# A note on univoque self-sturmian numbers

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2008)

- Volume: 42, Issue: 4, page 659-662
- ISSN: 0988-3754

## Access Full Article

top## Abstract

top## How to cite

topAllouche, Jean-Paul. "A note on univoque self-sturmian numbers." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 42.4 (2008): 659-662. <http://eudml.org/doc/245511>.

@article{Allouche2008,

abstract = {We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizes univoque real numbers; the other is a disguised version of the set of characteristic sturmian sequences. As a corollary to our study we obtain that a real number $\beta $ in $(1,2)$ is univoque and self-sturmian if and only if the $\beta $-expansion of $1$ is of the form $1v$, where $v$ is a characteristic sturmian sequence beginning itself in $1$.},

author = {Allouche, Jean-Paul},

journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},

keywords = {sturmian sequences; univoque numbers; self-sturmian numbers; Sturmian sequences; self-Sturmian numbers; kneading sequences},

language = {eng},

number = {4},

pages = {659-662},

publisher = {EDP-Sciences},

title = {A note on univoque self-sturmian numbers},

url = {http://eudml.org/doc/245511},

volume = {42},

year = {2008},

}

TY - JOUR

AU - Allouche, Jean-Paul

TI - A note on univoque self-sturmian numbers

JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

PY - 2008

PB - EDP-Sciences

VL - 42

IS - 4

SP - 659

EP - 662

AB - We compare two sets of (infinite) binary sequences whose suffixes satisfy extremal conditions: one occurs when studying iterations of unimodal continuous maps from the unit interval into itself, but it also characterizes univoque real numbers; the other is a disguised version of the set of characteristic sturmian sequences. As a corollary to our study we obtain that a real number $\beta $ in $(1,2)$ is univoque and self-sturmian if and only if the $\beta $-expansion of $1$ is of the form $1v$, where $v$ is a characteristic sturmian sequence beginning itself in $1$.

LA - eng

KW - sturmian sequences; univoque numbers; self-sturmian numbers; Sturmian sequences; self-Sturmian numbers; kneading sequences

UR - http://eudml.org/doc/245511

ER -

## References

top- [1] J.-P. Allouche, Théorie des nombres et automates. Thèse d’État, Université Bordeaux I (1983).
- [2] J.-P. Allouche and M. Cosnard, Itérations de fonctions unimodales et suites engendrées par automates. C. R. Acad. Sci. Paris Sér. I 296 (1983) 159–162. Zbl0547.58027MR693191
- [3] J.-P. Allouche and M. Cosnard, The Komornik-Loreti constant is transcendental. Amer. Math. Monthly 107 (2000) 448–449. Zbl0997.11052MR1763399
- [4] J.-P. Allouche and M. Cosnard, Non-integer bases, iteration of continuous real maps, and an arithmetic self-similar set. Acta Math. Hungar. 91 (2001) 325–332. Zbl1012.11007MR1912007
- [5] J.-P. Allouche, C. Frougny and K.G. Hare, On univoque Pisot numbers. Math. Comput. 76 (2007) 1639–1660. Zbl1182.11051MR2299792
- [6] J.-P. Allouche and A. Glen, Extremal properties of (epi)sturmian sequences and distribution modulo $1$, Preprint (2007). Zbl1254.68192
- [7] Y. Bugeaud and A. Dubickas, Fractional parts of powers and Sturmian words. C. R. Math. Acad. Sci. Paris 341 (2005) 69–74. Zbl1140.11318MR2153958
- [8] S. Bullett and P. Sentenac, Ordered orbits of the shift, square roots, and the devil’s staircase. Math. Proc. Cambridge 115 (1994) 451–481. Zbl0823.58012MR1269932
- [9] D.P. Chi and D. Kwon, Sturmian words, $\beta $-shifts, and transcendence. Theor. Comput. Sci. 321 (2004) 395–404. Zbl1068.68112MR2076154
- [10] M. Cosnard, Étude de la classification topologique des fonctions unimodales. Ann. Inst. Fourier 35 (1985) 59–77. Zbl0569.58004MR810668
- [11] P. Erdős, I. Joó and V. Komornik, Characterization of the unique expansions $1=\sum {q}^{-{n}_{i}}$ and related problems. Bull. Soc. Math. France 118 (1990) 377–390. Zbl0721.11005MR1078082
- [12] V. Komornik and P. Loreti, Unique developments in non-integer bases. Amer. Math. Monthly 105 (1998) 636–639. Zbl0918.11006MR1633077
- [13] M. Lothaire, Algebraic Combinatorics On Words, Encyclopedia of Mathematics and its Applications, Vol. 90. Cambridge University Press (2002). Zbl1001.68093MR1905123
- [14] G. Pirillo, Inequalities characterizing standard Sturmian words. Pure Math. Appl. 14 (2003) 141–144. Zbl1065.68081MR2054742
- [15] P. Veerman, Symbolic dynamics and rotation numbers. Physica A 134 (1986) 543–576. Zbl0655.58019MR861742
- [16] P. Veerman, Symbolic dynamics of order-preserving orbits. Physica D 29 (1987) 191–201. Zbl0625.28012MR923891

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.