Asymptotic behavior of the spectral radius of locally constant strongly irreducible cocycles

TL;DR

This paper investigates the asymptotic behavior of the spectral radius of certain matrix cocycles over subshifts, showing it converges to the top Lyapunov exponent under specific conditions, using large deviation techniques.

Contribution

It establishes conditions for exponential asymptotics of the spectral radius of GL(d,R)-valued cocycles, linking it to the top Lyapunov exponent, and employs large deviation estimates.

Findings

Spectral radius growth rate converges to the top Lyapunov exponent.

Provides conditions for exponential asymptotics of spectral radius.

Uses large deviation estimates for linear cocycles.

Abstract

We establish some conditions under which $\text{GL}(d,\mathbb{R})$-valued cocycles over a subshift of finite type, equipped with an equilibrium state, exhibit exponential asymptotics for the spectral radius. Specifically, we show that the exponential growth rate of the spectral radius converges to the top Lyapunov exponent of the cocycle. This result provides a partial answer to a question posed by Aoun and Sert in their paper "Law of large numbers for the spectral radius of random matrix products" (2021). Our approach relies on large deviation estimates for linear cocycles, which may be of independent interest.