Reducibility, differentiable rigidity and Lyapunov exponents for quasi-periodic cocycles on $\mathbf{T}timesSL(2,\mathbf{R})$

Given $\alpha$ in some set $\mathrm{\Sigma }$ of total (Haar) measure in ${\bf T}={\bf R}/{\bf Z}$,and$A\in C^{\infty}({\bf T},SL(2,{\bf R}))$ which is homotopic to the identity, we prove that if the fibered rotation number of the skew-product system $(\alpha ,A):\mathbf{T}\times SL(2,\mathbf{R})\to \mathbf{T}\times SL(2,\mathbf{R})$, $(\alpha ,A)(\theta ,y)=(\theta +\alpha ,A(\theta )y)$ is diophantine with respect to $\alpha$ and if the fibered products are uniformly bounded in the $C^0$-topology then the cocycle $(\alpha ,A)$ is $C^{\mathrm{\infty }}$-reducible --that is $A(\cdot )=B(\cdot +\alpha )A_{0}B(\cdot {)}^{-1}$, for some $A_0\in SL(2,\mathbf{R})$, $B\in C^{\mathrm{\infty }}(\mathbf{T},SL(2,\mathbf{R}))$. This result which can be seen as a non-pertubative version of a theorem by L.H. Eliasson has two interesting corollaries: the first one is a result of differentiable rigidity: if $\alpha \in \mathrm{\Sigma }$ and the cocycle $(\alpha ,A)$ is $C^0$-conjugated to a constant cocycle $(\alpha ,A_0)$ with $A_0$ in a set of total measure in $SL(2,\mathbf{R})$ then the conjugacy is $C^{\mathrm{\infty }}$; the second consequence is: if $\alpha \in \mathrm{\Sigma }$ is fixed then the set of $A\in C^\infty({\bf T},SL(2,{\bf R}))$forwhich$(\alpha,A)$haspositiveLyapunovexponentis$C^\infty$-dense. A similar result is true for the Schr"odinger cocycle and for 2-frequencies conservative differential equations in the plane.