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When are all closed subsets recurrent?

Published online by Cambridge University Press:  12 May 2016

JIE LI ,
PIOTR OPROCHA ,
XIANGDONG YE  and
RUIFENG ZHANG
JIE LI
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei, Anhui 230026, PR China School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, PR China email jiel0516@mail.ustc.edu.cn, yexd@ustc.edu.cn
PIOTR OPROCHA
Affiliation:
AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Kraków, Poland National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic email oprocha@agh.edu.pl
XIANGDONG YE
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei, Anhui 230026, PR China School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, PR China email jiel0516@mail.ustc.edu.cn, yexd@ustc.edu.cn
RUIFENG ZHANG
Affiliation:
School of Mathematics, Hefei University of Technology, Hefei, Anhui 230009, PR China email rfzhang@mail.ustc.edu.cn
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Abstract

In the paper we study relations of rigidity, equicontinuity and pointwise recurrence between an invertible topological dynamical system (t.d.s.) $(X,T)$ and the t.d.s. $(K(X),T_{K})$ induced on the hyperspace $K(X)$ of all compact subsets of $X$, and provide some characterizations. Among other examples, we construct a minimal, non-equicontinuous, distal and uniformly rigid t.d.s. and a weakly mixing t.d.s. which induces dense periodic points on the hyperspace $K(X)$ but itself does not have dense distal points, solving in that way a few open questions from earlier articles by Dong, and Li, Yan and Ye.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 7 , October 2017 , pp. 2223 - 2254
Copyright
© Cambridge University Press, 2016 

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