We give explicit $C^1$-open conditions (criterion) that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. This criterion implies 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 satisfied by a $C^1$-dense and open subset of the set of a diffeomorphisms having a robust cycle (this condition can be also formulated -mutatis mutandi- in terms of chain recurrence classes or robustly transitive sets). As a corollary we get the existence of 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 (we state this condition in the simples case of steps skew products).

This is a join work with Bochi and Bonatti.