A criterion for zero averages and full support of ergodic measures

2016/09/25 by Bonatti, Christian, Lorenzo J. Díaz, Jairo Bochi +2
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Geometric and Algebraic Topology

paper · doi:10.48550/arxiv.1609.07764

Abstract

Consider a homeomorphism f defined on a compact metric space X and a continuous map ϕ\colon X → ℝ. We provide an abstract criterion, called control at any scale with a long sparse tail for a point x∈ X and the map ϕ, that guarantees that any weak∗ limit measure μ of the Birkhoff average of Dirac measures \frac1n∑₀^n-1δ(fⁱ(x)) is such that μ-almost every point y has a dense orbit in X and the Birkhoff average of ϕ along the orbit of y is zero. As an illustration of the strength of this criterion, we prove that the diffeomorphisms with nonhyperbolic ergodic measures form a C¹-open and dense subset of the set of robustly transitive partially hyperbolic diffeomorphisms with one dimensional nonhyperbolic central direction. We also obtain applications for nonhyperbolic homoclinic classes.

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