Valérie Berthé

Frequencies of Sturmian series factors

Resume Dekking a explicite les frequences des facteurs de la suite de Fibonacci en utilisant le graphe des mots. Nous generalisons ce resultat aux suites sturmiennes en montrant, egalement par le graphe des mots, que les frequences des facteurs de meme longueur dune suite sturmienne prennent au plus 3 valeurs. Nous explicitons ces valeurs et donnons, pour chacune delles, le nombre de facteurs ayant cette frequence en fonction du developpement en fraction continue de langle α de la suite sturmienne.

Two-dimensional iterated morphisms and discrete planes

Iterated morphisms of the free monoid are very simple combinatorial objects which produce infinite sequences by replacing iteratively letters by words. The aim of this paper is to introduce a formalism for a notion of two-dimensional morphisms; we show that they can be iterated by using local rules, and that they generate two-dimensional patterns related to discrete approximations of irrational planes with algebraic parameters. We associate such a two-dimensional morphism with any usual Pisot unimodular one-dimensional iterated morphism over a three-letter alphabet.

On converting numbers to the double-base number system

This paper is an attempt to bring some theory on the top of some previously unproved experimental statements about the double-base number system (DBNS). We use results from diophantine approximation to address the problem of converting integers into DBNS. Although the material presented in this article is mainly theoretical, the proposed algorithm could lead to very efficient implementations.

Balance properties of multi-dimensional words

A word u is called 1-balanced if for any two factors v and w of u of equal length, we have 1|v|i|w|i1 for each letter i, where |v|i denotes the number of occurrences of i in the factor v. The aim of this paper is to extend the notion of balance to multi-dimensional words. We first characterize all 1-balanced words on &Zn. In particular, we prove they are fully periodic for n>1. We then give a quantitative measure of non-balancedness for some words on &Z2 with irrational density, including two-dimensional Sturmian words.

Smooth words over arbitrary alphabets

Smooth infinite words over ∑ = {1, 2} are connected to the Kolakoski word K = 221121 ..., defined as the fixpoint of the function Δ that counts the length of the runs of 1s and 2s. In this paper we extend the notion of smooth words to arbitrary alphabets and study some of their combinatorial properties. Using the run-length encoding Δ, every word is represented by a word obtained from the iterations of Δ. In particular we provide a new representation of the infinite Fibonacci word F as an eventually periodic word. On the other hand, the Thue-Morse word is represented by a finite one.

Covering numbers: Arithmetics and dynamics for rotations and interval exchanges

We study a particular case of the two-dimensional Steinhaus theorem, giving estimates of the possible distances between points of the formkα andkα+β on the unit circle, through an approximation algorithm of β by the pointskα. This allows us to compute covering numbers (maximal measures of Rokhlin stacks having certain prescribed regularity properties) for the symbolic dynamical systems associated to the rotation of argument α, acting on the partition of the circle by the points 0, β. We can the compute topological and measure-theoretic covering numbers for exchange of three intervals; in this way, we prove that every ergodic exchange of three intervals has simple spectrum and build a new class of three-interval exchanges which are not of rank one.

Brun expansions of stepped surfaces

Dual maps have been introduced as a generalization to higher dimensions of word substitutions and free group morphisms. In this paper, we study the action of these dual maps on particular discrete planes and surfaces, namely stepped planes and stepped surfaces. We show that dual maps can be seen as discretizations of toral automorphisms. We then provide a connection between stepped planes and the Brun multi-dimensional continued fraction algorithm, based on a desubstitution process defined on local geometric configurations of stepped planes. By extending this connection to stepped surfaces, we obtain an effective characterization of stepped planes (more exactly, stepped quasi-planes) among stepped surfaces.

An arithmetic and combinatorial approach to three-dimensional discrete lines

The aim of this paper is to discuss from an arithmetic and combinatorial viewpoint a simple algorithmic method of generation of discrete segments in the three-dimensional space. We consider discrete segments that connect the origin to a given point (u1, u2, u3) with coprime nonnegative integer coordinates. This generation method is based on generalized three-dimensional Euclids algorithms acting on the triple (u1, u2, u3). We associate with the steps of the algorithm substitutions, that is, rules that replace letters by words, which allow us to generate the Freeman coding of a discrete segment. We introduce a dual viewpoint on these objects in order to measure the quality of approximation of these discrete segments with respect to the corresponding Euclidean segment. This viewpoint allows us to relate our discrete segments to finite patches that generate arithmetic discrete planes in a periodic way.

Odometers on Regular Languages

AbstractOdometers or adding machines are usually introduced in then context of positional numeration systems built on a strictlyn increasing sequence of integers. We generalize this notion ton systems defined on an arbitrary infinite regular language. Inn this latter situation, if (A,<) is a totally ordered alphabet,n then enumerating the words of a regular language L over A withn respect to the induced genealogical ordering gives a one-to-onen correspondence between ℕ and L. In this general settingn the odometer is not defined on a set of sequences of digits but onn a set of pairs of sequences where the first (resp. the second)n component of the pair is an infinite word over A (resp. ann infinite sequence of states of the minimal automaton of L). Wen study some properties of the odometer such as continuity,n injectivity, surjectivity, minimality, .... We then study somen particular cases: we show the equivalence of this new functionn with the classical odometer built upon a sequence of integersn whenever the set of greedy representations of all the integers isn a regular language; we also consider substitution numerationn systems as well as the connection with β-numerations.

Lattices and multi-dimensional words

In the present paper we develop a formalism to generate multi-dimensional words using lattices which generalizes the construction of real numbers (one-dimensional words) from a sequence of partial quotients using continued fractions. The construction was introduced in a special case by Simpson and Tijdeman in order to derive a multi-dimensional generalization of the theorem of Fine and Wilf. We show that the produced multi-dimensional words are intrinsically connected with k-dimensional Sturmian words.