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 $h({\mathcal{T}}_M)={\max }_{1\le i\le q}h({\mathcal{T}}_{M_i})$ 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.