Communications in Mathematical Physics

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

Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures

  • Jairo BochiEmail author
  • Christian Bonatti
  • Lorenzo J. Díaz


We give explicit C 1-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 C 1-dense and open subset of the set of diffeomorphisms having a robust cycle. As a corollary, there exists a C 1-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.


Lyapunov Exponent Periodic Point Positive Entropy Homoclinic Tangency Homoclinic Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Jairo Bochi
    • 1
    Email author
  • 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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