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Transactions of the American Mathematical Society

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

 
 

 

Sensitivity, proximal extension and higher order almost automorphy


Authors: Xiangdong Ye and Tao Yu
Journal: Trans. Amer. Math. Soc. 370 (2018), 3639-3662
MSC (2010): Primary 37B05; Secondary 54H20
DOI: https://doi.org/10.1090/tran/7100
Published electronically: November 15, 2017
MathSciNet review: 3766861
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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

Keywords: Sensitivity, minimality, infinite step nilfactor, distal factor
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).
Article copyright: © Copyright 2017 American Mathematical Society