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Tilings associated with non-Pisot matrices. (English. French summary) Zbl 1142.15015

A tiling substitution \(\Theta\), mapping tiles in \(\mathbb{R}^2\) to finite tiling patches, induces a mapping \(\theta = \partial\Theta\) on the boundaries, which in turn has associated to it a structure matrix \(A\). This matrix is frequently hyperbolic with a two-dimensional expanding subspace.
This paper seeks to reverse this process. Given a hyperbolic matrix \(A\) in \(\text{GL}_d(\mathbb{Z})\) having a 2-dimensional expanding subspace and satisfying certain conditions, the authors construct a 1-dimensional substitution \(\theta\) that has A as its structure matrix, and a tiling substitution \(\Theta\) with \(\theta = \partial\Theta\). These conditions on \(A\) do not include a Pisot condition but rather a condition on what they call the oriented compound of \(A\), a matrix associated to \(A\). The authors use the Ammann matrix to illustrate the theory throughout the paper.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
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