Billingsley-Type Theorem of Weighted Bowen Topological Entropy for Amenable Group Actions

Abstract

Let $(X_i,d_i)$ be a compact metric space with metric $d_i, i=1,2\dots ,k$, and G be a discrete infinitely countable amenable group. This paper is based on continuous actions $G\curvearrowright X_i$ on compact metric spaces $(X_i,d_i)$. Firstly, we introduce the concept of weighted Bowen balls, and then use the concept of weighted Bowen balls to introduce the corresponding lower (upper) weighted local entropy, as well as propose the concept of weighted Bowen topological entropy defined in terms of Hausdorff dimension by weighted Bowen balls, and prove Billingsley-type theorem between these two types of entropies by using the equivalent definition of weighted Bowen topological entropy.

1. Introduction

1.1. Entropy

In order to study dynamical systems on topological spaces, the concept of ‘topological entropy’ was first proposed by Adler, Konheim, and McAndrew in 1965 [1]. In metric spaces, another definition of topological entropy was proposed by Bowen in 1971, and later independently developed by Dinaburg [2,3]. In 1958, the measure-theoretic entropy of a measure-preserving system was introduced by Kolmogorov [4]. The variational principle establishes the relationship between measure-theoretic entropy and topological entropy [5,6]. In 1984, the concept of topological pressure was extended by Pesin and Pitskel to arbitrary sets [7]. In smooth systems, the relationship between measure-theoretic entropy and box dimension characteristics was introduced by Brin and Katok [8]. Ma and Wen extended the Billingsley theorem [9,10]. In the dynamical systems for amenable group actions, Huang, Lian, and Zhu established the Billingsley-type theorem between local measure-theoretic entropy and topological entropy [11]. Ji and Wang established a Billingsley-type theorem for weighted Bowen topological entropy [12]. Zhong and Chen extended Feng Huang’s variational principle to packing pressure and obtained two new variational principles for the Pesin-Pitskel pressure and the packing pressure, respectively. Moreover, they obtained the Billingsley-type theorem concerning packing pressure and measure-theoretic upper local pressure of measures, and proposed a variational principle for the packing pressure of the set of generic points of any invariant ergodic Borel probability measure [13]. Zhang, Meng, and Peng generalized Ma and Wen’s result to the case between Bowen polynomial topological entropy and the local polynomial entropy of measures for fixed-point free flows [14]. Liu and Peng proved a Billingsley-type theorem to characterize the BS dimension using measure-theoretic quantities, and established a variational principle that connects the topological and measure-theoretic versions of the BS dimension [15]. Xiao, Jia, and Yin obtained the Billingsley-type theorem on scaled packing pressures in the G-action system, and derived a variational principle relating scaled packing pressure to scaled measure-theoretic upper local pressures [16]. Chen and Miao obtained the variational principles for Bowen and packing topological pressures by establishing Billingsley-type theorems for Bowen and packing topological pressures [17].

Because entropy is one of the most fundamental characteristics of a dynamical system, calculating topological entropy is crucial. However, the calculation of topological entropy is particularly difficult, which has led to increasing attention from scholars on estimating topological entropy [18], with the introduction of this estimate aimed at extending the results of classical topological entropy to non-compact cases. This type of entropy, defined in a manner similar to the Hausdorff dimension, can be determined by the Billingsley theorem [9]. Ma and Wen proved the existence of an analogue of the Billingsley-type theorem [10]. In 2016, Feng and Huang, inspired by the theory of dimensions of tori affine invariant subsets, introduced several generalized concepts such as weighted Bowen topological entropy (pressure) and weighted measure-theoretic entropy, and established the corresponding variational principles for them [19]. In particular, they drew on Bowen’s approach to defining topological entropy [18], replacing Bowen balls with weighted Bowen balls, and introduced the concept of weighted Bowen topological entropy.

Various weighted topological (measure-theoretic) entropies were defined by Wang and Huang and they investigated their relationships [20]. Shen, Xu, and Zhou introduced the weighted topological entropy of a flow on non-compact sets and the weighted measure-theoretic entropy of a flow, established the corresponding variational principle, and also studied the relationship between weighted entropy and the classical weighted entropy of a time-one map [21]. In response to the questions raised by Feng and Huang, Yang, Chen, Lin, and others provided an affirmative answer in the case of random dynamical systems with ergodic and compact driving systems [22]. Sarkooh proposed the concept of neutralized weighted Bowen topological entropy for finitely generated free semigroup actions on compact metric spaces [23]. Xie, Chen, and Yang established the conclusion that the two types of weighted topological entropy defined by the FK metric and the Bowen metric are equal [24]. Zhu defined the local concepts of weighted topological entropy and weighted measure-theoretic entropy [25]. Zhang, He, Shang, and others proposed topological dispersion entropy and weighted topological dispersion entropy based on dispersion patterns in order to better use the topology permutation sequence entropy based on ordinal partition networks to approximate the topological entropy of low-dimensional chaotic systems, aiming to characterize the complexity of the system [26]. Nie and Huang extended the concept of weighted entropy to weighted topological sequence entropy and weighted measure-theoretic sequence entropy [27].

Zhang and Liu extended the classical Bowen entropy to obtain the Bowen topological entropy of non-autonomous dynamical systems, and they proved that the Bowen topological entropy can be determined through the local entropy of measures of non-autonomous dynamical systems, which extends the results of Ma and Wen [28]. The Billingsley-type theorem for weighted Bowen topological entropy is obtained from Ji and Wang [12]. In a system for a free semigroup action, Liu and Zhao introduced the lower local polynomial entropy, obtained some properties related to Bowen polynomial topological entropy, and also established the relationship between Bowen topological entropy and Bowen polynomial topological entropy for free semigroup actions, as well as the relationship between the lower local polynomial entropy of Borel probability measures and Bowen polynomial topological entropy of free semigroup actions [29]. For any Borel subset Z of X, Chen and Li proved that the scaled packing topological entropy of Z is equal to the supremum of the upper local entropies of all Borel probability measures for which Z has full measure [30]. Inspired by the research methods of the aforementioned experts, this article will propose the Billingsley-type theorem of weighted Bowen topological entropy for amenable group actions.

1.2. Background

In this section, we first recall self-affine Sierpinski gaskets, which makes it easier to understand weighted Bowen topological entropy as defined by Wang and Huang [20]. Let the torus ${\mathbb{T}}^2={\mathbb{R}}^2/{\mathbb{Z}}^2$, consider an endmorphism T on ${\mathbb{T}}^2$ represented by a matrix $A=diag(m_1,m_2)$ with $m_2>m_1\ge 2$; self-affine Sierpinski gaskets are

$$K(T,D)=\left\{\sum _{n=1}^{\infty }{A}^{-n}{v}_n\in {\mathbb{T}}^2|v_n\in D,n\ge 1\right\},$$

where D runs over the non-empty subsets of

$$\{{(i,j)}^T:i=0,1,2,\dots ,m_1-1,j=0,1,2,\dots ,m_2-1\},$$

where ${(i,j)}^T$ represent transpose of matrix. $K(T,D)$ is compact and T-invariant. There exists a T invariant measure $\mu$ supported on $K(T,D)$ such that the Hausdorff dimension of $K(T,D)$ equals $di{m}_H\left(\mu \right)$, where $di{m}_H$ denotes the Hausdorff dimension of $\mu$.

Inspired by the dimension theory of affine invariant subsets of tori and Bowen’s ‘dimension’ approach for arbitrary subsets, Wang and Huang defined the weighted Bowen topological entropy in a new way [18].

