Distal strongly ergodic actions

TL;DR

The paper constructs a family of strongly ergodic, distal actions of the free group on three generators on compact metric spaces, demonstrating the existence of such actions with arbitrary rank and answering an open question.

Contribution

It introduces a method to produce strongly ergodic distal actions of free groups with arbitrary rank, expanding understanding of ergodic actions beyond compact systems.

Findings

Existence of strongly ergodic distal actions of $F_3$ with arbitrary rank.

Construction of actions that are strongly ergodic but not compact.

Answering an open question about the existence of such actions.

Abstract

Let $\eta$ be an arbitrary countable ordinal. Using results of Bourgain and Gamburd on compact systems with spectral gap we show the existence of an action of the free group on three generators $F_3$ on a compact metric space $X$, admitting an invariant probability measure $\mu$, such that the resulting dynamical system $(X, \mu, F_3)$ is strongly ergodic and distal of rank $\eta$. In particular this shows that there is a $F_3$ system which is strongly ergodic but not compact. This result answers the open question whether such actions exist.