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Lowering topological entropy over subsets revisited


Authors: Wen Huang, Xiangdong Ye and Guohua Zhang
Journal: Trans. Amer. Math. Soc. 366 (2014), 4423-4442
MSC (2010): Primary 37B40, 37A35, 37B10, 37A05
DOI: https://doi.org/10.1090/S0002-9947-2014-06117-5
Published electronically: March 31, 2014
MathSciNet review: 3206465
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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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Additional Information

Wen Huang
Affiliation: Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China
Email: wenh@mail.ustc.edu.cn

Xiangdong Ye
Affiliation: Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China
Email: yexd@ustc.edu.cn

Guohua Zhang
Affiliation: School of Mathematical Sciences and LMNS, Fudan University, Shanghai 200433, China
Email: zhanggh@fudan.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2014-06117-5
Keywords: Entropy, principal extension, lowerable, hereditarily lowerable
Received by editor(s): July 11, 2012
Received by editor(s) in revised form: December 4, 2012
Published electronically: March 31, 2014
Additional Notes: The first author was supported by NNSF of China (11225105), the Fok Ying Tung Education Foundation and the Fundamental Research Funds for the Central Universities
The first and second authors were supported by NNSF of China (11071231)
The third author was supported by FANEDD (201018) and NSFC (11271078).
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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