Partner orbits and action differences on compact factors of the hyperbolic plane. I: Sieber–Richter pairs

Abstract

Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions. In this paper we consider the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. We prove the existence of a periodic partner orbit for a given periodic orbit which has a small-angle self-crossing in configuration space which is a ‘2-encounter’; such configurations are called ‘Sieber–Richter pairs’ in physics literature. Furthermore, we derive an estimate for the action difference of the partners. In the second part of this paper (Huynh, submitted), an inductive argument is provided to deal with higher-order encounters.