On doubly minimal systems and a question regarding product recurrence

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.