On the periodic orbits of a strongly chaotic system (Q1113508)

A point particle sliding freely on a two-dimensional surface of constant negative curvature (Hadamard-Gutzwiller model) exemplifies the simplest chaotic Hamiltonian system. Exploiting the close connection between hyperbolic geometry and the group \(\mathrm{SU}(1,1)/\{\pm 1\}\), we construct an algorithm (symbolic dynamics), which generates the periodic orbits of the system. For the simplest compact Riemann surface having as its fundamental group the ``octagon group'', we present an enumeration of more than 206 million periodic orbits. For the length of the \(n\)th primitive periodic orbit we find a simple expression in terms of algebraic numbers of the form \(m+\sqrt{2}n\) (\(m,n\in\mathbb N\) are governed by a particular Beatty sequence), which reveals a strange arithmetical structure of chaos. Knowledge of the length spectrum is crucial for quantization via the Selberg trace formula (periodic orbit theory), which in turn is expected to unravel the mystery of quantum chaos.