The complexity of tangent words

In a previous paper, we described the set of words that appear in the coding of smooth (resp. analytic) curves at arbitrary small scale. The aim of this paper is to compute the complexity of those languages.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Geometric representation of interval exchange maps over algebraic number fields

We consider the restriction of interval exchange transformations to algebraic number fields, which leads to maps on lattices. We characterize renormalizability arithmetically, and study its relationships with a geometrical quantity that we call the drift vector. We exhibit some examples of renormalizable interval exchange maps with zero and non-zero drift vector, and carry out some investigations of their properties. In particular, we look for evidence of the finite decomposition property: each lattice is the union of finitely many orbits.Comment: 34 pages, 8 postscript figure

Computing the kk-binomial complexity of the Thue--Morse word

Two words are $k$-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most $k$ with the same multiplicities. This is a refinement of both abelian equivalence and the Simon congruence. The $k$-binomial complexity of an infinite word $\mathbf{x}$ maps the integer $n$ to the number of classes in the quotient, by this $k$-binomial equivalence relation, of the set of factors of length $n$ occurring in $\mathbf{x}$. This complexity measure has not been investigated very much. In this paper, we characterize the $k$-binomial complexity of the Thue--Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue--Morse word is aperiodic, its $k$-binomial complexity eventually takes only two values. In this paper, we first obtain general results about the number of occurrences of subwords appearing in iterates of the form $\Psi^\ell(w)$ for an arbitrary morphism $\Psi$. We also thoroughly describe the factors of the Thue--Morse word by introducing a relevant new equivalence relation

Substitutions par des motifs en dimension 1

Une substitution est un morphisme de monoïdes libres : chaque lettre a pour image un mot, et l'image d'un mot est la concaténation des images de ses lettres. Cet article introduit une généralisation de la notion de substitution, où l'image d'une lettre n'est plus un mot mais un motif, c'est-à-dire un “mot à trous”, l'image d'un mot étant obtenue en raccordant les motifs correspondant à chacune de ses lettres à l'aide de règles locales. On caractérise complètement les substitutions par des motifs qui sont définies sur toute suite biinfinie, et on explique comment les construire. On montre que toute suite biinfinie qui est point fixe d'une substitution par des motifs est substitutive, c'est-à-dire est l'image, par un morphisme lettre à lettre, d'un point fixe de substitution (au sens usuel)

On The Final Coalgebra Of Automatic Sequences

Abstract. Streams are omnipresent in both mathematics and theoretical computer science. Automatic sequences form a particularly interesting class of streams that live in both worlds at the same time: they are defined in terms of finite automata, which are basic computational structures in computer science; and they appear in mathematics in many different ways, for instance in number theory. Examples of automatic sequences include the celebrated Thue-Morse sequence and the Rudin-Shapiro sequence. In this paper, we apply the coalgebraic perspective on streams to automatic sequences. We show that the set of automatic sequences carries a final coalgebra structure, consisting of the operations of head, even, and odd. This will allow us to show that automatic sequences are to (general) streams what rational languages are to (arbitrary) languages. With all our best wishes to Dexter Kozen, on the occasion of his 60th birthday.

Optimierung der Hochfrequenztrocknung von Cellulosefasern Schlussbericht

SIGLEAvailable from TIB Hannover: F97B248 / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekArbeitsgemeinschaft Industrieller Forschungsvereinigungen e.V., Koeln (Germany)DEGerman

Symbolic coding of linear complexity for generic translations of the torus, using continued fractions

In this paper, we prove that almost every translation of $\mathbb{T}^2$ admits a symbolic coding which has linear complexity $2n+1$. The partitions are constructed with Rauzy fractals associated with sequences of substitutions, which are produced by a particular extended continued fraction algorithm in projective dimension $2$. More generally, in dimension $d\geq 1$, we study extended measured continued fraction algorithms, which associate to each direction a subshift generated by substitutions, called $S$-adic subshift. We give some conditions which imply the existence, for almost every direction, of a translation of the torus $\mathbb{T}^d$ and a nice generating partition, such that the associated coding is a conjugacy with the subshift