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The Upper Capacity Topological Entropy of Free Semigroup Actions for Certain Non-compact Sets

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Abstract

In this paper, we first introduce some new notions of ‘periodic-like’ points, such as almost periodic points, weakly almost periodic points, quasi-weakly almost periodic points, of free semigroup actions. We find that the corresponding sets and gap-sets of these points carry full upper capacity topological entropy of free semigroup actions under certain conditions. Furthermore, \(\phi \)-irregular set acting on free semigroup actions is introduced and it also carries full upper capacity topological entropy in the system with specification property. Finally, we introduce the level set for local recurrence of free semigroup actions and analyze its connections with upper capacity topological entropy. Our analysis generalizes the results obtained by Tian (Different asymptotic behavior versus same dynamical complexity: recurrence & (ir)regularity. Adv. Math. 288:464–526, 2016), Chen et al. (Topological entropy for divergence points. Ergodic Theory Dynam Syst. 25:1173–1208, 2005) and Lau and Shu (The spectrum of Poincaré recurrence. Ergodic Theory Dynam Syst 28:1917–1943, 2007) etc.

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Acknowledgements

The authors really appreciate the referees’ valuable remarks and suggestions that helped a lot. The work was supported by National Natural Science Foundation of China (grant no.11771149) and Guangdong Natural Science Foundation 2018B0303110005.

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  1. School of Mathematics, South China University of Technology, Guangzhou, 510641, P.R. China

    Li Zhu & Dongkui Ma

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  1. Li Zhu
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  2. Dongkui Ma
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Correspondence to Dongkui Ma.

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Communicated by Alessandro Giuliani.

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Zhu, L., Ma, D. The Upper Capacity Topological Entropy of Free Semigroup Actions for Certain Non-compact Sets. J Stat Phys 182, 19 (2021). https://doi.org/10.1007/s10955-020-02693-y

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