}})=-\int \sum_{i} \mu(R_{i}|\mathscr{F}_{\mathcal{E}})\log \mu (R_{i}|\mathscr{F}_{\mathcal{E}}) d \mathbb{P}(\omega)=\int H_{\mu_{\omega}}(\mathcal{R}(\omega)) d \mathbb{P}(\omega), \end{align*}

where $H_{\mu_{\omega}}(P)$ denotes the usual partition entropy of P. Let $\mu\in M_{\mathbb{P}}(\mathcal{E})$, $\xi \in {\mathcal{P}}_{{\mathcal{E}}}$ and define

\begin{align*} h_{\mu}^\textbf{r}({T}, \xi)&=\lim\limits_{n\rightarrow \infty}\frac{1}{n}H_{\mu}\left( \bigvee_{i=0}^{n-1}(\Theta^{i})^{-1}\xi|\mathscr{F}_{\mathcal{E}}\right) \\ &=\lim\limits_{n\rightarrow \infty} \dfrac{1}{n} \int H_{\mu_{\omega}}\left( \bigvee_{i=0}^{n-1}(T_{\omega}^{i})^{-1} \xi(\theta^{i}\omega)\right) d {\mathbb{P}}(\omega), \end{align*}

where the limit exists due to the subadditivity of conditional entropy[Reference Kifer15]. If $\mathbb{P}$ is ergodic, then $ h_{\mu}^\textbf{r}(T,\xi)=\lim\limits_{n\rightarrow \infty} \dfrac{1}{n} H_{\mu_{\omega}}\left( \bigvee_{i=0}^{n-1}(T_{\omega}^{i})^{-1} \xi(\theta^{i}\omega)\right) $ for $\mathbb{P}$-a.e. ω.

Let $\mathcal{U} \in {C}_{\mathcal{E}}^{0}$ and $\mu\in M_{{\mathbb{P}}}({\mathcal{E}})$. We define the measure-theoretic entropy of open cover $\mathcal{U}$ w.r.t. µ as

\begin{align*} h_{\mu}^\textbf{r}(T, \mathcal{U})=\inf_{\alpha\succeq \mathcal{U}, \alpha \in \mathcal{P}_{\mathcal{E}}} h_{\mu}^\textbf{r}(T, \alpha). \end{align*}

For each $\mathcal{U}\in {C}_{\mathcal{E}}^{0'}$, it is not difficult to verify (see [Reference Bogenschutz2, Reference Dooley and Zhang6, Reference Kifer15]) that infimum above can only take over the partitions Q of $\mathcal{E}$ into sets Q_i of the form $Q_{i}=(\Omega\times P_{i})\cap \mathcal{E}$, where $\mathcal{P}=\left\lbrace P_{i}\right\rbrace $ is a finite partition of X.

Let $\mathcal{U}\in C_{\mathcal{E}}^{0}$, $n\in {\mathbb{N}}$ and $\omega \in \Omega$. Put

\begin{align*} N(T, \omega, \mathcal{U}, n)=\min \left\lbrace \#F: F ~\text{is a finite subcover of }~\bigvee_{i=0}^{n-1}(T_{\omega}^{i})^{-1}\mathcal{U}(\theta^{i}\omega)~\text{over}~\mathcal{E}_{\omega}\right\rbrace, \end{align*}

By the proof of [Reference Kifer16, proposition 1.6], the quantity $N(T,\omega,\mathcal{U}, n)$ is measurable with respect to ω. The Kingman’s subadditive ergodic theorem then gives us the following:

(3.1)\begin{align} h_{top}^\textbf{r}(T, \mathcal{U}):&=\int\lim\limits_{n\rightarrow \infty}\frac{1}{n} \log N(T, \omega, \mathcal{U}, n) d \mathbb{P}(\omega) \end{align}

\begin{align*} =\lim\limits_{n\rightarrow \infty} \frac{1}{n}\int \log N(T, \omega, \mathcal{U}, n)d \mathbb{P}(\omega), \end{align*}

and (3.1) remains valid for $\mathbb{P}$-a.e ω without taking the integral on the right-hand side if $\mathbb{P}$ is ergodic.

The proof of the variational principle I as stated in Theorem 3.2 is based on the random version of the local variational principle for entropy of a fixed open cover. Local entropy theory for deterministic dynamical systems has been studied by Romagnoli [Reference Romagnoli28] and proved by Glasner and Weiss [Reference Glasner, Weiss and Hasselblatt8]. In the case of random dynamical systems, authors [Reference Dooley and Zhang6, Reference Ma and Chen26] established the following local variational principle.

Theorem 3.1. Let T be a homeomorphic bundle RDS on ${\mathcal{E}}$ over a measure-preserving system $(\Omega, \mathscr{F},{\mathbb{P}}, \theta)$. If $\mathcal{U}\in C_{\mathcal{E}}^{0'}$, then

\begin{align*} h_{top}^\textbf{r}(T, \mathcal{U})=\max\limits_{\mu\in M_{\mathbb{P}}({\mathcal{E}})}h_{\mu}^\textbf{r}(T, \mathcal{U}). \end{align*}

Additionally, if $ \mathbb{P}$ is ergodic, then

\begin{align*} h_{top}^\textbf{r}(T, \mathcal{U})=\sup_{\mu \in E_{\mathbb{P}}({\mathcal{E}})}h_{\mu}^\textbf{r}(T, \mathcal{U}). \end{align*}

Given a finite open cover ${\mathcal{U}}$ of X, We define ${\rm diam}({\mathcal{U}})$ as the diameter of ${\mathcal{U}}$, i.e., the maximal diameter of the elements of ${\mathcal{U}}$. The Lebesgue number of ${\mathcal{U}}$, denoted by ${\rm Leb}({\mathcal{U}})$, is the largest positive number δ with the property that every open ball of X with radius δ is contained in an element of ${\mathcal{U}}$.

Lemma 3.2. Let $\sigma=\left\lbrace A_{i}\right\rbrace $ be a finite open cover of X. Let $\mathcal{U}=(\Omega\times \sigma)_{\mathcal{E}}=\left\lbrace (\Omega\times A_{i}) \cap \mathcal{E}: A_i\in \sigma \right\rbrace $ be a finite open cover of $\mathcal{E}$. Then for each fixed ω,

(3.2)\begin{align} S({\mathcal{E}}, \omega,{\rm diam}(\sigma) ) \leq \lim\limits_{n\rightarrow \infty}\frac{1}{n}\log N(T, \omega, \mathcal{U},n) \leq S({\mathcal{E}}, \omega, \rm{Leb}(\sigma)). \end{align}

Proof. One can obtain the desired result by using

\begin{equation*}{\rm sep}({\mathcal{E}}, \omega, {\rm diam}(\sigma), n)\leq N(T, \omega,\mathcal{U}, n) \leq {\rm sep} ({\mathcal{E}}, \omega, {\rm Leb}(\mathcal{\sigma}), n).\end{equation*}

Theorem 3.2. Let T be a homeomorphic bundle RDS on ${\mathcal{E}}$ over a measure-preserving system $(\Omega, \mathscr{F},{\mathbb{P}}, \theta)$. Then

\begin{align*} &{\mathbb{E}}{{\rm\overline{mdim}_M}}(T,{\mathcal{E}},d)=\limsup\limits_{\epsilon\rightarrow 0} \frac{1}{|\log \epsilon|} \sup_{\mu \in {M}_{\mathbb{P}}({\mathcal{E}})} \inf_{\substack{{\rm diam}(\alpha)\leq \epsilon,\\ \alpha\in{\mathcal{P}}_{X}}} h_{\mu}^\textbf{r}(T, (\Omega\times\alpha)_{\mathcal{E}}),\\ & {\mathbb{E}}{{\rm\underline{mdim}_M}}(T,{\mathcal{E}},d)=\liminf\limits_{\epsilon\rightarrow 0} \frac{1}{|\log \epsilon|} \sup_{\mu \in {M}_{\mathbb{P}}({\mathcal{E}})} \inf_{\substack{{\rm diam}(\alpha)\leq {\epsilon},\\ \alpha\in{\mathcal{P}}_{X}}} h_{\mu}^\textbf{r}(T, (\Omega\times \alpha)_{\mathcal{E}}). \end{align*}

Additionally, if $(\Omega, \mathscr{F}, {\mathbb{P}}, \theta)$ is ergodic, then the results are also valid by changing the supremum into $\sup_{\mu \in {E}_{\mathbb{P}}({\mathcal{E}})}$.

Proof. It suffices to show the variational principles hold for ${\mathbb{E}}{{\rm\overline{mdim}_M}}(T,{\mathcal{E}}, d)$. Let ϵ > 0. From Lemma 3.1, there exists a finite open cover $\mathcal{U}$ of X such that ${\rm diam}({\mathcal{U}})\leq\epsilon$ and ${\rm Leb}({\mathcal{U}})\geq \frac{\epsilon}{4}$.

