Shrinking targets in fast mixing flows and the geodesic flow on negatively curved manifolds

摘要

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 manifolds with variable negative sectional curvature, establishing a logarithm law for such kind of flow.