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

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