Quantitative lower bounds on the Lyapunov exponent from multivariate matrix inequalities

Abstract

Lyapunov exponents characterize the asymptotic behavior of long matrix products. In this work we introduce a new technique that yields quantitative lower bounds on the top Lyapunov exponent in terms of an efficiently computable matrix sum in ergodic situations. Our approach rests on two results from matrix analysis—the n-matrix extension of the Golden–Thompson inequality and an effective version of the Avalanche Principle. While applications of this method are currently restricted to uniformly hyperbolic cocycles, we include specific examples of ergodic Schrödinger cocycles of polymer type for which outside of the spectrum our bounds are substantially stronger than the standard Combes–Thomas estimates. We also show that these techniques yield short proofs of quantitative stability results for the top Lyapunov exponent which are known from more dynamical approaches. We also discuss the problem of finding stable bounds on the Lyapunov exponent for almost-commuting matrices.