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 ε \varepsilon , quantifying strength leakage between two nearly invariant regions. We show that system exhibits metastability, and identify second Lyapunov exponent alttext="lamda 2 Superscript epsilon"> λ encoding="application/x-tex">\varepsilon ^2|\log \varepsilon | . approximation agrees with naive prediction provided by time-dependent two-state Markov chain. Furthermore, it is shown 1 Baseline equals 0"> 1 = 0 _1^\varepsilon =0 are simple, only exceptional exponents magnitude greater than alttext="minus plus upper O left-parenthesis StartFraction Over EndFraction slash right-parenthesis"> −<!-- − <mml:mo>+ O maxsize="1.623em" minsize="1.623em">( fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/ minsize="1.623em">) encoding="application/x-tex">-\log 2+ O\Big (\log \log \frac 1\varepsilon \big /\log \Big )