Degenerations of $\mathrm{SL}(2,ℂ)$ representations and Lyapunov exponents

Dujardin, Romain; Favre, Charles

Annales Henri Lebesgue, Tome 2 (2019), pp. 515-565.

Nous étudions le comportement asymptotique de l’exposant de Lyapunov dans une famille méromorphe de produits aléatoires de matrices dans $\mathrm{SL}(2,ℂ)$, lorsque le paramètre tend vers un pôle. Nous montrons que la divergence de l’exposant de Lyapunov est régie par une quantité qui peut être interprétée comme un exposant de Lyapunov non-archimédien. Nous décrivons également la limite de la famille correspondante de mesures stationnaires sur ${\mathbb{P}}^1\left(ℂ\right)$.

We study the asymptotic behavior of the Lyapunov exponent in a meromorphic family of random products of matrices in $\mathrm{SL}(2,ℂ)$, as the parameter converges to a pole. We show that the blow-up of the Lyapunov exponent is governed by a quantity which can be interpreted as the non-Archimedean Lyapunov exponent of the family. We also describe the limit of the corresponding family of stationary measures on ${\mathbb{P}}^1\left(ℂ\right)$.

Classification : 37H15, 37P50, 60B20, 60B15, 37F30

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