  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.

Dynamical Systems