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Article Dans Une Revue Communications in Mathematical Physics Année : 2016

## Robust criterion for the existence of nonhyperbolic ergodic measures

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Jairo Bochi
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Christian Bonatti
Lorenzo J. Díaz
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#### Résumé

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 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.

#### Domaines

Mathématiques [math] Systèmes dynamiques [math.DS]

### Dates et versions

hal-01407967 , version 1 (02-12-2016)

### Identifiants

• HAL Id : hal-01407967 , version 1
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Jairo Bochi, Christian Bonatti, Lorenzo J. Díaz. Robust criterion for the existence of nonhyperbolic ergodic measures. Communications in Mathematical Physics, 2016, 344 (3), pp. 751-795 ⟨10.1007/s00220-016-2644-5⟩. ⟨hal-01407967⟩

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