Abstract

In this paper we study conditions under which a free minimal ℤd-action on the Cantor set is a topological extension of the action of d rotations, either on the product Td of d 1-tori or on a single 1-torus T1. We extend the notion of linearly recurrent systems defined for ℤ-actions on the Cantor set to ℤd-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. © 2006 American Mathematical Society.