Note that, for any finite Borel partition α of X satisfying $\alpha\succeq \mathcal{U}$, we have ${\rm diam}(\alpha)\leq\epsilon$. By Theorem 3.1, we obtain

(3.3)\begin{align} \sup_{\mu \in {M}_{\mathbb{P}}({\mathcal{E}})} \inf_{\substack{{\rm diam}(\alpha)\leq\epsilon, \\ \alpha\in \mathcal{P}_{X}}} h_{\mu}^\textbf{r}(T, (\Omega\times\alpha)_{\mathcal{E}})\leq \sup_{\mu \in {M}_{\mathbb{P}}({\mathcal{E}})} \inf_{\substack{\alpha\succeq {\mathcal{U}},\\\alpha\in \mathcal{P}_{X}}} h_{\mu}^\textbf{r}(T, (\Omega\times\alpha)_{\mathcal{E}})=h_{top}^\textbf{r}(T, (\Omega\times{\mathcal{U}})_{\mathcal{E}}). \end{align}

Using Lemma 3.2,

(3.4)\begin{align} h_{top}^\textbf{r}(T,(\Omega\times {\mathcal{U}})_{\mathcal{E}}) \leq \int S({\mathcal{E}}, \omega, {\rm Leb}(\mathcal{U})) d \mathbb{P}(\omega)\leq \int S({\mathcal{E}}, \omega, \frac{\epsilon}{4}) d\mathbb{P}(\omega). \end{align}

It follows from inequalities (3.3) and (3.4) that

\begin{align*} \sup_{\mu \in {M}_{\mathbb{P}}({\mathcal{E}})} \inf_{\substack{{\rm diam}(\alpha)\leq\epsilon, \\ \alpha\in \mathcal{P}_{X}}} h_{\mu}^\textbf{r}(T, (\Omega\times\alpha)_{\mathcal{E}})\leq \int S({\mathcal{E}}, \omega, \frac{\epsilon}{4}) d\mathbb{P}(\omega). \end{align*}

So we get

\begin{equation*} \limsup\limits_{\epsilon\rightarrow 0} \frac{1}{|\log \epsilon|} \sup_{\mu \in {M}_{\mathbb{P}}({\mathcal{E}})} \inf_{\substack{{\rm diam}(\alpha)\leq \epsilon, \\ \alpha\in{\mathcal{P}}_{X}}} h_{\mu}^\textbf{r}(T, (\Omega\times\alpha)_{\mathcal{E}}) \leq {\mathbb{E}}{{\rm\overline{mdim}_M}}(T,{\mathcal{E}},d).\end{equation*}

On the other hand, for every finite Borel partition α of X such that ${\rm diam}(\alpha)\leq \frac{\epsilon}{8}$, we have $\alpha \succ \mathcal{U}$. Then Theorem 3.1 and Lemma 3.2 give us

\begin{align*} \sup_{\mu\in {M}_{\mathbb{P}}({\mathcal{E}})}\inf_{\substack{{\rm diam}(\alpha)\leq \frac{\epsilon}{8},\\ \alpha\in{\mathcal{P}}_{X}}}h_{\mu}^\textbf{r}(T,(\Omega\times\alpha)_{\mathcal{E}})&\geq \sup_{\mu \in {M}_{\mathbb{P}}({\mathcal{E}})} \inf_{\substack{\alpha\succeq {\mathcal{U}},\\ a\in \mathcal{P}_{X}}}h_{\mu}^\textbf{r}(T, (\Omega\times\alpha)_{\mathcal{E}})\\ &=h_{top}^{\textbf{r}}(T, (\Omega\times\mathcal{U})_{{\mathcal{E}}})\\ & \geq \int S({\mathcal{E}}, \omega, {\rm diam} (\mathcal{U})) d \mathbb{P}(\omega)\geq \int S({\mathcal{E}}, \omega, \epsilon) d \mathbb{P}(\omega), \end{align*}

which yields the desired results. If $(\Omega, \mathscr{F}, {\mathbb{P}}, \theta)$ is ergodic, one can get the variational principles by the similar arguments.

3.2. Variational principle II: Shapira’s ϵ-entropy

In this subsection, we introduce the notion of Shapira’s entropy in the setting of random dynamical systems and prove Theorem 3.3, which reflects the relationship between Shapira’s entropy and the measure-theoretic entropy of a fixed finite open cover $\mathcal{U}$ for random dynamical systems. Using this result, we can establish the variational principle of Shapira’s ϵ-entropy.

Let $\mathcal{U}=\left\lbrace U_{i} \right\rbrace_{i=1}^{k} $ be a finite open cover of $ \mathcal{E}$ and $\mu\in E_{\mathbb{P}}(\mathcal{E})$. Given $\omega\in \Omega$ and $0 \lt \delta \lt 1$, we define

\begin{equation*}N_{\mu_{\omega}}(\mathcal{U}, \delta)=\min \left\lbrace \#I: \mu_{\omega}\left( \bigcup_{i\in I} U_{i}(\omega)\right) \gt 1-\delta \right\rbrace.\end{equation*}

In order to define Shapira’s entropy of random dynamical systems, we need to prove the measurability of $N_{\mu_{\omega}}({\mathcal{U}}, \delta)$.

Proof. For every q > 0, we have

\begin{align*} &\Omega_q:=\left\lbrace \omega: N_{\mu_{\omega}}({\mathcal{U}}, \delta)=q\right\rbrace \\ =&\!\bigcup_{\#I=q,\atop I \subset \!\left\lbrace 1,\cdots, \#{\mathcal{U}} \!\right\rbrace }\!\left\lbrace\omega:\mu_{\omega}\!\left(\bigcup_{i\in I} {U}_{i}(\omega) \right)\!\! \gt \!\! 1\!-\! \delta \!\right\rbrace\! \bigcap \!\left( \bigcap_{\#J \lt q, \atop J\subset \left\lbrace 1,\cdots, \#{\mathcal{U}}\right\rbrace}\left\lbrace \omega: \mu_{\omega}\!\left(\bigcup_{i\in J} {U}_{i}(\omega) \right)\!\leq\! 1\!-\!\delta \right\rbrace\!\! \right)\!. \end{align*}

For each $I\subset \{1,\cdots, \#{\mathcal{U}}\}$, the graph(A_I) $=\{(\omega,x):x\in \bigcup_{i\in I} {U}_{i}(\omega) \}=\cup_{i\in I}U_i \cap \mathcal{E}$ is a measurable set of $\Omega \times X$. By [Reference Crauel5, corollary 3.4], the map $\omega \rightarrow \mu_{\omega}(U_{i}(\omega))$ is measurable. Then $\Omega_q$ is a measurable set of Ω. This implies that $\omega\mapsto N_{\mu_{\omega}}({\mathcal{U}}, \delta)$ is measurable since the map only takes finite many values.

Using proposition 3.1, we can define Shapira’s entropy of $\mathcal{U}\in C_{{\mathcal{E}}}^{0}$ with respect to µ as

\begin{align*} &\overline{h}_{\mu}^{S}(T, \mathcal{U}):= \lim\limits_{\delta \rightarrow 0} \limsup\limits_{n\rightarrow \infty}\dfrac{1}{n}\int \log N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \delta) d \mathbb{P}(\omega).\\ & \underline{h}_{\mu}^{S}(T, \mathcal{U}):= \lim\limits_{\delta \rightarrow 0} \liminf\limits_{n\rightarrow \infty}\dfrac{1}{n}\int \log N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \delta) d \mathbb{P}(\omega). \end{align*}

If the limit supremum and infimum agree, we denote the common value by ${h}_{\mu}^{S}(T, \mathcal{U})$. By the above definition, the essence of Shapira’s entropy is the alternative of Katok’s entropy defined by open covers. The Lemma 3.3 states the well-known Shannon–McMillan–Breiman Theorem for RDSs [Reference Zhu39]. The result of topological dynamical systems can be seen in [Reference Shiedlds31]. For $\xi \in \mathcal{P}_{{\mathcal{E}}}$ and $n\in {\mathbb{N}}$, denote by $A_{\xi, \omega}^{n}(x)$ be the atom of $\bigvee_{i=0}^{n-1} (T_{\omega}^{i})^{-1}\xi(\theta^{i}{\omega})$ containing the point $x\in \mathcal{E}_{\omega}$.

Adapting the ideas from [Reference Shapira29] and [Reference Wu37, lemma 6.1], the following theorem establishes the bridge between Shapira’s entropy and measure-theoretic entropy of a fixed finite open cover $\mathcal{U}$ for random dynamical systems.

Theorem 3.3. Let T be a homeomorphic bundle RDS on ${\mathcal{E}}$ over a measure-preserving system $(\Omega, \mathscr{F},{\mathbb{P}}, \theta)$. Let $\mathcal{U}\in C_{\mathcal{E}}^{0}$ and $\mu\in E_{\mathbb{P}}({\mathcal{E}})$. Then

\begin{align*} \overline{h}_{\mu}^{S}(T, \mathcal{U})=\underline{h}_{\mu}^{S}(T, \mathcal{U})=h_{\mu}^{S}(T, {\mathcal{U}})=h_{\mu}^\textbf{r}(T, \mathcal{U}). \end{align*}

Proof. Step 1: We prove $h_{\mu}^\textbf{r}(T, \mathcal{U})\geq \overline{h}_{\mu}^{S}(T, \mathcal{U})$.

