Communications in Mathematical Physics

, Volume 344, Issue 3, pp 751–795 | Cite as

Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures

  • Jairo Bochi
  • Christian Bonatti
  • Lorenzo J. Díaz


We give explicit C1-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with one-dimensional center and positive topological entropy on which the center Lyapunov exponent vanishes uniformly.

The conditions of the criterion are met on a C1-dense and open subset of the set of diffeomorphisms having a robust cycle. As a corollary, there exists a C1-open and dense subset of the set of non-Anosov robustly transitive diffeomorphisms consisting of systems with nonhyperbolic ergodic measures with positive entropy.

The criterion is based on a notion of a blender defined dynamically in terms of strict invariance of a family of discs.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Jairo Bochi
    • 1
  • Christian Bonatti
    • 2
  • Lorenzo J. Díaz
    • 3
  1. 1.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile
  2. 2.Institut de Mathématiques de BourgogneDijonFrance
  3. 3.Departamento de MatemáticaPontifícia Universidade Católica do Rio de JaneiroRio de JaneiroBrazil

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