Lowering Topological Entropy over Subsets

Let (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; is hereditarily lowerable if each non-empty compact subset is lowerable; 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 hexpansive.