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Topological Entropy and Sequence Entropy for Hom Tree-Shifts on Unexpandable Trees

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Abstract

This article explores the topological entropy and topological sequence entropy of hom tree-shifts on unexpandable trees. Regarding topological entropy, we establish that the entropy, denoted as \(h({\mathcal {T}}_X)\) on an unexpandable tree, equals the entropy h(X) of the base shift X when X is a subshift satisfying the almost specification property. Additionally, we derive some fundamental properties such as entropy approximation and the denseness of entropy for subsystems. Concerning topological sequence entropy, we show that the set of sequence entropies of hom tree-shifts with a base shift is generated by an irreducible matrix A, forming a subset of \(\log {\mathbb {N}}\). Precisely, these entropies correspond to the logarithms of the largest cardinalities of the periodic classes of A.

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Notes

  1. That is, \(\left| \Delta _{n}\backslash \Delta _{n-1}\right| /\left| \Delta _{n}\right| \) does not tend to 0 as \(n\rightarrow \infty \), where \(\Delta _{n}=\cup _{i=0}^{n-1}T_{i}\). For a deeper discussion of the amenability of a group we refer the reader to [14].

  2. The value \(h_{top}^{\infty }(f)\) is also known as the maximal pattern entropy of f, introduced by Huang and Ye [26]. We refer the reader to [26] or [27] for more details and a complete bibliography.

  3. The sequence \({\mathcal {C}}=\{{\mathcal {C}}_{i}\}_{i=0}^{\infty }\) symbolizes the sequence \(S=\{s_{n}\}_{n=1}^{\infty }\) in \({\mathbb {N}}\) shifts.

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Acknowledgements

We sincerely thank the anonymous referee for their valuable feedback and insightful suggestions on the initial draft of this article. Their contributions have significantly enhanced the clarity and overall quality of the paper. Ban and Chang are partially supported by the National Science and Technology Council, ROC (Contract No NSTC 111-2115-M-004-005-MY3 and 112-2115-M-390-003) and National Center for Theoretical Sciences. Hu is partially supported by the National Natural Science Foundation of China (Grant No.12271381). Lai is partially supported by the National Science and Technology Council, ROC (Contract NSTC 111-2811-M-004-002-MY2).

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Authors and Affiliations

  1. Department of Mathematical Sciences, National Chengchi University, Taipei, 11605, Taiwan, ROC

    Jung-Chao Ban & Guan-Yu Lai

  2. Math. Division, National Center for Theoretical Science, National Taiwan University, Taipei, 10617, Taiwan, ROC

    Jung-Chao Ban

  3. Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 81148, Taiwan, ROC

    Chih-Hung Chang

  4. College of Mathematics, Sichuan University, Chengdu, 610064, China

    Wen-Guei Hu

  5. Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014, Oulu, Finland

    Yu-Liang Wu

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Ban and Chang are partially supported by the National Science and Technology Council, ROC (Contract No NSTC 111-2115-M-004-005-MY3 and 112-2115-M-390-003) and National Center for Theoretical Sciences. Hu is partially supported by the National Natural Science Foundation of China (Grant No.12271381). Lai is partially supported by the National Science and Technology Council, ROC (Contract NSTC 111-2811-M-004-002-MY2).

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Ban, JC., Chang, CH., Hu, WG. et al. Topological Entropy and Sequence Entropy for Hom Tree-Shifts on Unexpandable Trees. Qual. Theory Dyn. Syst. 23, 108 (2024). https://doi.org/10.1007/s12346-024-00967-4

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