Clustering of periodic orbits and ensembles of truncated unitary matrices

TL;DR

This paper links the clustering behavior of periodic orbits in chaotic systems to spectral properties of truncated unitary matrices, providing asymptotic estimates for cluster sizes based on spectral edge universality.

Contribution

It introduces a spectral approach to analyze cluster sizes of periodic orbits using ensembles of truncated unitary matrices, revealing asymptotic behavior.

Findings

Derived asymptotics for second moment of cluster distribution

Estimated average cluster size based on number of encounters

Connected clustering in chaotic systems to spectral edge universality

Abstract

Periodic orbits in chaotic systems form clusters, whose elements traverse approximately the same points of the phase space. The distribution of cluster sizes depends on the length n of orbits and the parameter p which controls closeness of orbits actions. We show that counting of cluster sizes in the baker's map can be turned into a spectral problem for an ensemble of truncated unitary matrices. Based on the conjecture of the universality for the eigenvalues distribution at the spectral edge of these ensembles, we obtain asymptotics of the second moment of cluster distribution in a regime where both n and p tend to infinity. This result allows us to estimate the average cluster size as a function of the number of encounters in periodic orbits.