On the Question of Ergodicity for Minimal Group Actions on the Circle

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.

Groups with infinitely many ends acting analytically on the circle

“…A decade ago or so, some of the authors, in collaboration with Bertrand Deroin, started a systematic study of locally discrete subgroups of . They introduced an auxiliary property, named (and , but we do not make a distinction here), under which groups behave roughly like Fuchsian groups , that is, discrete subgroups of .…”

“…Property was indeed introduced in as a property under which these conjectures can be established by somewhat standard techniques. More precisely, as done in , one defines Markov partition of the minimal set, with a non‐uniformly expanding map encoding the dynamics of (see § ).…”

“…Second, the set intersects only finitely many orbits (also, when is a Cantor set, there are only finitely many orbits of connected components of the complement ). This can be seen as a consequence of the work where an expansion procedure was introduced, and later improved by Filimonov and Kleptsyn in . In this latter work, the authors show that, if , the dynamics on the minimal set can be encoded by a ‘Markov’ dynamics.…”

“…The problem is that the notion of non‐expandable points is not a dynamical one. The following definition, introduced in , forces a conjugacy‐invariant condition (see [, Corollary 1.10]). Definition Let be a subgroup with no finite orbit, and let be its minimal invariant set.…”

“…Property , even in regularity, entails several strong properties for the dynamics of the group action. For a detailed discussion, the reader may consult or [, § 3.5]. Here we collect the results that are relevant to our purposes.…”