Take any finite measurable partition ξ of $\mathcal{E}$ such that $\xi \succeq \mathcal{U}$. According to Lemma 3.3, there exists $F\subset \mathcal{E}$ such that $\mu(F)=1$ and for each $(\omega, x)\in F$,

\begin{align*} \lim\limits_{n\rightarrow \infty} -\dfrac{1}{n} \log \mu_{\omega}(A_{\xi, \omega}^{n}(x))=h_{\mu}^\textbf{r}(T, \xi). \end{align*}

Fix $\omega \in \pi_{\Omega}(F)$ and let a > 0. Set

\begin{align*} L_{\omega, n}=\left\lbrace x\in \mathcal{E}_{\omega}: -\dfrac{1}{m}\log \mu_{\omega}(A_{\xi,\omega}^{m}(x))\leq h_{\mu}^\textbf{r}(T, \xi)+a, \forall m\geq n \right\rbrace. \end{align*}

By Lemma 3.3, $\mu_{\omega}(L_{\omega, n}) \gt 1-\delta$ for n sufficiently large. Fix n and choose a finite subset $G_{\omega, n}=\left\lbrace x_{1}, \cdots, x_{s_{\omega, n}} \right\rbrace $ of $L_{\omega, n}$ such that $L_{\omega, n}\subset \bigcup_{i=1}^{s_{\omega, n}}A_{\xi, \omega}^{n}(x_{i})$. Since the sets $A_{\xi, \omega}^{n}(x_{i})$ are distinct and µ_ω measure of each member of them is not less than $\exp(-n(h_{\mu}^\textbf{r}(T, \xi)+a))$, then

\begin{align*} \#G_{\omega, n}=s_{\omega, n}\leq \exp(n(h_{\mu}^\textbf{r}(T, \xi)+a)). \end{align*}

Note that $\mu_{\omega} (L_{\omega, n}) \gt 1-\delta$, we have

(3.5)\begin{align} N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \delta)\leq N_{\mu_{\omega}}(\xi^{n}, \delta)\leq \exp(n(h_{\mu}^\textbf{r}(T, \xi)+a)). \end{align}

Thus for any a > 0

\begin{align*} \limsup\limits_{n \rightarrow \infty}\frac{1}{n}\int \log N_{\mu_{\omega}}({\mathcal{U}}_{0}^{n-1}, \delta) d {\mathbb{P}}(\omega) \leq h_{\mu}^\textbf{r}(T, \xi)+a. \end{align*}

Letting $a\rightarrow 0$, we obtain

\begin{equation*}\limsup\limits_{n \rightarrow \infty}\frac{1}{n}\int \log N_{\mu_{\omega}}({\mathcal{U}}_{0}^{n-1}, \delta) d {\mathbb{P}}(\omega)\leq h_{\mu}^\textbf{r}(T, \xi).\end{equation*}

Taking infimum over $\xi\succeq \mathcal{U}$ and $\delta\rightarrow 0$, we have

\begin{align*} \overline{h}_{\mu}^{S}(T, \mathcal{U}) \leq h_{\mu}^\textbf{r}(T, \mathcal{U}). \end{align*}

Applying the approach from [Reference Wu37, lemma 6.1], we have

Claim 1.

For any $\mathcal{V} \in {C}_{\mathcal{E}}^{0}$ and $0 \lt \delta \lt 1$, there exists $\beta \in \mathcal{P}_{\mathcal{E}}$ such that $\beta \succeq \mathcal{V}$ and $N_{\mu_{\omega}}(\beta, \delta)\leq N_{\mu_{\omega}}(\mathcal{V}, \delta)$ for ${\mathbb{P}}$-a.e. $\omega\in \Omega$.

Proof. Let $\pi_{\Omega}: ({\mathcal{E}}, \mathscr{F}_{{\mathcal{E}}}, \mu, \Theta)\rightarrow (\Omega, \mathscr{F}, \mathbb{F}, \theta)$ be a factor map. Let $\mathcal{V}=\left\lbrace V_{1}, \cdots, V_{m} \right\rbrace \in C_{{\mathcal{E}}}^{0}$. For $\mathbb{P}$-a.e. $\omega \in \Omega$, there exists $I_{\omega}\subset\left\lbrace 1, \cdots, m \right\rbrace $ with cardinality $N_{\mu_{\omega}}(\mathcal{V}, \delta)$ such that $\mu_{\omega}(\bigcup_{i\in I_{\omega}} V_{j}(\omega))\geq 1-\delta.$ Hence we can find $\omega_{1}, \cdots, \omega_{s}\in \Omega$ such that for $\mathbb{P}$-a.e. $\omega\in \Omega$, $I_{\omega}=I_{\omega_{i}}$ for some $i\in \left\lbrace 1, \cdots, s\right\rbrace $. For $i=1, \cdots, s$, define

\begin{align*} \Omega_{i}=\left\lbrace \omega\in \Omega: \mu_{\omega}(\bigcup_{j\in I_{\omega_{i}}}V_{j}(\omega)) \geq 1-\delta\right\rbrace. \end{align*}

Let $C_{1}=\Omega_{1}$, $C_{i}=\Omega_{i}\backslash \bigcup_{j=1}^{i-1} \Omega_{j}$, $i=2, \cdots, s$. Fix $i\in \left\lbrace 1,\cdots, s\right\rbrace $. Assume that $I_{\omega_{i}}=\left\lbrace k_{1}, \cdots, k_{t_{i}} \right\rbrace $, where $t_{i}=N_{\mu_{\omega_{i}}}(\mathcal{V}, \delta)$. Take $\left\lbrace W_{1}^{\omega_{i}}, \cdots, W_{t_{i}}^{\omega_{i}} \right\rbrace $ such that

\begin{align*} W_{1}^{\omega_{i}}=V_{k_{1}}, W_{2}^{\omega_{i}}=V_{k_{2}}\setminus V_{k_{1}}, \cdots, W_{t_{i}}^{\omega_{i}}=V_{k_{t_{i}}}\setminus \cup_{j=1}^{t_{i}-1}V_{k_{j}}. \end{align*}

Define $A:=\mathcal{E}\setminus \left( \cup_{i=1}^{s}(\pi_{\Omega}^{-1}C_{i}\cap \cup_{j=1}^{t_{i}} W_{j}^{\omega_{j}}) \right) $. Set $A_{1}=A\cap V_{1}$, $A_{l}:=A\cap (V_{l}\setminus \cup_{j=1}^{l-1}V_{j})$, $l=2, \cdots, m$. Finally, take

\begin{align*} \beta=\left\lbrace \pi_{\Omega}^{-1}C_{1}\cap W_{1}^{\omega_{1}},\! \cdots\!, \pi_{\Omega}^{-1}C_{1}\cap W_{t_{1}}^{\omega_{1}},\! \cdots\!, \pi_{\Omega}^{-1}C_{s}\cap W_{1}^{\omega_{s}},\! \cdots\!, \pi_{\Omega}^{-1}C_{s}\cap W_{t_{s}}^{\omega_{s}}, A_{1},\! \cdots\!, A_{m}\right\rbrace. \end{align*}

Then $\beta \succeq \mathcal{V}$ and $N_{\mu_{\omega}}(\beta, \delta)\leq N_{\mu_{\omega}}(\mathcal{V}, \delta)$ for ${\mathbb{P}}$-a.e. ω.

Definition 3.1. A measure-preserving system $(X, {\mathcal{B}}, \mu, T)$ is said to be aperiodic, if for every $n\in {\mathbb{N}}$, $\mu(\{x| T^{n}x=x\})=0$.

Lemma 3.4 (Lemma 1.5.4 in [Reference Shi30])

If $\delta \lt \frac{1}{2}$, then $\sum\limits_{j\leq \delta K}\binom{K}{j}\leq 2^{H(\delta)}$, where $H(\delta)=-\delta\log \delta-(1-\delta)\log (1-\delta)$.

The following lemma is the strong Rohlin Lemma [Reference Shapira29, lemma 2.5].

Lemma 3.5. Let $(X, \mathcal{B}, \mu, T)$ be an ergodic, aperiodic system and let $\alpha\in \mathcal{P}_{X}$. Then for any δ > 0 and $n\in \mathbb{N}$, one can find a set $B\in \mathcal{B}$ such that B, TB, ⋯, $T^{n-1}B$ are mutually disjoint, $\mu\left( \bigcup_{i=0}^{n-1} T^{i} B\right) \gt 1-\delta $ and the distribution of α is the same as the distribution of the partition $\alpha|_{B}$ that α induces on B.

The data $(n, \delta, B, \alpha)$ will be called a strong Rohlin tower of height n and error δ with respect to α and with B as a base.

