Local Rigidity of Diophantine Translations in Higher-dimensional Tori

We prove a theorem asserting that, given a Diophantine rotation alpha in a torus T (d) equivalent to R (d) /Z (d) , any perturbation, small enough in the C-infinity topology, that does not destroy all orbits with rotation vector alpha is actually smoothly conjugate to the rigid rotation. The proof relies on a KAM 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-infinity category.