Leading Pollicott-Ruelle Resonances for Chaotic Area-Preserving Maps

TL;DR

This paper analytically calculates the leading Pollicott-Ruelle resonances for a broad class of chaotic area-preserving maps, revealing insights into decay rates of correlations in mixed dynamical systems.

Contribution

It provides the first analytical derivation of leading resonances for these maps, including both diffusive and angular correlation decay rates.

Findings

Analytical expressions for leading Pollicott-Ruelle resonances.

Comparison with existing numerical and theoretical results.

Identification of multiple decay rates in chaotic area-preserving maps.

Abstract

Recent investigations in nonlinear sciences show that not only hyperbolic but also mixed dynamical systems may exhibit exponential relaxation in the chaotic regime. The relaxation rates, which lead the decay of probability distributions and correlation functions, are related to the classical evolution resolvent (Perron-Frobenius operator) pole logarithm, the so called Pollicott-Ruelle resonances. In this Brief Report, the leading Pollicott-Ruelle resonances are calculated analytically for a general class of area-preserving maps. Besides the leading resonances related to the diffusive modes of momentum dynamics (slow rate), we also calculate the leading faster rate, related to the angular correlations. The analytical results are compared to the existing results in the literature.