Step 2: Our aim is to prove

\begin{equation*}h_{\mu}^\textbf{r}(T, \mathcal{U})\leq \underline{h}_{\mu}^{S}(T, \mathcal{U})\end{equation*}

where µ is an ergodic measure and $\mathcal{U}\in C_{\mathcal{E}}^{0}$.

Case 1: If the system $(\mathcal{E}, \mathscr{F}_{\mathcal{E}}, \mu, \Theta)$ is periodic, then µ is supported on a periodic point of Θ. In this case, it is straightforward to see that

\begin{align*} h_{\mu}^\textbf{r}(T, \mathcal{U})= \underline{h}_{\mu}^{S}(T, \mathcal{U})=0. \end{align*}

Now, we can assume that the system is aperiodic. Let ${\mathcal{U}}=\{U_{1}, \cdots, U_{M}\}$ be a open cover of ${\mathcal{E}}$. For fixed $n\in \mathbb{N}$, by claim 1, we can find a partition $\beta \in \mathcal{P}_{\mathcal{E}}$ such that $\beta \succeq \mathcal{U}_{0}^{n-1}$. There exists a subset A of $\mathcal{E}$ such that $\mu(A) \lt \rho$ and for any $(\omega, x)\in A$, we have $N_{\mu_{\omega}}(\beta, \rho)\leq N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \rho)$. Choose δ > 0 with $0 \lt \rho +\delta \lt 1/4$. By Lemma 3.5, we can construct a strong Rohlin tower with respect to β with height n and error less than δ. Let $\tilde{B}$ denote the base of tower and $B=\tilde{B}\setminus A$. That is

• • the sets $\{\tilde{B}, \Theta\tilde{B}, \cdots, \Theta^{n-1}\tilde{B}\}$ are disjoint and $\mu(\cup_{i=0}^{n-1}\Theta^{i} \tilde{B}) \gt 1-\delta$;

• • $\mu(A)=\mu_{\tilde{B}}(A)$ for any $A\in \beta$.

Note that β is constructed according to Claim 1, we get that A is the union of the atoms of β. Since the distribution of β is equal to the distribution of $\beta_{\tilde{B}}$, we have $\mu(B) \gt (1-\rho)\mu(\tilde{B})$. Define $E=\cup_{i=0}^{n-1}\Theta^{i}B$, then $\mu(\Theta^{i} B)\geq (1-\rho) \mu(\Theta^{i} \tilde{B})$ and hence $\mu(E) \gt (1-\delta)(1-\rho)=1-(\delta+\rho)+\delta\cdot\rho \gt 1-(\delta+\rho)$.

Since $\beta_{\tilde{B}}\succeq \mathcal{U}_{0}^{n-1}$, there exist sequences $i_{0}, \cdots, i_{n-1}$ and $B_{i_{0}, \cdots, i_{n-1}}\in \beta|_{\tilde{B}},$ such that $\Theta^{j} B_{i_{0}, \cdots, i_{n-1}}\subset U_{i_{j}}$ for every $0\leq j \leq n-1$. Let $\hat{\alpha}=\left\lbrace \hat{A}_{1}, \cdots, \hat{A}_{M}\right\rbrace $ be a partition of E defined by

\begin{align*} \hat{A}_{m}:=\bigcup\limits \left\lbrace \Theta^{j} B_{i_{0}, \cdots, i_{n-1}}: 0\leq j\leq n-1, i_{j}=m \right\rbrace. \end{align*}

Note that $\hat{A}_{m}\subset U_{m}$ for every $1\leq m \leq M$. Extend $\hat{\alpha}$ to a partition α of $\mathcal{E}$ in some way such that $\alpha\succeq \mathcal{U}$ and $\#\alpha=2M$.

Set $\eta^{4}=\rho+\delta$. For large enough k > n large enough, define

\begin{equation*}f_{k}(\omega, x)=\frac{1}{k}\sum_{i=0}^{k-1}1_{E}(\Theta^{i}(\omega, x))\end{equation*}

and $L_{k}:=\left\lbrace (\omega, x)\in \mathcal{E}: f_{k}(\omega, x) \gt 1-\eta^{2} \right\rbrace $. By Birkhoff ergodic theorem, we have $\int f_{k} d \mu \gt 1-\eta^{4}$ and

\begin{align*} \eta^{2}\mu(L_{k}^{c})\leq \int_{L_{k}^{c}} 1-f_{k} d \mu \leq \int_{\mathcal{E}} 1-f_{k} d \mu \leq \eta^{4}. \end{align*}

It follows that $\mu(L_{k})\geq 1-\eta^{2}$. Since E is measurable, L_k is measurable with respect to $(\omega, x)\in\mathcal{E}$. For all $j\geq k$, take

\begin{align*} &J_{k}=\left\lbrace(\omega, x)\in \mathcal{E}\!:\! \mu_{\omega}(A_{\alpha,\omega}^{j}(x)) \lt \exp(-(h_{\mu}^\textbf{r}(T, \alpha)-\eta)j) \right\rbrace \bigcap \\ & \left\lbrace\! (\omega,x)\!\in\! {\mathcal{E}}\!: \!\bigg|\frac{1}{j}\!\sum_{i=0}^{j-1}\!\log N_{\mu_{\theta^{i}\omega}}(\mathcal{U}_{0}^{n-1}, \rho)1_{B}(\Theta^{i}(\omega, x))\!-\!\!\int\!\! \log N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \rho)1_{B}(\omega, x) d \mu\bigg |\!\leq\! \eta\! \right\rbrace\!. \end{align*}

Applying Lemma 3.3 and the Birkhoff ergodic theorem, we conclude that $\mu(J_{k}) \gt 1-\eta^{2}$ for k sufficiently large k. Then by [Reference Crauel5, corollary 3.4], the set

\begin{align*} \left\lbrace(\omega, x)\in \mathcal{E}: \mu_{\omega}(A_{\alpha,\omega}^{j}(x)) \lt \exp(-(h_{\mu}^\textbf{r}(T, \alpha)-\eta)j), \forall j\geq k \right\rbrace, \end{align*}

is measurable. Since $N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \rho)$ is measurable with respect to $\omega\in \Omega$, then the function $N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \rho)$ is measurable with respect to $(\omega, x)\in\mathcal{E}$ by [Reference Crauel5, lemma 1.1]. And $1_{B}(\Theta^{i}(\omega,x))=1_{\Theta^{-i}B}(\omega,x)$ is a measurable function. Then J_k is measurable. Set $G_{k}=L_{k}\cap J_{k}$, then G_k is measurable and $\mu(G_{k}) \gt 1-2\eta^{2}$. Define $\tilde{G}_{k}=\left\lbrace (\omega, x)\in G_{k}: \mu_{\omega}(G_{k}(\omega))\geq 1-4\eta \right\rbrace $, we have

\begin{align*} \tilde{G}_{k}^{c}\cap \mathcal{E}=\left\lbrace (\omega, x)\in G_{k}: \mu_{\omega}(G_{k}^{c}(\omega)) \gt 4\eta \right\rbrace \cup (G_{k}^{c}\cap \mathcal{E}). \end{align*}

Therefore,

\begin{equation*} \mu(\tilde{G}_{k}^{c}\cap \mathcal{E})\cdot 4 \eta \leq \int \mu_{\omega} (G_{k}^{c}(\omega)) d \mathbb{P}+\mu(G_{k}^{c})=2 \mu(G_{k}^{c})\leq 4 \eta^{2}, \end{equation*}

i.e., $ \mu(\tilde{G}_{k}^{c}\cap \mathcal{E})\leq \eta$.

Fix $\omega\in \pi_{\Omega}(\tilde{G}_{k})$ and choose a sufficiently large k > n. Let $0\leq i_{1} \leq \cdots \leq i_{m} \leq k-n$, and set

\begin{align*} C_{\omega}=\{x\in \tilde{G}_{k}(\omega): T_{\omega}^{i_{1}}x\in B(\theta^{i_{1}}\omega), \cdots, T_{\omega}^{i_{m}}x\in B(\theta^{i_{m}}\omega)\}. \end{align*}

Because each element of this partition corresponds to a collection of subintervals of $[0,k-1]$ of length n, which covers all but at most $\eta^{2}k+2n$ elements of $[0, k-1]$ in a one-to-one manner, we have the number of elements in the partition of $\tilde{G}_{k}(\omega)$ is bounded above by

\begin{align*} \sum\limits_{j \lt \eta^{2}k+2n}\binom{k}{j}. \end{align*}

In the sequel, we will want to estimate the number of $\alpha_{0}^{k-1}(\omega)$-elements needed to cover it. If $0\leq i_{1}\leq \cdots \leq i_{m}\leq k-n$ are the times elements of C_ω visit $B(\theta^{i_{1}}\omega),\cdots ,B(\theta^{i_{m}}\omega)$, then we need at most $N_{\mu_{\theta^{i_{j}}\omega}}(\mathcal{U}_{0}^{n-1}, \rho)$ $\alpha_{i_{j}}^{i_{j}+n-1}(\omega)$-elements to cover C_ω. Because the size of $[0, k-1]\setminus \cup_{j}[i_{j}, i_{j}+n-1]$ is at most $\eta^{2}k+2n,$ we need at most $\Pi_{j=1}^{m} N_{\mu_{\theta^{j}\omega}}(\mathcal{U}_{0}^{n-1}, \rho)\cdot (2M)^{\eta^{2}k+2n}$ $\alpha_{0}^{k-1}(\omega)$-elements to cover C_ω. Since $\tilde{G}_{k}\subset J_{k}$, we know that $\tilde{G}_{k}(\omega)$ can be covered by no more than

