TY - JOUR

T1 - On Singularly Perturbed Linear Cocycles over Irrational Rotations

AU - Ivanov, Alexey V.

PY - 2021/1/1

Y1 - 2021/1/1

N2 - 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.

AB - 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.

KW - exponential dichotomy

KW - linear cocycle

KW - Lyapunov exponent

KW - reducibility

UR - http://www.scopus.com/inward/record.url?scp=85107115462&partnerID=8YFLogxK

U2 - 10.1134/S1560354721030011

DO - 10.1134/S1560354721030011

M3 - Article

AN - SCOPUS:85107115462

VL - 26

SP - 205

EP - 221

JO - Regular and Chaotic Dynamics

JF - Regular and Chaotic Dynamics

SN - 1560-3547

IS - 3

ER -