On the question of ergodicity for minimal group actions on the circle

TL;DR

This paper investigates conditions under which minimal, smooth group actions on the circle are ergodic, providing criteria, examples, and implications for measures and foliations, while leaving some questions open.

Contribution

It introduces a sufficient condition for ergodicity of minimal group actions on the circle and explores related measure-theoretic properties, extending to foliations.

Findings

Provided a sufficient condition for ergodicity with respect to Lebesgue measure.

Studied examples illustrating the ergodicity condition.

Analyzed zero Lebesgue measure for exceptional minimal sets.

Abstract

This work is devoted to the study of minimal, smooth actions of finitely generated groups on the circle. We provide a sufficient condition for such an action to be ergodic (with respect to the Lebesgue measure), and we illustrate this condition by studying two relevant examples. Under an analogous hypothesis, we also deal with the problem of the zero Lebesgue measure for exceptional minimal sets. This hypothesis leads to many other interesting conclusions, mainly concerning the stationary and conformal measures. Moreover, several questions are left open. The methods work as well for codimension-one foliations, though the results for this case are not explicitly stated.