(3.6)\begin{align} \sum\limits_{j \lt \eta^{2}k+2n}\binom{k}{j} \cdot (2M)^{\eta^{2}k+2n}\cdot e^{k(\int_{B} \log N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \rho)d \mu+\eta)} \end{align}

$\alpha_{0}^{k-1}(\omega)$ elements. By Lemma 3.4, the above equation (3.6) less than

\begin{align*} e^{kH(\eta^{2}+2n/k)}\cdot (2M)^{\eta^{2}k+2n}\cdot e^{k(\int_{B} \log N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \rho)d \mu+\eta)}. \end{align*}

Note that $\omega\in \pi_{\Omega}(\tilde{G}_{k})$, then we have

(3.7)\begin{align} &1-4\eta\leq \mu_{\omega}(\tilde{G}_{k}(\omega))\leq\cr & \exp(-(h_{\mu}^\textbf{r}(T, \alpha)-\eta)k)\cdot\exp(kH(\eta^{2}+2n/k))\cdot \nonumber\\ &(2M)^{\eta^{2}k+2n}\cdot \exp\left( k(\int \log N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \rho) 1_{B}(\omega, x)d \mu+\eta)\right). \end{align}

Combining with (3.7), we obtain

\begin{align*} h_{\mu}^\textbf{r}(T, \alpha) &\leq \eta +H(\eta^{2})+\eta^{2}\log (2M)+ \int \log N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \rho) 1_{B}(\omega, x) d \mu+\eta \\ &\leq 2\eta+ H(\eta^{2})+\eta^{2}\log (2M)+\frac{1}{n}\int \log N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \rho) 1_{B}(\omega, x) d \mu\\ & \leq 2\eta+ H(\eta^{2})+\eta^{2}\log (2M)+\frac{1}{n}\int \log N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \rho) 1_{\Omega\times X}(\omega, x) d \mu_{\omega} d \mathbb{P}(\omega)\\ & \leq 2\eta+ H(\eta^{2})+\eta^{2}\log (2M)+\frac{1}{n}\int \log N_{\mu_{\omega}}(\mathcal{U}_{0}^{n-1}, \rho) d \mathbb{P}(\omega). \end{align*}

Letting $n\rightarrow \infty$ and then $\rho \rightarrow 0$, we get

\begin{align*} h_{\mu}^\textbf{r}(T, \mathcal{U})\leq h_{\mu}^\textbf{r}(T, \alpha) \leq \underline{h}_{\mu}^{S}(T, \mathcal{U}). \end{align*}

Theorem 3.4. Let T be a homeomorphic bundle RDS on ${\mathcal{E}}$ over an ergodic measure-preserving system $(\Omega, \mathscr{F},{\mathbb{P}}, \theta)$. Then

\begin{align*} &{\mathbb{E}}{{\rm\overline{mdim}_M}}(T,{\mathcal{E}},d)=\limsup\limits_{\epsilon \rightarrow 0}\frac{1}{|\log \epsilon|}\sup_{\mu\in {E}_{\mathbb{P}}({\mathcal{E}})}\inf_{\substack{{\rm diam}({\mathcal{U}})\leq \epsilon,\\ {\mathcal{U}}\in {\mathcal{C}}_{X}^o }}h_{\mu}^{S}(T, (\Omega\times{\mathcal{U}})_{\mathcal{E}}).\\ & {\mathbb{E}}{{\rm\underline{mdim}_M}}(T,{\mathcal{E}}, d)=\liminf\limits_{\epsilon \rightarrow 0} \frac{1}{|\log \epsilon|}\sup_{\mu\in {E}_{\mathbb{P}}({\mathcal{E}})}\inf_{\substack{{\rm diam}({\mathcal{U}})\leq \epsilon,\\ {\mathcal{U}}\in {\mathcal{C}}_{X}^o }}h_{\mu}^{S}(T, (\Omega\times{\mathcal{U}})_{\mathcal{E}}). \end{align*}

Proof. Fix ϵ > 0 and $\mu \in E_{\mathbb{P}}({\mathcal{E}})$. Then by Theorem 3.3, we have

(3.8)\begin{align} \inf_{\substack{{\rm diam}({\mathcal{U}})\leq \epsilon,\\ {\mathcal{U}}\in {C}_{X}^o }} h_{\mu}^{S}(T, (\Omega\times{\mathcal{U}})_{\mathcal{E}}) &=\inf_{{\rm diam}({\mathcal{U}})\leq \epsilon, \atop {\mathcal{U}}\in {C}_{X}^o }h_{\mu}^\textbf{r}(T, (\Omega\times{\mathcal{U}})_{\mathcal{E}}), \nonumber\\ &=\inf_{{\rm diam}({\mathcal{U}})\leq \epsilon, \alpha \succeq {\mathcal{U}}}h_{\mu}^\textbf{r}(T, (\Omega\times\alpha)_{\mathcal{E}})\nonumber\\ &\geq \inf_{{\rm diam}(\alpha)\leq \epsilon, \atop \alpha\in {P}_{X}}h_{\mu}^\textbf{r}(T,(\Omega\times\alpha)_{\mathcal{E}}). \end{align}

By Lemma 3.1, we can choose a finite open cover $\mathcal{U}'$ of X with ${\rm diam}(\mathcal{U}')\leq \epsilon$ and ${\rm Leb}(\mathcal{U}')\geq \frac{\epsilon}{4}$. Then

(3.9)\begin{align} \inf_{\substack{{\rm diam}({\mathcal{U}})\leq \epsilon\\ {\mathcal{U}}\in {C}_{X}^o }} h_{\mu}^{S}(T, (\Omega\times{\mathcal{U}})_{\mathcal{E}}) &\leq h_{\mu}^{S}(T, (\Omega\times{\mathcal{U'}})_{\mathcal{E}})\nonumber\\ &=h_{\mu}^\textbf{r}(T, (\Omega\times{\mathcal{U'}})_{\mathcal{E}}),~\text{by Theorem}~ 3.3\nonumber\\ &=\inf_{\alpha \succeq {\mathcal{U}}^{'},{\mathcal{\alpha}}\in {P}_{X}}h_{\mu}^\textbf{r}(T, (\Omega\times\alpha)_{\mathcal{E}}) \nonumber\\ &\leq \inf_{{\rm diam}(\alpha)\leq \frac{\epsilon}{8}, \atop {\mathcal{\alpha}}\in {P}_{X}}h_{\mu}^\textbf{r}(T,(\Omega\times\alpha)_{\mathcal{E}}). \end{align}

We finally get the desired results by the inequalities (3.8), (3.9) and Theorem 3.2.

3.3. Variational principle III: Katok’s ϵ-entropy

In this subsection, we prove the third main result by replacing Shapira’s ϵ-entropy with Katok local ϵ-entropy. Given $\mu\in {M}_{\mathbb{P}}({\mathcal{E}})$, let

\begin{align*} N_{\mu_{\omega}}^{\delta}(n, \epsilon)=\min\left\lbrace j: \mu_{\omega}\left( \bigcup_{i=1}^{j}B_{d_{n}^{\omega}}(x_{i}, \epsilon)\right) \gt 1-\delta \right\rbrace. \end{align*}

Based on proposition 3.1, we similarly obtain the measurability of $N_{\mu_{\omega}}^{\delta}(n, \epsilon)$ and define the upper and lower Katok’s ϵ-entropies of µ as follows

\begin{align*} &\overline{h}_{\mu}^{K}(T, \epsilon)=\lim\limits_{\delta \to0}\limsup\limits_{n\rightarrow \infty}\frac{1}{n}\int \log N_{\mu_{\omega}}^{\delta}(n, \epsilon) d \mathbb{P}(\omega),\\ & \underline{h}_{\mu}^{K}(T, \epsilon)=\lim\limits_{\delta \to0}\liminf\limits_{n\rightarrow \infty}\frac{1}{n}\int \log N_{\mu_{\omega}}^{\delta}(n, \epsilon) d \mathbb{P}(\omega). \end{align*}

Theorem 3.5. Let T be a homeomorphic bundle RDS on ${\mathcal{E}}$ over an ergodic measure-preserving system $(\Omega, \mathscr{F},{\mathbb{P}}, \theta)$. Then

\begin{align*} &{\mathbb{E}}{{\rm\overline{mdim}_M}}(T,{\mathcal{E}},d)=\limsup\limits_{\epsilon \rightarrow 0}\frac{1}{|\log \epsilon|}\sup_{\mu\in {E}_{\mathbb{P}}({\mathcal{E}})}\overline{h}_{\mu}^{K}(T, \epsilon), \\ & {\mathbb{E}}{{\rm\underline{mdim}_M}}(T,{\mathcal{E}}, d)=\liminf\limits_{\epsilon\rightarrow 0} \frac{1}{|\log \epsilon|}\sup_{\mu\in {E}_{\mathbb{P}}({\mathcal{E}})}\overline{h}_{\mu}^{K}(T, \epsilon). \end{align*}

The results are valid if we change $\overline{h}_{\mu}^{K}(T, \epsilon)$ into $\underline{h}_{\mu}^{K}(T, \epsilon)$.

