Multipliers and Disjointness from Mixing

Abstract

In 2005, Parreau proved that if a measure preserving system is not strongly mixing then it contains a non-trivial factor that is disjoint from every strongly mixing system. Taking this construction as the starting point, we develop the complementary notions of $\mathcal{U}$-generated and $\mathcal{U}$-mixing systems, for a set $\mathcal{U}$ of ultrafilters, and use them to recover several classical results in ergodic theory as special cases of a unified framework. We prove that a system is $\mathcal{U}$-mixing if and only if it is disjoint from all $\mathcal{U}$-generated systems. In fact, we show that if $\mathcal{Y}$ is a $\mathcal{U}$-generated system and $\mathcal{Z}$ is disjoint from every $\mathcal{U}$-mixing system, then any joining of $\mathcal{Y}$ and $\mathcal{Z}$ remains disjoint from all $\mathcal{U}$-mixing systems. We also show that every partially rigid system is a finite extension of some $\mathcal{U}$-generated system.

arXiv commentThis is the journal version of the article with referee comments incorporated. AI was also used to find and fix some minor mistakes. We added parts (ii) and (iii) to Lemma 1 and part (iii) to Corollary 5