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2021 On the Topological Entropy of Subshifts of Finite Type on Free Semigroups
Jung-Chao Ban, Chih-Hung Chang
Taiwanese J. Math. Advance Publication 1-20 (2021). DOI: 10.11650/tjm/201210


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.


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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. Advance Publication 1 - 20, 2021.


Published: 2021
First available in Project Euclid: 4 January 2021

Digital Object Identifier: 10.11650/tjm/201210

Primary: 37A35, 37B10, 92B20

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


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