Certain words on the real projective line. (Quelques mots sur la droite projective réelle.) (French) Zbl 0839.11008

The authors give complete proofs of their results announced in [C. R. Acad. Sci., Paris, Sér. I 313, 483-485 (1991; Zbl 0742.11013)] and F. Laubie [ibid. 565-567 (1991; Zbl 0768.11024)]. This is a pleasant paper playing with words, namely Christoffel and Lyndon words. They link substitutions acting on infinite Christoffel words with the action of \(\text{SL}_2 (\mathbb{Z})\) on the projective real line.


11B85 Automata sequences
68R15 Combinatorics on words
11A55 Continued fractions
11A67 Other number representations
Full Text: DOI Numdam EuDML


[1] Bernoulli, J., Recueil pour les astronomes, Berlin, 1772.
[2] Berstel, J., Fibonacci words - a survey, in G. Rozenberg, A. Salomaa, ed., the Book of L, Springer Verlag (1986), 13-27. · Zbl 0589.68053
[3] Borel, J.-P., Laubie, F., Construction de mots de Christoffel, C. R. Acad. Sci. Paris, Série I 313 (1991), 483-485. · Zbl 0742.11013
[4] Brown, T.C., A characterisation of the quadratic irrationals, Canad. Math. Bull.34 (1991), 36-41. · Zbl 0688.10007
[5] Brown, T.C., Descriptions of the characteristic sequence of an irrational, Canad. Math. Bull.36 (1993), 15-21. · Zbl 0804.11021
[6] Crisp, D., Moran, W., Pollington, A., Shiue, P., Substitution invariant cutting sequences, Séminaire de Théorie des Nombres de Bordeaux, Série II, ce volume. · Zbl 0786.11041
[7] Christoffel, E.B., Observatio arithmetica, Annali di Matematica, Ser. 26 (1875), 145-152. · JFM 06.0113.02
[8] Hardy, G.H., Wright, E.M., An introduction to the theory of numbers, Oxford at the Clarendom Press, 4th ed., 1960. · Zbl 0086.25803
[9] Ito, S., Yasutomi, S., On continued fractions, substitutions and characteristic sequences [nx + y] - [(n - 1 )x + y], Japan J. Math.16 (1990), 287-306. · Zbl 0721.11009
[10] Laubie, F., Prolongements homographiques des substitutions de mots de Christoffel, C. R. Acad. Sci. Paris313 Série I (1991), 565-567. · Zbl 0768.11024
[11] Nishioka, K., Shiokawa, I., Tamura, J., Arithmetical properties of a certain power series, J. Number Theory42 (1992), 61-87. · Zbl 0770.11039
[12] Perrin, D., Factorization of Free Monoids, in M. Lothaire, Combinatorics on words, Encyclopedia of Math. and its Applications17, 63-99, Addison-Wesley, 1983.
[13] Raney, G.N., On Continued Fractions and Finite Automata, Math. Ann.206 (1973), 265-283. · Zbl 0251.10024
[14] Rauzy, G., Mots infinis en arithmétique, in M. Nivat et D. Perrin ed., Automata on infinite words, Lecture Notes inComputer Science192 (1985), 165-171. · Zbl 0613.10044
[15] Riss, J., Communication personnelle, 1974.
[16] Viennot, G., Algèbres de Lie libres et Monoïdes libres, 691 (1978). · Zbl 0395.17003
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