Robust criterion for the existence of nonhyperbolic ergodic measures

TL;DR

This paper establishes explicit $C^1$-open conditions that guarantee the existence of nonhyperbolic ergodic measures with positive entropy in certain diffeomorphisms, expanding understanding of complex dynamical behaviors.

Contribution

It introduces a new criterion based on blenders for identifying nonhyperbolic ergodic measures with positive entropy in $C^1$-generic settings.

Findings

Existence of nonhyperbolic ergodic measures with positive entropy under explicit conditions.

The criterion applies to a $C^1$-dense and open subset of diffeomorphisms with robust cycles.

Non-Anosov robustly transitive diffeomorphisms can have nonhyperbolic ergodic measures with positive entropy.

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.