On the ergodic theory of free group actions by real-analytic circle diffeomorphisms

• 成果类型: Article
• 署名作者: Deroin, Bertrand; Kleptsyn, Victor; Navas, Andres
• 署名单位: CY Cergy Paris Universite; Centre National de la Recherche Scientifique (CNRS); CY Cergy Paris Universite; Universite de Rennes; Centre National de la Recherche Scientifique (CNRS); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Rennes; Universidad de Santiago de Chile
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0779-4
• 发表日期: 2018
• 页码: 731-779
• 关键词: foliations; SUBGROUPS

摘要

We consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due to Ae. Ghys, G. Hector and D. Sullivan.