Hereditary Lowerability of Topological Dynamical Systems
Xiaochen Wang
Abstract
Let (X,T) be a topological dynamical system and let h(T,K) denote the topological entropy of a compact set K⊂ 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⊂ X and every 0≤ h≤ h(T,K), there is a compact set Kh⊂ K such that h(T,Kh)=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'.
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