×

Invariance properties of Sturmian words. (Propriétés d’invariance des mots sturmiens.) (French) Zbl 0904.11008

In this interesting paper the author studies Sturmian (i.e., billiard) binary sequences that are invariant under a non-trivial substitution. The case of a trajectory going through the origin has been first addressed by D. Crisp, W. Moran, A. Pollington and P. Shiue [J. Théor. Nombres Bordx. 5, 123-138 (1993; Zbl 0786.11041); see also J. Berstel and P. Séébold, Bull. Belg. Math. Soc. – Simon Stevin 1, 175-189 (1994; Zbl 0803.68095) and T. Komatsu and A. J. van der Poorten, Jap. J. Math., New Ser. 22, 349-354 (1996; Zbl 0868.11015)].
In the paper under review, the author calls “Sturm numbers” the slopes pointed out in the paper of Crisp, Moran, Pollington and Shiue. He proves that a necessary condition for any Sturmian sequence to be fixed by a non-trivial substitution is that the slope be a Sturm number. The condition is proved sufficient for a class of intercepts that are homographic functions of the slope.

MSC:

11B85 Automata sequences
68R15 Combinatorics on words
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML EMIS

References:

[1] Bernoulli, J., Recueil pour astronomes, Berlin, (1772).
[2] Berstel, J. et Séébold, P., Morphismes de Sturm, Bull. Belg. Math. Soc.1 (1994), 175-189. · Zbl 0803.68095
[3] Berstel, J. and Séébold, P., A remark on morphic Sturmian words, Rairo Informatique théorique et applications28 (1994), 255-263. · Zbl 0883.68104
[4] Borel, J.-P. et Laubie, F., Quelques mots sur la droite projective réelle, J. Théorie des Nombres de Bordeaux (1993), 23-52. · Zbl 0839.11008
[5] Brown, T.C., A characterization of the quadratic irrationals, Canad. Math. Bull.34 (1991), 36-41. · Zbl 0688.10007
[6] Crisp, D., Moran, W., Pollington, A. and Shiue, P., Substitution invariant cutting sequences, J. Théorie des Nombres de Bordeaux5 (1993), 123-138. · Zbl 0786.11041
[7] Ito, S., On a dynamical system related to sequences nx + y - (n - 1)x + y, Collection: Dynamical Systems and Related Topics, Nagoya (1990), 192-197.
[8] Ito, S. and Hitoshi, N., Approximations of real numbers by the sequence {nα} and their metrical theory, Acta Math. Hungar.52 (1988), 91-100. · Zbl 0657.10034
[9] Ito, S. and Mimachi, H., A characterization of real quadratic numbers by Diophantine algorithms, Tokyo J. Math.14 (1991), 251-267. · Zbl 0751.11034
[10] Ito, S. and Yasutomi, S., On Continued fractions, substitutions and characteristic sequences, Japan J. Math.16 (1990), 287-306. · Zbl 0721.11009
[11] Komatsu, T. and van der Poorten, A.J., Substitution invariant Beatty sequences, Japan J. Math. 22 (1996), 349-354. · Zbl 0868.11015
[12] Morse, M. and Hedlund, G.A., Symbolic dynamics, Amer. J. Math.60 (1938), 815-866.
[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, Lecture Notes inComputer Science192 (1985), 165-171. · Zbl 0613.10044
[15] Séébold, P. et Mignosi, F., Morphismes sturmiens et règles de Rauzy, J. Théorie des Nombres de Bordeaux (1993), 221-233. · Zbl 0797.11029
[16] Shallit, J., Characteristic words as fixed points of homomorphisms, University of Waterloo, Department of Computer Science CS-91-72 (1991).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.