\jyear

2021

[3]\surDongkui Ma

1]\orgdivSchool of Mathematics, \orgnameSouth China University of Technology, \cityGuangzhou, \postcode510641, \countryP.R. China 2]\orgdivSchool of Mathematics, \orgnameSouth China University of Technology, \cityGuangzhou, \postcode510641, \countryP.R. China [3]\orgdivSchool of Mathematics, \orgnameSouth China University of Technology, \cityGuangzhou, \postcode510641, \countryP.R. China

The upper capacity topological entropy of free semigroup actions for certain non-compact sets, II

\surYanjie Tang [email protected] \surXiaojiang Ye [email protected] [email protected] [ [ *

Abstract

This paper’s major purpose is to continue the work of Zhu and Ma MR4200965 . To begin, the $\mathbf{g}$ -almost product property, more general irregular and regular sets, and some new notions of the Banach upper density recurrent points and transitive points of free semigroup actions are introduced. Furthermore, under the $\mathbf{g}$ -almost product property and other conditions, we coordinate the Banach upper recurrence, transitivity with (ir)regularity, and obtain lots of generalized multifractal analyses for general observable functions of free semigroup actions. Finally, statistical $\omega$ -limit sets are used to consider the upper capacity topological entropy of the sets of Banach upper recurrent points and transitive points of free semigroup actions, respectively. Our analysis generalizes the results obtained by Huang, Tian and Wang MR3963890 , Pfister and Sullivan MR2322186 .

keywords:

Free semigroup actions; Upper capacity topological entropy; Multifractal analysis;

\mathbf{g}

-almost product property; Banach upper recurrence

pacs:

[

MSC Classification]37B40, 37B20, 37C45, 37B05

1 Introduction

A dynamical system $(X,d,f)$ means always that $(X,d)$ is a compact metric space and $f$ is a continuous self map on $X$ . One of the fundamental issues in dynamical systems is how points with similar asymptotic behavior influence or determine the system’s complexity. Topological entropy is a term used to describe the dynamical complexity of a dynamical system. Using a similar defining way as the Hausdorff dimension, Bowen MR338317 and Pesin MR1489237 extended the concept of topological entropy to non-compact sets. On the basis of this theory, researchers are paying more attention to study Hausdorff dimension or topological entropy for non-compact sets, for example, multifractal spectra and saturated sets of dynamical systems, see, MR2322186 ; MR3436391 ; MR1971209 ; MR1439805 ; MR2931333 ; MR1759398 ; MR3963890 ; MR3833343 ; MR2158401 . In particular, recent work about non-compact sets with full entropy has attracted greater attention than before, see MR4200965 ; MR3963890 ; MR3833343 ; MR3436391 ; MR2158401 . As a result, the size of the topological entropy of the ‘periodic-like’ recurrent level sets is always in focus, see MR3963890 ; DongandTian1 ; DongandTian2 ; MR3436391 ; MR4200965 .

Most of different recurrent level sets are considered. The concepts of periodic point, recurrent point, almost periodic point and non-wandering point are described, see MR648108 , the concepts of (quai)weakly almost periodic point (sometimes called (upper)lower density recurrent point) can be see in MR1223083 ; MR2039054 ; 1993Measure ; 1995Level . Moreover, Zhou 1995Level firstly linked the recurrent frequency with the support of invariant measures which are the limit points of the empirical measure, which provided a tremendous benefit to researchers.

Birkhoff ergodic average MR1439805 ; MR1489237 (or Lyapunov exponents MR2645746 ) is a well-known method to distinguish the asymptotic behavior of a dynamical orbit. Let $\varphi:X\to\mathbb{R}$ be a continuous function and $a$ a real number. Consider a level set of Birkhoff averages

R_{\varphi}(a):=\left\{x\in X:\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi\left(f^{j}(x)\right)=a\right\}.

These $R_{\varphi}(a)$ form a multifractal decomposition and the function

a\to h_{R_{\varphi}(a)}(f)

is a entropy spectrum, where $h_{R_{\varphi}(a)}(f)$ denotes the topological entropy of $R_{\varphi}(a)$ defined by MR338317 . Takens and Verbitskiy MR1971209 proved that transformations satisfying the specification has the following variational principle

h_{R_{\varphi}(a)}\left(f\right)=\sup\left\{h_{\mu}(f):\mu\text{ is invariant and }\int\varphi d\mu=a\right\}

(1)

where $h_{\mu}(f)$ is the measure-theoretic (or Kolmogorov-Sinai) entropy of the invariant measure $\mu$ . In MR2322186 , the formula (1) holds under the conditions of the $\mathbf{g}$ -almost product property which was introduced by Pfister and Sullivan. Consider $\varphi$ -irregular set as follows:

I_{\varphi}:=\left\{x\in X:\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi\left(f^{j}(x)\right)\text{ diverges}\right\}.

The irregular set is not detectable from the standpoint of invariant measures, according to Birkhoff’s ergodic theorem. However, in systems with certain properties, the irregular set may carry full topological entropy, see MR2931333 ; MR3833343 ; MR2158401 ; MR775933 ; MR1759398 and references therein.

Based on these results, Tian MR3436391 distinguished various periodic-like recurrences and discovered that they all carry full topological entropy, as do their gap-sets, under the $\mathbf{g}$ -almost product property and other conditions. Furthermore, the author MR3436391 combined the periodic-like recurrences with (ir)regularity to obtain a large number of generalized multifractal analyses for all continuous observable functions. Later on, Huang, Tian and Wang MR3963890 introduced an abstract version of multifractal analysis for possible applicability to more general function. Let $\alpha:\mathcal{M}(X,f)\to\mathbb{R}$ be a continuous function where $\mathcal{M}(X,f)$ denotes the set of all $f$ -invariant probability measures on $X$ . Define more general regular and irregular sets as follows

R_{\alpha}:=\left\{x\in X:\inf_{\mu\in M_{x}}\alpha(\mu)=\sup_{\mu\in M_{x}}\alpha(\mu)\right\},

I_{\alpha}:=\left\{x\in X:\inf_{\mu\in M_{x}}\alpha(\mu)<\sup_{\mu\in M_{x}}\alpha(\mu)\right\},

respectively, where $M_{x}$ stands for the set of all limit points of the empirical measure. Furthermore, the authors MR3963890 established an abstract framework on the combination of Banach upper recurrence, transitivity and multifractal analysis of general observable functions. Learned from MR1601486 for maps and MR1716564 for flows, Dong and Tian DongandTian1 defined the statistical $\omega$ -limit sets to describe different statistical structure of dynamical orbits using concepts of density, and considered multifractal analysis on various non-recurrence and Birkhoff ergodic averages. Furthermore, the authors MR3963890 used statistical $\omega$ -limit sets to obtain the results of topological entropy on the refined orbit distribution of Banach upper recurrence.

On the other hand, Ghys et al MR926526 proposed a definition of topological entropy for finitely generated pseudo-groups of continuous maps. Later on, the topological entropy of free semigroup actions on a compact metric space defined by Biś MR2083436 and Bufetov MR1681003 , respectively. Rodrigues and Varandas MR3503951 and Lin et al MR3774838 extended the work of Bufetov MR1681003 from various perspectives. Similar to the methods of Bowen MR338317 and Pesin MR1489237 , Ju et al MR3918203 introduced topological entropy and upper capacity topological entropy of free semigroup actions on non-compact sets which extended the results obtained by MR1681003 ; MR338317 ; MR1489237 . Zhu and Ma MR4200965 investigated the upper capacity topological entropy of free semigroup actions for certain non-compact sets. More relevant results are obtained, see MR3503951 ; MR3828742 ; MR3784991 ; MR3774838 ; MR3918203 ; MR4200965 ; MR3592853 ; MR1767945 .

The above results raise the question of whether similar sets exist in dynamical system of free semigroup actions. Further, one can ask if the sets have full topological entropy or full upper capacity topological entropy, and if an abstract framework of general observable functions can be established. To answer the above questions, we introduce different asymptotic behavior of points and more general irregular set. Our results show that various subsets characterized by distinct asymptotic behavior may carry full upper capacity topological entropy of free semigroup actions. Our analysis is to continue the work of MR4200965 and generalize the results obtained by MR3963890 and MR2322186 .

This paper is organized as follows. In Sec. 2, we give our main results. In Sec. 3, we give some preliminaries. In Sec. 4, we introduce some concepts of transitive points, quasiregular points, upper recurrent points and Banach upper recurrent points with respect to a certain orbit of free semigroup actions. On the other hand, the g-almost product property of free semigroup actions, which is weaker than the specification property of free semigroup actions, is introduced. In Sec. 5, the upper capacity topological entropy of free semigroup actions on the more general irregular and regular sets is considered, and some examples satisfying the assumptions of our main results are described. In Sec. 6, we give the proofs of our main results.

2 Statement of main results

Let $(X,d)$ be a compact metric space and $G$ the free semigroup generated by $m$ generators $f_{0},\cdots,f_{m-1}$ which are continuous maps on $X$ . Let $\Sigma^{+}_{m}$ denote the one-side symbol space generated by the digits $\{0,1,\cdots,m-1\}$ . Recall that the skew product transformation is given as follows:

F:\Sigma^{+}_{m}\times X\to\Sigma^{+}_{m}\times X,\>\,(\iota,x)\mapsto\big{(}\sigma(\iota),f_{i_{0}}(x)\big{)},

where $\iota=(i_{0},i_{1},\cdots)\in\Sigma_{m}^{+}$ , $x\in X$ and $\sigma$ is the shift map of $\Sigma^{+}_{m}$ . Let $\mathcal{M}(\Sigma_{m}^{+}\times X,F)$ denote the set of invariant measures of the skew product $F$ . Denote by $\mathrm{Tran}(\iota,G)$ , $\mathrm{QR}(\iota,G)$ , $\mathrm{QW}(\iota,G)$ and $\mathrm{BR}(\iota,G)$ the sets of the transitive points, the quasiregular points, the upper recurrent points and the Banach upper recurrent points of free semigroup action $G$ with respect to $\iota\in\Sigma_{m}^{+}$ , respectively (see Sec. 4).

Let $\alpha:\mathcal{M}(\Sigma_{m}^{+}\times X,F)\to\mathbb{R}$ be a continuous function. Given $(\iota,x)\in\Sigma_{m}^{+}\times X$ , let $M_{(\iota,x)}(F)$ be the set of all limit points of $\{\frac{1}{n}\sum_{j=0}^{n-1}\delta_{F^{j}(\iota,x)}\}$ in weak^∗ topology. Let $L_{\alpha}:=[\inf_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu),\,\sup_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)]$ and $\mathrm{Int}(L_{\alpha})$ denote the interior of interval $L_{\alpha}$ . For possibly applicability to more general functions, MR3963890 defined three conditions for $\alpha$ to introduce an abstract version of multifractal analysis. To understand our main results, we list the three conditions(see MR3963890 for details):

A.1

For any $\mu,\nu\in\mathcal{M}(\Sigma_{m}^{+}\times X,F)$ , $\beta(\theta):=\alpha(\theta\mu+(1-\theta)\nu)$ is strictly monotonic on $[0,1]$ when $\alpha(\mu)\neq\alpha(\nu)$ .
A.2

For any $\mu,\nu\in\mathcal{M}(\Sigma_{m}^{+}\times X,F)$ , $\beta(\theta):=\alpha(\theta\mu+(1-\theta)\nu)$ is constant on $[0,1]$ when $\alpha(\mu)=\alpha(\nu)$ .
A.3

For any $\mu,\nu\in\mathcal{M}(\Sigma_{m}^{+}\times X,F)$ , $\beta(\theta):=\alpha(\theta\mu+(1-\theta)\nu)$ is not constant over any subinterval of $[0,1]$ when $\alpha(\mu)\neq\alpha(\nu)$ .

Given $\iota\in\Sigma_{m}^{+}$ , we introduce the general regular set and irregular set with respect to $\iota$ as follows:

R_{\alpha}(\iota,G):=\left\{x\in X:\inf_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)=\sup_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)\right\},

I_{\alpha}(\iota,G):=\left\{x\in X:\inf_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)<\sup_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)\right\}.

Let $C_{(\iota,x)}=\overline{\cup_{\nu\in M_{(\iota,x)}(F)}S_{\nu}}$ , where $S_{\nu}$ denotes the support of measure $\nu$ . Let $\mathrm{BR}^{\#}(\iota,G):=\mathrm{BR}(\iota,G)\backslash\mathrm{QW}(\iota,G)$ . Fixed $\iota\in\Sigma_{m}^{+}$ , the following sets are introduced to more precisely characterize the recurrence of the orbit,

	$\displaystyle W(\iota,G)$	$\displaystyle:=\left\{x\in X:S_{\nu}=C_{(\iota,x)}\text{ for every }\nu\in M_{(\iota,x)}(F)\right\},$
	$\displaystyle V(\iota,G)$	$\displaystyle:=\left\{x\in X:\exists\nu\in M_{(\iota,x)}(F)\text{ such that }S_{\nu}=C_{(\iota,x)}\right\},$
	$\displaystyle S(\iota,G)$	$\displaystyle:=\left\{x\in X:\cap_{\nu\in M_{(\iota,x)}(F)}S_{\nu}\neq\emptyset\right\}.$

More specifically, $\mathrm{BR}^{\#}(\iota,G)$ is subdivided into the following several levels with different asymptotic behaviour:

		$\displaystyle\mathrm{BR}_{1}(\iota,G):=\mathrm{BR}^{\#}(\iota,G)\cap W(\iota,G),$
		$\displaystyle\mathrm{BR}_{2}(\iota,G):=\mathrm{BR}^{\#}(\iota,G)\cap(V(\iota,G)\cap S(\iota,G)),$
		$\displaystyle\mathrm{BR}_{3}(\iota,G):=\mathrm{BR}^{\#}(\iota,G)\cap V(\iota,G),$
		$\displaystyle\mathrm{BR}_{4}(\iota,G):=\mathrm{BR}^{\#}(\iota,G)\cap(V(\iota,G)\cup S(\iota,G)),$
		$\displaystyle\mathrm{BR}_{5}(\iota,G):=\mathrm{BR}^{\#}(\iota,G).$

Then $\mathrm{BR}_{1}(\iota,G)\subseteq\mathrm{BR}_{2}(\iota,G)\subseteq\mathrm{BR}_{3}(\iota,G)\subseteq\mathrm{BR}_{4}(\iota,G)\subseteq\mathrm{BR}_{5}(\iota,G)$ .

