Topological Entropy and Sequence Entropy for Hom Tree-Shifts on Unexpandable Trees
Ban, Jung-Chao; Chang, Chih-Hung; Hu, Wen-Guei; Lai, Guan-Yu; Wu, Yu-Liang (2024-02-16)
Springer
16.02.2024
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
https://rightsstatements.org/vocab/InC/1.0/
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024. This is a post-peer-review, pre-copyedit version of an article published in Qualitative Theory of Dynamical Systems. The final authenticated version is available online at: https://doi.org/10.1007/s12346-024-00967-4.
https://rightsstatements.org/vocab/InC/1.0/
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024. This is a post-peer-review, pre-copyedit version of an article published in Qualitative Theory of Dynamical Systems. The final authenticated version is available online at: https://doi.org/10.1007/s12346-024-00967-4.
https://rightsstatements.org/vocab/InC/1.0/
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:oulu-202402272008
https://urn.fi/URN:NBN:fi:oulu-202402272008
Tiivistelmä
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 \(A\).
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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