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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.

This paper also contains a valuable bibliography.

### MSC:

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 |