Lowering topological entropy over subsets

TL;DR

This paper investigates the properties of topological dynamical systems related to the entropy of their subsets, establishing conditions under which systems are lowerable or hereditarily uniformly lowerable, and linking these to asymptotic $h$-expansiveness.

Contribution

It proves that all systems with finite entropy are lowerable and characterizes hereditarily uniformly lowerable systems as exactly those that are asymptotically $h$-expansive.

Findings

Systems with finite entropy are lowerable.

Hereditarily uniformly lowerable systems are characterized as asymptotically $h$-expansive.

Provides conditions for the existence of subsets with prescribed entropy.

Abstract

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.