A criterion for ergodicity of non-uniformly hyperbolic diffeomorphisms

In this work we exhibit a new criterion for ergodicity of diffeomorphisms involving conditions on Lyapunov exponents and the general position of some invariant manifolds. On the one hand, we derive uniqueness of SRB measures for transitive surface diffeomorphisms. On the other hand, using recent results on the existence of blenders we give a positive answer, in the C¹ topology, to a conjecture of Pugh and Shub in the context of partially hyperbolic conservative diffeomorphisms with two dimensional center bundle.