Exponential dichotomy of linear cocycles over irrational rotations

We study a linear cocycle over irrational rotation σ
_ω
(x) = x + ω of a circle T
¹
. It is supposed that the cocycle is generated by a C1-map A
_ε
: T
₁
→ SL(2,R) that depends on a small parameter ε ≪ 1 and has the form of the Poincarémap corresponding to a singularly perturbed Schrödinger equation. Under assumption that the eigenvalues of A
_ε
(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.