PERIODIC-ORBITS ON THE REGULAR HYPERBOLIC OCTAGON

The length spectrum of closed geodesics on a compact Riemann surface corresponding to a regular octagon on the Poincare disc is investigated. The general form of the elements of the "octagon group", a discrete subgroup of SU(1,1)/{+/- 1}, in terms of 2 x 2 matrices is derived, and the previously conjectured law for the length of periodic orbits is proved analytically. An algorithm for the multiplicity of geodesics with a given length is developed, which leads to an efficient enumeration of the periodic orbits of this strongly chaotic system.