Renormalization of cocycles and linear ODE with almost-periodic coefficients

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Abstract

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 Cr-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.

Original languageEnglish (US)
Pages (from-to)173-206
Number of pages34
JournalInventiones Mathematicae
Volume110
Issue number1
DOIs
Publication statusPublished - Dec 1992

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ASJC Scopus subject areas

  • Mathematics(all)

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