Year: 2021
Type: article
Abstract: 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.
Source: Regular and Chaotic Dynamics
Author: Alexey V. Ivanov
Institution: St Petersburg University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3047500612