## On singularly perturbed linear cocycles over irrational rotations.(English)Zbl 1484.37057

Summary: 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.

### MSC:

 37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations 37C55 Periodic and quasi-periodic flows and diffeomorphisms 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
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