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

Abstract

This paper deals with the topological entropy for hom Markov shifts T M on d -tree. If M is a reducible adjacency matrix with q irreducible components M 1 , ⋯ , M q , we show that h ( T M ) = max 1 ≤ i ≤ q ⁡ h ( 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 ( T M ) : M is binary and irreducible } and { h ( T X ) : X 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 , ∞ ) , numerical experiments suggest its complement contain open intervals.