Analogously, $\mathrm{QW}(\iota,G)$ is also subdivided into the following several levels with varying asymptotic behaviour:

		$\displaystyle\mathrm{QW}_{1}(\iota,G):=\mathrm{QW}(\iota,G)\cap W(\iota,G),$
		$\displaystyle\mathrm{QW}_{2}(\iota,G):=\mathrm{QW}(\iota,G)\cap(V(\iota,G)\cap S(\iota,G)),$
		$\displaystyle\mathrm{QW}_{3}(\iota,G):=\mathrm{QW}(\iota,G)\cap V(\iota,G),$
		$\displaystyle\mathrm{QW}_{4}(\iota,G):=\mathrm{QW}(\iota,G)\cap(V(\iota,G)\cup S(\iota,G)),$
		$\displaystyle\mathrm{QW}_{5}(\iota,G):=\mathrm{QW}(\iota,G).$

Note that $\mathrm{QW}_{1}(\iota,G)\subseteq\mathrm{QW}_{2}(\iota,G)\subseteq\mathrm{QW}_{3}(\iota,G)\subseteq\mathrm{QW}_{4}(\iota,G)\subseteq\mathrm{QW}_{5}(\iota,G)$ .

Let $Z_{0}(\iota,G),Z_{1}(\iota,G),\cdots,Z_{k}(\iota,G)$ and $Y(\iota,G)$ be the subsets of $X$ with respect to $\iota\in\Sigma_{m}^{+}$ . Consider the sets, for $j=1,\cdots,k$ ,

M_{j}(Y):=\cup_{\iota\in\Sigma_{m}^{+}}\big{(}Y(\iota,G)\cap(Z_{j}(\iota,G)\setminus Z_{j-1}(\iota,G))\big{)},

then we say that the sets $M_{1}(Y),M_{2}(Y),\cdots,M_{k}(Y)$ is the unions of gaps of $\{Z_{0}(\iota,G),Z_{1}(\iota,G),\cdots,Z_{k}(\iota,G)\}$ with respect to $Y(\iota,G)$ for all $\iota\in\Sigma_{m}^{+}$ .

Now we start to state our main theorems.

Theorem 1.

Suppose that $G$ has the $\mathbf{g}$ -almost product property, there exists a $\mathbb{P}$ -stationary measure (see Sec. 3.3) on $X$ with full support where $\mathbb{P}$ is a Bernoulli measure on $\Sigma_{m}^{+}$ . Let $\alpha:\mathcal{M}(\Sigma_{m}^{+}\times X,F)\to\mathbb{R}$ be a continuous function.

(1)

If $\alpha$ satisfies A.3 and $\mathrm{Int}(L_{\alpha})\neq\emptyset$ , then the unions of gaps of

\left\{\mathrm{QR}(\iota,G),\mathrm{BR}_{1}(\iota,G),\mathrm{BR}_{2}(\iota,G),\mathrm{BR}_{3}(\iota,G),\mathrm{BR}_{4}(\iota,G),\mathrm{BR}_{5}(\iota,G)\right\}

with respect to $I_{\alpha}(\iota,G)\cap\mathrm{Tran}(\iota,G)$ for all $\iota\in\Sigma_{m}^{+}$ have full upper capacity topological entropy of free semigroup action $G$ .

(2)

If the skew product $F$ is not uniquely ergodic, then the unions of gaps of

\left\{\emptyset,\mathrm{QR}(\iota,G)\cap\mathrm{BR}_{1}(\iota,G),\mathrm{BR}_{1}(\iota,G),\mathrm{BR}_{2}(\iota,G),\mathrm{BR}_{3}(\iota,G),\mathrm{BR}_{4}(\iota,G),\mathrm{BR}_{5}(\iota,G)\right\}

with respect to $\mathrm{Tran}(\iota,G)$ for all $\iota\in\Sigma_{m}^{+}$ have full upper capacity topological entropy of free semigroup action $G$ .

(3)

If the skew product $F$ is not uniquely ergodic and $\alpha$ satisfies A.1 and A.2, then the unions of gaps of

\left\{\emptyset,\mathrm{QR}(\iota,G)\cap\mathrm{BR}_{1}(\iota,G),\mathrm{BR}_{1}(\iota,G),\mathrm{BR}_{2}(\iota,G),\mathrm{BR}_{3}(\iota,G),\mathrm{BR}_{4}(\iota,G),\mathrm{BR}_{5}(\iota,G)\right\}

with respect to $R_{\alpha}(\iota,G)\cap\mathrm{Tran}(\iota,G)$ for all $\iota\in\Sigma_{m}^{+}$ have full upper capacity topological entropy of free semigroup action $G$ .

Theorem 2.

Suppose that $G$ has the $\mathbf{g}$ -almost product property and positively expansive, there exists a $\mathbb{P}$ -stationary measure with full support on $X$ where $\mathbb{P}$ is a Bernoulli measure on $\Sigma_{m}^{+}$ . Let $\varphi:X\to\mathbb{R}$ be a continuous function. If the skew product $F$ is not uniquely ergodic, then the unions of gaps of

\left\{\emptyset,\mathrm{QR}(\iota,G)\cap\mathrm{QW}_{1}(\iota,G),\mathrm{QW}_{1}(\iota,G),\mathrm{QW}_{2}(\iota,G),\mathrm{QW}_{3}(\iota,G),\mathrm{QW}_{4}(\iota,G),\mathrm{QW}_{5}(\iota,G)\right\}

with respect to $\mathrm{Tran}(\iota,G)$ , $R_{\alpha}(\iota,G)\cap\mathrm{Tran}(\iota,G)$ for all $\iota\in\Sigma_{m}^{+}$ have full upper capacity topological entropy of free semigroup action $G$ , respectively. If $I_{\alpha}(\iota,G)$ is non-empty for some $\iota\in\Sigma_{m}^{+}$ , similar arguments hold with respect to $I_{\alpha}(\iota,G)\cap\mathrm{Tran}(\iota,G)$ .

For $S\subseteq\mathbb{N}$ , let $\overline{d}(S)$ , $\underline{d}(S)$ , $B^{*}(S)$ and $B_{*}(S)$ denote the upper density, lower density, Banach upper density and Banach lower density of $S$ , respectively. Given $(\iota,x)\in\Sigma_{m}^{+}\times X$ and $\xi\in\{\,\overline{d},\underline{d},B^{*},B_{*}\}$ , denote by $\omega_{\xi}\left((\iota,x),F\right)$ the $\xi$ - $\omega$ -limit set of $(\iota,x)$ . The notions will be given in more detail later in Section 3.1 (also see MR3963890 ; DongandTian1 ; DongandTian2 ). If $\omega_{B_{*}}\left((\iota,x),F\right)=\emptyset$ and $\omega_{B^{*}}\left((\iota,x),F\right)=\omega\left((\iota,x),F\right)$ , then from DongandTian1 one has that $(\iota,x)$ satisfies only one of the following six cases:

(1)

$\emptyset=\omega_{B_{*}}\left((\iota,x),F\right)\subsetneq\omega_{\underline{d}}\left((\iota,x),F\right)=\omega_{\overline{d}}\left((\iota,x),F\right)=\omega_{B^{*}}\left((\iota,x),F\right)=\omega\left((\iota,x),F\right)$ ;
(2)

$\emptyset=\omega_{B_{*}}\left((\iota,x),F\right)\subsetneq\omega_{\underline{d}}\left((\iota,x),F\right)=\omega_{\overline{d}}\left((\iota,x),F\right)\subsetneq\omega_{B^{*}}\left((\iota,x),F\right)=\omega\left((\iota,x),F\right)$ ;
(3)

$\emptyset=\omega_{B_{*}}\left((\iota,x),F\right)=\omega_{\underline{d}}\left((\iota,x),F\right)\subsetneq\omega_{\overline{d}}\left((\iota,x),F\right)=\omega_{B^{*}}\left((\iota,x),F\right)=\omega\left((\iota,x),F\right)$ ;
(4)

$\emptyset=\omega_{B_{*}}\left((\iota,x),F\right)\subsetneq\omega_{\underline{d}}\left((\iota,x),F\right)\subsetneq\omega_{\overline{d}}\left((\iota,x),F\right)=\omega_{B^{*}}\left((\iota,x),F\right)=\omega\left((\iota,x),F\right)$ ;
(5)

$\emptyset=\omega_{B_{*}}\left((\iota,x),F\right)=\omega_{\underline{d}}\left((\iota,x),F\right)\subsetneq\omega_{\overline{d}}\left((\iota,x),F\right)\subsetneq\omega_{B^{*}}\left((\iota,x),F\right)=\omega\left((\iota,x),F\right)$ ;
(6)

$\emptyset=\omega_{B_{*}}\left((\iota,x),F\right)\subsetneq\omega_{\underline{d}}\left((\iota,x),F\right)\subsetneq\omega_{\overline{d}}\left((\iota,x),F\right)\subsetneq\omega_{B^{*}}\left((\iota,x),F\right)=\omega\left((\iota,x),F\right)$ .

Consider the two sets as follows:

T_{j}(\iota,G):=\left\{x\in\mathrm{Tran}(\iota,G):(\iota,x)\text{ satisfies Case }(j)\right\},

B_{j}(\iota,G):=\left\{x\in\mathrm{BR}(\iota,G):(\iota,x)\text{ satisfies Case }(j)\right\},

where $j=1,\cdots,6$ . Let

T_{j}(G):=\cup_{\iota\in\Sigma_{m}^{+}}T_{j}(\iota,G),\quad B_{j}(G):=\cup_{\iota\in\Sigma_{m}^{+}}B_{j}(\iota,G).

Theorem 3.

Suppose that $G$ has the $\mathbf{g}$ -almost product property, there exists a $\mathbb{P}$ -stationary measure with full support on $X$ where $\mathbb{P}$ is a Bernoulli measure on $\Sigma_{m}^{+}$ . If the skew product $F$ is not uniquely ergodic, then $T_{j}(G)\neq\emptyset$ and $B_{j}(G)\neq\emptyset$ . Moreover, they all have full capacity topological entropy of free semigroup action $G$ , that is,

\overline{Ch}_{T_{j}(G)}(G)=\overline{Ch}_{X}(G)=h(G),

\overline{Ch}_{B_{j}(G)}(G)=\overline{Ch}_{X}(G)=h(G),

for all $j=1,\cdots,6$ , where $\overline{Ch}_{Z}(G)$ denotes the upper capacity topological entropy on any subset $Z\subseteq X$ in the sense of MR3918203 , $h(G)$ denotes the topological entropy in the sense of Bufetov MR1681003 . If $Z=X$ , we have $\overline{Ch}_{X}(G)=h(G)$ from Remark 5.1 of MR3918203 .

3 Preliminaries

3.1 Some notions

Let $(X,d)$ be a compact metric space and $f$ be a continuous map on $X$ . For $S\subseteq\mathbb{N}$ , the upper density and the Banach upper density of $S$ are defined by

\overline{d}(S):=\limsup_{n\rightarrow\infty}\frac{\sharp\left\{S\cap\{0,1,\cdots,n-1\}\right\}}{n},\quad B^{*}(S):=\limsup_{\sharp I\rightarrow\infty}\frac{\sharp\left\{S\cap I\right\}}{\sharp I},

respectively, where $\sharp Y$ denotes the cardinality of the set $Y$ and $I\subseteq\mathbb{N}$ is taken from finite continuous integer intervals. Similarly, one can define the lower density and the Banach lower density of $S$ , denoted as $\underline{d}(S)$ and $B_{*}(S)$ , respectively. Let $U\subset X$ be a nonempty open set and $x\in X$ . Define the set of visiting time,

N(x,U):=\left\{n\geq 0:f^{n}(x)\in U\right\}.

Recall that a point $x\in X$ is called to be Banach upper recurrent, if for any $\varepsilon>0$ , the set of visiting time $N(x,B(x,\varepsilon))$ has a positive Banach upper density where $B(x,\varepsilon)$ denotes the ball centered at $x$ with radius $\varepsilon$ . Similarly, one can call a point $x\in X$ upper recurrent, if for any $\varepsilon>0$ , the set of visiting time $N(x,B(x,\varepsilon))$ has a positive upper density. Let us denote by $\mathrm{BR}(f)$ and $\mathrm{QW}(f)$ the sets of the Banach upper recurrent points and the upper recurrent points of $f$ , respectively. It is immediate that

\mathrm{QW}(f)\subseteq\mathrm{BR}(f).

A point $x\in X$ is called transitive if its orbit $\{x,f(x),f^{2}(x),\cdots\}$ is dense in $X$ . Let us denote by $\mathrm{Tran}(f)$ the set of transitive points of $f$ .

We recall that several concepts were introduced in MR3963890 . For $x\in X$ and $\xi\in\{\,\overline{d},\underline{d},B_{*},B^{*}\}$ , a point $y\in X$ is called $x$ - $\xi$ -accessible, if for any $\varepsilon>0$ , the set of visiting time $N\left(x,B(y,\varepsilon)\right)$ has positive density with respect to $\xi$ . Let

\omega_{\xi}(x):=\{y\in X:y\text{ is }x\text{-}\xi\text{-accessible }\}.

For convenience, it is called the $\xi$ - $\omega$ -limit set of $x$ .