According to Feng–Huang’s definition, under certain conditions, the definition of the weighted Bowen topological entropy of $K(T,D)$ is equal to $di{m}_{H}K(T,D)$. The following content is derived from Feng–Huang’s concept by Ji and Wang [12].

Let $k\ge 2$. We assume that $(X_i,d_i)(i=1,\dots ,k)$ are k compact metric spaces. Consider each dynamical system $(X_i,T_i)(i=1,\dots ,k)$ as a compact metric space. For each $1\le i\le k-1$, we assume that $(X_{i+1},T_{i+1})$ is a factor of $(X_i,T_i)$ with a factor map ${\pi }_i:X_i\to X_{i+1}$; i.e., ${\pi }_1,\dots ,{\pi }_{k-1}$ are continuous maps and ${\pi }_{i-1}\circ T_{i-1}=T_i\circ {\pi }_{i-1}, (i=1,2,\dots ,k)$. Using the symbol ${\pi }_0$ to represent the identity map on $X_1$. Define ${\tau }_i:X_1\to X_{i+1}$ by ${\tau }_i={\pi }_i\circ {\pi }_{i-1}\circ \dots \circ {\pi }_0$ for $i=0,1,\dots ,k-1$.

For $x\in X_1,m\in \mathbb{N},\epsilon >0$. $\overrightarrow{a}$ weighted Bowen ball $B_m^{\overrightarrow{a}}(x,\epsilon )$ is defined as

$$\{y\in X_1:d_i(T_i^j{\tau }_{i-1}x,T_i^j{\tau }_{i-1}y)<\epsilon ,0\le j\le \lceil (a_1+\dots +a_i)m\rceil -1,i=1,\dots ,k\},$$

where $\lceil u\rceil$ denotes the least integer $\ge u$. Let $m\in \mathbb{N}$ and $\epsilon >0$, we define

$${\mathcal{T}}_{m,\epsilon }^{\overrightarrow{a}}:=\{Z\subseteq X_1:Zisa\mathrm{Borel}subsetof{B}_m^{\overrightarrow{a}}(x,\epsilon )forsomex\in X_1\}.$$

Let $Z\subseteq X_1$, $s\ge 0$, $N\in \mathbb{N}$, and we define

$${\Lambda }_{N,\epsilon }^{\overrightarrow{a},s}\left(Z\right)=inf\sum _{j}exp(-s{n}_j),$$

where the infinum is taken over all collections $\Gamma =\left\{(m_j,A_j)\right\}$ such that $m_j\ge N,A_j\in {\mathcal{T}}_{m,\epsilon }^{\overrightarrow{a}}$, and ${\bigcup }_{(m_j,A_j)\in \Gamma }{A}_j\supseteq Z$. Then

$${\Lambda }_{\epsilon }^{\overrightarrow{a},s}\left(Z\right)=\underset{N\to \infty }{lim}{\Lambda }_{N,\epsilon }^{\overrightarrow{a},s}\left(Z\right)$$

and

$${\Lambda }_{\epsilon }^{\overrightarrow{a},s}\left(Z\right)=\left\{\begin{array}{ccc}\hfill \infty ,& & s<h_{top}^{\overrightarrow{a}}(T_1,Z,\epsilon ),\hfill \\ \hfill 0.& & s>h_{top}^{\overrightarrow{a}}(T_1,Z,\epsilon ).\hfill \end{array}\right.$$

The $\overrightarrow{a}$ weighted topological entropy of $T_1$ restricted to Z is

$$h_{top}^{\overrightarrow{a}}(T_1,Z)=\underset{\epsilon \to 0}{lim}{h}_{top}^{\overrightarrow{a}}(T_1,Z,\epsilon ).$$

1.3. Preliminaries

The main object of this study in this paper is dynamical systems for amenable group actions, which play a crucial role in many branches of mathematics (such as ergodic theory, harmonic analysis, dynamical systems, probability theory, and statistics, etc.). Therefore, the following introduces some concepts used in this paper and their basic properties [31].

We use a mapping that satisfies ${\alpha }_s\left({\alpha }_t\left(x\right)\right)={\alpha }_{st}\left(x\right)$, ${\alpha }_e\left(x\right)=x$, for $x\in X$, $s,t\in G$, and e is the identity element of the group G to represent the action of a group G on a set X. This type of action is generally denoted as $G\curvearrowright X$. The image of the pair $(s,x)$ is $sx$. Let $G\curvearrowright X$ and $G\curvearrowright Y$ be two group actions; if for any $x\in X$, $s\in G$, there is $\phi :X\to Y$ such that $\phi \left(sx\right)=s\phi \left(x\right)$, then the mapping $\phi$ is said to be equivalent. If $G\curvearrowright X$ and $G\curvearrowright Y$ are two continuous actions on the compact Hausdorff spaces X and Y, and there exists an equivalent continuous bijection $\pi :X⟶Y$, then the first action is called an extension of the second action, or the second action is called a factor of the first action, where $\pi$ is referred to as the G factor map or the G extension map. Let $(X,d)$ be a compact metric space, and G be a topological group. The fundamental object of this study will be continuous actions on X. Let $F\left(G\right)$ denotes a family of sets composed of all non-empty finite subsets of G. If there exists a sequence ${\left\{F_n\right\}}_{n=1}^{\infty }$ composed of non-empty finite subsets in G, and this sequence satisfies the following conditions

$$\underset{n\to \infty }{lim}{\displaystyle \frac{|F_n\Delta g{F}_n|}{|F_n|}}=0,\forall g\in G,$$

then this group is called an amenable group, and the aforementioned sequence is referred to as the Følner sequence (which can be abbreviated as $\left\{F_n\right\}$ when there is no confusion as to what it is referencing).

Assuming that $\left\{F_n\right\}$ is a Følner sequence, if for any $n\ge 1$, $F_n\subseteq F_{n+1}$, this Følner sequence is referred to as nested.

Let $k\ge 2$, $(X_i,d_i),i=1,2,...,k$ be k compact metric spaces; for any $1\le i\le k-1,G\curvearrowright X_{i+1}$ is the factor mapping of $G\curvearrowright X_i$ with a factor map ${\pi }_i:X_i\to X_{i+1}$.

For research convenience, let ${\pi }_0$ represent the identity mapping on $X_1$ with a factor map ${\pi }_i:X_i\to X_{i+1}$.

$${\tau }_i:X_1\to X_{i+1},i=0,1,...,k-1$$

is defined through ${\tau }_i={\pi }_i\circ {\pi }_{i-1}\circ ···\circ {\pi }_0$.

Use $M(X_i,G)$ to represent the set of all G invariant Borel probability measures on $X_i$. For convenience in the subsequent content and to ensure it is easily understood by readers, we simply denote it as a sequence $\left\{F_n\right\}$ satisfying condition $\mathbf{F}$, if sequence $\left\{F_n\right\}$ is a sequence composed of finite subsets of G and satisfies $|F_n|\to \infty$ (not necessarily an Følner sequence).

To prove the main result, we need the following Theorem 1, which can be found in [32].

Theorem 1.

Let G be a countable amenable group. Then, there is a nested Følner sequence $\left\{F_n\right\}$ such that

$$\bigcup _{n=1}^{\infty }{F}_n=G.$$

2. Weighted Local Entropy and Weighted Bowen Topological Entropy

Let $\overrightarrow{a}=(a_1,a_2,...,a_k)\in {\mathbb{R}}^k$ with $a_1>0$ and $a_i\ge 0,i\ge 2$.

