Raul Ures: Partial hyperbolicity in dimension 3: the ergodicity conjecture

Abstract:

A diffeomorphism f is partially hyperbolic if the tangent bundle of the ambient manifold splits into three invariant bundles: one is contracted (stable), one is expanded (unstable), and the third one has an intermediate behavior (center). For conservative f, there is a famous conjecture of Pugh and Shub that says that an open dense diffeomorphisms of the partially hyperbolic ones are ergodic. This conjecture has already been proved by Hertz-Hertz-Ures (HHU) if the center dimension is 1. In the case that the manifold has dimension 3, the three bundles have dimension 1, and then we can ask if we can go further in describing the set of ergodic diffeomorphisms. About fifteen years ago HHU conjectured that, in dimension 3, all partially hyperbolic diffeomorphisms are ergodic except for some particular cases of ambient manifolds (which, in particular, have (virtually) solvable fundamental group). In this talk we plan to present the advances obtained in this conjecture, with emphasis on recent times where the study of this conjecture has been especially active.