Geodesic excursions into an embedded disc on a hyperbolic Riemann   surface
  We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point. 
Andrew Haas