An answer to Furstenberg's problem on topological disjointness

TL;DR

This paper characterizes transitive systems disjoint from all minimal systems, showing they are weakly mixing with a dense subset satisfying certain syndetic intersection properties, and explores implications for powers and hyperspace systems.

Contribution

It provides a complete characterization of systems disjoint from all minimal systems, including conditions involving weak mixing and dense subsets, and extends results to powers and hyperspace systems.

Findings

Disjointness from all minimal systems characterized by weak mixing and dense subset conditions.

Disjointness property preserved under taking powers of the system.

Disjointness from all minimal systems equivalent to disjointness of the hyperspace system.

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 nonempty open subset $U\subset X$, there is $x\in D\cap U$ satisfying that $\{n\in{ \mathbb Z}_+: T^nx\in U, S^ny\in V\}$ is syndetic. Some characterization for the general case is also described. 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.