Ergodicity of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds

Abstract

We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative C1+ partially hyperbolic in a hyperbolic 3-manifold must be ergodic, giving an affirmative answer to a conjecture of Hertz-Hertz-Ures in the context of hyperbolic 3-manifolds. We also get some results for general partially hyperbolic diffeomorphisms homotopic to the identity and in some isotopy classes on Seifert manifolds.