Renormalization of cocycles and linear ODE with almost-periodic coefficients

In the current paper we address the problem of classification of cocycles over an irrational rotation. We use the renormalization group approach. A cocycle means a C^r-mapping u:T →SU (2, ℂ) (r≧2). We fix an irrational rotation τ:T →T. Two cocycles u, v:T →SU(2, ℂ) are considered equivalent (or cohomologous) if there is a continuous map h:T→SU (2, ℂ) such that u·h{o script} τ=h·v. By definition, our problem is to classify cocycles up to this equivalence relation. By introducing a suitable renormalization map we are able to define a notion of a fiber rotation number for a class of cocycles which are in the basin of the attractor of the renormalization map. The attractor itself is built of algebraic Anosov maps on T². We present a number of results and conjecuters resulting from this approach. We show how this approach sheds some light upon the problem of classifying linear ODE with almost-periodic, skew-hermitian coefficient matrix.