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Closed geodesics with prescribed intersection numbers

Yann Chaubet

Abstract

Let $(Σ, g)$ be a closed, oriented, negatively curved surface, and fix pairwise disjoint simple closed geodesics $γ_{\star,1}, \dots γ_{\star, r}$. We give an asymptotic growth as $L \to +\infty$ of the number of primitive closed geodesic of length less than $L$ intersecting $γ_{\star,j}$ 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 $Σ$ along $γ_{\star,1}, \dots, γ_{\star, r}$ and by using the theory of Pollicott-Ruelle resonances for open systems.

Closed geodesics with prescribed intersection numbers

Abstract

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

Paper Structure

This paper contains 43 sections, 31 theorems, 304 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Geometrical preliminaries
  3. Structural equations
  4. The Anosov property
  5. A nice system of coordinates
  6. Cutting the surface along $\gamma_\star$
  7. The resolvent of the geodesic vector field for open systems
  8. Restriction of the resolvent on the geodesic boundary
  9. The scattering operator
  10. First definitions
  11. The scattering operator via the resolvent
  12. Composing the scattering maps
  13. The flat trace of the scattering operator
  14. A priori bounds on the growth of geodesics with fixed intersection number with $\gamma_\star$
  15. The case $\gamma_\star$ is not separating
  16. ...and 28 more sections

Key Result

Theorem 1

Let $\mathbf{n} = (n_1, \dots, n_r) \in \mathbb{N}^{r}$. If $N(\mathbf{n}, L) > 0$ for some $L > 0$, then there are $C_\mathbf{n} > 0, d_\mathbf{n} \in \mathbb{N}$ and $h_{\mathbf{n}} \in ]0, h[$ such that

Figures (3)

  • Figure 1: A closed geodesic $\gamma$ on $\Sigma$. Here we have $r = 5$, $q = 3$, and $\omega(\gamma) \sim (u,v)$ where $u = (1, 2, 4, 5, 4, 3, 2)$ and $v = (1, 1, 2, 3, 2, 3, 2)$ (the starting point of $\gamma$ is the red arrow).
  • Figure 2: The surfaces $\Sigma$ (on the left) and $\Sigma_\delta$ (on the right), in the case where $\gamma_\star$ is not separating. In $\Sigma$, the darker region corresponds to the neighborhood $\pi(U)$ of $\gamma_\star$.
  • Figure 3: Proof of Lemma \ref{['lem:0']}.

Theorems & Definitions (60)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1.1
  • Theorem 4
  • Corollary 5
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • ...and 50 more