The set of invariant measures under $f$ will be denote by $\mathcal{M}(X,f)$ . For $\mu\in\mathcal{M}(X,f)$ , a point $x\in X$ is $\mu$ -generic if

\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f^{j}(x)}=\mu

where $\delta_{y}$ denotes the Dirac measure on $y$ . We will use $G_{\mu}(f)$ to denote the set of $\mu$ -generic points. Let $\mathrm{QR}(f):=\bigcup_{\mu\in\mathcal{M}(X,f)}G_{\mu}(f)$ . The points in $\mathrm{QR}(f)$ are called quasiregular points of $f$ .

3.2 The topological entropy and others concepts of free semigroup actions

In this paper, we use the topological entropy and upper capacity topological entropy of free semigroup actions defined by MR1681003 and MR3918203 , respectively. Let $(X,d)$ be a compact metric space and $G$ the free semigroup action on $X$ generated by $f_{0},\cdots,f_{m-1}$ . For convenience, we first recall the notion of words.

Let $F_{m}^{+}$ be the set of all finite words of symbols $0,1,\cdots,m-1$ . For any $w\in F_{m}^{+}$ , $\lvert w\rvert$ stands for the length of $w$ , that is, the number of symbols in $w$ . Obviously, $F^{+}_{m}$ with respect to the law of composition is a free semigroup with $m$ generators. We write $w^{\prime}\leq w$ if there exists a word $w^{\prime\prime}\in F^{+}_{m}$ such that $w=w^{\prime\prime}w^{\prime}$ . Remark that $\emptyset\in F_{m}^{+}$ and $\emptyset\leq w$ . For $w=i_{0}i_{1}\cdots i_{k}\in F^{+}_{m}$ , denote $\overline{w}=i_{k}\cdots i_{1}i_{0}$ .

Denote by $\Sigma^{+}_{m}$ the set of all one-side infinite sequences of symbols $\{0,1,\cdots,m-1\}$ , that is,

\Sigma^{+}_{m}=\left\{\iota=(i_{0},i_{1},\cdots)\,:\,i_{k}\in\{0,1,\cdots,m-1\},\>k\in\mathbb{N}\right\}.

The metric on $\Sigma^{+}_{m}$ is given by

d^{\prime}(\iota,\iota^{\prime}):=2^{-j},\quad j=\inf\{n\,:\,i_{n}\neq i^{\prime}_{n}\}.

It is easy to check that $\Sigma^{+}_{m}$ is compact with respect to this metric. The shift $\sigma:\Sigma^{+}_{m}\to\Sigma^{+}_{m}$ is given by the formula, for each $\iota=(i_{0},i_{1},\cdots)\in\Sigma^{+}_{m}$ ,

\sigma(\iota)=(i_{1},i_{2},\cdots).

Suppose that $\iota\in\Sigma^{+}_{m}$ , and $a,b\in\mathbb{N}$ with $a\leq b$ . We write $\iota\lvert_{[a,b]}=w$ if $w=i_{a}i_{a+1}\cdots i_{b}$ .

To each $w\in F^{+}_{m}$ , $w=i_{0}\cdots i_{k-1}$ , let us write $f_{w}=f_{i_{0}}\circ\cdots\circ f_{i_{k-1}}$ if $k>0$ , and $f_{w}=\mathrm{Id}$ if $k=0$ , where $\mathrm{Id}$ is the identity map. Obviously, $f_{ww^{\prime}}=f_{w}f_{w^{\prime}}$ .

For $w\in F^{+}_{m}$ , we assign a metric $d_{w}$ on $X$ by setting

d_{w}(x_{1},x_{2})=\max_{w^{\prime}\leq\overline{w}}d\left(f_{w^{\prime}}(x_{1}),f_{w^{\prime}}(x_{2})\right).

Given a number $\delta>0$ and a point $x\in X$ , define the $(w,\delta)$ -Bowen ball at $x$ by

B_{w}(x,\delta):=\left\{y\in X:{d_{w}\left(x,y\right)<\delta}\right\}.

Recall that the positively expansive of the free semigroup actions means that if there exists $\delta>0$ , such that any $x,y\in X$ with $x\neq y$ , for any $\iota\in\Sigma_{m}^{+}$ there exists $n\geq 1$ satisfying $d\left(f_{\overline{\iota\lvert_{[0,n-1]}}}(x),f_{\overline{\iota\lvert_{[0,n-1]}}}(y)\right)\geq\delta$ , which was introduced by Zhu and Ma MR4200965 .

The specification property of free semigroup actions was introduced by Rodrigues and Varandas MR3503951 . We say that $G$ has the specification property if for any $\varepsilon>0$ , there exists $\mathfrak{p}(\varepsilon)>0$ , such that for any $k>0$ , any points $x_{1},\cdots,x_{k}\in X$ , any positive integers $n_{1},\cdots,n_{k}$ , any $p_{1},\cdots,p_{k}\geq\mathfrak{p}(\varepsilon)$ , any $w_{(p_{j})}\in F_{m}^{+}$ with $\lvert w_{(p_{j})}\rvert=p_{j}$ , $w_{(n_{j})}\in F_{m}^{+}$ with $\lvert w_{(n_{j})}\rvert=n_{j},1\leq j\leq k$ , one has

B_{w_{(n_{1})}}\left(x_{1},\varepsilon\right)\cap\left(\bigcap_{j=2}^{k}{f^{-1}_{\overline{w_{(p_{j-1})}}\,\overline{w_{(n_{j-1})}}\cdots\overline{w_{(p_{1})}}\,\overline{w_{(n_{1})}}}}B_{w_{(n_{j})}}\left(x_{j},\varepsilon\right)\right)\neq\emptyset.

If $m=1$ , the specification property of free semigroup actions coincides with the classical definition introduced by Bowen MR282372 .

We recall the definition of topological entropy for free semigroup actions introduced by MR1681003 . A subset $K$ of $X$ is called a $(w,\varepsilon,G)$ -separated subset if, for any $x_{1},x_{2}\in K$ with $x_{1}\neq x_{2}$ , one has $d_{w}\left(x_{1},x_{2}\right)\geq\varepsilon$ . The maximum cardinality of a $(w,\varepsilon,G)$ -separated subset of $X$ is denoted by $N(w,\varepsilon,G)$ . The topological entropy of free semigroup actions is defined by the formula

h(G):=\lim_{\varepsilon\rightarrow 0}\limsup_{n\rightarrow\infty}\frac{1}{n}\log\left(\frac{1}{m^{n}}\sum_{\lvert w\rvert=n}N(w,\varepsilon,G)\right).

Remark 1.

If $m=1$ , this definition coincides with the topological entropy of a single map defined by MR175106 . For more information, see Chapter 7 of MR648108 .

The dynamical systems given by free semigroup action have a strong connection with skew product which has been analyzed to obtain properties of free semigroup actions through fiber associated with the skew product (see for instance MR4200965 ; MR3784991 ). Recall that the skew product transformation is given by as follows:

F:\Sigma^{+}_{m}\times X\to\Sigma^{+}_{m}\times X,\>\,(\iota,x)\mapsto\big{(}\sigma(\iota),f_{i_{0}}(x)\big{)},

where $\iota=(i_{0},i_{1},\cdots)$ and $\sigma$ is the shift map of $\Sigma^{+}_{m}$ . The metric $D$ on $\Sigma^{+}_{m}\times X$ is given by the formula

D\left(\left(\iota,x\right),\left(\iota^{\prime},x^{\prime}\right)\right):=\max\left\{d^{\prime}\left(\iota,\iota^{\prime}\right),d\left(x,x^{\prime}\right)\right\}.

Theorem 4.

(MR1681003 ,Theorem 1) Topological entropy of the skew product transformation $F$ satisfies

h(F)=\log m+h(G),

where $h(F)$ denotes the topological entropy of $F$ .

Now, let us recall the topological entropy and upper capacity topological entropy of free semigroup actions for non-compact sets defined by MR3918203 . Fixed $\delta>0$ , we define the collection of subsets

\mathcal{F}:=\left\{B_{w}(x,\delta):\,x\in X,\,w\in F^{+}_{m},\,\lvert w\rvert=n\text{ and }n\in\mathbb{N}\right\}.

Given subset $Z\subset X$ , we define, for $\gamma\geq 0$ , $N>0$ and $w\in F_{m}^{+}$ with $\lvert w\rvert=N$ ,

{M}_{w}(Z,\gamma,\delta,N):=\inf_{\mathcal{G}_{w}}\left\{\sum_{B_{w^{\prime}}(x,\delta)\in\mathcal{G}_{w}}\exp{\left(-\gamma\cdot\left(\lvert w^{\prime}\rvert+1\right)\right)}\right\},

where the infimum is taken over all finite or countable subcollections $\mathcal{G}_{w}\subseteq\mathcal{F}$ covering $Z$ (i.e. for any $B_{w^{\prime}}(x,\delta)\in\mathcal{G}_{w}$ , $\overline{w}\leq\overline{w^{\prime}}$ and $\bigcup_{B_{w^{\prime}}(x,\delta)\in\mathcal{G}_{w}}B_{w^{\prime}}(x,\delta)\supseteq Z$ ). Let

{M}(Z,\gamma,\delta,N):=\frac{1}{m^{N}}\sum_{\lvert w\rvert=N}{M}_{w}(Z,\gamma,\delta,N).

It is easy to verify that the function ${M}(Z,\gamma,\delta,N)$ is non-decreasing as $N$ increases. Therefore, there exists the limit

{m}(Z,\gamma,\delta)=\lim_{N\rightarrow\infty}{M}(Z,\gamma,\delta,N).

Furthermore, we can define

{R}_{w}(Z,\gamma,\delta,N):=\inf_{\mathcal{G}_{w}}\left\{\sum_{B_{w}(x,\delta)\in\mathcal{G}_{w}}\exp{\left(-\gamma\cdot(N+1)\right)}\right\},

where the infimum is taken over all finite or countable subcollections $\mathcal{G}_{w}\subseteq\mathcal{F}$ covering $Z$ and the length word correspond to every ball in $\mathcal{G}_{w}$ are all equal to $N$ . Let

{R}(Z,\gamma,\delta,N):=\frac{1}{m^{N}}\sum_{\lvert w\rvert=N}{R}_{w}(Z,\gamma,\delta,N),

and set

\overline{r}(Z,\gamma,\delta):=\limsup_{N\rightarrow\infty}{R}(Z,\gamma,\delta,N).

The critical values $h_{Z}(G,\delta)$ and $\overline{Ch}_{Z}(G,\delta)$ are defined as

	$\displaystyle h_{Z}(G,\delta)$	$\displaystyle:=\inf\{\gamma:{m}(Z,\gamma,\delta)=0\}=\sup\{\gamma:{m}(Z,\gamma,\delta)=\infty\},$
	$\displaystyle\overline{Ch}_{Z}(G,\delta)$	$\displaystyle:=\inf\{\gamma:\,\,\overline{r}(Z,\gamma,\delta)=0\}=\sup\{\gamma:\,\,\overline{r}(Z,\gamma,\delta)=\infty\}.$

The topological entropy and upper capacity topological entropy of $Z$ of free semigroup action $G$ are then defined as

	$\displaystyle h_{Z}(G)$	$\displaystyle:=\lim_{\delta\rightarrow 0}h_{Z}(G,\delta),$
	$\displaystyle\overline{Ch}_{Z}(G)$	$\displaystyle:=\lim_{\delta\rightarrow 0}\overline{Ch}_{Z}(G,\delta).$

Remark 2.

Let $f:X\to X$ be a continuous transformation and $G$ the free semigroup generated by the map $f$ . Then $h_{Z}(G)=h_{Z}(f)$ , $\overline{Ch}_{Z}(G)=\overline{Ch}_{Z}(f)$ , for any set $Z\subset X$ , where $h_{Z}(f)$ and $\overline{Ch}_{Z}(f)$ are the topological entropy and upper capacity topological entropy defined by Pesin MR1489237 . If $Z=X$ , then $h(G)=h(f)=h_{X}(f)=\overline{Ch}_{X}(f)$ , i.e., the classical topological entropy defined by Adler et al MR175106 .

In MR3918203 , they proved that the upper capacity topological entropy of the skew product $F$ satisfies the following result for any subset $Z\subseteq X$ .

Theorem 5.

(MR3918203 , Theorem 5.1) For any subset $Z\subset X$ , then

\overline{Ch}_{\Sigma_{m}^{+}\times Z}(F)=\log m+\overline{Ch}_{Z}(G).

Remark 3.

If $Z=X,$ the authors of MR3918203 proved that $h(G)=\overline{Ch}_{X}(G)$ . Hence, we have

\overline{Ch}_{\Sigma_{m}^{+}\times X}(F)=\log m+\overline{Ch}_{X}(G).

Since $h(F)=h_{\Sigma_{m}^{+}\times X}(F)=\overline{Ch}_{\Sigma_{m}^{+}\times X}(F)$ , then Theorem 4 can be restated as

h_{\Sigma_{m}^{+}\times X}(F)=\log m+h(G).

3.3 Stationary measure

Let $\mathbf{p}:=(p_{0},\cdots,p_{m-1})$ be a probability vector with non-zero entries (i.e., $p_{j}>0$ for each $j$ and $\sum_{j=0}^{m-1}p_{j}=1$ ). The Bernoulli measure $\mathbb{P}$ on $\Sigma_{m}^{+}$ generated by the probability vector $\mathbf{p}$ is $\sigma$ -invariant and ergodic. Given a point $x\in X$ and measurable set $A\subseteq X$ , the transition probabilities are defined by the formula

\mathcal{P}(x,A)=\int\chi_{A}\left(f_{j}(x)\right)\mathbf{p}(\mathrm{d}j),

where $\chi_{A}$ denotes the indicator map corresponding with the set $A$ . Let $\mathcal{M}(X)$ denote the set of all probability measures on $X$ . For every probability measure $\mu\in\mathcal{M}(X)$ , the adjoint operator $\mathcal{P}^{*}$ is defined by the following way,

\displaystyle\mathcal{P}^{*}\mu(A)=\int\mathcal{P}\left(x,{A}\right)d\mu(x)=\int\int\chi_{A}\left(f_{j}(x)\right)\mathbf{p}(\mathrm{d}j)\mu(\mathrm{d}x)=\sum_{j=0}^{m-1}p_{j}\mu\left(f_{j}^{-1}A\right).

