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

TL;DR

This paper investigates the structure of topological entropy for Markov shifts on trees, revealing that entropy behavior is complex and differs from classical cases, with new characterizations and insights into the sets of possible entropy values.

Contribution

It provides a detailed analysis of the entropy structure for tree-shift of finite type, including conditions for entropy equality and the relationship between different entropy sets.

Findings

The maximum entropy property for reducible matrices generally fails.

The set of entropies for binary irreducible Markov shifts and one-sided shifts have the same closure.

The closure of these entropy sets contains the interval [d log 2, ∞), but may have gaps.

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.