Abstract: There is a famous conjecture of Pugh and Shub saying that there exists an open dense set of ergodic partially hyperbolic diffeomorphisms. This conjecture has already been proved by Hertz-Hertz-Ures if the center dimension is one. In the case that the manifold has dimension 3, in particular the center dimension is one, we can ask if we can go further in describing the set of ergodic diffeomorphisms. About fifteen years ago Hertz-Hertz-Ures conjectured that, in dimension 3, all partially hyperbolic diffeomorphisms are ergodic except for some particular cases of ambient manifolds. In this talk we plan to present the advances obtained in this conjecture. We give an affirmative answer to this conjecture in the absence of periodic points.
