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 \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 λ 2 ε \lambda _2^\varepsilon within an error of order ε 2 | log ⁡ ε | \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 λ 1 ε = 0 \lambda _1^\varepsilon =0 and λ 2 ε \lambda _2^\varepsilon are simple, and the only exceptional Lyapunov exponents of magnitude greater than − log ⁡ 2 + O ( log ⁡ log ⁡ 1 ε / log ⁡ 1 ε ) -\log 2+ O\Big (\log \log \frac 1\varepsilon \big /\log \frac 1\varepsilon \Big ) .