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.


11B85 Automata sequences
68R15 Combinatorics on words
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