On singularly perturbed linear cocyles over irrational rotations

TL;DR

This paper investigates the behavior of singularly perturbed linear cocycles over irrational rotations, revealing that exponential dichotomy typically occurs only when the cocycle is close to constant, with large Lyapunov exponents otherwise.

Contribution

It provides a detailed analysis of the conditions under which singularly perturbed cocycles exhibit exponential dichotomy, especially near the limit of small perturbation parameter.

Findings

Exponential dichotomy occurs only when the cocycle is exponentially close to constant.

Large Lyapunov exponents are typical when the cocycle is not close to constant.

The behavior of the cocycle is characterized in the limit as the perturbation parameter approaches zero.

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 $A_{\varepsilon}: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ which depends on a small parameter $\varepsilon\ll 1$ and has the form of the Poincar\'e map corresponding to a singularly perturbed Schr\"odinger equation. Under assumption the eigenvalues of $A_{\varepsilon}(x)$ to be of the form $\exp(\pm \lambda(x)/\varepsilon)$, 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. In contrary, if the cocycle is not close to a constant one it does not posesses ED, whereas the Lyapunov exponent is "typically" large.