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Rotation topological factors of minimal \(\mathbb Z^d\)-actions on the Cantor set. (English) Zbl 1113.54025

Let \((X,A)\) be a \({\mathbb Z}^d\)-action (by homeomorphisms) on a compact space \(X\). The action is free if \(A(N,x) = x\) for some \(N \in {\mathbb Z}^d\) and \(x \in X\) implies \(N = 0\), and it is minimal if the orbit \(\{A(N,x): \, N \in {\mathbb Z}^d\}\) of any point \(x \in X\) is dense in \(X\). Consider the following two kinds of “rotation type” \(\mathbb Z^d\)-actions (or factors):
\(\bullet\) The \(\mathbb Z^d\)-action generated by the rotations on the product \(d\)-torus \(\mathbb T^d\). Let \(A^d_\Theta :\mathbb Z^d \times \mathbb T^d \to \mathbb T^d\) be the map defined by \(A^d_\Theta (N,x) = x + (n_1 \theta_1,\dots,n_d \theta_d ) \pmod{\mathbb Z^d}\), where \(N = (n_1,\dots,n_d)\) and \(\Theta = (\theta_1,\dots,\theta_d)\). This yields a minimal \(\mathbb Z^d\)-action \((\mathbb O^d, A^d_\Theta )\) on the closure \(\mathbb O^d\) of the orbit of \(0\) in the \(d\)-torus. When the coordinates of \(\Theta\) are rationally independent, then \(\mathbb O^d = \mathbb T^d\) and the action is free.
\(\bullet\) The \(\mathbb Z^d\)-action generated on the torus \(\mathbb T^1\) by the map \(A^1_\Theta :(N,t) \mapsto t + \sum_{i=1}^d n_i\theta_i \in\mathbb T^1\) \(\pmod\mathbb Z\). The \(\mathbb Z^d\)-action (\(\mathbb O^1,A^1_\Theta )\) on the closure \(\mathbb O^1\) of the orbit of \(0\) in \(\mathbb T^1\) is again minimal. When the coordinates of \(\Theta\) are independent on \(\mathbb Q\), then \(\mathbb O^1 =\mathbb T^1\) and the action is free.
Assume that \(X\) is a Cantor set, i.e., it has a countable basis of closed and open sets and has no isolated points. In the paper under review, the authors provide a necessary and sufficient condition under which a free minimal \(\mathbb Z^d\)-action \((X,A)\) on the Cantor set \(X\) is a topological extension of the action \((\mathbb O^k, A^k_\Theta )\) for some \(\Theta\) \((k = 1\) or \(d\)). For this, they extend the notion of linearly recurrent systems defined for \(\mathbb Z\)-actions on the Cantor sets to \(\mathbb Z^d\)-actions. That condition involves a natural combinatorial data associated with the action. Their result extends a recent one due to Bressaud, Durand and Maass.

MSC:

54H20 Topological dynamics (MSC2010)
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B10 Symbolic dynamics
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
52C23 Quasicrystals and aperiodic tilings in discrete geometry

Citations:

Zbl 1095.54016

References:

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