Chaotic billiards generated by arithmetic groups

Abstract

It is known that statistical properties of the energy levels for various billiards on a constant-negative-curvature surface do not follow the universal random-matrix predictions. We show that nongeneric behavior of the systems investigated so far originates from the special arithmetic nature of their tiling groups, which produces an exponentially large degeneracy of lengths of periodic orbits. A semiclassical study of the two-point correlation function shows that the spectral fluctuations are close to Poisson-like ones, typical of integrable systems.