Proof. It suffices to show the results hold for ${\mathbb{E}}{{\rm\overline{mdim}_M}}(T,{\mathcal{E}},d)$ since the second one follows similarly. Fix ϵ > 0. Let $0 \lt \delta \lt 1$ and $\mu\in {E}_{\mathbb{P}}({\mathcal{E}})$. Let $\mathcal{U}=\left\lbrace U_{1},\cdots, U_{l} \right\rbrace $ be a finite open cover of X with ${\rm diam}(\mathcal{U}) \lt \epsilon$. Then the family $\mathcal{U}(\omega)$ formed by the sets $U\cap \mathcal{E}_{\omega}$ with $U\in \mathcal{U}$ is an open cover of ${\mathcal{E}}_{\omega}$. This implies that each element of $\bigvee_{i=0}^{n-1}(T_{\omega}^{i})^{-1}\mathcal{U}(\theta^{i}\omega)$ can be contained in an $(n, \epsilon, \omega)$-Bowen ball. So

\begin{align*} N_{\mu_{\omega}}^{\delta}(n, \epsilon) \leq N_{\mu_{\omega}}\left( \bigvee_{i=0}^{n-1}(T_{\omega}^{i})^{-1}\mathcal{U}(\theta^{i}\omega), \delta\right). \end{align*}

This shows

(3.10)\begin{align} \overline{h}_{\mu}^{K}(T, \epsilon)\leq \inf_{\substack{{\rm diam}({\mathcal{U}})\leq \epsilon,\\ {\mathcal{U}}\in {\mathcal{C}}_{X}^o }}\overline{h}_{\mu}^{S}(T, (\Omega\times\mathcal{U})_{{\mathcal{E}}}). \end{align}

By Lemma 3.1, we can choose a finite cover $\mathcal{U}$ of X such that ${\rm diam}({\mathcal{U}})\leq\epsilon$ and ${\rm Leb}({\mathcal{U}})\geq \frac{\epsilon}{4}$. Since each $(n, \frac{\epsilon}{4}, \omega)$-Bowen ball is contained in some element of $\bigvee_{i=0}^{n-1}(T_{\omega}^{i})^{-1}\mathcal{U}(\theta^{i}\omega)$, then $N_{\mu_{\omega}}(\bigvee_{i=0}^{n-1}(T_{\omega}^{i})^{-1}\mathcal{U}(\theta^{i}\omega), \delta)\leq N_{\mu_{\omega}}^{\delta}(n, \frac{\epsilon}{4})$. This shows

(3.11)\begin{align} \inf_{\substack{{\rm diam}({\mathcal{U}})\leq \epsilon,\\ {\mathcal{U}}\in {\mathcal{C}}_{X}^o }}\overline{h}_{\mu}^{S}(T, (\Omega\times\mathcal{U})_{{\mathcal{E}}}) \leq \overline{h}_{\mu}^{K}(T, \frac{\epsilon}{4}). \end{align}

Since $\mu\in E_{{\mathbb{P}}}({\mathcal{E}})$, we have $\overline{h}_{\mu}^{S}(T, (\Omega\times\mathcal{U})_{{\mathcal{E}}})={h}_{\mu}^{S}(T, (\Omega\times\mathcal{U})_{{\mathcal{E}}})$ by Theorem 3.3. Therefore, by inequalities (3.10), (3.11) and Theorem 3.4, we get the desired results.

3.4. Variational principle IV: Brin–Katok local ϵ-entropy

In this subsection, we borrow the Shannon-McMillan-Breiman theorem of random dynamical systems and Theorem 3.5 to establish the fourth variational principle for metric mean dimensions in terms of Brin–Katok local ϵ-entropy.

Let $\mu \in M_{\mathbb{P}}({\mathcal{E}})$, $\omega\in \Omega$ and $x\in {\mathcal{E}}_{\omega}$. Put

\begin{align*} &\overline{h}_{\mu_{\omega}}^{BK}(T,x, \epsilon)=\limsup\limits_{n\rightarrow \infty}-\frac{1}{n}\log \mu_{\omega}(B_{d_{n}^{\omega}}(x, \epsilon)),\\ & \underline{h}_{\mu_{\omega}}^{BK}(T,x, \epsilon)=\liminf\limits_{n\rightarrow \infty}-\frac{1}{n}\log \mu_{\omega}(B_{d_{n}^{\omega}}(x, \epsilon)). \end{align*}

We define the upper and lower Brin–Katok local ϵ-entropies of µ at x as

\begin{align*} \overline{h}_{\mu}^{BK}(T, \epsilon)&=\int \overline{h}_{\mu_{\omega}}^{BK}(T, x , \epsilon) d \mu(\omega, x),\\ \underline{h}_{\mu}^{BK}(T, \epsilon)&=\int \underline{h}_{\mu_{\omega}}^{BK}(T, x , \epsilon) d \mu(\omega, x). \end{align*}

The Brin–Katok’s entropy formula for RDS is given by Zhu in [Reference Zhu40, theorem 2.1][Reference Zhu39, theorem 2.1]. When $\mu\in M_{{\mathbb{P}}}({\mathcal{E}})$, they stated that

\begin{align*} \lim\limits_{\epsilon\rightarrow 0}\overline{h}_{\mu}^{BK}(T, \epsilon)=\lim\limits_{\epsilon\rightarrow 0}\underline{h}_{\mu}^{BK}(T, \epsilon)=h_{\mu}^\textbf{r}(T). \end{align*}

In particular, when µ is ergodic, $\lim\limits_{\epsilon\rightarrow 0}\overline{h}_{\mu_{\omega}}^{BK}(T,x, \epsilon)=\lim\limits_{\epsilon\rightarrow 0}\underline{h}_{\mu_{\omega}}^{BK}(T,x, \epsilon)=h_{\mu}^\textbf{r}(T).$ We give the following equalities for given ϵ > 0.

Proposition 3.2. Let T be a continuous bundle RDS on ${\mathcal{E}}$ over a measure-preserving system $(\Omega, \mathscr{F},{\mathbb{P}}, \theta)$. If $\mu\in {E}_{\mathbb{P}}({\mathcal{E}})$, then for every ϵ > 0,

(3.12)\begin{align} \overline{h}_{\mu_{\omega}}^{BK}(T, x, \epsilon)=\overline{h}_{\mu}^{BK}(T, \epsilon) ~\text{and}~ \underline{h}_{\mu_{\omega}}^{BK}(T, x, \epsilon)=\underline{h}_{\mu}^{BK}(T, \epsilon) \end{align}

for µ-a.e $(\omega, x)$.

Proof. Let $\mu\in E_{\mathbb{P}}(\mathcal{E})$. By [Reference Crauel5, proposition 3.6], µ can be disintegrated as $d \mu(\omega, x)=d \mu_{\omega}(x) d {\mathbb{P}}(\omega)$. Here the definition of µ_ω can see [Reference Crauel5, defintion 3.1]. Let $F(\omega,x):= \overline{h}_{\mu_{\omega}}^{BK}(T, x, \epsilon)$. Fix $n\in \mathbb{N}$ and $\omega\in \Omega$, we have

\begin{align*} B_{d_{n}^{\omega}}(x, \epsilon)&=\cap_{j=0}^{n-1}(T_\omega^j)^{-1}(B(T_\omega^jx,\epsilon)\cap \mathcal{E}_{\theta^j\omega})\\ &=T_\omega^{-1}\big(\cap_{j=1}^{n-1}(T_{\theta\omega}^{j-1})^{-1}(B(T_{\theta \omega}^{j-1}(T_\omega x),\epsilon)\cap \mathcal{E}_{\theta^{j-1}{\theta \omega}})\big)\cap (B(x, \epsilon)\cap \mathcal{E}_{\omega}) \\ &\subseteq T_\omega^{-1}B_{d_{n-1}^{\theta \omega}}(T_\omega x, \epsilon), \end{align*}

and hence $\mu_{\omega}(B_{d_{n}^{\omega}}(x, \epsilon))\leq\mu_{\omega}(T_\omega^{-1}B_{d_{n-1}^{\theta \omega}}(T_\omega x, \epsilon))=\mu_{\theta \omega}(B_{d_{n-1}^{\theta \omega}}(T_\omega x, \epsilon))$ for $\mathbb{P}$-a.e ω by using the fact $T_\omega\mu_\omega=\mu_{\theta \omega}.$ This shows for µ-a.e $(\omega, x)$

\begin{equation*}F(\omega,x)= \overline{h}_{\mu_{\omega}}^{BK}(T, x, \epsilon)\geq\overline{h}_{\mu_{\theta \omega}}^{BK}(T, T_\omega x, \epsilon)=F\circ\Theta (\omega,x).\end{equation*}

Since µ is ergodic, this shows for µ-a.e $(\omega, x)$ $\overline{h}_{\mu_{\omega}}^{BK}(T, x, \epsilon)=\overline{h}_{\mu}^{BK}(T, \epsilon)$.

