Nonuniform measure rigidity

Research output: Chapter in Book/Report/Conference proceedingChapter


We consider an ergodic invariant measure µ for a smooth action a 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)
Title of host publicationThe Collected Works of Anatole Katok
Subtitle of host publicationIn 2 Volumes
PublisherWorld Scientific Publishing Co.
Number of pages40
ISBN (Electronic)9789811238079
ISBN (Print)9789811238062
StatePublished - Jan 1 2024

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • General Engineering

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