Nonuniform measure rigidity

• 成果类型: Article
• 署名作者: Kalinin, Boris; Katok, Anatole; Rodriguez Hertz, Federico
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.174.1.10
• 发表日期: 2011
• 页码: 361-400
• 关键词: anosov z(k) actions; invariant-measures; toral automorphisms; hyperbolic actions; cocycle rigidity; global rigidity; abelian actions; Metric Entropy; local rigidity; cartan actions

摘要

We consider an ergodic invariant measure mu for a smooth action alpha of Z(k), k >= 2, on a (k + 1)-dimensional manifold or for a locally free smooth action of R-k, k >= 2, on a (2k +1)-dimensional manifold. We prove that if mu is hyperbolic with the Lyapunov hyperplanes in general position and if one element in Z(k) has positive entropy, then mu is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups.