Degenerations of SL(2,C) representations and Lyapunov exponents

TL;DR

This paper investigates how the Lyapunov exponent behaves asymptotically in a family of random SL(2,C) matrix products near poles, revealing a connection to non-Archimedean Lyapunov exponents and stationary measures.

Contribution

It introduces a novel analysis of Lyapunov exponent degenerations in complex matrix families, linking complex and non-Archimedean dynamics.

Findings

Lyapunov exponent blow-up is governed by a non-Archimedean Lyapunov exponent.

Describes the limit of stationary measures on the complex projective line.

Establishes a connection between complex degenerations and non-Archimedean dynamics.

Abstract

We study the asymptotic behavior of the Lyapunov exponent in a meromorphic family of random products of matrices in 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 P 1 (C).