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 {\it lowerable} if for each $0\le h\le h (T, X)$ there is a non-empty compact subset with entropy $h$; is {\it hereditarily lowerable} if each non-empty compact subset is lowerable; is {\it hereditarily uniformly lowerable} if for each non-empty compact subset $K$ and each $0\le h\le h (T, K)$ there is a non-empty compact subset $K_h\subseteq K$ 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. 
Wen Huang, Xiangdong Ye and Guohua Zhang