A Borel probability measure $\mu\in\mathcal{M}(X)$ is said to be $\mathbb{P}$ -stationary if, $\mathcal{P}^{*}\mu=\mu$ .

As $X$ is a compact metric space, the set of $\mathbb{P}$ -stationary probability measures is a nonempty compact convex set with respect to the weak^∗ topology for every $\mathbb{P}$ . Its extreme points are called $\mathbb{P}$ -ergodic. For more information, see MR884892 . When convenient, we will use the following criterium :

Proposition 6.

(MR884892 , Lemma I.2.3) Let $\mathbb{P}$ be a Bernoulli measure on $\Sigma_{m}^{+}$ , and $\mu$ be a probability measure on $X$ , then

(1)

$\mu$ is $\mathbb{P}$ -stationary if and only if the product probability measure $\mathbb{P}\times\mu$ is $F$ -invariant.
(2)

$\mu$ is $\mathbb{P}$ -stationary and ergodic if and only if the product probability measure $\mathbb{P}\times\mu$ is $F$ -invariant and ergodic.

4 Periodic-like recurrence and $\mathbf{g}$ -almost product property of free semigroup actions

In this section, we introduce the new concept of $\mathbf{g}$ -almost product property of free semigroup actions, and some concepts of transitive points, quasiregular points, upper recurrent points and Banach upper recurrent points with respect to a certain orbit of free semigroup actions. We obtain that the $\mathbf{g}$ -almost product property is weaker than the specification property under free semigroup actions. The results in this section are inspired by MR2322186 . Throughout this section we assume that $X$ is a compact metric space, $G$ is the free semigroup generated by $m$ generators $f_{0},\cdots,f_{m-1}$ which are continuous maps on $X$ and $F$ is the skew product map corresponding to the maps $f_{0},\cdots,f_{m-1}$ .

Let us introduce the definitions of recurrence for free semigroup actions.

Definition 1.

Given $\iota=(i_{0},i_{1},\cdots)\in\Sigma_{m}^{+}$ , a point $x\in X$ is called a transitive point with respect to $\iota$ of free semigroup action $G$ if the orbit of $x$ under $\iota$ ,

orb(x,\iota,G):=\{x,f_{i_{0}}(x),f_{i_{1}i_{0}}(x),\cdots\}

is dense in $X$ .

Definition 2.

Given $\iota=(i_{0},i_{1},\cdots)\in\Sigma_{m}^{+}$ , a point $x\in X$ is called a quasiregular point with respect to $\iota$ of free semigroup action $G$ if a sequence

\frac{1}{n}\sum_{j=0}^{n-1}\delta_{F^{j}(\iota,x)}

converges in the weak^∗ topology.

Denote by $\mathrm{Tran}(\iota,G)$ and $\mathrm{QR}(\iota,G)$ the sets of the transitive points and the quasiregular points with respect to $\iota$ of free semigroup action, respectively. We write $\mathrm{Tran}(G)$ and $\mathrm{QR}(G)$ for the union of $\mathrm{Tran}(\iota,G)$ and $\mathrm{QR}(\iota,G)$ for all $\iota$ , respectively.

Let $U\subset X$ be a nonempty open set and $x\in X$ , $\iota=(i_{0},i_{1},\cdots)\in\Sigma_{m}^{+}$ , the set of visiting time with respect to $\iota$ is defined by

N_{\iota}(x,U):=\left\{n\in\mathbb{N}:f_{i_{n-1}\cdots i_{0}}(x)\in U\right\}.

Definition 3.

Given $\iota=(i_{0},i_{1},\cdots)\in\Sigma_{m}^{+}$ , a point $x\in X$ is called a upper recurrent point with respect to $\iota$ of free semigroup action $G$ if for any $\varepsilon>0$ , the set of visiting time $N_{\iota}\left(x,B(x,\varepsilon)\right)$ has a positive upper density.

Definition 4.

Given $\iota=(i_{0},i_{1},\cdots)\in\Sigma_{m}^{+}$ , a point $x\in X$ is called a Banach upper recurrent point with respect to $\iota$ of free semigroup action $G$ if for any $\varepsilon>0$ , the set of visiting time $N_{\iota}\left(x,B(x,\varepsilon)\right)$ has a positive Banach upper density.

Denote by $\mathrm{QW}(\iota,G)$ and $\mathrm{BR}(\iota,G)$ the sets of the upper recurrent points and the Banach upper recurrent points with respect to $\iota$ of free semigroup action $G$ , respectively. Let

\mathrm{QW}(G):=\bigcup_{\iota\in\Sigma_{m}^{+}}\mathrm{QW}(\iota,G),\,\mathrm{BR}(G):=\bigcup_{\iota\in\Sigma_{m}^{+}}\mathrm{BR}(\iota,G).

Let us call $\mathrm{QW}(G)$ and $\mathrm{BR}(G)$ the sets of the upper recurrent points and the Banach upper recurrent points of free semigroup action, respectively. It is easy to check that $\mathrm{QW}(G)$ coincides with the set of the quasi-weakly almost periodic points of free semigroup action defined by Zhu and Ma MR4200965 . Clearly,

\mathrm{QW}(\iota,G)\subseteq\mathrm{BR}(\iota,G).

The notion of specification, introduced by Bowen MR282372 , says that one can always find a single orbit to interpolate between different pieces of orbits. In the case of $\beta$ -shifts it is known that the specification property holds for a set of $\beta$ of Lebesgue measure zero (see MR1452189 ). In MR2322186 , the authors studied a new condition, called $\mathbf{g}$ -almost product product property, which is weaker that specification property, and proved the $\mathbf{g}$ -almost product product property always holds for $\beta$ -shifts.

Next we introduce the concept of $\mathbf{g}$ -almost product property of free semigroup actions:

Definition 5.

Let $\mathbf{g}:\mathbb{N}\rightarrow\mathbb{N}$ be a given nondecreasing unbounded map with the properties

\mathbf{g}(n)<n\quad\text{ and }\quad\lim_{n\rightarrow\infty}\frac{\mathbf{g}(n)}{n}=0.

The function $\mathbf{g}$ is called blowup function.

Fixed $\varepsilon>0$ , $w\in F_{m}^{+}$ and $x\in X$ , define the $\mathbf{g}$ -blowup of $B_{w}(x,\varepsilon)$ as the closed set

B_{w}(\mathbf{g}\mathchar 24635\relax\;x,\varepsilon):=\Big{\{}y\in X:\sharp\left\{w^{\prime}\leq\overline{w}:d\left(f_{w^{\prime}}(x),f_{w^{\prime}}(y)\right)>\varepsilon\right\}<\mathbf{g}\left(\lvert w\rvert+1\right)\Big{\}}.

Definition 6.

We say $G$ satisfies the $\mathbf{g}$ -almost product property with the blowup function $\mathbf{g}$ , if there exists a nonincreasing function $\mathfrak{m}:\mathbb{R}^{+}\to\mathbb{N}$ , such that for $k\geq 2$ , any $k$ points $x_{1},\cdots,x_{k}\in X$ , any positive $\varepsilon_{1},\cdots,\varepsilon_{k}$ , and any words $w_{(\varepsilon_{1})},\cdots,w_{(\varepsilon_{k})}\in F_{m}^{+}$ with $\lvert w_{(\varepsilon_{1})}\rvert\geq\mathfrak{m}(\varepsilon_{1}),\cdots,\lvert w_{(\varepsilon_{k})}\rvert\geq\mathfrak{m}(\varepsilon_{k})$ ,

B_{w_{(\varepsilon_{1})}}(\mathbf{g}\mathchar 24635\relax\;x_{1},\varepsilon_{1})\cap\left(\bigcap_{j=2}^{k}f^{-1}_{\overline{w_{(\varepsilon_{j-1})}}\cdots\overline{w_{(\varepsilon_{1})}}}B_{w_{(\varepsilon_{j})}}\left(\mathbf{g}\mathchar 24635\relax\;x_{j},\varepsilon_{j}\right)\right)\neq\emptyset.

Under $\mathbf{g}$ -almost product property, the topological entropy of periodic-like recurrent sets has been studied in MR3963890 , but the topological entropy of such sets has not been studied in dynamical systems of free semigroup actions. In this paper, we focus on the topological entropy of similar sets of free semigroup actions and obtain more extensive results. Therefore, it is important and necessary to introduce the $\mathbf{g}$ -almost product property of free semigroup actions.

If $m=1$ , the $\mathbf{g}$ -almost product property of free semigroup actions coincides with the definition introduced by Pfister and Sullivan MR2322186 ; MR2109476 .

The next proposition asserts the relationship between specification property and $\mathbf{g}$ -almost product property of free semigroup actions.

Proposition 7.

Let $\mathbf{g}$ be any blowup function and $G$ satisfies the specification property. Then it has the $\mathbf{g}$ -almost product property.

Proof: This proof extends the method of Proposition 2.1 in MR2322186 to the free semigroup actions, but we provide the complete proof for the reader’s convenience.

Let $\mathfrak{p}(\varepsilon)$ be the positive integer in the definition of specification property of $G$ (see Sec. 3.2) for $\varepsilon>0$ . It is no restriction to suppose that the function $\mathfrak{p}(\varepsilon)$ is nonincreasing. Let $\left\{x_{1},\cdots,x_{k}\right\}$ and $\left\{\varepsilon_{1},\cdots,\varepsilon_{k}\right\}$ be given. Let $\delta_{r}:=2^{-r},r\in\mathbb{N}$ . Next, we may define a nonincreasing function $\mathfrak{m}:\mathbb{R}^{+}\to\mathbb{N}$ as follows:

\mathfrak{m}(\varepsilon):=\widetilde{\mathfrak{m}}(2\delta_{r}),

where $r=\min\{i:2\delta_{i}\leq\varepsilon\}$ and $\widetilde{\mathfrak{m}}(2\delta_{r}):=\min\left\{m:\mathbf{g}(m)\geq 2\mathfrak{p}(\delta_{r})\right\}$ .

It is sufficient to prove the statement for $\varepsilon_{j}$ of the form $2\delta_{r_{j}},j=1,\cdots,k$ , where, as above $r_{j}=\min\{i:2\delta_{i}\leq\varepsilon_{j}\}$ . Precisely, if $\varepsilon_{j}$ is not of that form, we change it into $2\delta_{r_{j}}$ . From now on we assume that, for all $j$ , $\varepsilon_{j}$ is of the form $2\delta_{r_{j}}$ . Let $w_{(\varepsilon_{1})},\cdots,w_{(\varepsilon_{k})}$ be the words with the length not less than $\mathfrak{m}(\varepsilon_{1}),\cdots,\mathfrak{m}(\varepsilon_{k})$ , respectively. Let $n_{1},\cdots,n_{k}$ denote the length of $w_{(\varepsilon_{1})},\cdots,w_{(\varepsilon_{k})}$ , respectively.

We prove the proposition by an iterative construction. Let $\Delta\left(x_{j}\right):=\varepsilon_{j}/2=\delta_{r_{j}}$ , $w(x_{j}):=w_{(\varepsilon_{j})}$ , $p(x_{j}):=\mathfrak{p}\left(\Delta(x_{j})\right)$ and $n(x_{j}):=n_{j}$ . The sequence $\{x_{1},\cdots,x_{k}\}$ is considered as an ordered sequence; its elements are called original points. The possible values of $\Delta\left(x_{j}\right)$ are rewritten $\Delta_{1}>\Delta_{2}>\cdots>\Delta_{q}$ . A level- $i$ point is defined by an original point $x_{j}$ such that $\Delta\left(x_{j}\right)=\Delta_{i}.$

At step 1 we consider the level-1 points labeled by

S_{1}:=\left\{j\in[1,k]:\Delta\left(x_{j}\right)=\Delta_{1}\right\}.

If $S_{1}=[1,k]$ , then by the specification property there exists $y$ such that

d\left(f_{\overline{w{(x_{1})}\lvert_{[0,i]}}}(x_{1}),f_{\overline{w{(x_{1})}\lvert_{[0,i]}}}(y)\right)\leq\Delta_{1},\quad i=p(x_{1}),\cdots,n(x_{1})-p(x_{1})-1,

d\big{(}f_{\overline{w{(x_{j})}\lvert_{[0,i]}}}(x_{j}),f_{\overline{w{(x_{j})}\lvert_{[0,i]}}\,\overline{w{(x_{j-1})}}\cdots\overline{w{(x_{1})}}}(y)\big{)}\leq\Delta_{1},i=p(x_{j}),\cdots,n(x_{j})-p(x_{j})-1,

where $j=2,\cdots,k$ , which proves this case by the definition of the function $\mathfrak{m}$ . If $S_{1}\neq[1,k]$ , then we decompose it into maximal subsets of consecutive points, called components. (The components are defined with respect to the whole sequence.) Let $J$ be a component, say $[r,s]$ with $r<s$ . By the specification property there exists $y$ such that

d\left(f_{\overline{w{(x_{r})}\lvert_{[0,i]}}}(x_{r}),f_{\overline{w{(x_{r})}\lvert_{[0,i]}}}(y)\right)\leq\Delta_{1},\quad i=p(x_{r}),\cdots,n(x_{r})-p(x_{r})-1,

d\big{(}f_{\overline{w{(x_{j})}\lvert_{[0,i]}}}(x_{j}),f_{\overline{w{(x_{j})}\lvert_{[0,i]}}\,\overline{w{(x_{j-1})}}\cdots\overline{w{(x_{r})}}}(y)\big{)}\leq\Delta_{1},i=p(x_{j}),\cdots,n(x_{j})-p(x_{j})-1,

where $j=r+1,\cdots,s$ . Hence,

y\in B_{w{(x_{r})}}(\mathbf{g}\mathchar 24635\relax\;x_{r},\varepsilon_{r})\cap\left(\bigcap_{j=r+1}^{s}f^{-1}_{\overline{w{(x_{j-1})}}\cdots\overline{w{(x_{r})}}}B_{w{(x_{j})}}\left(\mathbf{g}\mathchar 24635\relax\;x_{j},\varepsilon_{j}\right)\right).

