  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.
