Uniformly quasiconformal partially hyperbolic systems
[Systèmes partiellement hyperboliques uniformément quasi conformes]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 5, pp. 1085-1127.

Nous étudions les perturbations lisses préservant le volume de l'application temps-un du flot géodésique ψt d'une variété riemannienne fermée de dimension au moins égale à trois et de courbure négative constante. Nous montrons que pour une telle perturbation, les exposants de Lyapunov extrémaux relativement au volume coïncident à la fois dans les sous-espaces stables et instables si et seulement si cette perturbation se plonge comme temps-un d'un flot lisse préservant le volume et dont les orbites sont conjuguées de manière lisse à celles de ψt. Nos techniques s'appliquent plus généralement pour donner une classification essentiellement complète des difféomorphismes lisses, partiellement hyperboliques préservant le volume et vérifient une condition de quasi-conformalité uniforme le long de leurs fibrés stables et instables qui, soit possèdent un feuilletage central compact avec une holonomie triviale, soit sont obtenus comme perturbations de l'application temps-un d'un flot d'Anosov.

We study smooth volume-preserving perturbations of the time-1 map of the geodesic flow ψt of a closed Riemannian manifold of dimension at least three with constant negative curvature. We show that such a perturbation has equal extremal Lyapunov exponents with respect to volume within both the stable and unstable bundles if and only if it embeds as the time-1 map of a smooth volume-preserving flow that is smoothly orbit equivalent to ψt. Our techniques apply more generally to give an essentially complete classification of smooth, volume-preserving partially hyperbolic diffeomorphisms which satisfy a uniform quasiconformality condition on their stable and unstable bundles and have either compact center foliation with trivial holonomy or are obtained as perturbations of the time-1 map of an Anosov flow.

DOI : 10.24033/asens.2372
Classification : 37D30, 37C85, 30C65.
Keywords: Partially hyperbolic diffeomorphisms, Lyapunov exponent, quasiconformal mapping, rigidity.
Mot clés : Difféomorphismes partiellement hyperboliques, exposant de Liapounov, application quasi conforme, rigidité. 
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Butler, Clark; Xu, Disheng. Uniformly quasiconformal partially hyperbolic systems. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 5, pp. 1085-1127. doi : 10.24033/asens.2372. http://www.numdam.org/articles/10.24033/asens.2372/

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