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Closed geodesics with prescribed intersection numbers

Yann Chaubet

Geometry & Topology 28 (2024) 701–758
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 + 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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