Closed geodesics with prescribed intersection numbers

Abstract

Let $(\Sigma, g)$ be a closed, oriented, negatively curved surface, and fix pairwise disjoint simple closed geodesics $\gamma_{\star,1}, \dots \gamma_{\star, r}$. We give an asymptotic growth as $L \to +\infty$ of the number of primitive closed geodesic of length less than $L$ intersecting $\gamma_{\star,j}$ exactly $n_j$ times, where $n_1, \dots, n_r$ are fixed nonnegative integers. This is done by introducing a dynamical scattering operator associated to the surface with boundary obtained by cutting $\Sigma$ along $\gamma_{\star,1}, \dots, \gamma_{\star, r}$ and by using the theory of Pollicott-Ruelle resonances for open systems.