  This article deals with the set of closed geodesics on complete finite type
hyperbolic surfaces. For any non-negative integer $k$, we consider the set of
closed geodesics that self-intersect at least $k$ times, and investigate those
of minimal length. The main result is that, if the surface has at least one
cusp, their self-intersection numbers are exactly $k$ for large enough $k$.

Geometric Topology;Differential Geometry