Shrinking targets in fast mixing flows and the geodesic flow on   negatively curved manifolds
  We show that in a rapidly mixing flow with an invariant measure, the time which is needed to hit a given section is related to a sort of conditional dimension of the measure at the section. The result is applied to the geodesic flow of compact variable negative sectional curvature manifolds, establishing a logarithm law for such kind of flow. 
Stefano Galatolo and Isaia Nisoli
math.DS