Following the method definitions of Bowen [19] and Pesin–Pitskel [7], the topological entropy of non-compact subsets, where weighted Bowen balls replace Bowen balls, this paper defines the concept of weighted Bowen topological entropy in the context of the dynamical systems for an amenable group action.

Definition 1.

Let $\left\{F_n\right\}$ satisfy condition $\mathbf{F}$. Let $x\in X_1, n\in \mathbb{N}, ϵ>0$, write

$$B_{G,n}^{\overrightarrow{a}}(x,ϵ)=\{y\in X_1:d_i({\alpha }_s{\tau }_{i-1}x,{\alpha }_s{\tau }_{i-1}y)<ϵ,s\in F_{\lceil (a_1+a_2+...+a_i)n\rceil },1\le i\le k\},$$

where $\lceil u\rceil$ represents the smallest integer of $\ge u$; call $B_{G,n}^{\overrightarrow{a}}(x,ϵ)$ a $|F_n|$-dimensional $\overrightarrow{a}$ weighted Bowen ball (with x as the centre and ϵ as the radius).

Remark 1.

For classical dynamical systems, let $F_n=\{0,1,2,...,n-1\}$, then the corresponding Bowen ball is $B_n^{\overrightarrow{a}}(x,ϵ)$ (refer to [12]). When $k=1$ and $a_1=1$, $B_n^{\overrightarrow{a}}(x,ϵ)=B_n(x,ϵ)$ (refer to [10]).

Definition 2.

Let $\left\{F_n\right\}$ satisfy condition $\mathbf{F}$. Let $F\in F\left(G\right)$, and define the metric $d_n^{\overrightarrow{a}}$ on $X_1$ as follows,

$$d_n^{\overrightarrow{a}}(x,y)=sup\{d_i({\alpha }_s{\tau }_{i-1}x,{\alpha }_s{\tau }_{i-1}y):s\in F_{\lceil (a_1+a_2+...+a_i)n\rceil },1\le i\le k\}.$$

Definition 3.

Let $\left\{F_n\right\}$ satisfy condition $\mathbf{F}$. Let $x\in X_1,\mu \in M(X_1,G)$, then we define the lower weighted local entropy concerning μ at point x as

$${\underline{h}}_{\mu }^{\overrightarrow{a}}(G,x,\left\{F_n\right\})=\underset{ϵ\to 0}{lim}\underset{n\to \infty }{lim\; inf}-{\displaystyle \frac{1}{|F_n|}}log\mu \left(B_{G,n}^{\overrightarrow{a}}(x,ϵ)\right).$$

Similarly, the upper weighted local entropy concerning μ at point x is defined by

$${\overline{h}}_{\mu }^{\overrightarrow{a}}(G,x,\left\{F_n\right\})=\underset{ϵ\to 0}{lim}\underset{n\to \infty }{lim\; sup}-{\displaystyle \frac{1}{|F_n|}}log\mu \left(B_{G,n}^{\overrightarrow{a}}(x,ϵ)\right).$$

Remark 2.

For classical dynamical systems, let $F_n=\{0,1,2,...,n-1\}$, then

$${\underline{h}}_{\mu }^{\overrightarrow{a}}(G,x,\left\{F_n\right\})={\underline{h}}_{\mu }^{\overrightarrow{a}}(T_1,x)$$

and

$${\overline{h}}_{\mu }^{\overrightarrow{a}}(G,x,\left\{F_n\right\})={\overline{h}}_{\mu }^{\overrightarrow{a}}(T_1,x).$$

Definition 4.

Let $\left\{F_n\right\}$ satisfy condition $\mathbf{F}$. Let $n\in \mathbb{N}$ be determined, and $ϵ>0$,

$${\zeta }_{n,ϵ}^{\overrightarrow{a}}=\{A\subseteq X_1:\exists x\in X_{1}suchthatAisa\mathit{Borel}subsetof{B}_{G,n}^{\overrightarrow{a}}(x,ϵ)\}.$$

Definition 5.

Let $\left\{F_n\right\}$ satisfy condition $\mathbf{F}$. Let $ϵ>0$, $Z\subseteq X_1,s\ge 0$, $N\in \mathbb{N}$, write

$${\Lambda }^{\overrightarrow{a}}(Z,ϵ,s,N)=inf\sum _{j}exp(-s|F_{n_j}\left|\right),$$

where the infimum takes over all countable collections $\Gamma =\left\{(F_{n_j},A_j)\right\}$ such that

$$n_j\ge N,A_j\in {\zeta }_{n_j,ϵ}^{\overrightarrow{a}}\mathrm{and}\bigcup _{(F_{n_j},A_j)\in \Gamma }{A}_j\supseteq Z.$$

It can be seen from above that as N increases, ${\Lambda }^{\overrightarrow{a}}(Z,ϵ,s,N)$ is increasing; therefore,

$${\Lambda }^{\overrightarrow{a}}(Z,ϵ,s,\left\{F_n\right\})=\underset{N\to \infty }{lim}{\Lambda }^{\overrightarrow{a}}(Z,ϵ,s,N).$$

At this moment, there the parameter value is critical, and this value is $h_{top}^{\overrightarrow{a}}(G,Z,ϵ,\left\{F_n\right\})$, where ${\Lambda }^{\overrightarrow{a}}(Z,ϵ,s,\left\{F_n\right\})$ to jump from ∞ to 0; that is,

$${\Lambda }^{\overrightarrow{a}}(Z,ϵ,s,\left\{F_n\right\})=\left\{\begin{array}{ccc}\hfill \infty ,& & s<h_{top}^{\overrightarrow{a}}(G,Z,ϵ,\left\{F_n\right\}),\hfill \\ \hfill 0,& & s>h_{top}^{\overrightarrow{a}}(G,Z,ϵ,\left\{F_n\right\}).\hfill \end{array}\right.$$

It is clear that $h_{top}^{\overrightarrow{a}}(G,Z,ϵ,\left\{F_n\right\})$ increases as ϵ decreases; therefore, the following limit exists

$$h_{top}^{\overrightarrow{a}}(G,Z,\left\{F_n\right\})=\underset{ϵ\to 0}{lim}{h}_{top}^{\overrightarrow{a}}(G,Z,ϵ,\left\{F_n\right\}),$$

we call $h_{top}^{\overrightarrow{a}}(G,Z,\left\{F_n\right\})$ the $\overrightarrow{a}$ weighted Bowen topological entropy of $X_1$ restricted on Z.

Remark 3.

If $k=1$ and $a_1=1$, then $h_{top}^{\overrightarrow{a}}(G,Z,\left\{F_n\right\})=h_{top}^B(Z,\left\{F_n\right\})$ (refer to [33]). This paper mainly considers $k\ge 2$. For classical dynamic systems, $F_n=\{0,1,2,...,n-1\}$ can be taken, then $h_{top}^{\overrightarrow{a}}(G,Z,\left\{F_n\right\})=h_{top}^{\overrightarrow{a}}(T_1,Z)$ (this part refers to [12]).

