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.

52 pages, 3 figures

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