We study close-to-constants quasiperiodic cocycles in $\mathbb{ T} ^{ d} \times G$, where $d \in \mathbb{ N} ^{ *} $ and $G$ is a compact Lie group, under the assumption that the rotation in the basis satisfies a Diophantine condition. We prove differentiable rigidity for such cocycles: if such a cocycle is measurably conjugate to a constant one satisfying a Diophantine condition with respect to the rotation, then it is $C^{ \infty}$-conjugate to it, and the K.A.M. scheme actually produces a conjugation. We also derive a global differentiable rigidity theorem, assuming the convergence of the renormalization scheme for such dynamical systems.
Categories: math.DS