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 $ε$, 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 $λ_2^ε$ within an error of order $ε^2|\log ε|$. This approximation agrees with the naive prediction provided by a time-dependent two-state Markov chain. Furthermore, it is shown that $λ_1^ε=0$ and $λ_2^ε$ are simple, and are the only exceptional Lyapunov exponents of magnitude greater than $-\log2+ O(\log\log\tfrac 1ε/\log\tfrac 1ε)$.