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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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Sensitivity, proximal extension and higher order almost automorphy
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by Xiangdong Ye and Tao Yu PDF
Trans. Amer. Math. Soc. 370 (2018), 3639-3662 Request permission

Abstract:

Let $(X,T)$ be a topological dynamical system, and $\mathcal {F}$ be a family of subsets of $\mathbb {Z}_+$. $(X,T)$ is strongly $\mathcal {F}$-sensitive if there is $\delta >0$ such that for each non-empty open subset $U$ there are $x,y\in U$ with $\{n\in \mathbb {Z}_+: d(T^nx,T^ny)>\delta \}\in \mathcal {F}$. Let $\mathcal {F}_t$ (resp. $\mathcal {F}_{ip}$, $\mathcal {F}_{fip}$) consist of thick sets (resp. IP-sets, subsets containing arbitrarily long finite IP-sets).

The following Auslander-Yorke’s type dichotomy theorems are obtained: (1) a minimal system is either strongly $\mathcal {F}_{fip}$-sensitive or an almost one-to-one extension of its $\infty$-step nilfactor; (2) a minimal system is either strongly $\mathcal {F}_{ip}$-sensitive or an almost one-to-one extension of its maximal distal factor; (3) a minimal system is either strongly $\mathcal {F}_{t}$-sensitive or a proximal extension of its maximal distal factor.

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Additional Information
  • Xiangdong Ye
  • Affiliation: Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, 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
  • Tao Yu
  • Affiliation: Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China
  • MR Author ID: 870424
  • Email: ytnuo@mail.ustc.edu.cn
  • Received by editor(s): May 7, 2016
  • Received by editor(s) in revised form: August 19, 2016
  • Published electronically: November 15, 2017
  • Additional Notes: The authors were supported by NNSF of China (11371339, 11431012, 11571335).
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 3639-3662
  • MSC (2010): Primary 37B05; Secondary 54H20
  • DOI: https://doi.org/10.1090/tran/7100
  • MathSciNet review: 3766861