3. Another Definition of Weighted Bowen Topological Entropy

To prove Proposition 3, we need to introduce another definition of weighted topological entropy. In this section, we will use open covers to introduce another new definition of weighted Bowen topological entropy. In the paper, for any subset Z of $(X,d)$, we set $diam\left(Z\right)={sup}_{x,y\in Z}d(x,y)$. For a family of open covers ${\left\{{\mathcal{U}}_i\right\}}_{i=1}^k$ with ${\mathcal{U}}_i$ open cover of $(X_i,d_i)$, $diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right):={max}_{1\le i\le k}diam\left({\mathcal{U}}_i\right)$, where $diam\left({\mathcal{U}}_i\right)={max}_{U\in {\mathcal{U}}_i}diam\left(U\right)$.

Definition 6.

Let $\overrightarrow{a}=\{a_1,a_2,\dots ,a_k\}$ with $a_1>0$ and $a_i\ge 0$ for $2\le i\le k$. For $1\le i\le k$, we take arbitrary open covers ${\left\{{\mathcal{U}}_i\right\}}_{i=1}^k$ where ${\mathcal{U}}_i$ is a finite open cover of $X_i$. We define the string

$${\mathbf{U}}_n^{\overrightarrow{a}}=\left(\bigcap _{s\in F_{\lceil a_{1}n\rceil }}{s}^{-1}{U}_s^1\right)\bigcap {\tau }_1^{-1}\left(\bigcap _{s\in F_{\lceil (a_1+a_2)n\rceil }}{s}^{-1}{U}_s^2\right)\bigcap \dots \bigcap {\tau }_{k-1}^{-1}\left(\bigcap _{s\in F_{\lceil (a_1+a_2+\dots +a_k)n\rceil }}{s}^{-1}{U}_s^k\right),$$

where $U_s^i\in {\mathcal{U}}_i$ for $s\in F_{\lceil (a_1+a_2+\dots +a_i)n\rceil }$, $1\le i\le k$.

Definition 7.

Let $\overrightarrow{a}=\{a_1,a_2,\dots ,a_k\}$ with $a_1>0$ and $a_i\ge 0$ for $2\le i\le k$. For any $n\in \mathbb{N}$, we define

$${\Gamma }_{n,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k}^{\overrightarrow{a}}:=\{A\subseteq X_1:AisaBorelsubsetofsome{\mathbf{U}}_n^{\overrightarrow{a}}\}.$$

For $Z\subseteq X_1$, $s\ge 0$, $N\in \mathbb{N}$, define

$${\mathcal{M}}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,N)=\underset{\Lambda }{inf}{\Sigma }_{j}exp(-s|F_{n_j}\left|\right),$$

where the infimum takes over all countable collections

$$\Lambda =\left\{(F_{n_j},A_j)\right\},n_j\ge N,A_j\in {\Gamma }_{n_j,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k}^{\overrightarrow{a}}\mathit{and}\bigcup _{(F_{n_j},A_j)\in \Lambda }{A}_j\supseteq Z.$$

The quality ${\mathcal{M}}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,N)$ does not increase with N; hence, the following limit exists:

$$\begin{array}{cc}\hfill {\mathcal{M}}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,\left\{F_n\right\})& =\underset{N\to \infty }{lim}{\mathcal{M}}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,N)\hfill \\ \hfill & =\underset{N\in \mathbb{N}}{sup}{\mathcal{M}}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,N).\hfill \end{array}$$

There exists a critical value of the parameters, which we will denote by $h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})$, where ${\mathcal{M}}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,\left\{F_n\right\})$ jumps from ∞ to 0, i.e.,

$${\mathcal{M}}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,\left\{F_n\right\})=\left\{\begin{array}{ccc}\hfill \infty ,& & s<h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\}),\hfill \\ \hfill 0,& & s>h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\}).\hfill \end{array}\right.$$

Define

$$h^{\overrightarrow{a}}(G,Z,\left\{F_n\right\})=\underset{{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k}{sup}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\}),$$

where the supremum runs over all open-cover ${\left\{{\mathcal{U}}_i\right\}}_{i=1}^k$.

To make the following proof of Proposition 1 more concrete, the following definitions need to be introduced.

Definition 8.

Define

$$\begin{array}{cc}\hfill \underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)\to 0}{lim\; sup}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})& :=\underset{ϵ\to 0}{lim}\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)<ϵ}{sup}\left\{h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\right\}\hfill \\ \hfill & =\underset{ϵ\to 0}{inf}\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)<ϵ}{sup}\left\{h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\right\}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)\to 0}{lim\; inf}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})& :=\underset{ϵ\to 0}{lim}\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)<ϵ}{inf}\left\{h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\right\}\hfill \\ \hfill & =\underset{ϵ\to 0}{sup}\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)<ϵ}{inf}\left\{h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\right\}.\hfill \end{array}$$

If

$$\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)\to 0}{lim\; sup}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})=\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)\to 0}{lim\; inf}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\}),$$

then we say that the limit

$$\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)\to 0}{lim}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})$$

exists.

Remark 4.

We need to note that as ϵ decreases, ${sup}_{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)<ϵ}\left\{h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\right\}$ decreases and ${inf}_{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)<ϵ}\left\{h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\right\}$ increases; therefore, as stated in the above definition,

$$\underset{ϵ\to 0}{lim}\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)<ϵ}{sup}\left\{h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\right\}=\underset{ϵ\to 0}{inf}\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)<ϵ}{sup}\left\{h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\right\}$$

and

$$\underset{ϵ\to 0}{lim}\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)<ϵ}{inf}\left\{h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\right\}=\underset{ϵ\to 0}{sup}\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)<ϵ}{inf}\left\{h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\right\}$$

hold.

Proposition 1.

The limit

$$\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)\to 0}{lim}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})$$

exists and is equal to $h^{\overrightarrow{a}}(G,Z,\left\{F_n\right\})$.

Proof of Proposition 1.

Write $\mathcal{V}=\{{\mathcal{V}}_1,{\mathcal{V}}_2,\dots ,{\mathcal{V}}_k\}$ and $\mathcal{U}=\{{\mathcal{U}}_1,{\mathcal{U}}_2,\dots ,{\mathcal{U}}_k\}$, where ${\mathcal{V}}_i,{\mathcal{U}}_i$ are two open covers of $X_i$ for $1\le i\le k$. Denote $L\left({\mathcal{U}}_i\right)$ as the Lebesgue number of ${\mathcal{U}}_i$ with respect to the metric $d_i$ and $L\left(\mathcal{U}\right)={min}_{1\le i\le k}\left\{{\mathcal{U}}_1\right\}$. Set ${ϵ}_0={\displaystyle \frac{L\left(\mathcal{U}\right)}{2}}$ and $diam\left(\mathcal{V}\right)<{ϵ}_0$. Then, we have

$${\mathcal{M}}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,\left\{F_n\right\})\le {\mathcal{M}}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{V}}_i\right\}}_{i=1}^k,s,\left\{F_n\right\}),$$

thus

$$h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\le h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{V}}_i\right\}}_{i=1}^k,\left\{F_n\right\}),$$

and therefore

$$h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\le \underset{diam\left({\left\{{\mathcal{V}}_i\right\}}_{i=1}^k\right)\to 0}{lim\; inf}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{V}}_i\right\}}_{i=1}^k,\left\{F_n\right\}),$$

that is

$$\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)\to 0}{lim\; sup}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\le \underset{diam\left({\left\{{\mathcal{V}}_i\right\}}_{i=1}^k\right)\to 0}{lim\; inf}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{V}}_i\right\}}_{i=1}^k,\left\{F_n\right\})$$

and

$$\underset{{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k}{sup}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\le \underset{diam\left({\left\{{\mathcal{V}}_i\right\}}_{i=1}^k\right)\to 0}{lim\; inf}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{V}}_i\right\}}_{i=1}^k,\left\{F_n\right\}).$$

Since ${\mathcal{V}}_i$ is any open cover of $X_i$ with $diam\left(\mathcal{V}\right)<{ϵ}_0$, ${\mathcal{U}}_i$ is any open cover of $X_i$, we have

$$\underset{{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k}{sup}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\le \underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)\to 0}{lim\; inf}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\}).$$

By the fact

$$\underset{{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k}{sup}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\ge \underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)\to 0}{lim\; sup}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\}),$$

we can deduce

$$h^{\overrightarrow{a}}(G,Z,\left\{F_n\right\})=\underset{diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)\to 0}{lim}{h}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\}).$$

□

Proposition 2.

