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.
Cite this article
Artur Avila, Marcelo Viana, Amie Wilkinson, Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows. J. Eur. Math. Soc. 17 (2015), no. 6, pp. 1435–1462