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

TL;DR

This paper studies classical chaotic systems on hyperbolic surfaces, proving the existence of partner orbits with similar actions for orbits with small-angle self-crossings, aiding understanding of spectral fluctuations in quantum chaos.

Contribution

It establishes the existence of partner orbits for geodesic flows with small-angle crossings on hyperbolic surfaces, providing estimates for their action differences, and extends to higher-order encounters.

Findings

Existence of partner orbits for small-angle self-crossings.

Estimate of action differences between orbit pairs.

Framework applicable to higher-order encounters.

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 specialize to 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 the physics literature. Furthermore, we derive an estimate for the action difference of the partners. In the second part of this paper [13], an inductive argument is provided to deal with higher-order encounters.