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arXiv · 2103.16301

Closed geodesics with prescribed intersection numbers

Yann Chaubet
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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.

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Yann Chaubet. 2021-03-30. Closed geodesics with prescribed intersection numbers. https://doi.org/10.2140/gt.2024.28.701

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