Exponential dichotomy of linear cocycles over irrational rotations

Abstract

We study a linear cocycle over irrational rotation σω(x) = x+ω of a circle $\mathbb{T}^1$. It is supposed that the cocycle is generated by a $A_\varepsilon :\mathbb{T}^1 \to SL(2,\mathbb{R})$ that depends on a small parameter ε ≪ 1 and has the form of the Poincaré map corresponding to a singularly perturbed Schrödinger equation. Under assumption that the eigenvalues of Aε(x) are of the form exp (±}λ(x)/ε), where λ(x) is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter ε. We show that in the limit ε → 0 the cocycle exhibits ED for the most parameter values only if it is exponentially close to a constant cocycle. In the other case, when the cocycle is not close to a constant one and, thus, it does not possess ED, the Lyapunov exponent is typically large.