# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Sturmian sequences and the lexicographic worldHTML articles powered by AMS MathViewer

by Shaobo Gan
Proc. Amer. Math. Soc. 129 (2001), 1445-1451 Request permission

## Abstract:

In this paper, we give a complete description for the lexicographic world ${\mathcal {L}}=\{(x,y)\in \Sigma \times \Sigma : \Sigma _{xy}\not =\emptyset \} =\{(x,y):y\ge \phi (x)\}$, where $\Sigma =\{0,1\}^{\mathbf {N}}$, $\Sigma _{ab}=\{x\in \Sigma : a\le \sigma ^i(x)\le b,\text {for\ all\ }i\ge 0\}$, $\phi :\Sigma \to \Sigma$ is defined by $\phi (a)=\inf \{b:\Sigma _{ab}\not =\emptyset \}$ and the order $\le$ is the lexicographic order on $\Sigma$. The main result is that $b=\phi (a)$ for some $a=0x$ if and only if $b$ is the Sturmian sequence $b$ such that $\operatorname {Orb}(b)\subset [0x,1x]$ and $\sigma ^i(b)\le b$ for all $i\ge 0$. At the same time, a new description of Sturmian minimal sets is given. A minimal set $M$ is a Sturmian minimal set if and only if, for some $x\in \Sigma$, $M\subset [0x,1x]$. Moreover, for any $x\in \Sigma$, there exists a unique Sturmian minimal set in $[0x,1x]$.
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