We replace the sequence $\left\{x_{1},\cdots,x_{k}\right\}$ by the (ordered) sequence

\left\{x_{1},\cdots,x_{r-1},y,x_{s+1}\cdots,x_{k}\right\}

and set, for the concatenated point $y$ , let

\Delta(y):=\Delta_{1},\,p(y):=\mathfrak{p}(\Delta(y)),\,n(y):=n(x_{r})+\cdots+n(x_{s}),\,w(y):=w(x_{r})\cdots w(x_{s}).

We do this operation for all components which are not singletons. After these operations we have a new (ordered) sequence $\left\{z_{1},\cdots,z_{k_{1}}\right\},k_{1}\leq k$ , where the point $z_{i}$ is either a point of the original sequence, or a concatenated point. This ends the construction at step 1.

Let

S_{2}:=\left\{j\in\left[1,k_{1}\right]:\Delta\left(z_{j}\right)\geq\Delta_{2}\right\}.

We decompose this set into components. Let $[r,s]$ be a component which is not a singleton $(r<s)$ . We replace that component by a single concatenated point $y$ such that if $z_{r}$ is concatenated point of $S_{1}$ ,

d\left(f_{\overline{w{(z_{r})}\lvert_{[0,i]}}}(z_{r}),f_{\overline{w{(z_{r})}\lvert_{[0,i]}}}(y)\right)\leq\Delta_{2},\quad i=0,\cdots,n(z_{r})-1,

otherwise,

d\left(f_{\overline{w{(z_{r})}\lvert_{[0,i]}}}(z_{r}),f_{\overline{w{(z_{r})}\lvert_{[0,i]}}}(y)\right)\leq\Delta_{2},\quad i=p(z_{r}),\cdots,n(z_{r})-p(z_{r})-1\mathchar 24635\relax\;

and, for $j=r+1,\cdots,s$ , if $z_{j}$ is concatenated point of $S_{1}$ ,

d\left(f_{\overline{w{(z_{j})}\lvert_{[0,i]}}}(z_{j}),f_{\overline{w{(z_{j})}\lvert_{[0,i]}}\,\overline{w(z_{j-1})}\cdots\overline{w(z_{r})}}(y)\right)\leq\Delta_{2},\quad i=0,\cdots,n(z_{j})-1,

otherwise,

d\big{(}f_{\overline{w{(z_{j})}\lvert_{[0,i]}}}(z_{j}),f_{\overline{w{(z_{j})}\lvert_{[0,i]}}\,\overline{w(z_{j-1})}\cdots\overline{w(z_{r})}}(y)\big{)}\leq\Delta_{2},i=p(z_{j}),\cdots,n(z_{j})-p(z_{j})-1.

Existence of such a $y$ is a consequence of the specification property. We set

\Delta(y):=\Delta_{2},\,p(y):=\mathfrak{p}(\Delta(y)),\,n(y):=n(z_{r})+\cdots+n(z_{s}),\,w(y):=w(z_{r})\cdots w(z_{s}).

The construction of $y$ involves consecutive points of the original sequence (via the concatenated points), say points $x_{j},j\in[u,t].$ Since $\delta_{j}=\sum_{i>j}\delta_{i}$ ,

y\in B_{w{(x_{u})}}(\mathbf{g}\mathchar 24635\relax\;x_{u},\varepsilon_{u})\cap\left(\bigcap_{j=u+1}^{t}f^{-1}_{\overline{w{(x_{j-1})}}\cdots\overline{w{(x_{u})}}}B_{w{(x_{j})}}\left(\mathbf{g}\mathchar 24635\relax\;x_{j},\varepsilon_{j}\right)\right).

We do these operations for all components of $S_{2}$ , which are not singletons. We get a new ordered sequence, still denoted by $\left\{z_{1},\cdots,z_{k_{2}}\right\}$ . This ends the construction at level 2.

The construction at level 3 is similar to the construction at level 2, using

S_{3}:=\left\{j\in\left[1,k_{2}\right]:\Delta\left(z_{j}\right)\geq\Delta_{3}\right\}.

Once step $q$ is completed, we have a single concatenated point $y$ such that

d\left(f_{\overline{w{(x_{1})}\lvert_{[0,i]}}}(x_{1}),f_{\overline{w{(x_{1})}\lvert_{[0,i]}}}(y)\right)\leq\varepsilon_{1},\quad i=p(x_{1}),\cdots,n(x_{1})-p(x_{1})-1,

d\big{(}f_{\overline{w{(x_{j})}\lvert_{[0,i]}}}(x_{j}),f_{\overline{w{(x_{j})}\lvert_{[0,i]}}\,\overline{w{(x_{j-1})}}\cdots\overline{w{(x_{1})}}}(y)\big{)}\leq\varepsilon_{j},i=p(x_{j}),\cdots,n(x_{j})-p(x_{j})-1,

where $j=2,\cdots,k.$ Observe that, for all $j$ , $\mathbf{g}(n_{j})\geq 2p(x_{j})$ . As a consequence,

y\in B_{w_{(\varepsilon_{1})}}(\mathbf{g}\mathchar 24635\relax\;x_{1},\varepsilon_{1})\cap\left(\bigcap_{j=2}^{k}f^{-1}_{\overline{w_{(\varepsilon_{j-1})}}\cdots\overline{w_{(\varepsilon_{1})}}}B_{w_{(\varepsilon_{j})}}\left(\mathbf{g}\mathchar 24635\relax\;x_{j},\varepsilon_{j}\right)\right).

(2)

Remark 4.

If $m=1$ , it generates the Proposition 2.1 in MR2322186 .

In MR3503951 , Rodrigues and Varandas proved that if $X$ is a compact Riemannian manifold, and $G$ is free semigroup generated by ${f_{0},\cdots,f_{m-1}}$ which are all expanding maps, then $G$ satisfies the specification property, furthermore, it has the $\mathbf{g}$ -almost product property by Proposition 7.

Next, we describe an example to help us interpret the $\mathbf{g}$ -almost product property of free semigroup actions.

Example 1.

Let $M$ be a compact Riemannian manifold and $G$ the free semigroup generated by $f_{0},\cdots,f_{m-1}$ on $M$ which are $C^{1}$ -local diffeomorphisms such that for any $j=0,\cdots,m-1$ , $\|Df_{j}(x)v\|\geq\lambda_{j}\|v\|$ for all $x\in M$ and all $v\in T_{x}M$ , where $\lambda_{j}$ is a constant larger than 1. It follows from MR4200965 and Theorem 16 of MR3503951 that $G$ satisfies positively expansive and specification property. Let $\mathbf{g}$ be a blowup function. Consider the nonincreasing function $\mathbf{m}:\mathbb{R}^{+}\to\mathbb{N}$ given by Proposition 7. For $k\geq 2$ , let $x_{1},\cdots,x_{k}\in X$ , $\varepsilon_{1},\cdots,\varepsilon_{k}>0$ , and $w_{(\varepsilon_{1})},\cdots,w_{(\varepsilon_{k})}\in F_{m}^{+}$ with $\lvert w_{(\varepsilon_{1})}\rvert\geq\mathfrak{m}(\varepsilon_{1}),\cdots,\lvert w_{(\varepsilon_{k})}\rvert\geq\mathfrak{m}(\varepsilon_{k})$ be given. By the formula (2) in Proposition 7, we have that

B_{w_{(\varepsilon_{1})}}(\mathbf{g}\mathchar 24635\relax\;x_{1},\varepsilon_{1})\cap\left(\bigcap_{j=2}^{k}f^{-1}_{\overline{w_{(\varepsilon_{j-1})}}\cdots\overline{w_{(\varepsilon_{1})}}}B_{w_{(\varepsilon_{j})}}\left(\mathbf{g}\mathchar 24635\relax\;x_{j},\varepsilon_{j}\right)\right)\neq\emptyset.

Hence $G$ satisfies the $\mathbf{g}$ -almost product property for any blowup function $\mathbf{g}$ .

Proposition 8.

If $G$ satisfies the $\mathbf{g}$ -almost product property, then the skew product map $F$ corresponding to the maps $f_{0},\cdots,f_{m-1}$ has $2\mathbf{g}$ -almost product property.

Proof: The shift map $\sigma:\Sigma_{m}^{+}\to\Sigma_{m}^{+}$ has specification property (see MR0457675 ). Let $\mathfrak{p}(\varepsilon)$ be the positive integer in the definition of specification property of $\sigma$ for $\varepsilon>0$ . Let $\mathfrak{m}_{G}$ denote the function in the $\mathbf{g}$ -almost product property for $G$ . Let $\delta_{r}:=2^{-r},\,r\in\mathbb{N}$ . It is no restriction to suppose that $\mathfrak{p}(\delta_{r})>r$ and the function $\mathfrak{p}(\delta_{r})$ is increasing as $r$ increases. Next, we may define a nonincreasing function $\mathfrak{m}_{F}:\mathbb{R}^{+}\to\mathbb{N}$ as follows:

\mathfrak{m}_{F}(\varepsilon):=\widetilde{\mathfrak{m}}(2\delta_{r})

where $r=\min\{i:2\delta_{i}\leq\varepsilon\}$ and $\widetilde{\mathfrak{m}}(2\delta_{r}):=\min\left\{n:\mathbf{g}(n)\geq 2\mathfrak{p}\left(\delta_{r}\right)\,\text{and }n\geq\mathfrak{m}_{G}\left(\delta_{r}\right)\right\}$ .

For $k\geq 2$ , let $(\iota^{(1)},x_{1}),\cdots,(\iota^{(k)},x_{k})\in\Sigma_{m}^{+}\times X$ and $\varepsilon_{1},\cdots,\varepsilon_{k}>0$ be given. It is sufficient to prove the statement for $\varepsilon_{j}$ of the form $2\delta_{r_{j}},j=1,\cdots,k$ , where, as above $r_{j}=\min\{i:2\delta_{i}\leq\varepsilon_{j}\}$ . Precisely, if $\varepsilon_{j}$ is not of that form, we change it into $2\delta_{r_{j}}$ . From now on we assume that, for all $j,\varepsilon_{j}$ is of the form $2\delta_{r_{j}}$ . For convenience, write $p_{j}:=\mathfrak{p}(\delta_{r_{j}})$ for all $j=1,\cdots,k$ .

For any $n_{1}\geq\mathfrak{m}_{F}(\varepsilon_{1}),\cdots,n_{k}\geq\mathfrak{m}_{F}(\varepsilon_{k})$ , let $\iota\in\Sigma_{m}^{+}$ satisfy the following condition:

\iota\lvert_{[n_{1}+n_{2}+\cdots+n_{j-1},\,n_{1}+n_{2}+\cdots+n_{j}-1]}=\iota^{(j)}\lvert_{[0,\,n_{j}-1]},\quad j=1,\cdots,k,

where $n_{0}=0$ . We now apply the argument $p_{j}>r_{j}$ for all $j$ to obtain

d^{\prime}\left(\sigma^{n_{1}+n_{2}+\cdots+n_{j-1}+r}(\iota),\sigma^{r}(\iota^{(j)})\right)\leq\varepsilon_{j},\quad r=p_{j},p_{j}+1,\cdots,n_{j}-p_{j}-1.

(3)

Let

	$\displaystyle w_{(\varepsilon_{1})}:$	$\displaystyle=\iota\lvert_{[0,\,n_{1}-1]},$
	$\displaystyle w_{(\varepsilon_{2})}:$	$\displaystyle=\iota\lvert_{[n_{1},\,n_{1}+n_{2}-1]},$
		$\displaystyle\vdots$
	$\displaystyle w_{(\varepsilon_{k})}:$	$\displaystyle=\iota\lvert_{[n_{1}+\cdots+n_{k-1},\,n_{1}+\cdots+n_{k}-1]}.$

Observe that $\lvert w_{(\varepsilon_{j})}\rvert\geq\mathfrak{m}_{G}(\varepsilon_{j})$ for each $j=1,\cdots,k$ . The $\mathbf{g}$ -almost product property of $G$ implies that

B_{w_{(\varepsilon_{1})}}(\mathbf{g}\mathchar 24635\relax\;x_{1},\varepsilon_{1})\cap\left(\bigcap_{j=2}^{k}f^{-1}_{\overline{w_{(\varepsilon_{j-1})}}\cdots\overline{w_{(\varepsilon_{1})}}}B_{w_{(\varepsilon_{j})}}\left(\mathbf{g}\mathchar 24635\relax\;x_{j},\varepsilon_{j}\right)\right)\neq\emptyset.

Take an element $x$ from the left set. For $j=1,\cdots,k$ , define

\Gamma_{j}:=\bigg{\{}p_{j}\leq r<n_{j}-p_{j}:d\big{(}f_{\overline{w_{(\varepsilon_{j})}\lvert_{[0,r]}}\overline{w_{(\varepsilon_{j-1})}}\cdots\overline{w_{(\varepsilon_{1})}}}(x),f_{\overline{w_{(\varepsilon_{j})}\lvert_{[0,r]}}}(x_{j})\big{)}\leq\varepsilon_{j}\bigg{\}}.

To be more precise, for any $r\in\Gamma_{j}$ ,

d\left(f_{\overline{\iota\lvert_{[0,\,n_{1}+n_{2}+\cdots+n_{j-1}+r]}}}(x),f_{\overline{\iota^{(j)}\lvert_{[0,r]}}}(x_{j})\right)\leq\varepsilon_{j}.

