Closed geodesics with prescribed intersection numbers

Abstract

Let (Σ, g) be a closed, oriented, negatively curved surface, and fix pairwise disjoint simple closed geodesics γ ,1 , . . . γ ,r . We give an asymptotic growth as L → +∞ of the number of primitive closed geodesic of length less than L intersecting γ ,j exactly n j times, where n 1 , . . . , n r are fixed nonnegative integers. This is done by introducing a dynamical scattering operator associated to the surface with boundary obtained by cutting Σ along γ ,1 , . . . , γ ,r and by using the theory of Pollicott-Ruelle resonances for open systems.

Closed billiard trajectories with prescribed bounces

“…Acknowledgements. I thank Colin Guillarmou for fruitful discussions and for his relecture of the present work, as well as Frédéric Naud for suggesting to consider this problem, which is somehow analogous to the one considered in [Cha21]. Finally I thank Benjamin Küster, Philipp Schütte and Tobias Weich for important discussions about their recent work [KSW].…”

“…Our approach for proving (0.2) is reminiscent of a previous work [Cha21] about the asymptotic growth of the number of closed geodesics on negatively curved surfaces for which certain intersection numbers are prescribed. In particular we make use of the work of Dyatlov-Guillarmou [DG16] about the existence of Pollicott-Ruelle resonances for open hyperbolic systems (the recent work of Küster-Schütte-Weich [KSW] details how a hyperbolic billiard flow can be described by the framework of [DG16]).…”

Resonances and weighted zeta functions for obstacle scattering via smooth models

“…This problem initially provided the main motivation for us to start this project. Another application is given by Yann Chaubet's work [Cha21a] on counting asymptotics of periodic trajectories with a fixed reflection number at some fixed obstacle based on previous results in a non-billiard setting [Cha21b]. A further motivation for this paper was to write down in detail some constructions that have so far been used mostly implicitly in the literature such as smooth models, and to work in a general setting that clearly differentiates between the geometric and dynamical features of Riemannian obstacle scattering.…”