Intersection Numbers of Geodesic Arcs

TL;DR

This paper investigates the distribution and asymptotic behavior of intersection numbers of closed geodesics on negatively curved surfaces, providing exponential estimates and confirming a conjecture about typical self-intersection counts.

Contribution

It establishes exponential tail bounds for intersection number distributions and confirms a conjecture on the typical self-intersection count of geodesics.

Findings

Exponential decay in the tail distribution of normalized intersection numbers.

Asymptotic average of intersection numbers for large geodesics.

Most geodesics with length T have self-intersections close to a quadratic function of T.

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 $S$ with $l(\alpha)\leq T$ have roughly $\kappa l(\alpha)^2/(2\pi^2(g-1))$ self-intersections, when $T$ is large.