Multipliers and Disjointness from Mixing

TL;DR

This paper introduces the concepts of $U$-generated and $U$-mixing systems in ergodic theory, providing a unified framework that generalizes classical results and explores their disjointness properties.

Contribution

It develops a new framework based on ultrafilters to unify and extend classical ergodic theory results on mixing and disjointness.

Findings

A system is $U$-mixing iff it is disjoint from all $U$-generated systems.

Joins of $U$-generated and systems disjoint from $U$-mixing systems remain disjoint.

Every partially rigid system is a finite extension of a $U$-generated system.

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.