Closed geodesics with prescribed intersection numbers
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 PollicottRuelle resonances for open systems.
 Publication:

arXiv eprints
 Pub Date:
 March 2021
 DOI:
 10.48550/arXiv.2103.16301
 arXiv:
 arXiv:2103.16301
 Bibcode:
 2021arXiv210316301C
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Differential Geometry;
 53C22;
 37C30;
 37D40
 EPrint:
 52 pages, 3 figures