Random and mean Lyapunov exponents for $\mathrm{GL}_n(\mathbb{R})$

TL;DR

This paper investigates the relationship between the average logarithm of eigenvalue moduli and Lyapunov exponents for random matrix products in _n(\u211d), providing bounds and employing spherical polynomial theory.

Contribution

It introduces a lower bound connecting eigenvalue logs and Lyapunov exponents for orthogonally invariant measures on _n()), utilizing spherical polynomials in the proof.

Findings

Established a lower bound for eigenvalue logs in terms of Lyapunov exponents.

Applied spherical polynomial theory to analyze random matrix products.

Results extend understanding of spectral properties in random matrix theory.

Abstract

We consider orthogonally invariant probability measures on $\mathrm{GL}_n(\mathbb{R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices to the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu-Shub\cite{DS}. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.