Billiards in right triangles and orbit closures in genus zero strata

TL;DR

This paper classifies orbit closures in certain flat surfaces derived from rational triangles and polygons, and deduces the asymptotic behavior of billiard trajectories in these shapes.

Contribution

It computes orbit closures for unfoldings of rational right and isosceles triangles and classifies all rank at least two orbit closures in hyperelliptic strata.

Findings

Orbit closures of unfoldings of rational right and isosceles triangles are explicitly computed.

Asymptotic counts of periodic billiard trajectories are derived.

Orbit closures for various polygons are classified outside a discrete set.

Abstract

The orbit closure of the unfolding of every rational right and isosceles triangle is computed and the asymptotic number of periodic billiard trajectories in these triangles is deduced. This follows by classifying all orbit closures of rank at least two in hyperelliptic components of strata of Abelian and quadratic differentials. Additionally, given a fixed set of angles, the orbit closure of the unfolding of all unit area rational parallelograms, isosceles trapezoids, and right trapezoids outside of a discrete set is determined.