(4)

Accordingly,

		$\displaystyle D\Big{(}F^{n_{1}+n_{2}+\cdots+n_{j-1}+r}(\iota,x),F^{r}(\iota^{(j)},x_{j})\Big{)}$
	$\displaystyle=$	$\displaystyle D\Big{(}\big{(}\sigma^{n_{1}+n_{2}+\cdots+n_{j-1}+r}(\iota),\,f_{\overline{\iota\lvert_{[0,\,M_{j-1}+r]}}}(x)\big{)},\big{(}\sigma^{r}(\iota^{(j)}),f_{\overline{\iota^{(j)}\lvert_{[0,r]}}}(x_{j})\big{)}\Big{)}$
	$\displaystyle=$	$\displaystyle\max\Big{\{}d^{\prime}\big{(}\sigma^{n_{1}+n_{2}+\cdots+n_{j-1}+r}(\iota),\sigma^{r}(\iota_{j})\big{)},d\big{(}f_{\overline{\iota\lvert_{[0,\,n_{1}+n_{2}+\cdots+n_{j-1}+r]}}}(x),f_{\overline{\iota^{(j)}\lvert_{[0,r]}}}(x_{j})\big{)}\Big{\}}$
	$\displaystyle\leq$	$\displaystyle\varepsilon_{j},\quad\text{by these inequations (\ref{3.1}) and (\ref{3.2})}.$

Observe that $\sharp\left(\Gamma_{j}\right)\geq n_{j}-2p_{j}-\mathbf{g}(n_{j})\geq n_{j}-2\mathbf{g}(n_{j})$ . As a consequence,

(\iota,x)\in B_{n_{1}}\big{(}2\mathbf{g}\mathchar 24635\relax\;(\iota^{(1)},x_{1}),\varepsilon_{1}\big{)}\cap\bigg{(}\bigcap_{j=2}^{k}F^{-(n_{1}+n_{2}+\cdots+n_{j-1})}B_{n_{j}}\big{(}2\mathbf{g}\mathchar 24635\relax\;(\iota^{(j)},x_{j}),\varepsilon_{j}\big{)}\bigg{)}.

This proves that $F$ has 2 $\mathbf{g}$ -almost product property.

5 General (ir)regularity

In this section, we study the more general irregular and regular sets of free semigroup actions and calculate the upper capacity topological entropy of the irregular and regular sets of free semigroup actions. The results in this section are inspired by MR3963890 ; MR4200965 . Throughout this section we assume that $X$ is a compact metric space, $G$ is the free semigroup generated by $m$ generators $f_{0},\cdots,f_{m-1}$ which are continuous maps on $X$ and $F$ is the skew product map corresponding to the maps $f_{0},\cdots,f_{m-1}$ .

Let

R_{\alpha}(G):=\bigcup_{\iota\in\Sigma_{m}^{+}}R_{\alpha}(\iota,G),\quad I_{\alpha}(G):=\bigcup_{\iota\in\Sigma_{m}^{+}}I_{\alpha}(\iota,G).

Let us call $R_{\alpha}(G)$ and $I_{\alpha}(G)$ the $\alpha$ -regular set and $\alpha$ -irregular set of free semigroup actions, respectively.

Theorem 9.

Let $(X,d)$ be a compact metric space and $G$ the free semigroup action on $X$ generated by $f_{0},\cdots,f_{m-1}$ . Let $\alpha:\mathcal{M}(\Sigma_{m}^{+}\times X,F)\to\mathbb{R}$ be a continuous function. Then,

\overline{Ch}_{R_{\alpha}(G)}(G)=\overline{Ch}_{X}(G)=h(G).

Proof: Consider a set

R_{\alpha}(F):=\left\{(\iota,x)\in\Sigma_{m}^{+}\times X:\inf_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)=\sup_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)\right\}.

It follows from Theorem 4.1(4) of MR3963890 that

h_{R_{\alpha}(F)}(F)=h_{\Sigma_{m}^{+}\times X}(F).

(5)

For $(\iota,x)\in R_{\alpha}(F)$ , it is immediate that $x\in R_{\alpha}(\iota,G)$ , then $x\in R_{\alpha}(G)$ . This implies that

R_{\alpha}(F)\subseteq\Sigma_{m}^{+}\times R_{\alpha}(G)\subseteq\Sigma_{m}^{+}\times X.

In this way we conclude from the formula (5) that

h_{\Sigma_{m}^{+}\times X}(F)=h_{R_{\alpha}(F)}(F)\leq\overline{Ch}_{\Sigma_{m}^{+}\times R_{\alpha}(G)}(F).

(6)

From Theorem 4, we obtain that

\log m+h(G)=h_{\Sigma_{m}^{+}\times X}(F).

(7)

By Theorem 5, one has

\overline{Ch}_{\Sigma_{m}^{+}\times R_{\alpha}(G)}(F)=\log m+\overline{Ch}_{R_{\alpha}(G)}(G).

(8)

Combining the equations (6), (7) and (8), we get that

	$\displaystyle\log m+h(G)$	$\displaystyle=h_{R_{\alpha}(F)}(F)$
		$\displaystyle\leq\overline{Ch}_{\Sigma_{m}^{+}\times R_{\alpha}(G)}(F)$
		$\displaystyle=\log m+\overline{Ch}_{R_{\alpha}(G)}(G).$

Hence,

\overline{Ch}_{X}(G)=h(G)\leq\overline{Ch}_{R_{\alpha}(G)}(G).

Obviously,

\overline{Ch}_{R_{\alpha}(G)}(G)\leq\overline{Ch}_{X}(G)=h(G).

Consequently,

\overline{Ch}_{R_{\alpha}(G)}(G)=\overline{Ch}_{X}(G)=h(G).

Theorem 10.

Suppose that $G$ has the $\mathbf{g}$ -almost product property, there exists a $\mathbb{P}$ -stationary measure with full support where $\mathbb{P}$ is a Bernoulli measure on $\Sigma_{m}^{+}$ . Let $\alpha:\mathcal{M}(\Sigma_{m}^{+}\times X,F)\to\mathbb{R}$ be a continuous function satisfying the condition A.3. If $\inf_{\nu\in\mathcal{M}(\Sigma_{m}^{+}\times X,F)}\alpha(\nu)<\sup_{\nu\in\mathcal{M}(\Sigma_{m}^{+}\times X,F)}\alpha(\nu)$ , then

\overline{Ch}_{I_{\alpha}(G)}(G)=\overline{Ch}_{E(I_{\alpha},\mathrm{Tran})}(G)=\overline{Ch}_{X}(G)=h(G),

where $E(I_{\alpha},\mathrm{Tran}):=\cup_{\iota\in\Sigma_{m}^{+}}\left(I_{\alpha}(\iota,G)\cap\mathrm{Tran}(\iota,G)\right)$ . In particular,

\overline{Ch}_{I_{\alpha}(G)}(G)=\overline{Ch}_{I_{\alpha}(G)\cap\mathrm{Tran}(G)}(G)=\overline{Ch}_{X}(G)=h(G).

Proof: Suppose $\mu$ is the $\mathbb{P}$ -stationary measure with full support. Then, Proposition 6 ensures that $\mathbb{P}\times\mu$ is an invariant measure under the skew product $F$ with support $\Sigma_{m}^{+}\times X$ . From Proposition 8, the skew product $F$ has 2 $\mathbf{g}$ -almost product property. Consider a set

I_{\alpha}(F):=\left\{(\iota,x)\in\Sigma_{m}^{+}\times X:\inf_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)<\sup_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)\right\}.

Hence, from Theorem 4.1 (2) of MR3963890 , one has

h_{I_{\alpha}(F)}(F)=h_{I_{\alpha}(F)\cap\mathrm{Tran}(F)}(F)=h_{\Sigma_{m}^{+}\times X}(F).

(9)

It is clear that if $(\iota,x)\in I_{\alpha}(F)\cap\mathrm{Tran}(F)$ , then $x\in I_{\alpha}(\iota,G)\cap\mathrm{Tran}(\iota,G)$ . Accordingly, $x\in E(I_{\alpha},\mathrm{Tran})$ . This yields that

I_{\alpha}(F)\cap\mathrm{Tran}(F)\subseteq\Sigma_{m}^{+}\times E(I_{\alpha},\mathrm{Tran})\subseteq\Sigma_{m}^{+}\times X.

In this way we conclude from the formula (9) that

h_{\Sigma_{m}^{+}\times X}(F)=h_{I_{\alpha}(F)\cap\mathrm{Tran}(F)}(F)\leq\overline{Ch}_{\Sigma_{m}^{+}\times E(I_{\alpha},\mathrm{Tran})}(F).

(10)

By Theorem 5, one has

\overline{Ch}_{\Sigma_{m}^{+}\times E(I_{\alpha},\mathrm{Tran})}(F)=\log m+\overline{Ch}_{E(I_{\alpha},\mathrm{Tran})}(G).

(11)

Combining the equations (7), (10) and (11), we get that

	$\displaystyle\log m+h(G)$	$\displaystyle=h_{I_{\alpha}(F)\cap\mathrm{Tran}(F)}(F)$
		$\displaystyle\leq\overline{Ch}_{\Sigma_{m}^{+}\times E(I_{\alpha},\mathrm{Tran})}(F)$
		$\displaystyle=\log m+\overline{Ch}_{E(I_{\alpha},\mathrm{Tran})}(G).$

Hence,

\overline{Ch}_{X}(G)=h(G)\leq\overline{Ch}_{E(I_{\alpha},\mathrm{Tran})}(G).

Obviously,

\overline{Ch}_{E(I_{\alpha},\mathrm{Tran})}(G)\leq\overline{Ch}_{X}(G)=h(G).

Consequently,

\overline{Ch}_{E(I_{\alpha},\mathrm{Tran})}(G)=\overline{Ch}_{X}(G)=h(G).

We may obtain from $E(I_{\alpha},\mathrm{Tran})\subseteq I_{\alpha}(G)$ that

\overline{Ch}_{I_{\alpha}(G)}(G)=\overline{Ch}_{E(I_{\alpha},\mathrm{Tran})}(G)=\overline{Ch}_{X}(G)=h(G).

It is easy to check that

	$\displaystyle I_{\alpha}(G)\cap\mathrm{Tran}(G)$	$\displaystyle=\left(\bigcup_{\iota\in\Sigma_{m}^{+}}I_{\alpha}(\iota,G)\right)\cap\left(\bigcup_{\iota^{\prime}\in\Sigma_{m}^{+}}\mathrm{Tran}(\iota^{\prime},G)\right)$
		$\displaystyle=\bigcup_{\iota,\iota^{\prime}\in\Sigma_{m}^{+}}\left(I_{\alpha}(\iota,G)\cap\mathrm{Tran}(\iota^{\prime},G)\right)$
		$\displaystyle\supseteq\bigcup_{\iota\in\Sigma_{m}^{+}}\left(I_{\alpha}(\iota,G)\cap\mathrm{Tran}(\iota,G)\right)=E(I_{\alpha},\mathrm{Tran}).$

Hence,

\overline{Ch}_{I_{\alpha}(G)}(G)=\overline{Ch}_{I_{\alpha}(G)\cap\mathrm{Tran}(G)}(G)=\overline{Ch}_{X}(G)=h(G).

Remark 5.

Both Theorem 9 and 10 are extension of Theorem 4.1 of MR3963890 .

For $\iota=(i_{0},i_{1},\cdots)\in\Sigma^{+}_{m}$ , consider a set

I_{\varphi}(\iota,G):=\left\{x\in X:\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi\left(f_{i_{j-1}\cdots i_{0}}(x)\right)\text{ does not exist}\right\}.

Let $R_{\varphi}(\iota,G):=X\backslash I_{\varphi}(\iota,G)$ , and

R_{\varphi}(G):=\bigcup_{\iota\in\Sigma_{m}^{+}}R_{\varphi}(\iota,G),\quad I_{\varphi}(G):=\bigcup_{\iota\in\Sigma_{m}^{+}}I_{\varphi}(\iota,G).

It is easy to find that $I_{\varphi}(G)$ coincides with the $\varphi$ -irregular set of free semigroup action defined by Zhu and Ma MR4200965 . For convenience, we call $R_{\varphi}(G)$ to be $\varphi$ -regular set of free semigroup action.

For a continuous function $\varphi:X\to\mathbb{R}$ , consider a function $\psi:\Sigma_{m}^{+}\times X\to\mathbb{R}$ such that for any $\iota=(i_{0},i_{1},\cdots)\in\Sigma_{m}^{+}$ , the map $\psi$ satisfies $\psi(\iota,x)=\varphi(x)$ , then $\psi$ is continuous. The continuous function $\alpha:\mathcal{M}(\Sigma_{m}^{+}\times X,F)\to\mathbb{R}$ is given by $\alpha(\nu)=\int\psi\mathrm{d}\nu$ . It is easy to check that the function $\alpha$ satisfies the conditions A.1, A.2 and A.3. It follows from the definition of the function $\psi$ that the limit

\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi\left(f_{i_{j-1}\cdots i_{0}}(x)\right)

exists if and only if

\inf_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)=\sup_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu).

Hence, $R_{\alpha}(\iota,G)=R_{\varphi}(\iota,G)$ . Analogously, $I_{\alpha}(\iota,G)=I_{\varphi}(\iota,G)$ .

Corollary 1.

Let $(X,d)$ be a compact metric space and $G$ the free semigroup action on $X$ generated by $f_{0},\cdots,f_{m-1}$ . Then the $\varphi$ -regular set of free semigroup action carries full upper capacity topological entropy, that is,

\overline{Ch}_{R_{\varphi}(G)}(G)=\overline{Ch}_{X}(G)=h(G).

If $m=1$ , then the above corollary coincides with the result of Theorem 4.2 that Tian proved in MR3436391 .

Corollary 2.

