Lyapunov statistics and mixing rates for intermittent systems

TL;DR

This paper investigates the relationship between Lyapunov exponents and mixing rates in weakly chaotic systems, demonstrating that a recent conjecture about their correlation decay fails for certain maps and that tail probabilities offer limited insight into system dynamics.

Contribution

The paper provides a rigorous analysis showing the failure of a conjecture relating Lyapunov exponents and correlation decay in Pomeau-Manneville maps, challenging previous assumptions.

Findings

The conjecture does not hold for Pomeau-Manneville type maps.

Tail probabilities of Lyapunov exponents do not reliably indicate ergodicity.

Rigorous results and numerical simulations confirm the findings.

Abstract

We consider here a recent conjecture stating that correlation functions and tail probabilities of finite time Lyapunov exponents would have the same power law decay in weakly chaotic systems. We demonstrate that this conjecture fails for a generic class of maps of the Pomeau-Manneville type. We show further that, typically, the decay properties of such tail probabilities do not provide significant information on key aspects of weakly chaotic dynamics such as ergodicity and instability regimes. Our approaches are firmly based on rigorous results, particularly the Aaronson-Darling-Kac theorem, and are also confirmed by exhaustive numerical simulations.