The set of sequence entropies for a given space

Abstract

Let X be a compact metric space and T : X → X be continuous. Let h*(T) be the supremum of sequence entropies of T over all subsequences of and S(X) be the set of h*(T) for all continuous maps T on X. It is known that S(X) ⊏ {∞, 0, log 2, log 3, …}. In this paper it is proved that if X is a finite tree or the unit circle S1 then S(X) = {∞, 0, log 2}. Moreover, it is shown that if X = [0, 1] and T has zero topological entropy then the set of sequence entropy pairs is countable and any sequence entropy pair is asymptotic.All the possible sets of S(X) for zero-dimensional spaces X are determined. Moreover, it is shown that for each there is a continuum Xn with dimension n such that S(Xn) = {∞, 0, log 2, log 3, …}.