On the ergodicity of partially hyperbolic systems
  Pugh and Shub have conjectured that essential accessibility implies ergodicity, for a $C^2$, partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satsified by all partially hyperbolic systems with 1-dimensional center bundle. We also obtain ergodicity results for $C^{1+\gamma}$ partially hyperbolic systems. 
Keith Burns and Amie Wilkinson