On structure of topological entropy for tree-shift of finite type

Abstract: This paper deals with the topological entropy for hom Markov shifts $\mathcal{T}M$ on $d$-tree. If $M$ is a reducible adjacency matrix with $q$ irreducible components $M_1, \cdots, M_q$, we show that $h(\mathcal{T}{M})=\max_{1\leq i\leq q}h(\mathcal{T}{M{i}})$ fails generally, and present a case study with full characterization in terms of the equality. Though that it is likely the sets ${h(\mathcal{T}{M}):M\text{ is binary and irreducible}}$ and ${h(\mathcal{T}{X}):X\text{ is a one-sided shift}}$ are not coincident, we show the two sets share the common closure. Despite the fact that such closure is proved to contain the interval $[d \log 2, \infty)$, numerical experiments suggest its complement contain open intervals.