On étudie ici un procédé universel d’extraction de suites - extraction en un sens élargi qui sera précisé - consistant à piquer les chiffres de l’écriture en base des indices de la suite, cela suivant une partie de . On s’intéresse plus particulièrement à l’action de ce procédé sur les suites périodiques, en liaison avec la régularité de la partie , en termes de périodicité, de quasi-périodicité et d’automaticité. Ainsi (à une restriction évidente près), les procédés associés aux parties ultimement...
Topological and combinatorial properties of dynamical systems called odometers and arising from number systems are investigated. First, a topological classification is obtained. Then a rooted tree describing the carries in the addition of 1 is introduced and extensively studied. It yields a description of points of discontinuity and a notion of low scale, which is helpful in producing examples of what the dynamics of an odometer can look like. Density of the orbits is also discussed.
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 -numeration and its generalisations, abstract numeration systems and...
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