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The geometry of non-unit Pisot substitutions. (Géométrie des substitutions de type Pisot non unimodulaires.) (English. French summary) Zbl 1309.05045
G. Rauzy [Bull. Soc. Math. Fr. 110, 147–178 (1982; Zbl 0522.10032)] introduced a family of fractal tiles associated to a specific unit Pisot substitution, and P. Arnoux and S. Ito [Bull. Belg. Math. Soc. - Simon Stevin 8, No. 2, 181–207 (2001; Zbl 1007.37001)] systematically developed these tiling systems. Much of the subsequent developments concentrated on the unit Pisot case until A. Siegel [Ergodic Theory Dyn. Syst. 23, No. 4, 1247–1273 (2003; Zbl 1052.37009)] confirmed part of Rauzy’s conjectures using a representation space including a \(p\)-adic component to build tilings associated to certain Pisot substitutions without the unit assumption. Here, Rauzy crystals are defined and studied in several new situations. The Dumont-Thomas numeration system [J.-M. Dumont and A. Thomas, Theor. Comput. Sci. 65, No. 2, 153–169 (1989; Zbl 0679.10010)] (a generalization of the system of \(\beta\)-numeration) is reviewed, and a system of Rauzy fractals is defined as a natural geometric system associated to the numeration; the geometric realization of a substitution and its dual studied by P. Arnoux and S. Ito [loc. cit.] is extended to the non-unit Pisot case, and constructed as renormalized pieces of stepped hypersurfaces with \(p\)-adic factors; finally B. Sing’s construction [Pisot substitutions and beyond. Bielefeld: Univ. Bielefeld, Fakultät für Mathematik (PhD. Thesis) (2006; Zbl 1210.93006)] of Rauzy crystals using cut-and-project schemes is described using a graph directed iterated function system, in which setting the Rauzy fractals arise as dual prototiles of the associated model set. The different approaches are shown to be related, and the Rauzy fractals are shown to all be the same up to affine equivalence. Here the fractals are subsets of an open subring of the adele ring of the number field defined by the Pisot number. Further dynamical, geometric, and topological properties of Rauzy fractals in these different settings are studied, and the relations between Rauzy crystals and three different structures (subshifts defined using periodic points of the underlying substitution, adic transformations, and domain exchanges of pieces of Rauzy fractals) are discussed.

MSC:
05B45 Combinatorial aspects of tessellation and tiling problems
11A63 Radix representation; digital problems
11F85 \(p\)-adic theory, local fields
28A80 Fractals
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