Suppose that $G$ has the $\mathbf{g}$ -almost product property, there exists a $\mathbb{P}$ -stationary measure with full support where $\mathbb{P}$ is a Bernoulli measure on $\Sigma_{m}^{+}$ . Let $\varphi:X\to\mathbb{R}$ be a continuous function. If $I_{\varphi}(\iota,G)$ is non-empty for some $\iota\in\Sigma_{m}^{+}$ , then

\overline{Ch}_{I_{\varphi}(G)}(G)=\overline{Ch}_{E(I_{\varphi},\mathrm{Tran})}(G)=\overline{Ch}_{X}(G)=h(G),

where ${E(I_{\varphi},\mathrm{Tran})}:=\cup_{\iota\in\Sigma_{m}^{+}}\left(I_{\varphi}(\iota,G)\cap\mathrm{Tran}(\iota,G)\right)$ . In particular,

\overline{Ch}_{I_{\varphi}(G)}(G)=\overline{Ch}_{I_{\varphi}(G)\cap\mathrm{Tran}(G)}(G)=\overline{Ch}_{X}(G)=h(G).

Remark 6.

The previous corollary generalizes Theorem 2 obtained by Zhu and Ma MR4200965 . Indeed, from Lemma 3.3 of MR4200965 , if the free semigroup action $G$ has specification property, then the skew product $F$ has specification property. This yields from MR646049 that there exists a $F$ -invariant probability measures on $\Sigma_{m}^{+}\times X$ with full support. On the other hand, from Proposition 7 we know that specification implies the $\mathbf{g}$ -almost product property for free semigroup actions. Therefore, the specification property of the free semigroup $G$ implies the hypothesis of Corollary 2, as we wanted to prove.

We provide an example that satisfies the assumptions of Theorem 1, 2 and 3.

Example 2.

Given $q\in\mathbb{N}$ , let $A_{j}\in\mathrm{GL}(q,\mathbb{Z})$ be the integer coefficients matrix whose the determinant is different from zero and eigenvalues have absolute value bigger than one, for $j=0,\cdots,m-1$ . Let $f_{A_{j}}:\mathbb{T}^{q}\to\mathbb{T}^{q}$ be the linear endomorphism of the torus induced by the matrix $A_{j}$ . Then the transformations $f_{A_{0}},f_{A_{1}},\cdots,f_{A_{m-1}}$ are all expanding (see Sec. 11.1 of MR3558990 for details). Let $G$ be the free semigroup action generated by $f_{A_{0}},f_{A_{1}},\cdots,f_{A_{m-1}}$ . It follows from MR3503951 that $G$ is positively expansive with $\mathbf{g}$ -almost product property.

Suppose that $F:\Sigma_{m}^{+}\times X\to\Sigma_{m}^{+}\times X$ is the skew product map corresponding to the maps $f_{A_{0}},f_{A_{1}},\cdots,f_{A_{m-1}}$ . From Section 4.2.5 of MR3558990 , we have that $f_{A_{j}}$ preserves the Lebesgue measure $\mu$ on $\mathbb{T}^{q}$ for all $j=0,\cdots,m-1$ . Hence the Lebesgue measure is stationary with respect to any Bernoulli measure, so the skew product $F$ is not uniquely ergodic. This may be seen as follows. Let $\mathbb{P}_{a}$ and $\mathbb{P}_{b}$ be two Bernoulli measures on $\Sigma_{m}^{+}$ generated by different probability vectors $a=(a_{0},a_{1},\cdots,a_{m-1})$ and $b=(b_{0},b_{1},\cdots,b_{m-1})$ , respectively. It follows from Proposition 6 that the different product measures both $\mathbb{P}_{a}\times\mu$ and $\mathbb{P}_{b}\times\mu$ are invariant under the skew product $F$ . This shows that $F$ is not uniquely ergodic. Hence, it satisfies the hypothesis of Theorem 1, 2 and 3, as we wanted to prove.

6 Proofs of the main results

In this section, we complete the proofs of Theorem 1, 2 and 3. Let $(X,f)$ be a dynamical system and $Z_{1},Z_{2},\cdots,Z_{k}\subseteq X(k\geq 2)$ be a collection of subsets of $X$ . Recall that $\left\{Z_{i}\right\}$ has full entropy gaps with respect to $Y\subseteq X$ if

h_{(Z_{i+1}\backslash Z_{i})\cap Y}(f)=h_{Y}(f)\text{ for all }1\leq i<k.

Next throughout this section we assume that $X$ is a compact metric space, $G$ is the free semigroup generated by $m$ generators $f_{0},\cdots,f_{m-1}$ which are continuous maps on $X$ , and $F$ is the skew product map corresponding to the maps $f_{0},\cdots,f_{m-1}$ .

From MR3963890 , Tian defined the recurrent level sets of the upper Banach recurrent points with respect to a single map. For the skew product map $F$ , let $\mathrm{BR}^{\#}(F):=\mathrm{BR}(F)\backslash\mathrm{QW}(F)$ ,

	$\displaystyle W(F)$	$\displaystyle:=\left\{(\iota,x)\in\Sigma_{m}^{+}\times X:S_{\mu}=C_{(\iota,x)}\text{ for every }\mu\in M_{(\iota,x)}(F)\right\},$
	$\displaystyle V(F)$	$\displaystyle:=\left\{(\iota,x)\in\Sigma_{m}^{+}\times X:\exists\mu\in M_{(\iota,x)}(F)\text{ such that }S_{\mu}=C_{(\iota,x)}\right\},$
	$\displaystyle S(F)$	$\displaystyle:=\left\{(\iota,x)\in\Sigma_{m}^{+}\times X:\cap_{\mu\in M_{(\iota,x)}(F)}S_{\mu}\neq\emptyset\right\}.$

More precisely, $\mathrm{BR}^{\#}(F)$ is divided into the following several levels with different asymptotic behaviour:

		$\displaystyle\mathrm{BR}_{1}(F):=\mathrm{BR}^{\#}(F)\cap W(F),$
		$\displaystyle\mathrm{BR}_{2}(F):=\mathrm{BR}^{\#}(F)\cap(V(F)\cap S(F)),$
		$\displaystyle\mathrm{BR}_{3}(F):=\mathrm{BR}^{\#}(F)\cap V(F),$
		$\displaystyle\mathrm{BR}_{4}(F):=\mathrm{BR}^{\#}(F)\cap(V(F)\cup S(F)),$
		$\displaystyle\mathrm{BR}_{5}(F):=\mathrm{BR}^{\#}(F).$

Then $\mathrm{BR}_{1}(F)\subseteq\mathrm{BR}_{2}(F)\subseteq\mathrm{BR}_{3}(F)\subseteq\mathrm{BR}_{4}(F)\subseteq\mathrm{BR}_{5}(F)$ .

Proof: [Proof of Theorem 1] Suppose $\mu$ is the $\mathbb{P}$ -stationary measure with full support. Then, Proposition 6 ensures that $\mathbb{P}\times\mu$ is an invariant measure under the skew product $F$ with support $\Sigma_{m}^{+}\times X$ . From Lemma 8, the skew product $F$ has 2 $\mathbf{g}$ -almost product property.

(1) Consider a set

I_{\alpha}(F):=\left\{(\iota,x)\in\Sigma_{m}^{+}\times X:\inf_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)<\sup_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)\right\}.

If $\alpha$ satisfies A.3 and $\mathrm{Int}(L_{\alpha})\neq\emptyset$ , it follows from Theorem 6.1(1) of MR3963890 that

\left\{\mathrm{QR}(F),\mathrm{BR}_{1}(F),\mathrm{BR}_{2}(F),\mathrm{BR}_{3}(F),\mathrm{BR}_{4}(F),\mathrm{BR}_{5}(F)\right\}

has full entropy gaps with respect to $I_{\alpha}(F)\cap\mathrm{Tran}(F)$ . Hence,

h_{I_{\alpha}(F)\cap\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{1}(F)\setminus\mathrm{QR}(F)\right)}(F)=h_{I_{\alpha}(F)\cap\mathrm{Tran}(F)}(F)=h_{\Sigma_{m}^{+}\times X}(F)

and

h_{I_{\alpha}(F)\cap\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{j}(F)\setminus\mathrm{BR}_{j-1}(F)\right)}(F)=h_{I_{\alpha}(F)\cap\mathrm{Tran}(F)}(F)=h_{\Sigma_{m}^{+}\times X}(F),

for $j=2,\cdots,5$ . From the Theorem 4, we get that

\log m+h(G)=h_{I_{\alpha}(F)\cap\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{1}(F)\setminus\mathrm{QR}(F)\right)}(F),

(12)

and

\log m+h(G)=h_{I_{\alpha}(F)\cap\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{j}(F)\setminus\mathrm{BR}_{j-1}\right)}(F)\quad\text{for }j=2,\cdots,5.

(13)

By the definitions of the sets, if $(\iota,x)\in I_{\alpha}(F)\cap\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{1}(F)\setminus\mathrm{QR}(F)\right)$ then

x\in I_{\alpha}(\iota,G)\cap\mathrm{Tran}(\iota,G)\cap\left(\mathrm{BR}_{1}(\iota,G)\setminus\mathrm{QR}(\iota,G)\right).

This shows that

I_{\alpha}(F)\cap\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{1}(F)\setminus\mathrm{QR}(F)\right)\subseteq\Sigma_{m}^{+}\times M_{1}(I_{\alpha},\mathrm{Tran})\subseteq\Sigma_{m}^{+}\times X,

(14)

where

M_{1}(I_{\alpha},\mathrm{Tran}):=\cup_{\iota\in\Sigma_{m}^{+}}\big{\{}I_{\alpha}(\iota,G)\cap\mathrm{Tran}(\iota,G)\cap\left(\mathrm{BR}_{1}(\iota,G)\setminus\mathrm{QR}(\iota,G)\right)\big{\}}.

It follows using the formula (14) and Theorem 5 that

$\displaystyle h_{\Sigma_{m}^{+}\times X}(F)$	$\displaystyle=h_{I_{\alpha}(F)\cap\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{1}(F)\setminus\mathrm{QR}(F)\right)}(F)$	(15)
	$\displaystyle\leq\overline{Ch}_{\Sigma_{m}^{+}\times M_{1}(I_{\alpha},\mathrm{Tran})}(F)$
	$\displaystyle=\log m+\overline{Ch}_{M_{1}(I_{\alpha},\mathrm{Tran})}(G).$

Combining these two relations (12) and (15), we find that

\overline{Ch}_{M_{1}(I_{\alpha},\mathrm{Tran})}(G)=\overline{Ch}_{X}(G)=h(G).

Denote

M_{j}(I_{\alpha},\mathrm{Tran}):=\cup_{\iota\in\Sigma_{m}^{+}}\left\{I_{\alpha}(\iota,G)\cap\mathrm{Tran}(\iota,G)\cap\left(\mathrm{BR}_{j}(\iota,G)\setminus\mathrm{BR}_{j-1}(\iota,G)\right)\right\},

for $j=2,\cdots,5.$ Similarly, if $(\iota,x)\in I_{\alpha}(F)\cap\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{j}(F)\setminus\mathrm{BR}_{j-1}(F)\right)$ , we obtain that $x\in I_{\alpha}(\iota,G)\cap\mathrm{Tran}(\iota,G)\cap\left(\mathrm{BR}_{j}(\iota,G)\setminus\mathrm{BR}_{j-1}(\iota,G)\right)$ . This shows that

I_{\alpha}(F)\cap\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{j}(F)\setminus\mathrm{BR}_{j-1}(F)\right)\subseteq\Sigma_{m}^{+}\times M_{j}(I_{\alpha},\mathrm{Tran})\subseteq\Sigma_{m}^{+}\times X.

It follows using the Theorem 5 that

$\displaystyle h_{\Sigma_{m}^{+}\times X}(F)$	$\displaystyle=h_{I_{\alpha}(F)\cap\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{j}(F)\setminus\mathrm{BR}_{j-1}(F)\right)}(F)$	(16)
	$\displaystyle\leq\overline{Ch}_{\Sigma_{m}^{+}\times M_{j}(I_{\alpha},\mathrm{Tran})}(F)$
	$\displaystyle=\log m+\overline{Ch}_{M_{j}(I_{\alpha},\mathrm{Tran})}(G).$

Combining these two relations (12) and (16), we find that

\overline{Ch}_{M_{j}(I_{\alpha},\mathrm{Tran})}(G)=\overline{Ch}_{X}(G)=h(G),\,\text{for }j=2,\cdots,5.

This completes the proof.

(2) By the hypothesis about the skew product, it follows from Theorem 6.1(3) of MR3963890 that

\left\{\emptyset,\mathrm{QR}(F)\cap\mathrm{BR}_{1}(F),\mathrm{BR}_{1}(F),\mathrm{BR}_{2}(F),\mathrm{BR}_{3}(F),\mathrm{BR}_{4}(F),\mathrm{BR}_{5}(F)\right\}

has full entropy gaps with respect to $\mathrm{Tran}(F)$ . Hence,

h_{\mathrm{Tran}(F)\cap\mathrm{QR}(F)\cap\mathrm{BR}_{1}(F)}(F)=h_{\mathrm{Tran}(F)}(F)=h_{\Sigma_{m}^{+}\times X}(F),

h_{\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{1}(F)\setminus\mathrm{QR}(F)\right)}(F)=h_{\mathrm{Tran}(F)}(F)=h_{\Sigma_{m}^{+}\times X}(F),

h_{\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{j}(F)\setminus\mathrm{BR}_{j-1}(F)\right)}(F)=h_{\mathrm{Tran}(F)}(F)=h_{\Sigma_{m}^{+}\times X}(F),\,\text{for }j=2,\cdots,5.

Applying Theorem 4, we obtain that

\log m+h(G)=h_{\mathrm{Tran}(F)\cap\mathrm{QR}(F)\cap\mathrm{BR}_{1}(F)}(F),

(17)

\log m+h(G)=h_{\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{1}(F)\setminus\mathrm{QR}(F)\right)}(F),

(18)

\log m+h(G)=h_{\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{j}(F)\setminus\mathrm{BR}_{j-1}(F)\right)}(F),\,\text{for }j=2,\cdots,5.

