Intersection Numbers of Geodesic Curves in a Surface

TL;DR

This paper investigates the statistical distribution of intersection numbers of geodesic curves on negatively curved surfaces, providing exponential tail estimates and asymptotic behaviors for intersection counts and self-intersections.

Contribution

It offers new exponential estimates for the distribution tails of normalized intersection numbers and self-intersections of geodesics on hyperbolic surfaces, extending previous asymptotic results.

Findings

Exponential decay in the tail distribution of normalized intersection numbers.

Asymptotic average of intersection numbers for pairs of geodesics.

Most long geodesics have self-intersection counts close to a quadratic function of length.

Abstract

For a compact surface $S$ with constant negative curvature $-\kappa$ (for some $\kappa>0$) and genus $g\geq2$, we show that the tails of the distribution of $i(\alpha,\beta)/l(\alpha)l(\beta)$ (where $i(\alpha,\beta)$ is the intersection number of the closed geodesics and $l(\cdot)$ denotes the geometric length) are estimated by a decreasing exponential function. As a consequence, we find the asymptotic normalized average of the intersection numbers of pairs of closed geodesics on $S$. In addition, we prove that the size of the sets of geodesics whose $T$-self-intersection number is not close to $\kappa T^2/(2\pi^2(g-1))$ is also estimated by a decreasing exponential function. And, as a corollary of the latter, we obtain a result of S. Lalley which states that most of the closed geodesics $\alpha$ on $S$ of length $\leq T$ have roughly $\kappa l(\alpha)^2/(2\pi^2(g-1))$ self-intersections, when $T$ is large.