An isolated point in the Heinis spectrum
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Special issue dedicated to the 15th "Journées Montoises d'Informatique Théorique", Tome 50 (2016) no. 1, pp. 21-38.

This paper continues the study initiated by Alex Heinis of the set H of pairs (α,β) obtained as the lower and upper limit of the ratio of complexity and length for an infinite word. Heinis proved that this set contains no point under a certain curve. We extend this result by proving that there are only three points on this curve, namely (1,1), (3/2, 5/3) and (2,2), and moreover the point (3/2, 5/3) is an isolated point in the set H. For this, we use Rauzy graphs, generalizing techniques of Ali Aberkane.

Reçu le :
Accepté le :
DOI : 10.1051/ita/2016004
Classification : 68R15
Mots-clés : Rauzy graph, Sturmian words, complexity function
Turki, Ramzi 1, 2

1 Institut de mathématiques de Marseille, Université d’Aix-Marseille, Case 907, 163, avenue de Luminy, 13288 Marseille, cedex 9, France
2 Faculté des Sciences de Sfax, Département de Mathématiques, Route de la Soukra km 3.5, B.P. 1171, 3000 Sfax, Tunisie
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Turki, Ramzi. An isolated point in the Heinis spectrum. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Special issue dedicated to the 15th "Journées Montoises d'Informatique Théorique", Tome 50 (2016) no. 1, pp. 21-38. doi : 10.1051/ita/2016004. http://archive.numdam.org/articles/10.1051/ita/2016004/

A. Aberkane, Exemples de suites de complexité inférieure à 2n. Bull. Belg. Math. Soc. 8 (2001) 161–180. | MR | Zbl

A. Aberkane, Utilisation des graphes de Rauzy dans la caractérisation de certaines familles de suites de faible complexité, Ph. D. thesis. Université d’Aix-Marseille 2 (2002).

P. Arnoux and G. Rauzy, Représentation géometrique des suites de complexité 2n+1. Bull. Soc. Math. France 119 (1991) 199–215. | DOI | Numdam | MR | Zbl

V. Berthé and M. Rigo, Combinatorics, Automata, and Number Theory. Cambridge University Press, Cambridge (2010). | MR | Zbl

J. Cassaigne, Complexité et facteurs speciaux. Bull. Belg. Math. Soc. 4 (1997) 67–88. | MR | Zbl

D. Damanik, Local symmetries in the period doubling sequence. Discrete Appl. Math. 100 (2000) 115–121. | DOI | MR | Zbl

A. Heinis, Arithmetics and combinatorics of words of low complexity. Ph. D. thesis, University of Leiden (2001). | Zbl

A. Heinis, The P(n)/n-function for bi-infinite words. Theoret. Comput. Sci. 273 (2002) 35–46. | DOI | MR | Zbl

J. Leroy, An S-adic characterization of minimal subshifts with first difference of complexity 1p(n+1)-p(n)2. Discrete Math. Theor. Comput. Sci. 16 (2014) 233–286. | MR | Zbl

M. Lothaire, Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002). | Zbl

M. Morse, G.A. Hedlund, Symbolic dynamics. Amer. J. Math. 60 (1938) 815–866. | DOI | JFM | MR

G. Rote, Sequences with subword complexity 2n. J. Number Theory 46 (1994) 196–213. | DOI | MR | Zbl

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