An abstract continuity theorem of the Lyapunov exponents

TL;DR

This paper introduces a versatile, abstract method to prove the continuity of Lyapunov exponents across various types of linear cocycles, relying on large deviation estimates and an inductive Avalanche Principle approach.

Contribution

It provides a unified, quantitative framework for establishing Lyapunov exponent continuity applicable to diverse cocycle classes with weaker assumptions.

Findings

Applicable to quasi-periodic and random cocycles

Provides explicit modulus of continuity based on LDT estimates

Uses an inductive Avalanche Principle for quantitative results

Abstract

We devise an abstract, modular scheme to prove continuity of the Lyapunov exponents for a general class of linear cocycles. The main assumption is the availability of appropriate large deviation type (LDT) estimates which are uniform in the data. We provide a modulus of continuity that depends explicitly on the sharpness of the LDT estimate. Our method uses an inductive procedure based on the deterministic, general Avalanche Principle in [4]. The main advantage of this approach, besides the fact that it provides quantitative estimates, is its versatility, as it applies to quasi-periodic cocycles (one and multivariable torus translations), to random cocycles (i.i.d. and Markov systems) and to any other types of base dynamics as long as appropriate LDT estimates are satisfied. Moreover, compared to other available quantitative results for quasi-periodic or random cocycles, this method allows for weaker assumptions. This is a draft of a chapter in our forthcoming research monograph [4].