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Null systems in the non-minimal case

Part of: Connections with other structures, applications Topological dynamics

Published online by Cambridge University Press:  17 June 2019

JIAHAO QIU  and
JIANJIE ZHAO
JIAHAO QIU
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics USTC, Chinese Academy of Sciences and School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China email qiujh@mail.ustc.edu.cn, zjianjie@mail.ustc.edu.cn
JIANJIE ZHAO
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics USTC, Chinese Academy of Sciences and School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China email qiujh@mail.ustc.edu.cn, zjianjie@mail.ustc.edu.cn
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Abstract

In this paper, it is shown that if a dynamical system is null and distal, then it is equicontinuous. It turns out that a null system with closed proximal relation is mean equicontinuous. As a direct application, it follows that a null dynamical system with dense minimal points is also mean equicontinuous. Meanwhile, a distal system with trivial $\text{Ind}_{\text{fip}}$-pairs and a non-trivial regionally proximal relation of order $\infty$ are constructed.

Keywords

null systemsmean equicontinuityregionally proximal relation of order d

MSC classification

Primary: 54H20: Topological dynamics
Secondary: 37B40: Topological entropy
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 12 , December 2020 , pp. 3420 - 3437
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Copyright
© Cambridge University Press, 2019

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