On singularly perturbed linear cocyles over irrational rotations

Abstract

We study a linear cocycle over irrational rotation ${\sigma }_{\omega }(x)=x+\omega$ of a circle ${\mathbb{T}}^1$. It is supposed the cocycle is generated by a $C^1$-map $A_{\epsilon }:{\mathbb{T}}^1\to SL(2,\mathbb{R})$ which depends on a small parameter $\epsilon \ll 1$ and has the form of the Poincaré map corresponding to a singularly perturbed Schrödinger equation. Under assumption the eigenvalues of $A_{\epsilon }(x)$ to be of the form $\exp (\pm \lambda (x)/\epsilon )$, 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 $\epsilon$. We show that in the limit $\epsilon \to 0$ 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.

Cited by in corpus (1)

• On $SL(2,\mathbb{R})$-cocycles over irrational rotations with secondary collisions