Intersection Numbers of Geodesic Arcs
Yoe Alexander Herrera Jaramillo

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 with constant negative curvature (for some ) and genus , we show that the tails of the distribution of (where is the intersection number of the closed geodesics and 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 . In addition, we prove that the size of the sets of geodesics whose -self-intersection number is not close to 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 on with have roughly …
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Mathematical Dynamics and Fractals · Geometry and complex manifolds
