TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 9, September 1998, Pages 3523–3535 S 0002-9947(98)02089-3 PERIODIC BILLIARD ORBITS ARE DENSE IN RATIONAL POLYGONS M. BOSHERNITZAN, G. GALPERIN, T. KR ¨ UGER, AND S. TROUBETZKOY 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 π. 1. Introduction A billiard ball, i.e. a point mass, moves inside a polygon Q with unit speed along a straight line until it reaches the boundary ∂Q of the polygon, then instantaneously changes direction according to the mirror law: “the angle of incidence is equal to the angle of reflection,” and continues along the new line (Fig. 1(a)). Despite the simplicity of this description there is much that is unknown about the existence and the description of periodic orbits in arbitrary polygons. On the other hand, quite a bit is known about a special class of polygons, namely, rational polygons. A polygon is called rational if the angle between each pair of sides is a rational multiple of π. The main theorem we will prove is Theorem 1. For rational polygons, periodic points of the billiard flow are dense in the phase space M of the billiard flow. Theorem 1 is a strengthening of Masur’s theorem [M], which says that any rational polygon has “many” periodic billiard trajectories; more precisely, the set of directions of the periodic trajectories is dense in the set of velocity directions S 1 . We will also prove some refinements of Theorem 1: the “well distribution” of periodic orbits in the polygon and the residuality of the points q ∈ Q with a dense set of periodic directions (precise statements will be given in section 3). The structure of the article is as follows. In section 2 we give a brief description of billiards in polygons and some results related to Theorem 1. In section 3 we state the strengthenings of Theorem 1, and the proofs will be given in section 4. 2. Description of billiards in polygons The trace of a moving billiard ball is called a billiard trajectory or orbit. If a billiard trajectory hits a vertex of the polygon, then it is called singular. For convenience we will define such billiard trajectories by continuity from the left (with respect to a fixed orientation of the boundary); thus every trajectory is infinite and Received by the editors July 29, 1996. 1991 Mathematics Subject Classification. Primary 58F05. MB is partially supported by NSF-DMS-9224667. GG thanks the Alexander von Humboldt Stiftung for their support. ST thanks the Deutsche Forschungsgemeinschaft for their support. c 1998 American Mathematical Society 3523 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use