Lowering topological entropy over subsets

Abstract

AbstractLet (X,T) be a topological dynamical system (TDS), and h(T,K) the topological entropy of a subset K of X. (X,T) is lowerable if for each 0≤h≤h(T,X) there is a non-empty compact subset with entropy h; it is hereditarily lowerable if each non-empty compact subset is lowerable; it is hereditarily uniformly lowerable if for each non-empty compact subset K and each 0≤h≤h(T,K) there is a non-empty compact subset Kh⊆K with h(T,Kh)=h and Kh has at most one limit point. It is shown that each TDS with finite entropy is lowerable, and that a TDS (X,T) is hereditarily uniformly lowerable if and only if it is asymptotically h-expansive.