Topological entropy for shifts of finite type over [formula omitted] and trees

Abstract

We study the topological entropy of hom tree-shifts and show that, although the topological entropy is not a conjugacy invariant for tree-shifts in general, it remains invariant for hom tree higher block shifts. In [16,17], Petersen and Salama demonstrated the existence of topological entropy for tree-shifts and h(TX)≥h(X), where TX is the hom tree-shift derived from X. We characterize a necessary and sufficient condition when the equality holds for the case where X is a shift of finite type. Additionally, two novel phenomena have been revealed for tree-shifts. There is a gap in the set of topological entropy of hom tree-shifts of finite type, making such a set not dense. Last but not least, the topological entropy of a reducible hom tree-shift of finite type can be strictly larger than that of its maximal irreducible component.