Some algorithms to compute the conjugates of episturmian morphisms

Richomme, Gwenael

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003), p. 85-104 / Harvested from Numdam

Résumé

Episturmian morphisms generalize sturmian morphisms. They are defined as compositions of exchange morphisms and two particular morphisms $L$, and $\mathbb {D}$. Epistandard morphisms are the morphisms obtained without considering $\mathbb {D}$. In [14], a general study of these morphims and of conjugacy of morphisms is given. Here, given a decomposition of an Episturmian morphism $f$ over exchange morphisms and $\lbrace L, \mathbb {D}\rbrace $, we consider two problems: how to compute a decomposition of one conjugate of $f$; how to compute a list of decompositions of all the conjugates of $f$ when $f$ is epistandard. For each problem, we give several algorithms. Although the proposed methods are fundamently different, we show that some of these lead to the same result. We also give other algorithms, using the same input, to compute for instance the length of the morphism, or its number of conjugates.

Détail

Publié le : 2003-01-01

MR 1991753 | Zbl 1084.68094

DOI : https://doi.org/10.1051/ita:2003009
Classification:  68R15

Bibliographie

[1] P. Arnoux and G. Rauzy, Représentation géométrique de suites de complexités $2n+1$. Bull. Soc. Math. France 119 (1991) 199-215. | Numdam | MR 1116845 | Zbl 0789.28011

[2] J. Berstel and P. Séébold, Sturmian words, Chap. 2, edited by M. Lothaire. Cambridge Mathematical Library, Algebraic Combinatorics on Words 90 (2002). | MR 1905123

[3] V. Berthé and L. Vuillon, Tilings and rotations on the torus: A two dimensional generalization of Sturmian sequences. Discrete Math. 223 (2000) 27-53. | MR 1782038 | Zbl 0970.68124

[4] M.G. Castelli, F. Mignosi and A. Restivo, Fine and Wilf's theorem for three periods and a generalization of Sturmian words. Theoret. Comput. Sci. 218 (1999) 83-94. | Zbl 0916.68114

[5] X. Droubay, J. Justin and G. Pirillo, Episturmian words and some constructions of de Luca and Rauzy. Theoret. Comput. Sci. 255 (2001) 539-553. | MR 1819089 | Zbl 0981.68126

[6] P. Hubert, Suites équilibrées. Theoret. Comput. Sci. 242 (2000) 91-108. | MR 1769142 | Zbl 0944.68149

[7] J. Justin, On a paper by Castelli, Mignosi, Restivo. RAIRO: Theoret. Informatics Appl. 34 (2000) 373-377. | Numdam | MR 1829233 | Zbl 0987.68056

[8] J. Justin, Episturmian words and morphisms (results and conjectures), edited by H. Crapo and D. Senato. Springer-Verlag, Algebraic Combinatorics and Comput. Sci. (2001) 533-539. | MR 1854492 | Zbl 0971.68125

[9] J. Justin and G. Pirillo, Episturmian words and Episturmian morphisms. Theoret. Comput. Sci. 276 (2002) 281-313. | MR 1896357 | Zbl 1002.68116

[10] J. Justin and L. Vuillon, Return words in Sturmian and Episturmian words. RAIRO: Theoret. Informatics Appl. 34 (2000) 343-356. | Numdam | MR 1829231 | Zbl 0987.68055

[11] F. Levé and P. Séébold, Conjugation of standard morphisms and a generalization of singular words2002).

[12] M. Morse and G.A. Hedlund, Symbolic Dynamics II: Sturmian trajectories. Amer. J. Math. 61 (1940) 1-42. | JFM 66.0188.03 | MR 745

[13] G. Rauzy, Suites à termes dans un alphabet fini, in Séminaire de théorie des Nombres de Bordeaux. Exposé 25 (1983). | MR 750326 | Zbl 0547.10048

[14] G. Richomme, Conjugacy and Episturmian morphisms, Technical Report 2001-03. LaRIA, Theoret. Comput. Sci. (to appear). | MR 1981940 | Zbl 1044.68142

[15] P. Séébold, Fibonacci morphisms and Sturmian words. Theoret. Comput. Sci. 88 (1991) 365-384. | MR 1131075 | Zbl 0737.68068

[16] P. Séébold, On the conjugation of standard morphisms. Theoret. Comput. Sci. 195 (1998) 91-109. | MR 1603835 | Zbl 0981.68104

[17] Z.X. Wen and Y. Zhang, Some remarks on invertible substitutions on three letter alphabet. Chin. Sci. Bulletin 44 (1999) 1755-1760. | MR 1737516 | Zbl 1040.20504