Lowering topological entropy over subsets revisited

Huang, Wen; Ye, Xiangdong; Zhang, Guohua

doi:10.1090/S0002-9947-2014-06117-5

Lowering topological entropy over subsets revisited
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by Wen Huang, Xiangdong Ye and Guohua Zhang PDF

Trans. Amer. Math. Soc. 366 (2014), 4423-4442 Request permission

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-lowerable (resp. 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. $h (T, K_h)= h$) and is called D-hereditarily lowerable (resp. 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.

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