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

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 $h(mathcal{T}_{M})=max_{1leq ileq q}h(mathcal{T}_{M_{i}})$ fails generally, and present a case study with full characterization in terms of the equality. Though that it is likely the sets ${h(mathcal{T}_{M}):Mtext{ is binary and irreducible}}$ and ${h(mathcal{T}_{X}):Xtext{ is a one-sided shift}}$ 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.