# Some algorithms to compute the conjugates of episturmian morphisms

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2003)

- Volume: 37, Issue: 1, page 85-104
- ISSN: 0988-3754

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topRichomme, Gwenael. "Some algorithms to compute the conjugates of episturmian morphisms." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 37.1 (2003): 85-104. <http://eudml.org/doc/245427>.

@article{Richomme2003,

abstract = {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.},

author = {Richomme, Gwenael},

journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},

keywords = {combinatorics on words; sturmian morphisms; conjugacy; algorithms; Combinatorics on words; Sturmian morphisms},

language = {eng},

number = {1},

pages = {85-104},

publisher = {EDP-Sciences},

title = {Some algorithms to compute the conjugates of episturmian morphisms},

url = {http://eudml.org/doc/245427},

volume = {37},

year = {2003},

}

TY - JOUR

AU - Richomme, Gwenael

TI - Some algorithms to compute the conjugates of episturmian morphisms

JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

PY - 2003

PB - EDP-Sciences

VL - 37

IS - 1

SP - 85

EP - 104

AB - 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.

LA - eng

KW - combinatorics on words; sturmian morphisms; conjugacy; algorithms; Combinatorics on words; Sturmian morphisms

UR - http://eudml.org/doc/245427

ER -

## References

top- [1] P. Arnoux and G. Rauzy, Représentation géométrique de suites de complexités $2n+1$. Bull. Soc. Math. France 119 (1991) 199-215. Zbl0789.28011MR1116845
- [2] J. Berstel and P. Séébold, Sturmian words, Chap. 2, edited by M. Lothaire. Cambridge Mathematical Library, Algebraic Combinatorics on Words 90 (2002). Zbl0883.68104MR1905123
- [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. Zbl0970.68124MR1782038
- [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. Zbl0916.68114
- [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. Zbl0981.68126MR1819089
- [6] P. Hubert, Suites équilibrées. Theoret. Comput. Sci. 242 (2000) 91-108. Zbl0944.68149MR1769142
- [7] J. Justin, On a paper by Castelli, Mignosi, Restivo. RAIRO: Theoret. Informatics Appl. 34 (2000) 373-377. Zbl0987.68056MR1829233
- [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. Zbl0971.68125MR1854492
- [9] J. Justin and G. Pirillo, Episturmian words and Episturmian morphisms. Theoret. Comput. Sci. 276 (2002) 281-313. Zbl1002.68116MR1896357
- [10] J. Justin and L. Vuillon, Return words in Sturmian and Episturmian words. RAIRO: Theoret. Informatics Appl. 34 (2000) 343-356. Zbl0987.68055MR1829231
- [11] F. Levé and P. Séébold, Conjugation of standard morphisms and a generalization of singular words, in Proc. of the 9${}^{th}$ international conference Journées Montoises d’Informatique Théorique. Montpellier, France (2002). Zbl1076.68056
- [12] M. Morse and G.A. Hedlund, Symbolic Dynamics II: Sturmian trajectories. Amer. J. Math. 61 (1940) 1-42. Zbl0022.34003MR745JFM66.0188.03
- [13] G. Rauzy, Suites à termes dans un alphabet fini, in Séminaire de théorie des Nombres de Bordeaux. Exposé 25 (1983). Zbl0547.10048MR750326
- [14] G. Richomme, Conjugacy and Episturmian morphisms, Technical Report 2001-03. LaRIA, Theoret. Comput. Sci. (to appear). Zbl1044.68142MR1981940
- [15] P. Séébold, Fibonacci morphisms and Sturmian words. Theoret. Comput. Sci. 88 (1991) 365-384. Zbl0737.68068MR1131075
- [16] P. Séébold, On the conjugation of standard morphisms. Theoret. Comput. Sci. 195 (1998) 91-109. Zbl0981.68104MR1603835
- [17] Z.X. Wen and Y. Zhang, Some remarks on invertible substitutions on three letter alphabet. Chin. Sci. Bulletin 44 (1999) 1755-1760. Zbl1040.20504MR1737516

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