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An answer to Furstenberg’s problem on topological disjointness

Part of: Connections with other structures, applications Topological dynamics

Published online by Cambridge University Press:  10 April 2019

WEN HUANG ,
SONG SHAO  and
XIANGDONG YE
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, PR China email wenh@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn
SONG SHAO
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, PR China email wenh@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@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, PR China email wenh@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn
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Abstract

In this paper we give an answer to Furstenberg’s problem on topological disjointness. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable dense subset $D$ of $X$ such that for any minimal system $(Y,S)$, any point $y\in Y$ and any open neighbourhood $V$ of $y$, and for any non-empty open subset $U\subset X$, there is $x\in D\cap U$ such that $\{n\in \mathbb{Z}_{+}:T^{n}x\in U,S^{n}y\in V\}$ is syndetic. Some characterization for the general case is also given. By way of application we show that if a transitive system $(X,T)$ is disjoint from all minimal systems, then so are $(X^{n},T^{(n)})$ and $(X,T^{n})$ for any $n\in \mathbb{N}$. It turns out that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system $(K(X),T_{K})$ is disjoint from all minimal systems.

Keywords

topological dynamicsjoiningdisjointweak mixingcentral set

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 54H20: Topological dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 9 , September 2020 , pp. 2467 - 2481
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Copyright
© Cambridge University Press, 2019

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