On a class of Thue-Morse type sequences.
On low-complexity bi-infinite words and their factors
In this paper we study bi-infinite words on two letters. We say that such a word has stiffness if the number of different subwords of length equals for all sufficiently large. The word is called -balanced if the numbers of occurrences of the symbol a in any two subwords of the same length differ by at most . In the present paper we give a complete description of the class of bi-infinite words of stiffness and show that the number of subwords of length from this class has growth order...
On substitution invariant Sturmian words: an application of Rauzy fractals
Sturmian words are infinite words that have exactly n+1 factors of length n for every positive integer n. A Sturmian word sα,p is also defined as a coding over a two-letter alphabet of the orbit of point ρ under the action of the irrational rotation Rα : x → x + α (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...
On the complexity of infinite sequences. (Sur la complexité des suites infinies.)
On the conjectures of Rauzy and Shallit for infinite words
We show a connection between a recent conjecture of Shallit and an older conjecture of Rauzy for infinite words on a finite alphabet. More precisely we show that a Rauzy-like conjecture is equivalent to Shallit's. In passing we correct a misprint in Rauzy's conjecture.
On the number of Arnoux-Rauzy words
On the real roots of generalized Thue-Morse polynomials
On the recognizability of self-generating sets.
On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon
The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of a new conjecture that we call Aubry-Godrèche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the -rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner;...
On the Thue-Morse measure
Opérations sur les mots de Christoffel
On peut définir la pente d'un mot écrit avec des 0 et des 1 comme le nombre de 1 divisé par le nombre de 0, et généraliser cette définition aux mots de longueur infinie. Considérant le lien entre les mots de Christoffel et les fractions continues, on se propose d'étudier le comportement de tels mots lorsqu'on additionne leurs pentes, ou qu'on les multiplie par un entier positif. Après un bref exposé des différentes notions liées aux mots de Christoffel, l'étude de la somme et de la multiplication...