Rotation topological factors of minimal $\mathbb{Z}^{d}$-actions on the Cantor set

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