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

Karaliolios Nikolaos 1
1 Théorie Ergodique
LPMA - Laboratoire de Probabilités et Modèles Aléatoires
Abstract : In this PhD thesis 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. We will firstly define two invariants of the dynamics, which we will call energy and degree and which give a preliminary distinction between 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 analyse these perturbations using methods inspired by K.A.M. theory. In this context we will prove that if a $C^{\infty }$ cocycle is measurably reducible to a diophantine constant, it is actually $C^{\infty }$-reducible.
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Karaliolios Nikolaos. Global aspects of the reducibility of quasiperiodic cocycles in semisimple compact Lie groups. Dynamical Systems [math.DS]. Université Pierre et Marie Curie - Paris VI, 2013. English. ⟨tel-00777911⟩

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