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arXiv · 2006.13415

Topological Entropy for Shifts of Finite Type Over $\mathbb{Z}$ and Trees

Jung-Chao Ban·Chih-Hung Chang·Wen-Guei Hu·Yu-Liang Wu
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Abstract

We study the topological entropy of hom tree-shifts and show that, although the topological entropy is not a conjugacy invariant for tree-shifts in general, it remains invariant for hom tree higher block shifts. In doi:10.1016/j.tcs.2018.05.034 and doi:10.3934/dcds.2020186, Petersen and Salama demonstrated the existence of topological entropy for tree-shifts and $h(\mathcal{T}_X) \geq h(X)$, where $\mathcal{T}_X$ is the hom tree-shift derived from $X$. We characterize a necessary and sufficient condition when the equality holds for the case where $X$ is a shift of finite type. In addition, two novel phenomena have been revealed for tree-shifts. There is a gap in the set of topological entropy of hom tree-shifts of finite type, which makes such a set not dense. Last but not least, the topological entropy of a reducible hom tree-shift of finite type is equal to or larger than that of its maximal irreducible component.

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Jung-Chao Ban, Chih-Hung Chang, Wen-Guei Hu, Yu-Liang Wu. 2020-06-24. Topological Entropy for Shifts of Finite Type Over $\mathbb{Z}$ and Trees. https://arxiv.org/abs/2006.13415

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