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

TL;DR

This paper investigates the topological entropy of hom tree-shifts, revealing invariance properties, conditions for equality with base shifts, and novel phenomena such as entropy gaps and relationships between reducible shifts and their components.

Contribution

It characterizes when topological entropy remains invariant under hom tree higher block shifts for shifts of finite type and uncovers new phenomena in the entropy set for tree-shifts.

Findings

Topological entropy is invariant for hom tree higher block shifts.

A necessary and sufficient condition for entropy equality in shifts of finite type.

Existence of a gap in the set of topological entropy values for hom tree-shifts.

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.