Self-affine tiles generated by a finite number of matrices. (English) Zbl 1538.28024
Summary: We study self-affine tiles generated by iterated function systems consisting of affine mappings whose linear parts are defined by different matrices. We obtain an interior theorem for these tiles. We prove a tiling theorem by showing that for such a self-affine tile, there always exists a tiling set. We also obtain a more complete interior theorem for reptiles, which are tiles obtained when the matrices in the iterated function system are similarities. Our results extend some of the classical ones by J. C. Lagarias and Y. Wang [Adv. Math. 121, No. 1, 21–49 (1996; Zbl 0893.52013)], where the IFS maps are defined by a single matrix.
MSC:
| 28A80 | Fractals |
| 52C22 | Tilings in \(n\) dimensions (aspects of discrete geometry) |
| 05B45 | Combinatorial aspects of tessellation and tiling problems |
Citations:
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