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Fonction ζ de Carlitz et automates

Valérie Berthé — 1993

Journal de théorie des nombres de Bordeaux

Carlitz a défini sur 𝔽 q une fonction ζ et une série formelle I I , analogues respectivement à la fonction ζ de Riemann et au réel π . Yu a montré, en utilisant les modules de Drinfeld, que ζ ( s ) / I I 3 est transcendant pour tout s non divisible par q - 1 . Nous donnons ici une preuve «automatique» de la transcendance de ζ ( s ) / I I 3 pour 1 s q - 2 , en utilisant le théorème de Christol, Kamae, Mendès France et Rauzy.

Complexité et automates cellulaires linéaires

Valérie Berthé — 2010

RAIRO - Theoretical Informatics and Applications

The aim of this paper is to evaluate the growth order of the complexity function (in rectangles) for two-dimensional sequences generated by a linear cellular automaton with coefficients in / l , and polynomial initial condition. We prove that the complexity function is quadratic when is a prime and that it increases with respect to the number of distinct prime factors of .

Suites doubles de basse complexité

Valérie BerthéLaurent Vuillon — 2000

Journal de théorie des nombres de Bordeaux

Nous donnons une représentation géométrique des suites doubles uniformément récurrentes de fonction de complexité rectangulaire m n + n . Nous montrons que ces suites codent l’action d’une 2 -action définie par deux rotations irrationnelles sur le cercle unité. La preuve repose sur une étude des suites doubles dont les lignes sont des suite sturmiennes de même langage.

Connectedness of fractals associated with Arnoux–Rauzy substitutions

Valérie BerthéTimo JolivetAnne Siegel — 2014

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Rauzy fractals are compact sets with fractal boundary that can be associated with any unimodular Pisot irreducible substitution. These fractals can be defined as the Hausdorff limit of a sequence of compact sets, where each set is a renormalized projection of a finite union of faces of unit cubes. We exploit this combinatorial definition to prove the connectedness of the Rauzy fractal associated with any finite product of three-letter Arnoux–Rauzy substitutions.

Discrete planes, 2 -actions, Jacobi-Perron algorithm and substitutions

Pierre ArnouxValérie BerthéShunji Ito — 2002

Annales de l’institut Fourier

We introduce two-dimensional substitutions generating two-dimensional sequences related to discrete approximations of irrational planes. These two-dimensional substitutions are produced by the classical Jacobi-Perron continued fraction algorithm, by the way of induction of a 2 -action by rotations on the circle. This gives a new geometric interpretation of the Jacobi-Perron algorithm, as a map operating on the parameter space of 2 -actions by rotations.

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

Pierre ArnouxValérie BerthéArnaud HilionAnne Siegel — 2006

Annales de l’institut Fourier

In this paper, we extend to automorphisms of free groups some results and constructions that classically hold for morphisms of the free monoid, i.e., the so-called substitutions. A geometric representation of the attractive lamination of a class of automorphisms of the free group (irreducible with irreducible powers () 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,...

Dynamical directions in numeration

Guy BaratValérie BerthéPierre LiardetJörg Thuswaldner — 2006

Annales de l’institut Fourier

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 β -numeration and its generalisations, abstract numeration systems and...

On substitution invariant Sturmian words: an application of Rauzy fractals

Valérie BerthéHiromi EiShunji ItoHui Rao — 2007

RAIRO - Theoretical Informatics and Applications

Sturmian words are infinite words that have exactly factors of length for every positive integer . A Sturmian word is also defined as a coding over a two-letter alphabet of the orbit of point under the action of the irrational rotation (mod 1). A substitution fixes a Sturmian word if and only if it is invertible. The main object of the present paper is to investigate Rauzy fractals associated with two-letter invertible substitutions. As an application, we give an alternative geometric...

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