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.


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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