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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Communicated by K. Khanin
The authors received support from CNPq, FAPERJ, PRONEX, and Ciência Sem Fronteiras CAPES (Brazil); Balzan–Palis Project, ANR (France); Fondecyt and PIA-Conicyt (Anillo ACT 1103 DySyRF) (Chile), and EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS). The authors acknowledge the kind hospitality of Institut de Mathématiques de Bourgogne and Departamento de Matemática PUC-Rio while preparing this paper.
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Bochi, J., Bonatti, C. & Díaz, L.J. Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures. Commun. Math. Phys. 344, 751–795 (2016). https://doi.org/10.1007/s00220-016-2644-5
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DOI: https://doi.org/10.1007/s00220-016-2644-5