Sturmian sequences and the lexicographic world (Q2701640)

For two sequences \(a\) and \(b\) in \(\Sigma:= \{0,1\}^{\mathbb{N}}\), the author defines \(\Sigma_{a,b}:= \{x\in\Sigma\), \(\forall i\geq 0\), \(a\leq \sigma^ix\leq b\}\), where \(\sigma\) is the left one-sided shift, and \(\leq\) the usual lexicographic order. The author then defines the ``lexicographic world'' \({\mathfrak L}\) by \({\mathfrak L}:= \{(x,y)\in \Sigma^2\), \(\Sigma_{xy}\neq \emptyset\}\). NEWLINENEWLINENEWLINEFinally let, for each sequence \(a\) in \(\Sigma\), \(\varphi(a):= \inf\{b\in \Sigma\); \(\Sigma_{a,b}\neq \emptyset\}\) (hence \({\mathfrak L}= \{(x,y)\in \Sigma^2\), \(y\geq \varphi(x)\}\)). NEWLINENEWLINENEWLINEThe main result of the paper is the following: NEWLINENEWLINENEWLINETheorem: \(b=\varphi(a)\) for some \(a= 0x\in \Sigma\) if and only if \(b\) is the Sturmian sequence whose orbit under the shift is contained in \([0x,1x]\) and \(\sigma^ib\leq b\) \(\forall i\geq 0\). NEWLINENEWLINENEWLINENote that \textit{A. Borel} and \textit{F. Laubie} proved among other things in 1993 [J. Théor. Nombres Bordx. 5, No. 1, 23-51 (1993; Zbl 0839.11008)] that for each characteristic Sturmian sequence \(c\) one has \(0c\leq \sigma^ic\leq 1c\) \(\forall i\geq 0\). NEWLINENEWLINENEWLINEAlso note that in Reference [2], in the URL for the quoted preprint, the sign ``\(\sim\)'' is missing before ``pub-off''.