Theorem 3.6. Let T be a homeomorphic bundle RDS on ${\mathcal{E}}$ over an ergodic measure-preserving $(\Omega, \mathscr{F},{\mathbb{P}}, \theta)$. Then

\begin{align*} &{\mathbb{E}}{{\rm\overline{mdim}_M}}(T,{\mathcal{E}},d)=\limsup\limits_{\epsilon \rightarrow 0}\frac{1}{|\log \epsilon|}\sup_{\mu\in {M}_{\mathbb{P}}({\mathcal{E}})}\overline{h}_{\mu}^{BK}(T, \epsilon),\\ & {\mathbb{E}}{{\rm\underline{mdim}_M}}(T, {\mathcal{E}}, d)=\liminf\limits_{\epsilon\rightarrow 0} \frac{1}{|\log \epsilon|}\sup_{\mu\in {M}_{\mathbb{P}}({\mathcal{E}})}\overline{h}_{\mu}^{BK}(T, \epsilon). \end{align*}

Moreover, both ${M}_{\mathbb{P}}({\mathcal{E}})$ in the above equalities can be replaced by ${E}_{\mathbb{P}}({\mathcal{E}})$.

Proof. Given ϵ > 0 and $\mu\in M_{{\mathbb{P}}}({\mathcal{E}})$, choose a finite Borel partition ξ of X with ${\rm diam} \xi\leq\epsilon$. By Lemma 3.3, we have

\begin{align*} \int \lim\limits_{n\rightarrow \infty}-\frac{1}{n} \log \mu_{\omega}(A_{(\Omega\times\xi)_{{\mathcal{E}}},\omega}^{n}(x)) d \mu(\omega, x)=h_{\mu}^\textbf{r}(T, (\Omega\times\xi)_{{\mathcal{E}}}). \end{align*}

It is clear that $ A_{(\Omega\times\xi)_{{\mathcal{E}}},\omega}^{n}(x)\subset B_{d_{n}^{\omega}}(x, \epsilon)$ for every $n\in {\mathbb{N}}$. Then

\begin{align*} &\int \limsup\limits_{n \rightarrow \infty} -\frac{1}{n}\log \mu_{\omega}(B_{d_{n}^{\omega}}(x, \epsilon)) d\mu (\omega, x)\leq \int \lim\limits_{n\rightarrow \infty}-\frac{1}{n} \log \mu_{\omega}(A_{(\Omega\times\xi)_{{\mathcal{E}}},\omega}^{n}(x)) d \mu(\omega, x). \end{align*}

Moreover

\begin{equation*}\overline{h}_{\mu}^{BK}(T,\epsilon)\leq h_{\mu}^\textbf{r}(T, (\Omega\times\xi)_{{\mathcal{E}}}).\end{equation*}

Therefore,

\begin{align*} {\mathbb{E}}{{\rm\overline{mdim}_M}}(T,{\mathcal{E}}, d)&=\limsup_{\epsilon \rightarrow 0}\frac{1}{|\log \epsilon|}\sup_{\mu \in {M}_{\mathbb{P}}({\mathcal{E}})}\inf_{{\rm diam }\xi \leq \epsilon \atop \xi \in \mathcal{P}_X} h_{\mu}^\textbf{r}(T,(\Omega\times \xi)_{{\mathcal{E}}}),\text{by Theorem}~ 3.2 \\ & \geq\limsup_{\epsilon \rightarrow 0}\frac{1}{|\log \epsilon|} \sup_{\mu \in {M}_{\mathbb{P}}({\mathcal{E}})} \overline{h}_{\mu}^{BK}(T, \epsilon). \end{align*}

We next verify the converse direction. By (3.12), there exists a µ-full measure set $E\subset \Omega\times X$ so that for $(\omega,x)\in E$,

\begin{align*} \limsup_{n \rightarrow \infty}-\frac{1}{n} \log \mu_{\omega}(B_{d_{n}^{\omega}}(x, \epsilon))=\overline{h}_{\mu}^{BK}(T, \epsilon). \end{align*}

Note that $\mathbb{P}(\pi_{\Omega} E)=1$ and $\mu(E)=\int_{\pi_{\Omega} E}\mu_{\omega}(E(\omega))d\mathbb{P}(\omega)=1,$ where $E(\omega)=\{x\in \mathcal{E}_\omega:(\omega,x)\in E\}$. Then we can obtain that $\mu_{\omega}(E(\omega))=1$ for all $\omega\in \pi_{\Omega} E$. Given $\omega\in \pi_{\Omega} E$, ρ > 0 and $n\in {\mathbb{N}}$, set

\begin{align*} G_{n, \rho}^{\omega}=\left\lbrace x\in {E}(\omega): -\frac{1}{n}\log \mu_{\omega}(B_{d_{n}^{\omega}}(x, \epsilon)) \lt \overline{h}_{\mu}^{BK}(T, \epsilon)+\rho\right\rbrace . \end{align*}

Let $0 \lt \delta \lt 1$. Then for all sufficiently large $n\in {\mathbb{N}}$ (depending on $\delta,\omega,\rho$), one has $\mu_{\omega}(G_{n, \rho}^{\omega}) \gt 1-\delta$. Let H_n be a maximal $(\omega, 2\epsilon, n )$-separated subset of $G_{n, \rho}^{\omega}$. Therefore it is also an $(\omega, 2\epsilon, n)$-spanning subset of $G_{n,\rho}^{\omega}$ and the family $\left\lbrace B_{d_{n}^{\omega}}(x, \epsilon): x\in H_{n} \right\rbrace $ is pairwise disjoint. It follows that $\mu_{\omega}(\bigcup_{x\in H_{n}} B_{d_{n}^{\omega}}(x, 2\epsilon))\geq \mu_{\omega}(G_{n, \rho}^{\omega}) \gt 1-\delta$ and

\begin{align*} \# H_{n}\cdot e^{-n(\overline{h}_{\mu}^{BK}(T, \epsilon)+\rho)}\leq \sum_{x\in H_{n}} \mu_{\omega}(B_{d_{n}^{\omega}}(x, \epsilon))=\mu_{\omega}(\bigcup_{x\in H_{n}} B_{d_{n}^{\omega}}(x, \epsilon))\leq 1. \end{align*}

Then $N_{\mu_{\omega}}^{\delta}(n, 2\epsilon)\leq \#H_{n}\leq e^{n(\overline{h}_{\mu}^{BK}(T, \epsilon)+\rho)}$. This yields that

\begin{align*} \overline{h}_{\mu}^{BK}(T,\epsilon)+\rho&\geq \int_{\pi_{\Omega} E} \limsup_{n \rightarrow \infty} \frac{1}{n} \log N_{\mu_{\omega}}^{\delta}(n, 2\epsilon) d {\mathbb{P}}(\omega)\\ &= \int \limsup_{n \rightarrow \infty} \frac{1}{n} \log N_{\mu_{\omega}}^{\delta}(n, 2\epsilon) d {\mathbb{P}}(\omega)\\ &\geq \limsup_{n \rightarrow \infty} \frac{1}{n} \int \log N_{\mu_{\omega}}^{\delta}(n, 2\epsilon) d {\mathbb{P}}(\omega),\text{by Fatou's lemma}. \end{align*}

Letting $\delta \rightarrow 0$ and then letting $\rho \rightarrow 0$, we obtain $\overline{h}_{\mu}^{K}(T, 2\epsilon) \leq \overline{h}_{\mu}^{BK}(T, \epsilon)$ for every $\mu\in E_{{\mathbb{P}}}({\mathcal{E}})$. Then by Theorem 3.5, we have

\begin{align*} {\mathbb{E}}{{\rm\overline{mdim}_M}}(T,{\mathcal{E}}, d)=\limsup\limits_{\epsilon \rightarrow 0} \frac{1}{|\log \epsilon|}\sup_{\mu \in {E}_{\mathbb{P}}({\mathcal{E}})} \overline{h}_{\mu}^{K}(T, \epsilon)\leq \limsup\limits_{\epsilon \rightarrow 0} \frac{1}{|\log \epsilon|}\sup_{\mu \in {E}_{\mathbb{P}}({\mathcal{E}})}\overline{h}_{\mu}^{BK}(T, \epsilon). \end{align*}

To illustrate our main theorem, we discuss the following example.

