Closed geodesics with prescribed intersection numbers

Chaubet, Yann

doi:10.2140/gt.2024.28.701

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Closed geodesics with prescribed intersection numbers

Yann Chaubet

Geometry & Topology 28 (2024) 701–758

DOI: 10.2140/gt.2024.28.701

Abstract

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

Keywords

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

Mathematical Subject Classification

Primary: 37D40

References

Publication

Received: 27 August 2021

Revised: 4 May 2022

Accepted: 20 July 2022

Published: 13 March 2024

Proposed: Benson Farb

Seconded: Mladen Bestvina, Dmitri Burago

Authors

Yann Chaubet
Institut de Mathématiques d’Orsay Université Paris-Saclay Orsay France	Laboratoire de Mathématiques Jean Leray Université de Nantes Nantes France

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Volume 28, issue 2 (2024)