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Episturmian morphisms and a Galois theorem on continued fractions

Published online by Cambridge University Press:  15 March 2005

Jacques Justin
Jacques Justin*
Affiliation:
Present address: 19 rue de Bagneux, 92330 Sceaux, France. LIAFA, ERS 586, Université Paris VII, case 7014, 2 place Jussieu, 75251 Paris Cedex 5, France; [email protected]
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Abstract

We associate with a word w on a finite alphabet A an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of w. Then when |A|=2 we deduce, using the Sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other.

Keywords

Episturmian morphismArnoux-Rauzy morphismpalindromecontinued fractionSturmian word.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 39 , Issue 1: Imre Simon, the tropical computer scientist , January 2005 , pp. 207 - 215
Copyright
© EDP Sciences, 2005

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References

Allauzen, C., Une caractérisation simple des nombres de Sturm. J. Th. Nombres Bordeaux 10 (1998) 237–241. CrossRef
Arnoux, P. and Rauzy, G., Représentation géometrique de suites de complexité 2n + 1. Bull. Soc. Math. France 119 (1991) 199–215. CrossRef
Berstel, J., Recent results on extensions of Sturmian words. Internat. J. Algebra Comput. 12 (2002) 371–385. CrossRef
Berthé, V., Autour du système de numération d'Ostrowski. Bull. Belg. Math. Soc. 8 (2001) 209–239.
E. Cahen, Théorie des Nombres. Tome 2, Librairie Scient. A. Hermann, Paris (1924).
A. Carpi and A. de Luca, Harmonic and Gold Sturmian Words, preprint, Dipart. di Mat. G. Castelnuovo, Università degli Studi di Roma La Sapienza, 22/2003 (2003).
Castelli, M.G., Mignosi, F. and Restivo, A., Fine and Wilf's theorem for three periods and a generalization of Sturmian words. Theor. Comput. Sci. 218 (2001) 83–94. CrossRef
Droubay, X., Justin, J. and Pirillo, G., Episturmian words and some constructions of de Luca and Rauzy. Theor. Comput. Sci. 255 (2001) 539–553. CrossRef
Galois, E., Démonstration d'un théorème sur les fractions continues périodiques. Ann. Math. Pures Appl. de M. Gergonne 19 (1829) 294–301.
Justin, J., On a paper by Castelli, Mignosi, Restivo. Theor. Inform. Appl. 34 (2000) 373–377. CrossRef
Justin, J. and Pirillo, G., Episturmian words and episturmian morphisms. Theor. Comput. Sci. 276 (2002) 281–313. CrossRef
Justin, J. and Pirillo, G., Episturmian words: shifts, morphisms and numeration systems. Intern. J. Foundat. Comput. Sci. 15 (2004) 329–348. CrossRef
M. Lothaire, Algebraic Combinatorics on Words, edited by M. Lothaire. Cambridge University Press. Encyclopedia of Mathematics 90 (2002).
Mignosi, F. and Zamboni, L.Q., On the number of Arnoux-Rauzy words. Acta Arith. 101 (2002) 121–129. CrossRef
Morse, M. and Hedlund, G.A., Symbolic dynamics II: Sturmian trajectories. Amer. J. Math. 62 (1940) 1–42. CrossRef
Rauzy, G., Nombres algébriques et substitutions. Bull. Soc. Math. France 110 (1982) 147–178. CrossRef
Rauzy, G., Mots infinis en arithmétique, in Automata on infinite words, edited by M. Nivat and D. Perrin. Lect. Notes Comput. Sci. 192 (1985) 165–171.
Risley, R.N. and Zamboni, L.Q., A generalization of Sturmian sequences, combinatorial structure and transcendence. Acta Arithmetica 95 (2000) 167–184.
Wozny, N.N. and Zamboni, L.Q., Frequencies of factors in Arnoux-Rauzy sequences. Acta Arithmetica 96 (2001) 261–278. CrossRef
Zamboni, L.Q., Une généralisation du théorème de Lagrange sur le développement en fraction continue. C. R. Acad. Sci. Paris I 327 (1998) 527–530. CrossRef