Geodesic
Lyapunov exponents for transfer operator cocycles of~metastable maps: a quarantine approach
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 82 (2021) no. 1, pp. 79-92.

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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 $-\log2+ O\Big(\log\log\frac 1\varepsilon\big/\log\frac 1\varepsilon\Big)$. 
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C. González-Tokman; A. Quas. Lyapunov exponents for transfer operator cocycles of~metastable maps: a quarantine approach. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 82 (2021) no. 1, pp. 79-92. http://geodesic.mathdoc.fr/item/MMO_2021_82_1_a3/

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