Closed geodesics with prescribed intersection numbers

Abstract

Let.†; g/ be a closed oriented negatively curved surface, and fix a simple closed geodesic Ɣ?. We give the asymptotic growth as L ! C1 of the number of primitive closed geodesics of length less than L intersecting Ɣ? exactly n times, where n is fixed positive integer. This is done by introducing a dynamical scattering operator associated to the surface with boundary obtained by cutting † along Ɣ? and by using the theory of Pollicott–Ruelle resonances for open systems.