Nonuniform Center Bunching and the Genericity of Ergodicity among $C^1$   Partially Hyperbolic Symplectomorphisms
  We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns--Wilkinson and Avila--Santamaria--Viana. Combining this new technique with other constructions, we prove that $C^1$-generic partially hyperbolic symplectomorphisms are ergodic. We also construct new examples of stably ergodic partially hyperbolic diffeomorphisms. 
Artur Avila, Jairo Bochi, Amie Wilkinson