Robust criterion for the existence of nonhyperbolic ergodic measures

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 a 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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• Variational principle for nonhyperbolic ergodic measures: Skew products and elliptic cocycles
• Extremal exponents of random products of conservative diffeomorphisms
• Disintegrations of non-hyperbolic ergodic measures along the center foliation of DA maps
• Blender-producing mechanisms and a dichotomy for local dynamics for heterodimensional cycles
• Cocycles with Quasi-Conformality I: Stability and abundance