Regularity of invariant distributions and rigidity of partially hyperbolic diffeomorphisms

The stable, unstable (and central) distributions of (partially) hyperbolic dynamics are a priori only Hölder continuous, and several works seem to suggest that their lack of regularity is in fact the only obstacle to their rigidity. Concerning contact-Anosov flows, successive works of Ghys (in dimension three) and Benoist-Foulon-Labourie (in higher dimensions) have for instance proved that the smoothness of the stable and unstable distributions forces the system to be algebraic. In this talk, I will present an analog rigidity result for three-dimensional volume-preserving partially hyperbolic diffeomorphisms whose stable, unstable and central distributions are smooth, and whose stable-unstable plane field is a contact distribution.