Example 3. Let $(\Omega,\mathcal{F}, \mathbb{P}, \theta)$ be an ergodic measure-preserving system. Define the standard metric on torus $\mathbb{T}$ as follows:

\begin{align*} ||x-y||_{{\mathbb{T}}}=\min\{|x-y-n|: n\in \mathbb{N}\} \end{align*}

for each $x,y\in \mathbb{T}$. Set $\mathbb{T}^{\mathbb{Z}}$ equipped with the metric

\begin{align*} d(x,y)=\sum_{n\in \mathbb{Z}}2^{-|n|}||x_{n}-y_{n}||_{\mathbb{T}} \end{align*}

and let $\sigma: \mathbb{T}^{\mathbb{Z}}\rightarrow \mathbb{T}^{\mathbb{Z}}$ be the shift on $\mathbb{T}$. Assume that $h:\Omega \rightarrow \mathbb{T}^{\mathbb{Z}}$ is a measurable map. We consider the $T_{\omega}(x)$ generated by $T_{\omega}(x)=\sigma(x)+h(\omega)$ for $\omega\in \Omega$, $x\in \mathbb{T}^{\mathbb{Z}}$. Observer that for any $0 \leq j \lt n$,

\begin{align*} T_{\omega}^{j}(x)=\sigma^{j}(x)+ \sum_{k=0}^{j-1}\sigma^{j-1-m}h(\theta^{k}\omega). \end{align*}

The skew product is defined by $\Theta^{n}(w, x)\rightarrow (\theta^{n} w, T_{\omega}^{n} x)$ for $n\in \mathbb{Z}$. Let $0 \lt \epsilon \lt \frac{1}{2}$, set $l = \lceil\log_2(4/\varepsilon)\rceil$. Then $\sum_{|n| \gt l} 2^{-|n|} \leq \varepsilon/2$. Consider the cover of $\mathbb{T}$ by

\begin{equation*}I_k = \left(\frac{(k - 1)\varepsilon}{12}, \frac{(k + 1)\varepsilon}{12}\right),\quad 0 \leq k \leq \lfloor12/\varepsilon\rfloor.\end{equation*}

I_k has length $\varepsilon/6$. For $n \geq 1$, consider

\begin{equation*} \mathbb{T}^{\mathbb{Z}} = \bigcup_{0 \leq k_{-l}, \ldots, k_{n+l} \leq \lfloor 12/\varepsilon \rfloor} \left\{x \mid ||x_{-l}-\frac{k_{\ell}\epsilon}{12}||_{{\mathbb{T}}} \lt \frac{\epsilon}{12} \ldots, \\||x_{n+\ell}-\frac{k_{n+\ell}\epsilon}{12}||_{{\mathbb{T}}} \lt \frac{\epsilon}{12} \right\}.\end{equation*}

Each open set in the right-hand side has diameter less than ɛ with respect to the distance d_n. Hence

(3)\begin{equation} \#(\mathbb{T}^{\mathbb{Z}}, d_n, \varepsilon) \leq (1 + \lfloor 12/\varepsilon \rfloor)^{n + 2l + 1} = (1 + \lfloor 12/\varepsilon \rfloor)^{n + 2\lceil \log_2(4/\varepsilon) \rceil + 1} \end{equation}

On the other hand, any two distinct points in the sets

\begin{equation*} \{x \in \mathbb{T}^{\mathbb{Z}} \mid x_m \in \{0, \varepsilon, 2\varepsilon, \ldots, \lfloor 1/\varepsilon \rfloor \varepsilon\} \text{for all } 0 \leq m \lt n\} \end{equation*}

have distance $\geq \varepsilon$ with respect to d_n. It follows that

\begin{equation*} \#(\mathbb{T}^{\mathbb{Z}}, d_n, \varepsilon) \geq (1 + \lfloor 1/\varepsilon \rfloor)^n. \end{equation*}

Thus

\begin{equation*}\rm mdim_M(\mathbb{T}^{\mathbb{Z}}, \sigma, d) = 1. \end{equation*}

Let $d_{\Omega}$ be the metric on Ω. Take the metric dʹ on $\Omega\times [0, 1]^{{\mathbb{Z}}}$ as follows:

\begin{align*} d'((\omega_{1}, x), (\omega_{2}, y))=d(x,y)+d_{\Omega}(w_{1}, w_{2}), ~~\forall \omega_{1}, \omega_{2}\in \Omega, ~x, y\in \mathbb{T}^{{\mathbb{Z}}}. \end{align*}

Note that for any $\omega\in \Omega$, $k\geq 0$ and $x,y\in \mathbb{T}^{{\mathbb{Z}}}$,

\begin{align*} &d'(\Theta^{k}(\omega, x), \Theta^{k}(\omega, y))\\ &= d'((\theta^{k}\omega, T_{\omega}^{n}x), (\theta^{k}\omega, T_{\omega}^{n}y))\\ &= d'((\theta^{k}\omega, \sigma^{j}(x)+ \sum_{k=0}^{j-1}\sigma^{j-1-m}h(\theta^{k}\omega)), (\theta^{k}\omega, \sigma^{j}(y)+ \sum_{k=0}^{j-1}\sigma^{j-1-m}h(\theta^{k}\omega))\\ &=d(\sigma^{k}x, \sigma^{k}y). \end{align*}

Hence for any ϵ > 0, and $n\geq 1$, one has

\begin{align*} {\rm sep}(\Omega\times \mathbb{T}^{{\mathbb{Z}}}, \omega, \epsilon, n)={\rm sep}(\mathbb{T}^{{\mathbb{Z}}}, \epsilon, n), ~~\forall \omega \in \Omega. \end{align*}

Thus

\begin{align*} &{\mathbb{E}}{{\rm mdim}}_{M}(T,\Omega\times \mathbb{T}^{{\mathbb{Z}}},d)\\ &=\lim\limits_{\epsilon\rightarrow 0}\int_{\Omega}\frac{1}{|\log \epsilon|}\limsup_{n \rightarrow \infty}\frac{1}{n}\log {\rm sep}(\Omega\times \mathbb{T}^{\mathbb{Z}}, \omega, \epsilon, n) d {\mathbb{P}}(\omega)\\ &=\lim\limits_{\epsilon\rightarrow 0}\frac{1}{|\log \epsilon|}\limsup_{n \rightarrow \infty}\frac{1}{n}\log {\rm sep}( \mathbb{T}^{\mathbb{Z}}, \epsilon, n)\\ &={\rm mdim}_{M}(\mathbb{T}^{{\mathbb{Z}}}, \sigma,d)=1. \end{align*}

Let $\mathcal{L}$ be the Lebesgue measure on $\mathbb{T}$ and $\mu_{\omega}=\mathcal{L}^{\bigotimes \mathbb{Z}}$ for $\mathbb{P}$ a.e ω. Set $d \mu (\omega, x)= d \mu_{\omega}(x) d {\mathbb{P}}(\omega)$. Let $0 \lt \epsilon \lt \frac{1}{2}$, $x\in \mathbb{T}^{\mathbb{Z}}$. Take $\ell=\lceil\log_{2}\frac{4}{\epsilon} \rceil $ such that $\sum_{|n| \gt \ell}\frac{1}{2^{|n|}}\leq \frac{\epsilon}{2}$. Let

\begin{align*} I_{n}(x, \epsilon)=\left\lbrace y\in \mathbb{T}^{\mathbb{Z}}: ||y_{k}-x_{k}||_{\mathbb{T}}\leq \frac{\epsilon}{6}, ~\forall -\ell \leq k \leq n+\ell\right\rbrace, \end{align*}

and

\begin{align*} J_{n}(x, \epsilon)=\left\lbrace y\in \mathbb{T}^{\mathbb{Z}}:||y_{k}-x_{k}|_{\mathbb{T}}\leq \epsilon, ~\forall 0 \leq k \lt n\right\rbrace. \end{align*}

It is easy to see that for any $\omega\in \Omega$,

\begin{align*} I_{n}(x, \epsilon)\subset B_{d_n^{\omega}}(x, \epsilon)\subset J_{n}(x, \epsilon). \end{align*}

Since $\mu_{\omega}(I_{n}(x, \epsilon))\geq \left( \frac{\epsilon}{6}\right)^{n+\ell}$ and $\mu_{\omega}(J_{n}(x, \epsilon))\leq (4\epsilon)^{n}$, we obtain that

\begin{align*} \log \frac{1}{4\epsilon} \leq h_{\mu_{\omega}}^{BK}(x, \epsilon)\leq \log \frac{3}{\epsilon} \end{align*}

for $\forall \omega\in \Omega$. Therefore

\begin{align*} \lim\limits_{\epsilon \rightarrow 0} \frac{\int h_{\mu_{\omega}}^{BK}(x, \epsilon) d \mu(\omega, x) }{\left| \log \epsilon\right| }=1=\mathbb{E}{{{\rm mdim}}}_{M}(T,\Omega\times\mathbb{T}^{{\mathbb{Z}}},d). \end{align*}