Topological entropy of free semigroup actions for noncompact sets

Yujun Ju; Dongkui Ma; Yupan Wang

doi:10.3934/dcds.2019041

Article Contents

2019, Volume 39, Issue 2: 995-1017. Doi: 10.3934/dcds.2019041

This issue Previous Article A Billingsley-type theorem for the pressure of an action of an amenable group Next Article Intermediate Lyapunov exponents for systems with periodic orbit gluing property

Topological entropy of free semigroup actions for noncompact sets

1.
School of Mathematics, South China University of Technology, Guangzhou 510641, China
2.
School of Computer Science and Engineering, South China University of Technology, Guangzhou 510641, China

* Corresponding author: Dongkui Ma
* Corresponding author: Dongkui Ma

Received: February 28, 2018

Revised: July 31, 2018

Published: January 2019

Project supported by National Natural Science Foundation of China grant no.11771149, 11671149

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Abstract

In this paper we introduce the topological entropy and lower and upper capacity topological entropies of a free semigroup action, which extends the notion of the topological entropy of a free semigroup action defined by Bufetov [10], by using the Carathéodory-Pesin structure (C-P structure). We provide some properties of these notions and give three main results. The first is the relationship between the upper capacity topological entropy of a skew-product transformation and the upper capacity topological entropy of a free semigroup action with respect to arbitrary subset. The second are a lower and a upper estimations of the topological entropy of a free semigroup action by local entropies. The third is that for any free semigroup action with $m$ generators of Lipschitz maps, topological entropy for any subset is upper bounded by the Hausdorff dimension of the subset multiplied by the maximum logarithm of the Lipschitz constants. The results of this paper generalize results of Bufetov [10], Ma et al. [26], and Misiurewicz [27].

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Mathematics Subject Classification: Primary: 37B40, 37A35; Secondary: 37C45.

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