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 Poincaré 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.