Sturmian sequences and the lexicographic world
HTML articles powered by AMS MathViewer
- by Shaobo Gan PDF
- 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 $[0x,1x]$.References
- Ethan M. Coven and G. A. Hedlund, Sequences with minimal block growth, Math. Systems Theory 7 (1973), 138–153. MR 322838, DOI 10.1007/BF01762232
- R. Labarca and S. Plaza, Bifurcation of discontinuous maps of the interval and palindromicnumbers, ICTP preprint IC/98/165, http://www.ictp.trieste.it/ pub_off/
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
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
- Email: gansb@sxx0.math.pku.edu.cn
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1445-1451
- MSC (2000): Primary 37B10
- DOI: https://doi.org/10.1090/S0002-9939-00-05950-5
- MathSciNet review: 1814171