Let $Z\subseteq X_1$, then

$$h^{\overrightarrow{a}}(G,Z,\left\{F_n\right\})=h_{top}^{\overrightarrow{a}}(G,Z,\left\{F_n\right\}).$$

Proof of Proposition 2.

Let $s>h_{top}^{\overrightarrow{a}}(G,Z,\left\{F_n\right\})$, ${\left\{{\mathcal{U}}_i\right\}}_{i=1}^k$ be a family of open covers for $X_i(1\le i\le k)$. For any open-cover $\mathcal{U}$, let $\delta \left(\mathcal{U}\right)$ denote the Lebesgue number of $\mathcal{U}$ and we define

$$\delta \left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right):=min\{\delta \left({\mathcal{U}}_i\right):1\le i\le k\}.$$

Let $0<\gamma <{\displaystyle \frac{\delta \left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)}{2}}$, N be sufficiently large numbers that satisfy

$${\Lambda }^{\overrightarrow{a}}(Z,\gamma ,s,N)\le {\displaystyle \frac{1}{2}}.$$

Then, by the definition of ${\Lambda }^{\overrightarrow{a}}(Z,\gamma ,s,N)$, we can find a $\Gamma =\left\{(F_{n_j},A_j)\right\}$ that satisfies the following conditions

$$n_j\ge N,A_j\in {\Gamma }_{n_j,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k}^{\overrightarrow{a}}\mathrm{satisfying}\bigcup _{(F_{n_j},A_j)\in \Gamma }{A}_j\supseteq Z$$

and

$$\sum _{(F_{n_j},A_j)\in \Gamma }exp(-s|F_{n_j}\left|\right)\le 1.$$

Since $\gamma <{\displaystyle \frac{\delta \left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)}{2}}$ and $(F_{n_j},A_j)\in \Gamma$ hold, there exists ${\mathbf{U}}_{n_j}^{\overrightarrow{a}}$ that satisfies $A_j\subseteq B_{G,n}^{\overrightarrow{a}}(x,\gamma )\subseteq {\mathbf{U}}_{n_j}^{\overrightarrow{a}}$. So,

$${\mathcal{M}}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,N)\le \sum _{(F_{n_j},A_j)\in \Gamma }exp(-s|F_{n_j}\left|\right)\le 1.$$

Let $N\to \infty$,

$${\mathcal{M}}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,\left\{F_n\right\})\le 1.$$

Therefore,

$$h^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\le s.$$

And because Proposition 1 holds, it follows that $h^{\overrightarrow{a}}(G,Z,\left\{F_n\right\})\le h_{top}^{\overrightarrow{a}}(G,Z,\left\{F_n\right\})$.

Conversely, consider any $s>h^{\overrightarrow{a}}(G,Z,\left\{F_n\right\}).$ Let any $\epsilon >0$, we can obtain a family of open-cover ${\left\{{\mathcal{U}}_i\right\}}_{i=1}^k$ that satisfies $diam\left({\left\{{\mathcal{U}}_i\right\}}_{i=1}^k\right)<ϵ$ and a sufficiently large positive integer number N such that

$${\mathcal{M}}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,N)<1.$$

We consider each ${\mathbf{U}}_{n_j}^{\overrightarrow{a}}\bigcap Z\ne \varnothing$ for ${\left\{{\mathcal{U}}_i\right\}}_{i=1}^k$. Picking an element $x_{{\mathbf{U}}_{n_j}^{\overrightarrow{a}}}$ from ${\mathbf{U}}_{n_j}^{\overrightarrow{a}}\bigcap Z$ such that

$${\mathbf{U}}_{n_j}^{\overrightarrow{a}}\subseteq B_{G,n}^{\overrightarrow{a}}(x_{{\mathbf{U}}_{n_j}^{\overrightarrow{a}}},\epsilon ).$$

Then, we can obtain

$${\Lambda }^{\overrightarrow{a}}(Z,\epsilon ,s,N)\le {\mathcal{M}}^{\overrightarrow{a}}(Z,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,N)<1.$$

Letting $N\to \infty$, we have ${\Lambda }^{\overrightarrow{a}}(Z,\epsilon ,s,\left\{F_n\right\})\le 1$, which yields that

$$h_{top}^{\overrightarrow{a}}(G,Z,\epsilon ,\left\{F_n\right\})\le s.$$

Further, let $\epsilon \to 0$; then, we have

$$h_{top}^{\overrightarrow{a}}(G,Z,\left\{F_n\right\})\le s.$$

That implies

$$h^{\overrightarrow{a}}(G,Z,\left\{F_n\right\})\ge h_{top}^{\overrightarrow{a}}(G,Z,\left\{F_n\right\}).$$

□

4. Main Results

According to the definition of the weighted Bowen topological entropy, we can easily obtain the following properties of the weighted Bowen topological entropy.

Proposition 3.

Let $\left\{F_n\right\}$ satisfy condition $\mathbf{F}$, $Z_1,Z_2,···\subseteq X_1$, then

• If $Z_1\subseteq Z_2$, $h_{top}^{\overrightarrow{a}}(G,Z_1,\left\{F_n\right\})\le h_{top}^{\overrightarrow{a}}(G,Z_2,\left\{F_n\right\})$;
• $h_{top}^{\overrightarrow{a}}(G,{\cup }_{i\in \mathbb{N}}{Z}_i,\left\{F_n\right\})={sup}_{i\in \mathbb{N}}{h}_{top}^{\overrightarrow{a}}(G,Z_i,\left\{F_n\right\}).$

Proof of Proposition 3.

The first property of Proposition 3 follows directly from the definition of the $\overrightarrow{a}$ weighted Bowen topological entropy.

