Nonuniform measure rigidity

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Abstract

We consider an ergodic invariant measure μ for a smooth action α of Zk, k ≥ 2, on a (k + 1)-dimensional manifold or for a locally free smooth action of Rk, k ≥ 2, on a (2k+1)-dimensional manifold. We prove that if μ is hyperbolic with the Lyapunov hyperplanes in general position and if one element in Zk has positive entropy, then μ 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.

Original languageEnglish (US)
Pages (from-to)361-400
Number of pages40
JournalAnnals of Mathematics
Volume174
Issue number1
DOIs
StatePublished - Jul 1 2011

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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