Lowering topological entropy over subsets revisited

TL;DR

This paper investigates the entropy properties of subsets within topological dynamical systems, proving that all such systems are D-lowerable and that asymptotically h-expansive systems are D-hereditarily lowerable, with a minimal counterexample provided.

Contribution

It establishes that every topological dynamical system is D-lowerable and that asymptotically h-expansive systems are D-hereditarily lowerable, advancing understanding of entropy distribution in dynamical systems.

Findings

All topological dynamical systems are D-lowerable.

Asymptotically h-expansive systems are D-hereditarily lowerable.

Existence of a minimal system that is lowerable but not hereditarily lowerable.

Abstract

Let $(X, T)$ be a topological dynamical system. Denote by $h (T, K)$ and $h^B (T, K)$ the covering entropy and dimensional entropy of $K\subseteq X$, respectively. $(X, T)$ is called D-{\it lowerable} (resp. {\it lowerable}) if for each $0\le h\le h (T, X)$ there is a subset (resp. closed subset) $K_h$ with $h^B (T, K_h)= h$ (resp. $h (T, K_h)= h$); is called D-{\it hereditarily lowerable} (resp. {\it hereditarily lowerable}) if each Souslin subset (resp. closed subset) is D-lowerable (resp. lowerable). In this paper it is proved that each topological dynamical system is not only lowerable but also D-lowerable, and each asymptotically $h$-expansive system is D-hereditarily lowerable. A minimal system which is lowerable and not hereditarily lowerable is demonstrated.