Periodic orbits of an almost integrable system

Abstract

The authors enumerate and classify all the periodic orbits of the ( pi /3)-rhombus billiard. The billiard flow for this system resides on a classically invariant surface of genus two. After transforming the rhombus billiard trajectories on an exactly equivalent set of trajectories on a barrier billiard (with barrier-to-gap ratio equal to 2), they present a criterion facilitating the complete enumeration of periodic billiard trajectories. Having enumerated the trajectories, they classify them distinctly and provide the underlying number-theoretic rationale for the same. Their analysis involves a 'polar construction' which facilitates the determination of lengths of periodic orbits and the areas of the bands in which they occur. It is clear from the analysis that there are no isolated periodic trajectories implying that all the periodic orbits close after an even number of reflections from the boundaries. These results are used to study the family-counting function F(x) (i.e. the number of families of periodic orbits of length less than x) as a function of x.