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

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 $-\log 2+O(\log \log \frac{1}{\epsilon }/\log \frac{1}{\epsilon })$.