Représentation géométrique de suites de complexité $2n+1$

Arnoux, Pierre; Rauzy, Gérard

doi:10.24033/bsmf.2164

Cited by 275 publications

(425 citation statements)

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“…There are a number of recent papers on powers of words occurring in Sturmian sequences (see for instance [1,2,3,6,8,9,16,17,27,32,38,41]). Quantities of interest include the supremum of powers of factors of a sequence (the index or critical exponent of the sequence), and the limit superior of powers of longer and longer factors of the sequence.…”

Section: Introductionmentioning

confidence: 99%

Initial powers of Sturmian sequences

2006

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Section: Introductionmentioning

confidence: 99%

Initial powers of Sturmian sequences

2006

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“…However balanced words over a higher-alphabet do not seem to be good candidates for describing discrete segments in the space, as shown in [13]. The family of Arnoux-Rauzy words [4] provides a third fruitful way of generalizing Sturmian words. They have a linear number of factors of a given length (2n + 1 factors of length n), and can be described in terms of a multi-dimensional continued fraction algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…It is well known that the reduction of a two-dimensional integer vector by using Euclid's algorithm allows one to construct the discrete segment (also called Christoffel word) which is such that no integer point is in the interior of the gray region. Its Freeman coding w = aaabaaabaaabaab can be obtained by applying on the letter b the substitutions associated with the steps of Euclid's algorithm performed on (11,4).…”

Section: Introductionmentioning

confidence: 99%

An Arithmetic and Combinatorial Approach to Three-Dimensional Discrete Lines

Berthé

Labbé

2011

Discrete Geometry for Computer Imagery

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Abstract. 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 Euclid's 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.

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“…Two very interesting generalizations are very close: the Arnoux-Rauzy sequences (e.g., see [2,14,23,30]) and episturmian words (e.g., see [5,13,15]). The first of these two families is a particular subclass of the second one.…”

Section: Introductionmentioning

confidence: 99%

Directive words of episturmian words: equivalences and normalization

Glen

Levé

Richomme

2008

RAIRO-Theor. Inf. Appl.

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Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing the same episturmian word. We also propose a way to uniquely define any episturmian word through a normalization of its directive words. As a consequence of these results, we characterize episturmian words having a unique directive word.

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Représentation géométrique de suites de complexité $2n+1$

Abstract: Représentation géométrique de suites de complexité 2n + 1

Cited by 275 publications

References 0 publications

Initial powers of Sturmian sequences

Initial powers of Sturmian sequences

An Arithmetic and Combinatorial Approach to Three-Dimensional Discrete Lines

Directive words of episturmian words: equivalences and normalization

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