Periodic trajectories in right-triangle billiards

Abstract

Billiard problems are simple examples of Hamiltonian dynamical systems. These problems have been used as model systems to study the link betwen classical and quantum chaos. The heart of this linkage is provided by the periodic orbits in the classical system. In this article we will show that for an arbitrary right triangle, almost all trajectories that begin perpendicular to a side are periodic, that is, the set of points on the sides of a right triangle from which nonperiodic (perpendicular) trajectories begin is a set of measure zero. Our proof incorporates the previous result for rational right triangles (where the angles are rational multiples of \ensuremath{\pi}), while extending the result to nonrational right triangles.