Partner orbits and action differences on compact factors of the hyperbolic plane. II: Higher-order encounters

TL;DR

This paper extends the understanding of classical chaotic systems by analyzing higher-order orbit encounters in hyperbolic plane flows, revealing the number and action differences of partner orbits.

Contribution

It provides an inductive method to identify multiple partner orbits for higher-order encounters, advancing the mathematical understanding of orbit pairings in chaotic systems.

Findings

Number of partner orbits for an L-encounter is (L-1)! - 1.

Constructs partner orbits and estimates their action differences.

Extends previous work on 2-encounters 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. We specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. The companion paper [Huynh and Kunze, 2015] proved the existence of a unique periodic partner orbit for a given periodic orbit with a small-angle self-crossing in configuration space that is a 2-encounter and derived an estimate for the action difference of the orbit pair. In this paper, we provide an inductive argument to deal with higher-order encounters: we prove that a given periodic orbit including an $L$-parallel encounter has $(L-1)!-1$ partner orbits; we construct partner orbits and give estimates for the action differences between orbit pairs.