Billiards on rational-angled triangles

Abstract

Let P be a polygon in the plane. A point moves along a billiard trajectory in P if it moves with constant speed along a straight line in the interior of P and if it reflects off edges so that the speed is unchanged and the angle of incidence is equal to the angle of reflection. If a billiard trajectory hits a vertex then we do not define the path further. We are interested in statistical properties of trajectories. When are they periodic? If they are not periodic to what extent do they “fill up” the polygon? Billiard trajectories are projections of orbits of a flow on a four dimensional phase space which we can think of as the tangent bundle of P . We can make this flow continuous away from the corners by identifying the appropriate inward and outward pointing vectors over the edges of P . We leave the flow undefined over the vertices of P . The billiard flow can be thought of as a geodesic flow and in particular as a Hamiltonian flow ( a general reference for Hamiltonian flows is [Ad]). Hamiltonian flows can be quite simple or quite complex; a first step in analyzing the complexity of such a flow is to look for “integrals of motion”. These are functions on the phase space which are invariant under the flow. For billiard flows the length of a tangent vector provides one such integral of motion. We have a second integral of motion in the special case in which the polygon P is rational i.e. each of its corner angles is a rational multiple of 2π. The existence of these two integrals of motion mean that the billiard flow is effectively taking place on