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Atomic surfaces, tilings and coincidence. I: Irreducible case. (English) Zbl 1143.37013
An irreducible and unimodular Pisot substitution can be associated with a graph-directed iterated function system. The invariant sets of this iterated function system are called the partial atomic surfaces, the union of these invariant sets is called the atomic surface of the substitution. Several types of tilings have been associated to the class of irreducible and unimodular Pisot substitutions. In the paper under review the authors construct a quasiperiodic collection which consists of translations of partial atomic surfaces. It is investigated when this collection is a tiling. Basing on the coincidence condition by F. M. Dekking [Z. Wahrscheinlichkeitstheor. Verw. Geb. 41, 221–239 (1978; Zbl 0348.54034)], the authors introduce the super-coincidence condition for substitutions and study its role for atomic surfaces.

##### MSC:
 37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry)
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