Periodic billiard orbits are dense in rational polygons

Abstract

We show that periodic orbits are dense in the phase space for billiards in polygons for which the angle between each pair of sides is a rational multiple of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi period"> <mml:semantics> <mml:mrow> <mml:mi>π</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\pi .</mml:annotation> </mml:semantics> </mml:math> </inline-formula>