Local Rigidity of Diophantine translations in higher dimensional tori

We prove a theorem asserting that, given a Diophantine rotation $\alpha$ in a torus ${\text{T}}^d\equiv {\mathbb{R}}^d\mathrm{/}{\mathbb{Z}}^d$, any perturbation, small enough in the $C^{\mathrm{\infty }}$ topology, that does not destroy all orbits with rotation vector $\alpha$ is actually smoothly conjugate to the rigid rotation. The proof relies on a K.A.M. scheme (named after Kolmogorov-Arnol’d-Moser), where at each step the existence of an invariant measure with rotation vector $\alpha$ assures that we can linearize the equations around the same rotation $\alpha$. The proof of the convergence of the scheme is carried out in the $C^{\mathrm{\infty }}$ category.