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June, 2021 On the Topological Entropy of Subshifts of Finite Type on Free Semigroups
Jung-Chao Ban, Chih-Hung Chang
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Taiwanese J. Math. 25(3): 579-598 (June, 2021). DOI: 10.11650/tjm/201210

Abstract

In this paper, we provide an effective method to compute the topological entropies of $G$-subshifts of finite type ($G$-SFTs) with $G = F_{d}$ and $S_{d}$, the free group and free semigroup with $d$ generators respectively. We develop the entropy formula by analyzing the corresponding systems of nonlinear recursive equations (SNREs). Four types of SNREs of $S_{2}$-SFTs, namely the types $\mathbf{E}$, $\mathbf{D}$, $\mathbf{C}$ and $\mathbf{O}$, are introduced, and we could compute their entropies explicitly. This enables us to give the complete characterization of $S_{2}$-SFTs on two symbols. That is, the set of entropies of $S_{2}$-SFTs on two symbols is equal to $\mathbf{E} \cup \mathbf{D} \cup \mathbf{C} \cup \mathbf{O}$. The methods developed in $S_{d}$-SFTs will also apply to the study of the entropy theory of $F_{d}$-SFTs. The entropy formulae of $S_{d}$-, $F_{d}$-golden mean shifts and $k$-colored chessboards are also presented herein.

Funding Statement

This work is partially supported by the Ministry of Science and Technology, Taiwan (Contract No. MOST 109-2115-M-004-002-MY2 and 109-2115-M-390-003-MY3).

Acknowledgments

We want to express our gratitude to the anonymous referees. Their valuable comments and suggestions have significantly improved the readability and quality of this paper. Also, some inspired further study is under preparation.

Citation

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Jung-Chao Ban. Chih-Hung Chang. "On the Topological Entropy of Subshifts of Finite Type on Free Semigroups." Taiwanese J. Math. 25 (3) 579 - 598, June, 2021. https://doi.org/10.11650/tjm/201210

Information

Received: 6 March 2020; Revised: 25 July 2020; Accepted: 21 December 2020; Published: June, 2021
First available in Project Euclid: 4 January 2021

Digital Object Identifier: 10.11650/tjm/201210

Subjects:
Primary: 37A35, 37B10, 92B20

Rights: Copyright © 2021 The Mathematical Society of the Republic of China

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Vol.25 • No. 3 • June, 2021
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