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Tilings associated with beta-numeration and substitutions. (English) Zbl 1139.37008
Summary: This paper surveys different constructions and properties of some multiple tilings (that is, finite-to-one coverings) of the space that can be associated with beta-numeration and substitutions. It is indeed possible, generalizing Rauzy’s and Thurston’s constructions, to associate in a natural way either with a Pisot number \(\beta\) (of degree \(d\)) or with a Pisot substitution \(\sigma\) (on \(d\) letters) some compact basic tiles that are the closure of their interior, that have non-zero measure and a fractal boundary; they are attractors of some graphdirected Iterated Function System. We know that some translates of these prototiles under a Delone set \(\Gamma\) (provided by \(\beta\) or \(\sigma\)) cover \(\mathbb R^{d-1}\); it is conjectured that this multiple tiling is indeed a tiling (which might be either periodic or self-replicating according to the translation set \(\Gamma\)). This conjecture is known as the Pisot conjecture and can also be reformulated in spectral terms: the associated dynamical systems have pure discrete spectrum. We detail here the known constructions for these tilings, their main properties, some applications, and focus on some equivalent formulations of the Pisot conjecture, in the theory of quasicrystals for instance. We state in particular for Pisot substitutions a finiteness property analogous to the well-known (F) property in beta-numeration, which is a sufficient condition to get a tiling.
This paper also contains a valuable bibliography.

37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
37B10 Symbolic dynamics
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
52C23 Quasicrystals and aperiodic tilings in discrete geometry
68R15 Combinatorics on words
11B83 Special sequences and polynomials
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