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Disk-like self-affine tiles in \(\mathbb{R}^2\). (English) Zbl 1020.52018

The authors give simple necessary and sufficient conditions for self-affine \({\mathbb Z}^2\)-tiles in \({\mathbb R}^2\) to be homeomorphic to a disk. These conditions are formulated in terms of \({\mathcal F}\)-connectedness of the digit set of a tile, where \({\mathcal F}\) is a (sub-)set of neighbors of a tile.

MSC:

52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
28A80 Fractals
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