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On Sturmian sequences which are invariant under some substitutions. (English) Zbl 0971.11007

Kanemitsu, Shigeru (ed.) et al., Number theory and its applications. Proceedings of the conference held at the RIMS, Kyoto, Japan, November 10-14, 1997. Dordrecht: Kluwer Academic Publishers. Dev. Math. 2, 347-373 (1999).
Sturmian sequences can be defined as sequences \((\mu_n)_{n\geq 0}\) for which there exist an irrational number \(\alpha\in (0,1)\) and a real number \(\beta\) such that either \(\forall n\geq 0\) \(u_n= \lfloor \alpha(n+1)+ \beta\rfloor- \lfloor \alpha n+\beta\rfloor\) or \(\forall n\geq 0\) \(u_n= \lceil \alpha(n+1)+ \beta\rceil- \lceil \alpha n+\beta\rceil\).
D. Crisp, W. Moran, A. Pollington and P. Shiue [J. Théor. Nombres Bordx. 5, 123-137 (1993; Zbl 0786.11041)] gave a characterization of those Sturmian sequences that are fixed points of morphisms in case \(\beta= 0\). The author extends their result in case either the sequence is obtained from shifting a Sturmian sequence having \(\beta=0\), or for Sturmian sequences for which \(\beta\neq \alpha x+y\), \(x,y\in \mathbb{Z}\). Note that a partial result of B. Parvaix for these sequences [J. Théor. Nombres Bordx. 9, 351-369 (1997; Zbl 0904.11008)] and a result of B. Parvaix for Sturmian bisequences [J. Théor. Nombres Bordx. 11, 201-210 (1999; Zbl 0978.11005)] are not quoted by the author. Also note that a recent paper of T. Komatsu gives similar results [Tokyo J. Math. 22, 235-243 (1999; Zbl 0940.11012)].
For the entire collection see [Zbl 0932.00040].

MSC:

11B85 Automata sequences
68R15 Combinatorics on words
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