Classification of Perturbations of Diophantine $\mathbb{Z}^m$ Actions on Tori of Arbitrary Dimension

We generalize results of Moser [17] on the circle to $\mathbb{T}^d$: we show that a smooth sufficiently small perturbation of a $\mathbb Z^m$ action, $m \geqslant 2$, on the torus $\mathbb{T}^d$ by simultaneously Diophantine translations, is smoothly conjugate to the unperturbed action under a natural condition on the rotation sets of diffeomorphisms isotopic to identity and we answer the question Moser posed in [17] by proving the existence of a continuum of $m$-tuples of simultaneously Diophantine vectors such that every element of the induced $\mathbb Z^m$ action is Liouville.