On Sturmian sequences which are invariant under some substitutions (Q2707579)

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 \(\forall n\geq 0\) \(u_n= \lceil \alpha(n+1)+ \beta\rceil- \lceil \alpha n+\beta\rceil\). NEWLINENEWLINENEWLINE\textit{D. Crisp, W. Moran, A. Pollington} and \textit{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 \textit{B. Parvaix} for these sequences [J. Théor. Nombres Bordx. 9, 351-369 (1997; Zbl 0904.11008)] and a result of \textit{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 \textit{T. Komatsu} gives similar results [Tokyo J. Math. 22, 235-243 (1999; Zbl 0940.11012)].NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].