Lyapunov exponents for transfer operator cocycles of metastable maps: a quarantine approach

TL;DR

This paper analyzes the Lyapunov spectrum of transfer operator cocycles for metastable maps, revealing how the second Lyapunov exponent depends on leakage strength and confirming predictions from a Markov chain model.

Contribution

It provides a precise approximation of the second Lyapunov exponent for metastable maps and demonstrates its simplicity and relation to metastability phenomena.

Findings

Second Lyapunov exponent approximated within rac{psilon^2}{|psilon|} error

Second exponent is simple and uniquely significant among exponents

Approximation aligns with a two-state Markov chain prediction

Abstract

This works investigates the Lyapunov-Oseledets spectrum of transfer operator cocycles associated to one-dimensional random paired tent maps depending on a parameter $\epsilon$, quantifying the strength of the \emph{leakage} between two nearly invariant regions. We show that the system exhibits metastability, and identify the second Lyapunov exponent $\lambda_2^\epsilon$ within an error of order $\epsilon^2|\log \epsilon|$. This approximation agrees with the naive prediction provided by a time-dependent two-state Markov chain. Furthermore, it is shown that $\lambda_1^\epsilon=0$ and $\lambda_2^\epsilon$ are simple, and are the only exceptional Lyapunov exponents of magnitude greater than $-\log2+ O(\log\log\tfrac 1\epsilon/\log\tfrac 1\epsilon)$.