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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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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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
  • MR Author ID: 677726
  • 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
  • MR Author ID: 266004
  • Email: yexd@ustc.edu.cn
  • Guohua Zhang
  • Affiliation: School of Mathematical Sciences and LMNS, Fudan University, Shanghai 200433, China
  • Email: zhanggh@fudan.edu.cn
  • 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).
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 3206465