论文标题
在非理性旋转的奇异扰动线性共晶上
On singularly perturbed linear cocyles over irrational rotations
论文作者
论文摘要
我们在非理性旋转上研究了一个线性旋转$σ_Ω（x）= x +ω$的圆圈$ \ mathbb {t}^{1} $。 It is supposed the cocycle is generated by a $C^{1}$-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干扰的schrödinger方程。在假设下，$ a _ {\ varepsilon}（x）$的特征值属于$ \ exp（\pmpmλ（x）/\ varepsilon）$，其中$λ（x）$是一个积极的函数，我们检查了与cocycle comparmote of ed $ ed $ commote $ agams $ commote commoge的属性。我们表明，在限制$ \ varepsilon \至0 $中，“通常” cocycle仅在成倍接近恒定的共生时才显示出ED。相反，如果旋子不接近一个常数，则不会摆姿势，而Lyapunov指数通常“通常”大。
We study a linear cocycle over irrational rotation $σ_ω(x) = x + ω$ of a circle $\mathbb{T}^{1}$. It is supposed the cocycle is generated by a $C^{1}$-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 Schrödinger equation. Under assumption the eigenvalues of $A_{\varepsilon}(x)$ to be of the form $\exp(\pm λ(x)/\varepsilon)$, where $λ(x)$ 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. In contrary, if the cocycle is not close to a constant one it does not posesses ED, whereas the Lyapunov exponent is "typically" large.
