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

Yasutomi, Shin-Ichi

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].

Reviewer: Jean-Paul Allouche (Orsay)

MSC:

11B85	Automata sequences
68R15	Combinatorics on words

Keywords:

Sturmian sequences; morphisms; substitution sequences

Citations:

Zbl 0786.11041; Zbl 0904.11008; Zbl 0978.11005; Zbl 0940.11012