Dynamical directions in numeration

Barat, Guy; Berthé, Valérie; Liardet, Pierre; Thuswaldner, Jörg

Annales de l'Institut Fourier, Tome 56 (2006) no. 7, pp. 1987-2092.

Le but de ce survol est d’aborder définitions et propriétés concernant la numération d’un point de vue dynamique : nous nous concentrons sur les systèmes de numération, leur compactification, et les systèmes dynamiques qui peuvent être définis dessus. La notion de système de numération fibré unifie la présentation. De nombreux exemples sont étudiés. Plusieurs numérations sur les entiers naturels, relatifs, les nombres réels ou les nombres complexes sont présentées. Nous portons une attention spéciale à la $\beta$-numération ainsi qu’à ses généralisations, aux systèmes de numération abstraits, aux systèmes dits “shift radix”, de même qu’aux $G$-échelles et aux odomètres. Un paragraphe d’applications conclut ce survol.

This survey aims at giving a consistent presentation of numeration from a dynamical viewpoint: we focus on numeration systems, their associated compactification, and dynamical systems that can be naturally defined on them. The exposition is unified by the fibred numeration system concept. Many examples are discussed. Various numerations on rational integers, real or complex numbers are presented with special attention paid to $\beta$-numeration and its generalisations, abstract numeration systems and shift radix systems, as well as $G$-scales and odometers. A section of applications ends the paper.

Classification : 37B10, 11A63, 11J70, 11K55, 11R06, 37A45, 68Q45, 68R15
Keywords: Numeration, fibred systems, symbolic dynamics, odometers, numeration scales, subshifts, $f$-expansions, $\beta $-numeration, sum-of-digits function, abstract number systems, canonical numeration systems, shift radix systems, additive functions, tilings, Rauzy fractals, substitutive dynamical systems.
Mot clés : Numération, systèmes fibrés, dynamique symbolique, odomètres, échelles de numération, sous-shifts, $f$-développements, $\beta $-numération, fonction somme des chiffres, systèmes de numération abstraits, systèmes de numération canoniques, systèmes shift radix, fonctions additives, pavages, fractals de Rauzy, systèmes dynamiques substitutifs.

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