Robust Criterion for the Existence of Nonhyperbolic Ergodic Measures

Abstract

International audienceWe 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

The structure of the space of ergodic measures of transitive partially hyperbolic sets

“…• g 2 (x) = x + 1 mod 2 (or any appropriate map preserving {0, 1} and interchanging the interior of the intervals (0, 1) and (1,2)). These maps are depicted in Figure 3.…”

“…Properties of the space of measures are summarized in the next theorem. Given N ⊂ M, its closed convex hull is the smallest closed convex set containing N. The fact that there are ergodic measures with zero Lyapunov exponent and positive entropy in M erg (Λ core ) can be shown using methods in [1], we refrain from discussing this here. We also refrain from studying how such measures are approached by hyperbolic ergodic measures in M erg (Λ core ) as this is much more elaborate and will be part of an ongoing project (see [10] for techniques in a slightly different but technically simpler context).…”

Blender-horseshoes in Center-unstable Hénon-like Families

“…Since, also by (3.2), we have that |q ξ,µ −p ξ,µ | ∈ [31. 4,35,6] we derive a contradiction, completing the proof of the lemma.…”

“…Later, blenders were used in several dynamical contexts: Generation of robust heterodimensional cycles and homoclinic tangencies, stable ergodicity, Arnold diffusion, and construction of nonhyperbolic measures, among others. Each of these applications involves a specific type of blender such as blender-horseshoes [8], symbolic blenders [20,2], dynamical blenders [4] and super-blenders [1].…”

Historic behavior in nonhyperbolic homoclinic classes

“…Dynamical blender. In [BBD16], the authors introduce the notion of a strictly invariant family of discs as a criterion to obtain a blender, which we explain in what follows.…”