Weak containment rigidity for distal actions

TL;DR

This paper establishes that measure distal actions weakly contained in strongly ergodic actions are factors, revealing that the weak equivalence class of such actions encodes the maximal distal factor's structure.

Contribution

It proves a rigidity result linking weak containment and factor relations for distal actions within ergodic theory.

Findings

Weak containment implies factor relation for measure distal actions.

Weak equivalence class determines the maximal distal factor.

Results apply to compact actions as a special case.

Abstract

We prove that if a measure distal action $\alpha$ of a countable group $\Gamma$ is weakly contained in a strongly ergodic probability measure preserving action $\beta$ of $\Gamma$, then $\alpha$ is a factor of $\beta$. In particular, this applies when $\alpha$ is a compact action. As a consequence, we show that the weak equivalence class of any strongly ergodic action completely remembers the weak isomorphism class of the maximal distal factor arising in the Furstenberg-Zimmer Structure Theorem.