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; 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 K(h) subset of K with h(T, K(h)) = h and K(h) 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.