On Singularly Perturbed Linear Cocycles over Irrational Rotations

We study a linear cocycle over the irrational rotation $\sigma_{\omega}(x) = x + \omega$ of the circle~$\mathbb{T}^{1}$. It is supposed that the cocycle is generated by a $C^{2}$-map $A_{\varepsilon}: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ which depends on a small parameter $\varepsilon\ll 1$ and has the form of the Poincar\'e map corresponding to a singularly perturbed Hill equation with quasi-periodic potential. Under the assumption that the norm of the matrix $A_{\varepsilon}(x)$ is of order $\exp(\pm \lambda(x)/\varepsilon)$, where $\lambda(x)$ is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter~$\varepsilon$. We show that in the limit $\varepsilon\to 0$ the cocycle ``typically'' exhibits ED only if it is exponentially close to a constant cocycle. Conversely, if the cocycle is not close to a constant one, it does not possess ED, whereas the Lyapunov exponent is ``typically'' large.