The understanding of invariant foliations is very important in the theory of uniformly and partially hyperbolic dynamics. The main theme of this talk is to study transitive Anosov (or uniformly hyperbolic) systems having a decomposition of the form E^s + E^c + E^u, where E^c expands uniformly. There are two foliations that we will consider, the (center)unstable foliation W^{cs} and the strong unstable foliation W^u, tangent to E^c + E^u and E^u, respectively.

The foliation W^{cu} is very well understood. It is known that the foliation is minimal, i.e. every leaf is dense, and that there is only one ergodic invariant measure "compatible" with that foliation, the so-called SRB measure. However, the strong unstable foliation is not well understood. In this talk, I will survey some recent progress in the direction of understanding topological and ergodic properties of the strong unstable foliation. Then, I will talk about a recent result with Sylvain Crovisier and Mauricio Poletti, where we show that in a certain class of Anosov systems, generically there is only one ergodic measure "compatible" with the strong unstable foliation (the so-called u-Gibbs measures) and that the strong unstable foliation is minimal. Speaker(s): Davi Obata (U Chicago)