For another property of Proposition 3, by Proposition 2, we only need to show that

$$h^{\overrightarrow{a}}(G,{\cup }_{i\in \mathbb{N}}{Z}_i,\left\{F_n\right\})\le \underset{i\in \mathbb{N}}{sup}{h}_{top}^{\overrightarrow{a}}(G,Z_i,\left\{F_n\right\}).$$

Let ${sup}_{i\in \mathbb{N}}{h}_{top}^{\overrightarrow{a}}(G,Z_i,\left\{F_n\right\})<\infty$ and s is any real number with

$$s>\underset{i\in \mathbb{N}}{sup}{h}_{top}^{\overrightarrow{a}}(G,Z_i,\left\{F_n\right\}).$$

In what follows, we will show that

$$h^{\overrightarrow{a}}(G,{\cup }_{i\in \mathbb{N}}{Z}_i,\left\{F_n\right\})\le s.$$

We assume that ${\mathcal{U}}_i$ is an open cover of $X_i$, so we will show that

$$h^{\overrightarrow{a}}({\cup }_{i\in \mathbb{N}}{Z}_i,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\le s$$

by Definition 7. As

$$s>h^{\overrightarrow{a}}(Z_i,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\}),\forall i\ge 1,$$

we know that

$${\mathcal{M}}^{\overrightarrow{a}}(Z_i,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,N)=0,\forall N\in \mathbb{N}.$$

Let $N\in \mathbb{N}$, $\delta >0$. For any i, there is a collection of string ${\Gamma }_i=\left\{(F_{n_{j_i}},A_{j_i})\right\}$, $n_{j_i}\ge N$, $A_{j_i}\in {\Gamma }_{n_{j_i},{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k}^{\overrightarrow{a}}$, which cover $Z_i$ and satisfy

$$\sum _{j_i}exp(-s|F_{n_{j_i}}\left|\right)<{\displaystyle \frac{\delta }{2^{i+1}}}.$$

Then, we have

$$\begin{array}{cc}\hfill {\mathcal{M}}^{\overrightarrow{a}}({\cup }_{i\in \mathbb{N}}{Z}_i,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,N)& \le \sum _{i\ge 1}\sum _{j_i}exp(-s|F_{n_{j_i}}\left|\right)\hfill \\ \hfill & <\sum _{i\ge 1}{\displaystyle \frac{\delta }{2^{i+1}}}\hfill \\ \hfill & <\delta .\hfill \end{array}$$

It follows that

$${\mathcal{M}}^{\overrightarrow{a}}({\cup }_{i\in \mathbb{N}}{Z}_i,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,N)=0,$$

which means

$${\mathcal{M}}^{\overrightarrow{a}}({\cup }_{i\in \mathbb{N}}{Z}_i,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,s,\left\{F_n\right\})=0.$$

Thus, we have that

$$h^{\overrightarrow{a}}({\cup }_{i\in \mathbb{N}}{Z}_i,{\left\{{\mathcal{U}}_i\right\}}_{i=1}^k,\left\{F_n\right\})\le s.$$

So,

$$h^{\overrightarrow{a}}(G,{\cup }_{i\in \mathbb{N}}{Z}_i,\left\{F_n\right\})\le s.$$

Since s is any real number with $s>{sup}_{i\in \mathbb{N}}{h}_{top}^{\overrightarrow{a}}(G,Z_i,\left\{F_n\right\})$, we get that

$$h_{top}^{\overrightarrow{a}}(G,{\cup }_{i\in \mathbb{N}}{Z}_i,\left\{F_n\right\})\le \underset{i\in \mathbb{N}}{sup}{h}_{top}^{\overrightarrow{a}}(G,Z_i,\left\{F_n\right\}).$$

□

Next, this paper aims to prove the Billingsley-type theorem for weighted Bowen topological entropy in the dynamical systems for amenable group actions. However, before proving the main result, we need the following Lemma 1. In the dynamical systems for amenable group actions, the following Lemma 1, similar to the Vitali covering lemma in classical dynamical systems, holds. Its proof can be obtained by slightly modifying the proof of Lemma 4.7 in [11], so the proof of the following lemma is omitted here.

Lemma 1.

Let $G\curvearrowright X$ be a continuous action, ρ be a continuous pseudometric, and let $\left\{F_n\right\}$ be a nested sequence composed of finite subsets of G. Let $\delta >0$,

$$B\left(\delta \right)=\{B_{G,n}^{\overrightarrow{a}}(x,\delta ):x\in X,n=1,2,...\}$$

for any family $\mathcal{F}\subseteq B\left(\delta \right)$, there exists a countable subfamily $\mathcal{G}\subseteq \mathcal{F}$ of pairwise disjoint elements such that

$$\bigcup _{B\in \mathcal{F}}B\subseteq \bigcup _{B_{G,n}^{\overrightarrow{a}}(x,\delta )\in \mathcal{G}}{B}_{G,n}^{\overrightarrow{a}}(x,3\delta ).$$

Theorem 2.

Let G be a countable discrete amenable group, and $(X_1,d_1)$ a metric space. $G\curvearrowright X_1$ is a continuous action, and $\left\{F_n\right\}$ is a nested sequence composed and satisfying condition $\mathbf{F}$. μ is a Borel probability measure on $X_1$, E is a Borel subset of $X_1$, and $0<s<\infty$; then,

• If for any $x\in E$ and ${\underline{h}}_{\mu }^{\overrightarrow{a}}(G,x,\left\{F_n\right\})\le s$, then $h_{top}^{\overrightarrow{a}}(G,E,\left\{F_n\right\})\le s$;
• If for any $x\in E,{\underline{h}}_{\mu }^{\overrightarrow{a}}(G,x,\left\{F_n\right\})\ge s$ and $\mu \left(E\right)>0$, then $h_{top}^{\overrightarrow{a}}(G,E,\left\{F_n\right\})\ge s$.

Proof of Theorem 2.

For any $\gamma >0$, let

$$E_k=\{x\in E:\underset{n\to \infty }{lim\; inf}-{\displaystyle \frac{1}{|F_n|}}log\mu \left(B_{G,n}^{\overrightarrow{a}}(x,r)\right)<s+\gamma ,\forall r\in (0,{\displaystyle \frac{1}{k}})\}.$$

Because for any $x\in E,{\underline{h}}_{\mu }^{\overrightarrow{a}}(G,x,\left\{F_n\right\})\le s$, we have $E={\bigcup }_{k=1}^{\infty }{E}_k.$

Let $k\ge 1$ and $0<r<{\displaystyle \frac{1}{3k}}$, for any $x\in E_k$ there is a strictly increasing sequence ${\left\{n_j\left(x\right)\right\}}_{j=1}^{\infty }$ with

$$\mu \left(B_{G,n_j\left(x\right)}^{\overrightarrow{a}}(x,r)\right)\ge exp(-(s+\gamma )|F_{n_j\left(x\right)}\left|\right),j\ge 1.$$

Therefore, for any $N\ge 1$,

$$E_k\subseteq \bigcup _{x\in E_k,n_j\left(x\right)\ge N}{B}_{G,n_j\left(x\right)}^{\overrightarrow{a}}(x,r).$$

According to Lemma 1, there is a subclass $\mathcal{G}={\left\{B_{G,n_i\left(x\right)}^{\overrightarrow{a}}(x_i,r)\right\}}_{i\in I}$ and elements in $\mathcal{G}$ that are pairwise disjoint and

$$E_k\subseteq \bigcup _{i\in I}{B}_{G,n_i}^{\overrightarrow{a}}(x_i,3r)$$

and

$$\mu \left(B_{G,n_i}^{\overrightarrow{a}}(x_i,r)\right)\ge exp(-(s+\gamma )|F_{n_i}\left|\right),i\in I,$$

where $\mathcal{F}=\{B_{G,n_j\left(x\right)}^{\overrightarrow{a}}(x,r):x\in E_k,n_j\left(x\right)\ge N\}$.

