A model-theoretic approach to rigidity of strongly ergodic, distal actions

TL;DR

This paper introduces a model-theoretic framework to analyze distal factors in strongly ergodic, measure-preserving dynamical systems, revealing their algebraic closure properties and establishing new rigidity results.

Contribution

It develops a novel model-theoretic approach to study distal factors, proving their algebraic closure and deriving new rigidity theorems for strongly ergodic, distal systems.

Findings

All such factors are in the algebraic closure of the empty set.

Strongly ergodic, distal systems are coalescent.

Any two weakly equivalent such systems are isomorphic.

Abstract

We develop a model-theoretic framework for the study of distal factors of strongly ergodic, measure-preserving dynamical systems of countable groups. Our main result is that all such factors are contained in the (existential) algebraic closure of the empty set. This allows us to recover some rigidity results of Ioana and Tucker-Drob as well as prove some new ones: for example, that strongly ergodic, distal systems are coalescent and that every two such systems that are weakly equivalent are isomorphic. We also prove the existence of a universal distal, ergodic system that contains any other distal, ergodic system of the group as a factor.