Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows
  We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic. 
Artur Avila, Marcelo Viana and Amie Wilkinson
math.DS