Because $\mu$ is a probability measure, the elements in $\mathcal{G}$ are pairwise disjointed, and the index set I can have at most countably many elements. Furthermore, every element in $\mathcal{G}$ has a positive $\mu$ measure and

$${\Lambda }^{\overrightarrow{a}}(E_k,3r,s+\gamma ,N)\le \sum _{i\in I}exp(-(s+\gamma )|F_{n_i}|)\le \sum _{i\in I}\mu \left(B_{G,n_i}^{\overrightarrow{a}}(x_i,r)\right)\le 1,$$

the last inequality holds because the elements in ${\left\{B_{G,n_i\left(x\right)}^{\overrightarrow{a}}(x_i,r)\right\}}_{i\in I}$ do not intersect with each other; thus,

$${\Lambda }^{\overrightarrow{a}}(E_k,3r,s+\gamma ,\left\{F_n\right\})=\underset{N\to \infty }{lim}{\Lambda }^{\overrightarrow{a}}(E_k,3r,s+\gamma ,N)$$

and therefore $h_{top}^{\overrightarrow{a}}(G,E_k,3r,\left\{F_n\right\})\le s+\gamma ,\forall 0<r<{\displaystyle \frac{1}{3k}}$.

Let $r\to 0$, we obtain

$$h_{top}^{\overrightarrow{a}}(G,E_k,\left\{F_n\right\})\le s+\gamma ,\forall k\ge 1.$$

From Proposition 3, we can get

$$h_{top}^{\overrightarrow{a}}(G,E,\left\{F_n\right\})=h_{top}^{\overrightarrow{a}}(G,\bigcup _{k=1}^{\infty }{E}_k,\left\{F_n\right\})=\underset{k\ge 1}{sup}{h}_{top}^{\overrightarrow{a}}(G,E_k,\left\{F_n\right\})\le s+\gamma .$$

Due to the arbitrariness of $\gamma >0$, we can know

$$h_{top}^{\overrightarrow{a}}(G,E,\left\{F_n\right\})\le s.$$

In addition, let $\alpha >0$, and for any $k\ge 1$. Let

$$\begin{array}{c}\hfill E_k=\left\{x\in E:\underset{n\to \infty }{lim\; inf}-{\displaystyle \frac{1}{|F_n|}}log\mu \left(B_{G,n}^{\overrightarrow{a}}(x,r)\right)>s-\alpha ,\forall r\in (0,{\displaystyle \frac{1}{k}})\right\}.\end{array}$$

(1)

Because for any $x\in E,{\underline{h}}_{\mu }^{\overrightarrow{a}}(G,x,\left\{F_n\right\})\ge s$, we have $E_k\subseteq E_{k+1}$ and $E={\bigcup }_{k\ge 1}{E}_k$. It can be seen from the continuity of the measure that

$$\underset{k\to \infty }{lim}\mu \left(E_k\right)=\mu \left(E\right)>0,$$

thus, take $k_0\ge 1$ so that

$$\begin{array}{c}\hfill \mu \left(E_{k_0}\right)>{\displaystyle \frac{1}{2}}\mu \left(E\right)>0.\end{array}$$

(2)

For any $N\ge 1$, let

$$\begin{array}{c}\hfill E_{k_0,N}=\left\{x\in E_k:\underset{n\to \infty }{lim\; inf}-{\displaystyle \frac{1}{|F_n|}}log\mu \left(B_{G,n}^{\overrightarrow{a}}(x,r)\right)>s-\alpha ,\forall n\ge N,r\in (0,{\displaystyle \frac{1}{k_0}})\right\},\end{array}$$

(3)

then, based on the construction characteristics of (1) and (3), it can be seen that ${\left\{E_{k_0,N}\right\}}_{N=1}^{\infty }$ increases to $E_{k_0}$ as N increases; therefore, $M\ge 1$ can be taken to satisfy

$$\begin{array}{c}\hfill \mu \left(E_{k_0,M}\right)>{\displaystyle \frac{1}{2}}\mu \left(E_{k_0}\right).\end{array}$$

(4)

From $D=E_{k_0,M},\beta ={\displaystyle \frac{1}{k_0}}$, we can obtain $\mu \left(D\right)>0$ from (2) and (4), and for any $x\in D,0<r\le \beta$, $n\ge M$, there is

$$\begin{array}{c}\hfill \mu \left(B_{G,n}^{\overrightarrow{a}}(x,r)\right)\le exp(-(s-\alpha )|F_n\left|\right).\end{array}$$

(5)

For any $i\ge 1,0<r\le \beta$, let $\Gamma ={\{B_{G,n_i}^{\overrightarrow{a}}(y_i,{\displaystyle \frac{r}{2}}):n_i\ge N\}}_{i\ge 1}$ such that

$$\bigcup B_{G,n_i}^{\overrightarrow{a}}(y_i,{\displaystyle \frac{r}{2}})\subseteq DandD\cap B_{G,n_i}^{\overrightarrow{a}}(y_i,{\displaystyle \frac{r}{2}})\ne \varnothing ,n_i>N\ge M.$$

For any $i\ge 1$, take $x_i\in D\cap B_{G,n_i}^{\overrightarrow{a}}(y_i,{\displaystyle \frac{r}{2}})$ satisfying

$$B_{G,n_i}^{\overrightarrow{a}}(y_i,{\displaystyle \frac{r}{2}})\subseteq B_{G,n_i}^{\overrightarrow{a}}(x_i,r).$$

From (5), we can deduce

$$\sum _{i\ge 1}exp(-(s-\alpha )|F_{n_i}\left|\right)\ge \sum _{i\ge 1}\mu \left(B_{G,n_i}^{\overrightarrow{a}}(x_i,r)\right)\ge \mu \left(D\right),$$

then

$${\Lambda }^{\overrightarrow{a}}(D,{\displaystyle \frac{r}{2}},s-\alpha ,N)\ge \mu \left(D\right)>0,\forall N>M,$$

and therefore, we can obtain

$${\Lambda }^{\overrightarrow{a}}(D,{\displaystyle \frac{r}{2}},s-\alpha ,\left\{F_n\right\})=\underset{N\to \infty }{lim}{\Lambda }^{\overrightarrow{a}}(D,{\displaystyle \frac{r}{2}},s-\alpha ,N),$$

that is,

$$h_{top}^{\overrightarrow{a}}(G,D,{\displaystyle \frac{r}{2}},\left\{F_n\right\})\ge s-\alpha .$$

Let $r\to 0$, there is

$$h_{top}^{\overrightarrow{a}}(G,D,\left\{F_n\right\})\ge s-\alpha ,$$

combining Proposition 3, we can obtain

$$h_{top}^{\overrightarrow{a}}(G,E,\left\{F_n\right\})\ge h^{\overrightarrow{a}}(G,D,\left\{F_n\right\})\ge s-\alpha ,$$

then, by combining the arbitrariness of $\alpha$, we can obtain

$$h_{top}^{\overrightarrow{a}}(G,E,\left\{F_n\right\})\ge s.$$

□

Theorem 3.

Let G be a countable discrete amenable group, and $(X_1,d_1)$ a metric space. $G\curvearrowright X_1$ is a continuous action, $\left\{F_n\right\}$ is a nested sequence composed and satisfying condition $\mathbf{F}$. μ is a Borel probability measure on $X_1$, E is a Borel subset of $X_1$, and $0<s<\infty$, then

• If for any $x\in E$ and ${\overline{h}}_{\mu }^{\overrightarrow{a}}(G,x,\left\{F_n\right\})\le s$, then $h_{top}^{\overrightarrow{a}}(G,E,\left\{F_n\right\})\le s$;
• If for any $x\in E,{\overline{h}}_{\mu }^{\overrightarrow{a}}(G,x,\left\{F_n\right\})\ge s$ and $\mu \left(E\right)>0$, then $h_{top}^{\overrightarrow{a}}(G,E,\left\{F_n\right\})\ge s$.

Proof of Theorem 3.

