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arXiv · 1112.1002

Tame dynamics and robust transitivity

Christian Bonatti·Sylvain Crovisier·Nicolas Gourmelon·Rafael Potrie
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Abstract

One main task of smooth dynamical systems consists in finding a good decomposition into elementary pieces of the dynamics. This paper contributes to the study of chain-recurrence classes. It is known that $C^1$-generically, each chain-recurrence class containing a periodic orbit is equal to the homoclinic class of this orbit. Our result implies that in general this property is fragile. We build a C1-open set U of tame diffeomorphisms (their dynamics only splits into finitely many chain-recurrence classes) such that for any diffeomorphism in a C-infinity-dense subset of U, one of the chain-recurrence classes is not transitive (and has an isolated point). Moreover, these dynamics are obtained among partially hyperbolic systems with one-dimensional center. Résumé : Dynamique modérée et transitivité robuste. L'un des buts des systèmes dynamiques consiste à trouver une bonne décomposition de la dynamique en pièces élémentaires. Cet article contribue à l'étude des classes de récurrence par chaînes. On sait que C1-génériquement, chaque classe de récurrence par chaînes contenant une orbite périodique coincide avec la classe homocline de cette orbite. Notre résultat montre que cette propriété est en générale fragile. Nous construisons un ouvert U de difféomorphismes modérés (leur dynamique ne se décompose qu'en un nombre fini de classes de récurrence par chaînes) tel que pour tout difféomorphisme appartenant à un sous-ensemble C-infini-dense de U, une des classes de récurrence par chaînes n'est pas transitive (elle a un point isolé). De plus, ces dynamiques sont obtenues comme systèmes partiellement hyperboliques avec une direction centrale uni-dimensionnelle.

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Christian Bonatti, Sylvain Crovisier, Nicolas Gourmelon, Rafael Potrie. 2011-12-05. Tame dynamics and robust transitivity. https://arxiv.org/abs/1112.1002

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