Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures

By Jairo Bochi, C. Bonatti, Lorenzo J. Díaz. 2015

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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.

Subfields: Control and Systems Engineering, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/19c431c5f4bd2c0da0aeaa8c04fa37f6eb6c7652