Let $\epsilon >0$, $\kappa >0$. Given $e_0\in E$, we have ${\overline{h}}_{\mu }^{\overrightarrow{a}}(G,e_0,\left\{F_n\right\})\le s$ and

$${\overline{h}}_{\mu }^{\overrightarrow{a}}(G,e_0,\left\{F_n\right\})=\underset{\kappa \to 0}{lim}\underset{n\to \infty }{lim\; sup}-{\displaystyle \frac{1}{|F_n|}}log\mu \left(B_{G,n}^{\overrightarrow{a}}(e_0,\epsilon )\right)<s+\kappa .$$

Denote

$$\phi (e_0,r)=\underset{n\to \infty }{lim\; sup}-{\displaystyle \frac{1}{|F_n|}}log\mu \left(B_{G,n}^{\overrightarrow{a}}(e_0,r)\right).$$

It is easy to get

$$\phi (e_0,r_1)\ge \phi (e_0,r_2),0<r_1<r_2.$$

Since

$$\underset{r\to 0}{lim}\phi (e_0,r)={\overline{h}}_{\mu }^{\overrightarrow{a}}(G,e_0,\left\{F_n\right\})\le s+\kappa ,$$

there exists $0<r_0<ϵ$ such that $\phi (e_0,r_0)<s+\kappa$; this implies that

$$\phi (e_0,ϵ)<\phi (e_0,r_0)<s+\kappa .$$

(6)

Using Theorem 1 and the fact that G is a countably infinite amenable group, there exists a nested Følner sequence $\left\{F_n\right\}$ of G. Let $n\in \mathbb{N}$. For each $e\in E$ with ${\overline{h}}_{\mu }^{\overrightarrow{a}}(G,e,\left\{F_n\right\})\le s$. By (6), we have

$$\underset{n\to \infty }{lim\; sup}-{\displaystyle \frac{1}{|F_n|}}log\mu \left(B_{G,n}^{\overrightarrow{a}}(e,\epsilon )\right)<s+\kappa .$$

Thus, there exists $N\left(e\right)$, which may depend on e such that $N\left(e\right)\ge n$ and

$$\underset{m\ge N\left(e\right)}{sup}-{\displaystyle \frac{1}{|F_m|}}log\mu \left(B_{G,m}^{\overrightarrow{a}}(e,\epsilon )\right)<s+\kappa .$$

(7)

From (7), there is $m\left(z\right)\in \mathbb{N}$, such that

$$\mu \left(B_{G,m\left(e\right)}^{\overrightarrow{a}}(e,\epsilon )\right)\ge e^{-(s+\kappa )|F_{m\left(e\right)}|}.$$

(8)

Define a collection of weighted Bowen balls by

$$\mathcal{F}=\{B_{G,m\left(e\right)}^{\overrightarrow{a}}(e,\epsilon ):e\in E\}.$$

Recalling that $\left\{F_n\right\}$ is nested, by Lemma 1, there is a subcollection

$$\mathcal{G}={\left\{B_{G,m_i}^{\overrightarrow{a}}(e_i,\epsilon )\right\}}_{i\in \mathcal{I}}\subseteq \mathcal{F},$$

(where $m_i=m_{e_i},i\in \mathcal{I}$) of pairwise disjoint Bowen balls satisfying

$$Z\subseteq \bigcup _{i\in \mathcal{I}}{B}_{G,m_i}^{\overrightarrow{a}}(e_i,3\epsilon ).$$

(9)

According to the inequality (2), we have

$$\mu \left(B_{G,m_i}^{\overrightarrow{a}}(e_i,\epsilon )\right)\ge e^{-(s+\kappa )|F_{m_i}|},\forall i\in \mathcal{I}.$$

(10)

So the definition of ${\Lambda }^{\overrightarrow{a}}(E,3\epsilon ,s+\kappa ,\left\{F_n\right\})$ yield that

$$\begin{array}{cc}\hfill {\Lambda }^{\overrightarrow{a}}(E,3\epsilon ,s+\kappa ,\left\{F_n\right\})& \le \sum _{i}exp(-(s+\kappa )|F_{m_i}\left|\right)\hfill \\ \hfill & <\sum _i\mu \left(B_{G,m_i}^{\overrightarrow{a}}(e_i,\epsilon )\right)\hfill \\ \hfill & \le \mu \left(X_1\right)=1.\hfill \end{array}$$

From the above inequality, it follows that

$$h_{top}^{\overrightarrow{a}}(G,E,3\epsilon ,\left\{F_n\right\})\le s+\kappa .$$

Taking $\epsilon \to 0$, we get

$$h_{top}^{\overrightarrow{a}}(G,E,\left\{F_n\right\})\le s+\kappa .$$

The arbitrariness of $\kappa$ implies that

$$h_{top}^{\overrightarrow{a}}(G,E,\left\{F_n\right\})\le s.$$

The result of the second part of Theorem 3 directly follows from the validity of Theorem 2. □

The following corollaries can be derived from Theorems 2 and 3.

Corollary 1.

Let G be a countable discrete amenable group, $(X_1,d_1)$ a metric space. $G\curvearrowright X_1$ be a continuous action, $\left\{F_n\right\}$ be a nested sequence composed and satisfying condition $\mathbf{F}$. μ is a Borel probability measure on $X_1$, E is a Borel subset of $X_1$, and $0<s<\infty$; then,

• If for any $x\in E$ and ${\overline{h}}_{\mu }^{\overrightarrow{a}}(G,x,\left\{F_n\right\})\le s$, then $h_{top}^{\overrightarrow{a}}(G,E,\left\{F_n\right\})\le s$;
• If for any $x\in E,{\underline{h}}_{\mu }^{\overrightarrow{a}}(G,x,\left\{F_n\right\})\ge s$ and $\mu \left(E\right)>0$, then $h_{top}^{\overrightarrow{a}}(G,E,\left\{F_n\right\})\ge s$.

Remark 5.

From Remark 1–3, it can be seen the main results of this paper reduce to the cases in classical dynamical systems when $G=\mathbb{Z},F_n=\{0,1,2,\dots ,n-1\}$. The following Corollary 2 can serve as an application of Theorems 2 and 3 in this paper, and Corollary 2 can be found in [12].

Corollary 2.

μ is a Borel probability measure on $X_1$, E is a Borel subset of $X_1$, and $0<s<\infty$, then

• If ${\overline{h}}_{\mu }^{\overrightarrow{a}}(T,x)\le s$ for all $x\in E$, then $h_{top}^{\overrightarrow{a}}(T,E)\le s$;
• If ${\overline{h}}_{\mu }^{\overrightarrow{a}}(T,x)\ge s$ for all $x\in E$,and $\mu \left(E\right)>0$, then $h_{top}^{\overrightarrow{a}}(T,E)\ge s$.

5. Conclusions

This article studies the relationship between weighted Bowen topological entropy and lower (upper) weighted local entropy in the dynamical systems for amenable group actions. By defining weighted Bowen topological entropy and weighted local measure entropy, we derive a corresponding weighted version of the Billingsley-type theorem using the properties of weighted Bowen topological entropy, and achieve the effect of estimating the overall weighted entropy by weighted local entropy. However, the theory of weighted entropy in dynamical systems with group actions is not yet complete (for example, extensions or modifications of the results of the present work to $\sigma$-finite measure spaces and the entropy formula for the weighted version in such systems have not been resolved), so further exploration of the properties of weighted entropy in dynamical systems for amenable group actions remains a very challenging topic.