Local Rigidity of Diophantine Translations in Higher-dimensional Tori

Abstract

We prove a theorem asserting that, given a Diophantine rotation α in a torus T^d ≡ R^d/Z^d, any perturbation, small enough in the C^∞ topology, that does not destroy all orbits with rotation vector α 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 α assures that we can linearize the equations around the same rotation α. The proof of the convergence of the scheme is carried out in the C^∞ category.