Local Rigidity for Simultaneous Diophantine Translations on Tori of Arbitrary Dimension

Abstract

We show that a smooth sufficiently small perturbation of a ${\mathbb{Z}}^m$ action on the torus by simultaneously Diophantine translations, is smoothly conjugate to the unperturbed action under a natural condition on the rotation sets. This generalizes recent result of Karaliolios [4] of the action generators to higher rank abelian actions, and the result of Moser [8] to higher dimensional tori.

13 pages