Limiting Distributions for Geodesics Excursions on the Modular Surface

0. Introduction We begin by recalling a well known result of Dennis Sullivan for geodesic excursions on finite area non-compact surfaces V . Let v(t) be position reached after time t > 0 by a vector traveling at unit speed along the geodesic whose initial unit tangent vector was v(0). This following result describes the behaviour of the distance from v(t) to v(0) for typical geodesics as t tends to infinity. Theorem 1 (Sullivan) [17,18]. For almost all initial vectors v(t) we have that