Illumination in Rational Billiards

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.