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Self-affine tiles in $$\mathbb{R}^n$$. (English) Zbl 0893.52013
A set $$T$$ in $$\mathbb{R}^n$$ is called a self-affine tile, if it is compact, with non-empty interior, and if there are essentially disjoint translates of $$T$$ whose union is an affine image of $$T$$; i.e. there exists a finite set $$D=\{{\underset \sim d}_1, \dots, {\underset \sim d}_m\}$$ of “digits”, and a linear transformation $$A$$ with all eigenvalues of absolute value greater than 1, such that $$A(T)= \bigcup^m_{i=1} (T+ {\underset \sim d}_i)$$. This is a generalization of self-similar tiles, where $$A$$ is a similarity.
For a self-affine tile, $$D$$ and $$A$$ are not unique. Conversely, given $$D$$ and $$A$$ (such that $$|\text{det} A|=m)$$ there is a unique compact set $$T=\{\sum^\infty_{j=1} A^{-j} {\underset \sim d}_{i_j}$$: each $${\underset \sim d}_{i_j}\in D\}$$. A first theorem gives conditions on $$A$$ and $$D$$ for $$T$$ to have non-empty interior; then $$T$$ is the self-affine tile associated with $$A$$ and $$D$$.
A second theorem reproves that every self-affine tile gives a tiling of $$\mathbb{R}^n$$ by translations, it also shows that every self-affine tile can be used as a prototile for a self-replicating of $$\mathbb{R}^n$$ in the sense of Kenyon. The third theorem adds a converse to Kenyon’s rigidity theorem concerning quasiperiodic self-replicating tilings. The paper closes with some open problems and conjectures.
Reviewer: J.Schaer (Calgary)

MSC:
 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry)
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