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Sturmian sequences and the lexicographic world

Author: Shaobo Gan
Journal: Proc. Amer. Math. Soc. 129 (2001), 1445-1451
MSC (2000): Primary 37B10
Published electronically: December 13, 2000
MathSciNet review: 1814171
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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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Additional Information

Shaobo Gan
Affiliation: School of Mathematical Science, Peking University, Beijing 100871, China
Address at time of publication: The Abdus Salam International Centre for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy

Keywords: Sturmian sequences, lexicographic world
Received by editor(s): August 21, 1999
Published electronically: December 13, 2000
Additional Notes: This research was supported by the NSFC (No. 10001003) and Scientific Foundation for Returned Overseas Chinese Scholars, Ministry of Education
Communicated by: Michael Handel
Article copyright: © Copyright 2000 American Mathematical Society