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

Deroin, Bertrand; Kleptsyn, Victor; Navas, Andrés

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

Authors

Deroin, Bertrand

Kleptsyn, Victor

Navas, Andrés

Description

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.
41 pages, 7 figures; submitted to Moscow Mathematical Journal