First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity

Anatoly Katok, Ralf J. Spatzier

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88 Scopus citations

Abstract

This is the first in a series of papers exploring rigidity properties of hyperbolic actions of Z k or R k for k ≥ 2. We show that for all known irreducible examples, the cohomology of smooth cocycles over these actions is trivial. We also obtain similar Hölder and C 1 results via a generalization of the Livshitz theorem for Anosov flows. As a consequence, there are only trivial smooth or Hölder time changes for these actions (up to an automorphism). Furthermore, small perturbations of these actions are Hölder conjugate and preserve a smooth volume.

Original languageEnglish (US)
Pages (from-to)131-156
Number of pages26
JournalPublications Mathématiques de l'Institut des Hautes Scientifiques
Volume79
Issue number1
DOIs
StatePublished - Dec 1 1994

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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