Counting of Sieber-Richter pairs of periodic orbits

TL;DR

This paper investigates the distribution and clustering of periodic orbits in chaotic systems using symbolic dynamics, deriving probabilities related to orbit degeneracies and cluster sizes in the context of semiclassical spectral correlations.

Contribution

It introduces a novel ultrametric clustering approach for periodic orbits and derives asymptotic probabilities for cluster sizes and degeneracies in the length spectrum.

Findings

Derived probability that k orbits belong to the same cluster

Analyzed asymptotic behavior of the largest cluster size

Established connection to degeneracies in de Bruijn graph spectra

Abstract

In the framework of the semiclassical approach the universal spectral correlations in the Hamiltonian systems with classical chaotic dynamics can be attributed to the systematic correlations between actions of periodic orbits which (up to the switch in the momentum direction) pass through approximately the same points of the phase space. By considering symbolic dynamics of the system one can introduce a natural ultrametric distance between periodic orbits and organize them into clusters. Each cluster consists of orbits approaching closely each other in the phase space. We study the distribution of cluster sizes for the backer's map in the asymptotic limit of long trajectories. This problem is equivalent to the one of counting degeneracies in the length spectrum of the {\it de Bruijn} graphs. Based on this fact, we derive the probability $\P_k$ that $k$ randomly chosen periodic orbits belong to the same cluster. Furthermore, we find asymptotic behaviour of the largest cluster size $|\Cll_{\max}|$ and derive the probability $P(t)$ that a random periodic orbit belongs to a cluster of the size smaller than $t|\Cll_{\max}|$, $t\in[0,1]$.