Rotation topological factors of minimal ${\text{ZM}}^d$-actions on the cantor set

In this paper we study conditions under which a free minimal ${\text{mz}}^d$-action on the Cantor set is a topological extension of the action of $d$ rotations, either on the product ${\text{mt}}^d$ of $d$ 1-tori or on a single 1-torus ${\text{mt}}^1$. We extend the notion of {\it linearly recurrent} systems defined for $\text{mz}$-actions on the Cantor set to ${\text{mz}}^d$-actions and we derive in this more general setting, a necessary and sufficient condition, which involves natural combinatorial data associated with the action, allowing the existence of a rotation topological factor of one these two types.