Hereditary Lowerability of Topological Dynamical Systems

Abstract

Let $(X,T)$ be a topological dynamical system and let $h(T,K)$ denote the topological entropy of a compact set $K\subset X$. We settle a question and a conjecture raised by Huang, Ye, and Zhang (2014) concerning hereditary lowerability. First, we show that every system with finite topological entropy is hereditarily lowerable: for every nonempty compact set $K\subset X$ and every $0\le h\le h(T,K)$, there is a compact set $K_h\subset K$ such that $h(T,K_h)=h$. This gives a negative answer to their Question 2'. Second, we prove that if $(X,T)$ admits an ergodic invariant measure with infinite entropy, then $(X,T)$ is not hereditarily lowerable. More precisely, we construct a compact set $K$ with infinite entropy such that every compact subset of $K$ has entropy either zero or infinity. This proves the conjecture stated immediately after Question 2'.