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 $\varepsilon$, 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 $-\log 2+ O\Big (\log \log \frac 1\varepsilon \big /\log \frac 1\varepsilon \Big )$.