Quantitative recurrence and large deviations for Teichmuller geodesic   flow
  We prove quantitative recurrence and large deviations results for the Teichmuller geodesci flow on a connected component of a stratum of the moduli space $Q_g$ of holomorphic unit-area quadratic differentials on a compact genus $g \geq 2$ surface. 
Jayadev S. Athreya