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.

我们研究了圆\(\mathbb{T}^{1}\)的无理旋转\(\sigma_{\omega}(x)=x+\omega\)上的线性共循环。假设共环是由\(C^{2}\) -map \(A_{\varepsilon}:\mathbb{T}^{1}\to SL(2,\mathbb{R}) \)取决于一个小参数\(\varepsilon\ll 1\)并且具有 Poincaré 映射的形式，对应于具有准周期势的奇异扰动 Hill 方程。假设矩阵\(A_{\varepsilon}(x)\)的范数是\(\exp(\pm\lambda(x)/\varepsilon)\)，其中\(\lambda(x) )\)是一个正函数，我们检查了关于参数\(\varepsilon\)的 cocycle 具有指数二分法 (ED) 的属性。我们表明，在极限\(\varepsilon\to 0\) 中，共环“通常”仅在它以指数方式接近恒定共环时才表现出 ED。相反，如果共环不接近常数，则它不具有 ED，而李雅普诺夫指数“通常”很大。