Periodic geodesics on translation surfaces

Abstract

As shown by Masur in 80s, for any translation surface there exists a periodic geodesic of bounded length, the directions of periodic geodesics are dense in the unit circle, and the number of cylinders of periodic geodesics of length at most T admits upper and lower quadratic estimates. The main goal of this paper is to generalize the above results, with special interest in obtaining effective estimates. In addition, we apply results of Eskin and Masur to establish some properties of periodic geodesics on generic translation surfaces.