Global aspects of the reducibility of quasiperiodic cocycles in semisimple compact Lie groups

Abstract

In this mémoire we study quasiperiodic cocycles in semi-simple compact Lie groups. For the greatest part of our study, we will focus ourselves to one-frequency cocyles. We will prove that $C^{\infty}$ reducible cocycles are dense in the $C^{\infty}$ topology, for a full measure set of frequencies. Moreover, we will show that every cocycle (or an appropriate iterate of it, if homotopy appears as an obstruction) is almost torus-reducible (i.e. can be conjugated arbitrarily close to cocycles taking values in an abelian subgroup of G). In the course of the proof we will firstly define two invariants of the dynamics, which we will call energy and degree and which give a preliminary distinction between (almost-)reducible and non-reducible cocycles. We will then take up the proof of the density theorem. We will show that an algorithm of renormalization converges to perturbations of simple models, indexed by the degree. Finally, we will analyze these perturbations using methods inspired by K.A.M. theory.