Valérie Berthé: Fractals and substitutive dynamical systems

Substitutions (i.e., morphisms of the free monoid) are very simple combinatorial objects which replace a letter by a word and which produce sequences by iteration. Substitutive dynamical systems (that is, symbolic dynamical systems generated by substitutions) have a rich structure as shown by the natural interactions with combinatorics on words, ergodic theory, spectral theory, geometry of tilings, Diophantine approximation, transcendence, theoretical computer science, and so on. It is possible to associate in a natural way with substitutions with some prescribed algebraic properties (Pisot type) some tilings of the space with fractal boudary, called Rauzy fractals. These fractal prototiles are also defined as acceptation windows for some cut-and-project schemes and satisfy a graph-directed Iterated Function System equation. We consider in this survey lecture two families of applications for these tilings in discrete geometry, number theory, and beta-numeration: first, Galois' type theorems, characterizing purely periodic orbits, and second, discrepancy results for translations in compact Abelian groups.

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