On Anosov diffeomorphisms with asymptotically conformal periodic data

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Abstract

We consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We prove various properties of such systems, including strong pinching, C1+β smoothness of the Anosov splitting, and C1 smoothness of measurable invariant conformal structures and distributions. We apply these results to volume-preserving diffeomorphisms with two-dimensional stable and unstable distributions and diagonalizable derivatives of the return maps at periodic points. We show that a finite cover of such a diffeomorphism is smoothly conjugate to an Anosov automorphism of T4; as a corollary, we obtain local rigidity for such diffeomorphisms. We also establish a local rigidity result for Anosov diffeomorphisms in dimension three.

Original languageEnglish (US)
Pages (from-to)117-136
Number of pages20
JournalErgodic Theory and Dynamical Systems
Volume29
Issue number1
DOIs
StatePublished - Feb 1 2009

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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