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

TL;DR

This paper extends the study of topological entropy in free semigroup actions by introducing new properties and analyzing the multifractal structure of various sets under these actions, generalizing previous results.

Contribution

It introduces the $ extbf{g}$-almost product property and new notions of recurrence and transitivity, advancing the understanding of entropy and multifractal analysis in non-compact sets for free semigroup actions.

Findings

Established generalized multifractal analysis for free semigroup actions.

Connected Banach upper recurrence and transitivity with irregularity.

Extended entropy results to broader classes of sets and actions.

Abstract

This paper's major purpose is to continue the work of Zhu and Ma[1]. 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[2], Pfister and Sullivan [3].