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We prove that if the frequency of the quasi-periodic $\mathrm{SL}(2,\R)$ cocycle is Diophantine, then the following properties are dense in the subcritical regime: for any $\frac{1}{2}

We provide a general argument for the failure of Anosov-Katok-like constructions (as in \cite{AFKo2015} and \cite{NKInvDist}) to produce Cohomologically Rigid diffeomorphisms in manifolds other than tori. A $C^{\infty }$ smooth diffeomorphism $f $ of a compact manifold $M$ is Cohomologically Rigid iff the equation, known as Linear Cohomological one, \begin{equation*} \psi \circ f - \psi = \varphi \end{equation*} admits a $C^{\infty }$ smooth solution $\psi$ for every $\varphi$ in a codimension $1$ closed subspace of $C^{\infty } (M, \mathbb{C} )$. As an application, we show that no Cohomologically Rigid diffeomorphisms exist in the Almost Reducibility regime for quasi-periodic cocycles in homogeneous spaces of compact type, even though the Linear Cohomological equation over a generic such system admits a solution for a dense subset of functions $\varphi$. We thus confirm a conjecture by M. Herman and A. Katok in that context and provide some insight in the mechanism obstructing the construction of counterexamples., Comment: 43 pages. The proof of corollary B provides a more precise statement and some minor errors have been corrected

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., Cette thèse porte sur l'étude des cocycles quasi-périodiques à valeurs dans des groupes de Lie compacts semi-simples. Nous nous restreindrons au cas des cocycles à une fréquence. Nous démontrons que, pour un ensemble de mesure de Lebesgue pleine de fréquences, l'ensemble des cocycles $C^{\infty }$ qui sont $C^{\infty }$-réductibles sont $C^{\infty }$-denses. Le premier pas sera l'obtention de deux invariants de la dynamique, qu'on appellera énergie et degré, qui distinguent en particulier les cocycles réductibles des cocycles non-réductibles. On entamera ensuite la démonstration du théorème principal. Nous démontrons dans un second temps qu'un algorithme dit de renormalisation permet de ramener l'étude de tout cocycle à celle des perturbations de modèles simples indexés par le degré. Nous analysons ensuite ces perturbations par des méthodes inspirés de la théorie K.A.M.. En particulier, nous démontrons qu'un cocycle $C^{\infty }$ mesurablement réductible à une constante diophantienne est alors $C^{\infty }$-réductible.