An answer of Furstenberg' problem on topological disjointness for transitive systems

Wen Huang, Song Shao, Xiangdong Ye

Published 2018-07-25Version 1

In this paper we give an answer of Furstenberg' problem on topological disjointness for transitive systems. 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 nonempty open subset $U\subset X$, there is $x\in D\cap U$ satisfying that $\{n\in { \mathbb Z}_+: T^n x\in U, S^n y\in V\}$ is syndetic. As applications 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.