Topological Entropy and Sequence Entropy for Hom Tree-Shifts on Unexpandable Trees

This article explores the topological entropy and topological sequence entropy of hom tree-shifts on unexpandable trees. Regarding topological entropy, we establish that the entropy, denoted as h(T_X) on an unexpandable tree, equals the entropy h(X) of the base shift X when X is a subshift satisfying the almost specification property. Additionally, we derive some fundamental properties such as entropy approximation and the denseness of entropy for subsystems. Concerning topological sequence entropy, we show that the set of sequence entropies of hom tree-shifts with a base shift is generated by an irreducible matrix A, forming a subset of logN. Precisely, these entropies correspond to the logarithms of the largest cardinalities of the periodic classes of A.