Asymptotic expansion in measure and strong ergodicity

TL;DR

This paper introduces the concept of asymptotic expansion in measure for measurable actions, linking it to strong ergodicity, and provides structural theorems and applications, including characterizations via local spectral gaps.

Contribution

It generalizes expansion in measure, offers a new perspective on strong ergodicity, and establishes structural and spectral characterizations of asymptotically expanding actions.

Findings

Asymptotic expansion in measure generalizes classical expansion.

Homogeneous strongly ergodic actions are always expanding in measure.

Connections established between asymptotic expansion and asymptotic expanders.

Abstract

In this paper, we introduce and study a notion of asymptotic expansion in measure for measurable actions. This generalises expansion in measure and provides a new perspective on the classical notion of strong ergodicity. Moreover, we obtain structure theorems for asymptotically expanding actions, showing that they admit exhaustions by domains of expansion. As an application, we recover a recent result of Marrakchi, characterising strong ergodicity in terms of local spectral gaps. We also show that homogeneous strongly ergodic actions are always expanding in measure and establish a connection between asymptotic expansion in measure and asymptotic expanders by means of approximating spaces.