(19)

By the definitions of the sets, if $(\iota,x)\in\mathrm{Tran}(F)\cap\mathrm{QR}(F)\cap\mathrm{BR}_{1}(F)$ , then $x\in\mathrm{Tran}(\iota,G)\cap\mathrm{QR}(\iota,G)\cap\mathrm{BR}_{1}(\iota,G)$ . This shows that

\mathrm{Tran}(F)\cap\mathrm{QR}(F)\cap\mathrm{BR}_{1}(F)\subseteq\Sigma_{m}^{+}\times M_{0}(\mathrm{Tran})\subseteq\Sigma_{m}^{+}\times X,

where $M_{0}(\mathrm{Tran}):=\bigcup_{\iota\in\Sigma_{m}^{+}}\{\mathrm{Tran}(\iota,G)\cap\mathrm{QR}(\iota,G)\cap\mathrm{BR}_{1}(\iota,G)\}$ . It follows using the Theorem 5 that

$\displaystyle h_{\Sigma_{m}^{+}\times X}(F)$	$\displaystyle=h_{\mathrm{Tran}(F)\cap\mathrm{QR}(F)\cap\mathrm{BR}_{1}(F)}(F)$	(20)
	$\displaystyle\leq\overline{Ch}_{\Sigma_{m}^{+}\times M_{0}(\mathrm{Tran})}(F)$
	$\displaystyle=\log m+\overline{Ch}_{M_{0}(\mathrm{Tran})}(G).$

Combining these two relations (17) and (20), we find that

\overline{Ch}_{M_{0}(\mathrm{Tran})}(G)=\overline{Ch}_{X}(G)=h(G).

By the definitions of the sets, if $(\iota,x)\in\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{1}(F)\setminus\mathrm{QR}(F)\right)$ , then $x\in\mathrm{Tran}(\iota,G)\cap\left(\mathrm{BR}_{1}(\iota,G)\setminus\mathrm{QR}(\iota,G)\right)$ . This shows that

\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{1}(F)\setminus\mathrm{QR}(F)\right)\subseteq\Sigma_{m}^{+}\times M_{1}(\mathrm{Tran})\subseteq\Sigma_{m}^{+}\times X,

where $M_{1}(\mathrm{Tran}):=\bigcup_{\iota\in\Sigma_{m}^{+}}\left\{\mathrm{Tran}(\iota,G)\cap\left(\mathrm{BR}_{1}(\iota,G)\setminus\mathrm{QR}(\iota,G)\right)\right\}$ . It follows using the Theorem 5 that

$\displaystyle h_{\Sigma_{m}^{+}\times X}(F)$	$\displaystyle=h_{\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{1}(F)\setminus\mathrm{QR}(F)\right)}(F)$	(21)
	$\displaystyle\leq\overline{Ch}_{\Sigma_{m}^{+}\times M_{1}(\mathrm{Tran})}(F)$
	$\displaystyle=\log m+\overline{Ch}_{M_{1}(\mathrm{Tran})}(G).$

Combining these two relations (18) and (21), we find that

\overline{Ch}_{M_{1}(\mathrm{Tran})}(G)=\overline{Ch}_{X}(G)=h(G).

Denote

M_{j}(\mathrm{Tran}):=\cup_{\iota\in\Sigma_{m}^{+}}\left\{\mathrm{Tran}(\iota,G)\cap\left(\mathrm{BR}_{j}(\iota,G)\setminus\mathrm{BR}_{j-1}(\iota,G)\right)\right\},

for $j=2,\cdots,5$ . Similarly, if $(\iota,x)\in\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{j}(F)\setminus\mathrm{BR}_{j-1}(F)\right)$ , we obtain that $x\in\mathrm{Tran}(\iota,G)\cap\left(\mathrm{BR}_{j}(\iota,G)\setminus\mathrm{BR}_{j-1}(\iota,G)\right)$ . This shows that

\mathrm{Tran}(F)\cap\left(\mathrm{BR}_{j}(F)\setminus\mathrm{BR}_{j-1}(F)\right)\subseteq\Sigma_{m}^{+}\times M_{j}(\mathrm{Tran})\subseteq\Sigma_{m}^{+}\times X.

Analogously, using Theorem 5 and the formula 19, we conclude that

\overline{Ch}_{M_{j}(\mathrm{Tran})}(G)=\overline{Ch}_{X}(G)=h(G),\,\text{for }j=2,\cdots,5.

This completes the proof.

(3) Define a set as

R_{\alpha}(F):=\left\{(\iota,x)\in\Sigma_{m}^{+}\times X:\inf_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)=\sup_{\nu\in M_{(\iota,x)}(F)}\alpha(\nu)\right\}.

If the skew product $F$ is not uniquely ergodic and $\alpha$ satisfies A.1 and A.2, from Theorem 6.1(4) of MR3963890 , we obtain that

\left\{\emptyset,\mathrm{QR}(F)\cap\mathrm{BR}_{1}(F),\mathrm{BR}_{1}(F),\mathrm{BR}_{2}(F),\mathrm{BR}_{3}(F),\mathrm{BR}_{4}(F),\mathrm{BR}_{5}(F)\right\}

(22)

has full entropy gaps with respect to $R_{\alpha}(F)\cap\mathrm{Tran}(F)$ . Similar to Theorem 1 (1) and (2), one can adopt the proof to complete the proof for $R_{\alpha}(\iota,G)\cap\mathrm{Tran}(\iota,G)$ . Here we omit the details.

Proof: [Proof of Theorem 2] By Lemma 3.4 of MR4200965 , it follows that $G$ is expansive if and only if the skew product $F$ is expansive. In the same spirit, one can adapt the proof of Theorem 1 using Theorem 7.1 of MR3963890 to complete the proof. Here we omit the details.

Remark 7.

We extend the some results of MR3963890 to the dynamical systems of free semigroup actions.

Corollary 3.

Suppose that $G$ has the $\mathbf{g}$ -almost product property, there exists a $\mathbb{P}$ -stationary measure on $X$ with full support where $\mathbb{P}$ is a Bernoulli measure on $\Sigma_{m}^{+}$ . Let $\varphi:X\to\mathbb{R}$ be a continuous function.

(1)

If $I_{\varphi}(\iota,G)\neq\emptyset$ for some $\iota\in\Sigma_{m}^{+}$ , then the unions of gaps of

\left\{\mathrm{QR}(\iota,G),\mathrm{BR}_{1}(\iota,G),\mathrm{BR}_{2}(\iota,G),\mathrm{BR}_{3}(\iota,G),\mathrm{BR}_{4}(\iota,G),\mathrm{BR}_{5}(\iota,G)\right\}

with respect to $I_{\varphi}(\iota,G)\cap\mathrm{Tran}(\iota,G)$ for all $\iota\in\Sigma_{m}^{+}$ have full upper capacity topological entropy of free semigroup action $G$ .

(2)

If the skew product $F$ is not uniquely ergodic, then the unions of gaps of

\left\{\emptyset,\mathrm{QR}(\iota,G)\cap\mathrm{BR}_{1}(\iota,G),\mathrm{BR}_{1}(\iota,G),\mathrm{BR}_{2}(\iota,G),\mathrm{BR}_{3}(\iota,G),\mathrm{BR}_{4}(\iota,G),\mathrm{BR}_{5}(\iota,G)\right\}

with respect to $\mathrm{Tran}(\iota,G)$ , $R_{\varphi}(\iota,G)\cap\mathrm{Tran}(\iota,G)$ for all $\iota\in\Sigma_{m}^{+}$ have full upper capacity topological entropy of free semigroup action $G$ , respectively.

Corollary 4.

\left\{\emptyset,\mathrm{QW}_{1}(\iota,G),\mathrm{QW}_{2}(\iota,G),\mathrm{QW}_{3}(\iota,G),\mathrm{QW}_{4}(\iota,G),\mathrm{QW}_{5}(\iota,G)\right\}

with respect to $\mathrm{Tran}(\iota,G)$ , $R_{\varphi}(\iota,G)\cap\mathrm{Tran}(\iota,G)$ for all $\iota\in\Sigma^{+}_{m}$ have full upper capacity topological entropy of free semigroup action $G$ , respectively. If $I_{\varphi}(\iota,G)$ is non-empty for some $\iota\in\Sigma_{m}^{+}$ , similar arguments hold with respect to $I_{\varphi}(\iota,G)\cap\mathrm{Tran}(\iota,G)$ .

Proof: [Proof of Theorem 3] Suppose that $\mu$ is the $\mathbb{P}$ -stationary measure with full support. Then, Proposition 6 ensures that $\mathbb{P}\times\mu$ is an invariant measure under the skew product transformation $F$ with support $\Sigma_{m}^{+}\times X$ . From Lemma 8, the skew product $F$ has 2 $\mathbf{g}$ -almost product property. For $j=1,\cdots,6$ , let us denote:

T_{j}(F):=\left\{(\iota,x)\in\mathrm{Tran}(F):(\iota,x)\text{ satisfies Case }(j)\right\},

and

B_{j}(F):=\left\{(\iota,x)\in\mathrm{BR}(F):(\iota,x)\text{ satisfies Case }(j)\right\}.

It follows from Theorem 1.3 of MR3963890 that

T_{j}(F)\neq\emptyset,\quad B_{j}(F)\neq\emptyset,

and they all have full topological entropy for all $j=1,\cdots,6$ .

For $j=1,\cdots,6$ , if $(\iota,x)\in T_{j}(F)$ with $\iota=(i_{0},i_{1},\cdots)$ , then the orbit of $(\iota,x)$ under $F$ , that is, $\{F^{k}(\iota,x):k\in\mathbb{N}\}$ is dense in $\Sigma_{m}^{+}\times X$ . This implies that $orb(x,\iota,G)$ is dense in $X$ , hence $x\in\mathrm{Tran}(\iota,G)$ . Notice that $(\iota,x)$ satisfies Case $(j)$ . Hence $x\in T_{j}(G)$ and $T_{j}(G)\neq\emptyset$ . In particular, one has that

T_{j}(F)\subseteq\Sigma_{m}^{+}\times T_{j}(G)\subseteq\Sigma_{m}^{+}\times X.

Summing the two Theorems 4 and 5, we get that

	$\displaystyle\log m+h(G)=h_{\Sigma_{m}^{+}\times X}(F)$	$\displaystyle=h_{T_{j}(F)}(F)$
		$\displaystyle\leq\overline{Ch}_{\Sigma_{m}^{+}\times T_{j}(G)}(F)$
		$\displaystyle=\log m+\overline{Ch}_{T_{j}(G)}(G).$

Since $h(G)=\overline{Ch}_{X}(G)$ , this proves that

h(G)=\overline{Ch}_{X}(G)\leq\overline{Ch}_{T_{j}(G)}(G).

Finally, we conclude that

h(G)=\overline{Ch}_{X}(G)=\overline{Ch}_{T_{j}(G)}(G).

On the other hand, for $j=1,\cdots,6$ , if $(\iota,x)\in B_{j}(F)$ with $\iota=(i_{0},i_{1},\cdots)$ , then for any $\varepsilon>0$ , the set of visiting time $N\left((\iota,x),B\left((\iota,x),\varepsilon\right)\right)$ has a positive Banach upper density and so does $N_{\iota}(x,B(x,\varepsilon))$ . Hence, we get that $x\in\mathrm{BR}(\iota,G)$ . Using the fact that $(\iota,x)$ satisfies Case $(j)$ , hence $x\in B_{j}(G)$ , so $B_{j}(G)\neq\emptyset$ . In particular, one has that

B_{j}(F)\subseteq\Sigma_{m}^{+}\times B_{j}(G)\subseteq\Sigma_{m}^{+}\times X.

Summing the two Theorems 4 and 5, we get that

	$\displaystyle\log m+h(G)=h_{\Sigma_{m}^{+}\times X}(F)$	$\displaystyle=h_{B_{j}(F)}(F)$
		$\displaystyle\leq\overline{Ch}_{\Sigma_{m}^{+}\times B_{j}(G)}(F)$
		$\displaystyle=\log m+\overline{Ch}_{B_{j}(G)}(G).$

Since $h(G)=\overline{Ch}_{X}(G)$ , this proves that

h(G)=\overline{Ch}_{X}(G)\leq\overline{Ch}_{B_{j}(G)}(G).

Finally, we conclude that

h(G)=\overline{Ch}_{X}(G)=\overline{Ch}_{B_{j}(G)}(G).

Remark 8.

Theorem 3 is a generalization of Theorem 1.3 of MR3963890 .

Acknowledgements The authors really appreciate the referees’ valuable remarks and suggestions that helped a lot.

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The upper capacity topological entropy of free semigroup actions for certain non-compact sets, II

Abstract

keywords:

pacs:

1 Introduction

2 Statement of main results

Theorem 1.

Theorem 2.

Theorem 3.

3 Preliminaries

3.1 Some notions

3.2 The topological entropy and others concepts of free semigroup actions

Remark 1.

Theorem 4.

Remark 2.

Theorem 5.

Remark 3.

3.3 Stationary measure

Proposition 6.

4 Periodic-like recurrence and 𝐠\mathbf{g}-almost product property of free semigroup actions

Definition 1.

Definition 2.

Definition 3.

Definition 4.

Definition 5.

Definition 6.

Proposition 7.

Remark 4.

Example 1.

Proposition 8.

5 General (ir)regularity

Theorem 9.

Theorem 10.

Remark 5.

Corollary 1.

Corollary 2.

Remark 6.

Example 2.

6 Proofs of the main results

Remark 7.

Corollary 3.

Corollary 4.

Remark 8.

References

4 Periodic-like recurrence and $\mathbf{g}$ -almost product property of free semigroup actions