Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts

In this book we study the problem of reducibility (conjugacy to constants) of quasi-periodic skew-product systems with values in compact semisimple groups, as well as the existence of Floquet type solutions for linear differential quasi-periodic systems with values in compact semisimple algebras. The main result (chapter 6) is that for real one parameter families of quasi-periodic systems with values in the group of rotations of the 3-space, reducibility holds for almost all value of the parameter (provided the family is close enough to some family of constant systems). For the proof of this result, which relies on a resonance removing procedure due to L.H. Eliasson, we introduce a notion of transversality à la Pyartli, which enables us to keep on controlling the dependance of the eigenvalues on the parameter. We also use a positive measure reducibility theorem proven, in case the group is compact semisimple, in chapter 3. We also prove in chapter 5, again in the compact semisimple group case, that modulo some finite covering which depends only on the group, the set of reducible systems is dense near the constants. Chapter 4 is devoted to a normal form type theorem which enables us to recover the result of chapter 3. Finally, we give in chapter 2 a necessary and sufficient condition (modulo a finite covering) for reducibility of skew-product systems and study the centralizer of constant systems.