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

Cortez, Maria; Gambaudo, Jean-Marc; Maass, Alejandro

doi:10.1090/S0002-9947-06-04027-X

Rotation topological factors of minimal $\mathbb {Z}^{d}$-actions on the Cantor set
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by Maria Isabel Cortez, Jean-Marc Gambaudo and Alejandro Maass PDF

Trans. Amer. Math. Soc. 359 (2007), 2305-2315 Request permission

Abstract:

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.

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