Purely periodic beta-expansions in the Pisot non-unit case

It is well known that real numbers with a purely periodic decimal expansion are the rationals having, when reduced, a denominator coprime with 10. The aim of this paper is to extend this result to beta-expansions with a Pisot base beta which is not necessarily a unit: we characterize real numbers having a purely periodic expansion in such a base; this characterization is given in terms of an explicit set, called generalized Rauzy fractal, which is shown to be a graph-directed self-affine compact subset of non-zero measure which belongs to the direct product of Euclidean and p-adic spaces

Generalized Substitutions and Stepped Surfaces

A substitution is a non-erasing morphism of the free monoid. The notion of multidimensional substitution of non-constant length acting on multidimensional words introduced in [AI01,ABS04] is proved to be sell-defined on the set of two-dimensional words related to discrete approximations of irrational planes. Such a multidimensional substitution can be associated to any usual Pisot unimodular substitution. The aim of this paper is to try to extend the domain of definition of such multidimensional substitutions. In particular, we study an example of a multidimensional substitution which acts on a stepped surface in the sense of [Jam04,JP04]

Finiteness properties for Pisot SS-adic tilings

International audienceIn this paper, we will first formulate and prove some equivalent sufficient conditions to obtain the tiling property for a Pisot unimodular substitution. We will then apply these condition to the more general framework of adic systems, to extend to this more general (and non algebraic) case results already known for the substitutive case

Functional Stepped Surfaces, Flips and Generalized Substitutions

Remplace N° 06014 (2006) 23 [lirmm-00102710 ? version 1]International audienceA substitution is a non-erasing morphism of the free monoid. The notion of multidimensional substitution of non-constant length acting on multidimensional words is proved to be well-defined on the set of two-dimensional words related to discrete approximations of irrational planes. Such a multidimensional substitution can be associated with any usual unimodular substitution. The aim of this paper is to extend the domain of definition of such multidimensional substitutions to functional stepped surfaces. One central tool for this extension is the notion of flips acting on tilings by lozenges of the plane

Fractal representation of the attractive lamination of an automorphism of the free group

N°RR 05066 (2005)International audienceIn this paper, we extend to automorphisms of free groups some results and constructions that classically hold for morphisms of the free monoid, i.e., so-called substitutions. A geometric representation of the attractive lamination of a class of automorphisms of the free group (irreducible with irreducible powers ({\it iwip}) automorphisms) is given in the case where the dilation coefficient of the automorphism is a unit Pisot number. The shift map associated with the attractive symbolic lamination is, in this case, proved to be measure-theoretically isomorphic to a domain exchange on a self-similar Euclidean compact set. This set is called the central tile of the automorphism, and is inspired by Rauzy fractals associated with Pisot primitive substitutions. The central tile admits some specific symmetries, and is conjectured under the Pisot hypothesis to be a fundamental domain for a toral translation

Purely Periodic beta-Expansions in the Pisot Non-unit Case

International audienceIt is well known that real numbers with a purely periodic decimal expansion are rationals having, when reduced, a denominator coprime with $10$. The aim of this paper is to extend this result to beta-expansions with a Pisot base beta which is not necessarily a unit. We characterize real numbers having a purely periodic expansion in such a base. This characterization is given in terms of an explicit set, called a generalized Rauzy fractal, which is shown to be a graph-directed self-affine compact subset of non-zero measure which belongs to the direct product of Euclidean and $p$-adic spaces

Dynamical Directions in Numeration

International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper

Generalized Substitutions and Stepped Surfaces

A substitution is a non-erasing morphism of the free monoid. The notion of multidimensional substitution of non-constant length acting on multidimensional words introduced in [AI01,ABS04] is proved to be sell-defined on the set of two-dimensional words related to discrete approximations of irrational planes. Such a multidimensional substitution can be associated to any usual Pisot unimodular substitution. The aim of this paper is to try to extend the domain of definition of such multidimensional substitutions. In particular, we study an example of a multidimensional substitution which acts on a stepped surface in the sense of [Jam04,JP04]

Substitutions, Numeration and Tilings

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Basic notions on substitutions

International audienceThe aim of this chapter is to introduce some concepts and to fix the notation we will use throughout this book. We will first introduce some terminology in combinatorics on words. These notions have their counterpart in terms of symbolic dynamics. We shall illustrate these definitions through the example of a particular sequence, the Morse sequence, generated by an algorithmic process we shall study in details in this book, namely a substitution. After recalling some basic notions on substitutions, we shall focus on the concept of automatic sequences. We then introduce the first notions of ergodic theory and focus on the spectral description of discrete dynamical systems