Nonuniform center bunching and the genericity of ergodicity among $C^1$ partially hyperbolic symplectomorphisms

Avila, Artur; Bochi, Jairo; Wilkinson, Amie

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 6, pp. 931-979.

Nous introduisons une notion non-uniforme de resserrement central pour les difféomorphismes partiellement hyperboliques qui nous permet de généraliser quelques résultats de Burns-Wilkinson et Avila-Santamaria-Viana. Cette nouvelle technique est utilisée, en combinaison avec d’autres constructions, pour démontrer la généricité de l’ergodicité parmi les difféomorphismes symplectiques partiellement hyperboliques de classe $C^1$. De plus, nous obtenons de nouveaux exemples de dynamiques stablement ergodiques.

We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns-Wilkinson and Avila-Santamaria-Viana. Combining this new technique with other constructions we prove that $C^1$-generic partially hyperbolic symplectomorphisms are ergodic. We also construct new examples of stably ergodic partially hyperbolic diffeomorphisms.

Classification : 37D30, 37D25, 37J10
Keywords: partial hyperbolicity, center bunching, ergodicity, symplectic diffeomorphisms
Mot clés : hyperbolicité partielle, resserrement central, ergodicité, difféomorphismes symplectiques

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