Analytic Dependence of the Lyapunov Moment Function and the Projective Stationary Measure for Random Matrix Products

TL;DR

The paper analyzes the analytic dependence of the Lyapunov moment function and the projective stationary measure for products of i.i.d. random matrices drawn from a proximal-irreducible setup. By introducing the $q$-twisted Koopman operator $\\mathcal{P}_{q,\\mu}$ on Hölder spaces and proving its joint Fréchet analyticity in $(q,\\mu)$, it establishes a spectral gap with a simple dominant eigenvalue $e^{\\Lambda_\\mu(q)}$, linking the Lyapunov moment function to operator spectral data. A Kato-style perturbation framework then yields holomorphic dependence of the leading eigenvalue and eigenfunction on $(q,\\mu)$, which in turn implies the analyticity of $\\Lambda_\\mu(q)$ in a neighborhood and the analyticity of all higher Lyapunov moments $\\Lambda_\\mu^{(k)}(0)$ as well as the stationary measure $\\nu_\\mu$ on the projective space. Consequently, the paper extends known results on the analytic dependence of the top Lyapunov exponent to higher moments and the invariant measure, providing a robust infinite-dimensional perturbation approach for random matrix products. This deepens understanding of how noise distributions influence invariant quantities in random dynamical systems.

Abstract

We consider the product of i.i.d. random matrices sampled according to a probability measure $μ$ supported on a strongly irreducible and proximal subset of a compact set $S\subset GL(d,\mathbb{R})$. We establish the local analyticity of the Lyapunov moment function and the unique stationary measure on the projective space with respect to $μ$ in the total variation topology. As a consequence, we obtain the analyticity of the asymptotic variance and all higher-order Lyapunov moments.

Theorems & Definitions (23)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 3.1
• Lemma 3.2: RM_Book
• Lemma 3.3
• Definition 3.4