Pseudo-self-affine tilings in \(\mathbb R^d\). (English) Zbl 1085.52013

Solomyak, B. M.

Zap. Nauchn. Semin. POMI 326, 198-213 (2005); translated in J. Math. Sci., New York 140, No. 3, 452-460 (2007).

Pseudo-self-affine tilings were introduced by N. M. Priebe [Geom. Dedicata 79, 239–265 (2000; Zbl 1048.37014)]. It has been conjectured by E. A. Robinson, Jr. that pseudo-self-affine tilings are mutually locally derivable from self-affine tilings. This has been proved in the plane by the author and N. M. Priebe [Discrete Comput. Geom. 26, 289–306 (2001; Zbl 0997.52012)].
In this paper the conjecture is proved for all dimensions. The proof uses the theory of graph-directed iterated function systems developed by J. C. Lagarias and Y. Wang [Discrete Comput. Geom. 29, 175–209 (2003; Zbl 1037.52017)].

Reviewer: Konrad Swanepoel (Unisa)

MSC:

52C23	Quasicrystals and aperiodic tilings in discrete geometry
37B50	Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
52C22	Tilings in \(n\) dimensions (aspects of discrete geometry)

Keywords:

quasicrystals; self-affine tilings; Voronoi tesselations