Closed geodesics with prescribed intersection numbers

TL;DR

This paper studies the asymptotic growth of primitive closed geodesics on negatively curved surfaces with prescribed intersection numbers, using a dynamical scattering operator and Pollicott-Ruelle resonances.

Contribution

It introduces a novel dynamical scattering operator and applies resonance theory to count geodesics with fixed intersection patterns.

Findings

Derived asymptotic formulas for geodesic counts with fixed intersection numbers.

Connected scattering operator spectral properties to geodesic distribution.

Extended resonance techniques to open dynamical systems on surfaces.

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.