Illumination in Rational Billiards

TL;DR

This paper investigates illumination and blocking properties in rational polygonal billiards, establishing finiteness results and extending previous findings through advanced dynamical systems techniques on translation surfaces.

Contribution

It proves finiteness of non-illuminating point pairs in rational billiards and extends existing results on finitely blocked pairs using moduli space dynamics.

Findings

Finite set of non-illuminating point pairs in rational billiards

Extended results on finitely blocked pairs with specific blocking cardinalities

Applied advanced dynamics of translation surfaces to billiard problems

Abstract

We show that for a rational polygonal billiard, the set of pairs of points that do not illuminate each other (not connected by a billiard trajectory) is finite, and use the same method to extend the results of Leli\`evre, Monteil and Weiss, and of Apisa and Wright about the amount of pairs of points that are finitely blocked with a certain blocking cardinality. We rely on previous work about the blocking property in translation surfaces which ultimately stems from results of Eskin, Mirzakhani and Mohammadi on dynamics of moduli spaces of translation surfaces.