On doubly minimal systems and a question regarding product recurrence

TL;DR

This paper investigates the structure of doubly minimal systems in topological dynamics, showing a specific orbit closure property, and resolves a 2008 open problem on product recurrence.

Contribution

It establishes a new characterization of orbit closures in doubly minimal systems and answers a longstanding question on product recurrence.

Findings

Orbit closures are either the entire product or a specific graph of a factor map.

Doubly minimal systems exhibit a unique orbit closure structure.

The paper resolves a problem posed by Haddad and Ott in 2008.

Abstract

We show that a doubly minimal system $X$ has the property that for every minimal system $Y$ the orbit closure of any pair $(y,x) \in Y \times X$ is either $Y \times X$ or it has the form $\Gamma_\pi = \{(\pi(x),x) : x \in X\}$ for some factor map $\pi: X \to Y$. As a corollary we resolve a problem of Haddad and Ott from 2008 regarding product recurrence.