On Singularly Perturbed Linear Cocycles over Irrational Rotations

We study a linear cocycle over the irrational rotation sigma(omega) (x) = x + omega of the circle T-1. It is supposed that the cocycle is generated by a C-2 -map A(epsilon) : T-1 SL(2, R) which depends on a small parameter epsilon << 1 and has the form of the Poincare map corresponding to a singularly perturbed Hill equation with quasi-periodic potential. Under the assumption that the norm of the matrix A(epsilon)(x) is of order exp(+/-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 -> 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.

Automated summary

The study focuses on the properties of a linear cocycle on the circle T-1 generated by a C-2 map A(epsilon) depending on a small parameter epsilon and having the form of a Poincare map. It examines the behavior of the cocycle with an assumption on the norm of the matrix A(epsilon)(x) and discusses its exponential dichotomy with respect to the parameter epsilon. The results suggest that the cocycle typically exhibits an exponential dichotomy only when it is exponentially close to a constant cocycle in the limit epsilon -> 0.