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Diffeomorphisms with positive metric entropy

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Abstract

We obtain a dichotomy for \(C^{1}\)-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé.

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Authors and Affiliations

  1. CNRS, IMJ-PRG, UMR 7586, Univ. Paris Diderot, Sorbonne Paris Cité, Sorbonne Universités, UPMC Univ. Paris 06, 75013, Paris, France

    A. Avila

  2. IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brazil

    A. Avila

  3. CNRS, Laboratoire de Mathématiques d’Orsay, UMR 8628, Université Paris-Sud 11, 91405, Orsay Cedex, France

    S. Crovisier

  4. Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL, 60637, USA

    A. Wilkinson

Authors
  1. A. Avila
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  2. S. Crovisier
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  3. A. Wilkinson
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Corresponding author

Correspondence to A. Wilkinson.

Additional information

To the memory of Jean-Christophe Yoccoz

A.A. was partially supported by the ERC Starting Grant Quasiperiodic. A.A. and S.C. were partially supported by the Balzan Research Project of J. Palis. A.W. was supported by NSF grant DMS-1316534.

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Avila, A., Crovisier, S. & Wilkinson, A. Diffeomorphisms with positive metric entropy. Publ.math.IHES 124, 319–347 (2016). https://doi.org/10.1007/s10240-016-0086-4

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  • DOI: https://doi.org/10.1007/s10240-016-0086-4

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