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Variational principle for neutralized Bowen topological entropy on subsets of free semigroup actions

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Abstract

In this paper, we introduce the notions of neutralized Bowen topological entropy, neutralized weighted Bowen topological entropy, neutralized Brin–Katok’s lower local entropy, and neutralized Katok’s entropy for a finitely generated free semigroup action \(\mathcal {F}\) on a compact metric space (X, d). For any non-empty compact subset Z of X, we show that

$$\begin{aligned} \qquad h_\textrm{top}^\textrm{NB}(\mathcal {F},Z)=h_\textrm{top}^\textrm{NWB}(\mathcal {F},Z)= & {} \lim _{\epsilon \rightarrow 0}\sup \{\underline{h}_{\mu }^\textrm{NBK}(\mathcal {F},\epsilon ):\mu \in \mathcal {M}(X), \mu (Z)=1\}\\= & {} \lim _{\epsilon \rightarrow 0}\sup \{h_{\mu }^\textrm{NK}(\mathcal {F},\epsilon ):\mu \in \mathcal {M}(X), \mu (Z)=1\}, \end{aligned}$$

where \(\mathcal {M}(X)\), \(h_\textrm{top}^\textrm{NB}(\mathcal {F},Z)\), \(h_\textrm{top}^\textrm{NWB}(\mathcal {F},Z)\), \(\underline{h}_{\mu }^\textrm{NBK}(\mathcal {F},\epsilon )\), and \(h_{\mu }^\textrm{NK}(\mathcal {F},\epsilon )\) are the set of all Borel probability measures on X, neutralized Bowen topological entropy of Z, neutralized weighted Bowen topological entropy of Z, neutralized Brin–Katok’s lower local entropy of \(\mu \), and neutralized Katok’s entropy of \(\mu \), respectively.

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Acknowledgements

The authors would like to thank the respectful referee for his/her comments on the manuscript.

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  1. Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

    Javad Nazarian Sarkooh

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  1. Javad Nazarian Sarkooh
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Correspondence to Javad Nazarian Sarkooh.

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Communicated by Anish Ghosh.

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Sarkooh, J.N. Variational principle for neutralized Bowen topological entropy on subsets of free semigroup actions. Proc Math Sci 133, 35 (2023). https://doi.org/10.1007/s12044-023-00755-1

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