Robust criterion for the existence of nonhyperbolic ergodic measures

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Abstract : 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.
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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-01407967
Contributeur : Imb - Université de Bourgogne <>
Soumis le : vendredi 2 décembre 2016 - 18:17:23
Dernière modification le : vendredi 8 juin 2018 - 14:50:07

Citation

Jairo Bochi, Christian Bonatti, Lorenzo J. Díaz. Robust criterion for the existence of nonhyperbolic ergodic measures. Communications in Mathematical Physics, Springer Verlag, 2016, 344 (3), pp. 751-795 ⟨10.1007/s00220-016-2644-5 ⟩. ⟨hal-01407967⟩

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