Closed geodesics with prescribed intersection numbers

Chaubet, Yann¹

¹ Institut de Mathématiques d’Orsay, Université Paris-Saclay, Orsay, France, Laboratoire de Mathématiques Jean Leray, Université de Nantes, Nantes, France

Geometry & topology, Tome 28 (2024) no. 2, pp. 701-758.

Résumé

Let $(\mathrm{\Sigma },g)$ be a closed oriented negatively curved surface, and fix a simple closed geodesic ${\gamma }_{\star }$. We give the asymptotic growth as $L\to +\infty$ of the number of primitive closed geodesics of length less than $L$ intersecting ${\gamma }_{\star }$ 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 $\mathrm{\Sigma }$ along ${\gamma }_{\star }$ and by using the theory of Pollicott–Ruelle resonances for open systems.

Détail

DOI : 10.2140/gt.2024.28.701

Keywords: closed geodesics, intersection numbers, microlocal analysis, Ruelle resonances, scattering

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