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A note on univoque self-Sturmian numbers

Published online by Cambridge University Press:  04 January 2008

Jean-Paul Allouche
Jean-Paul Allouche*
Affiliation:
CNRS, LRI, UMR 8623, Université Paris Sud, Bâtiment 490, 91405 Orsay Cedex, France; allouche@lri.fr
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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 characterizesunivoque 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 β in (1,2) is univoque and self-Sturmian if and only if the β-expansion of 1 is of the form 1v, where v is a characteristic Sturmian sequence beginning itself in 1.

Keywords

Sturmian sequencesunivoque numbersself-Sturmian numbers.

Information

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 42 , Issue 4: Fibonacci words , October 2008 , pp. 659 - 662
Copyright
© EDP Sciences, 2008

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References

J.-P. Allouche, Théorie des nombres et automates. Thèse d'État, Université Bordeaux I (1983).
Allouche, J.-P. and Cosnard, M., Itérations de fonctions unimodales et suites engendrées par automates. C. R. Acad. Sci. Paris Sér. I  296 (1983) 159–162.
Allouche, J.-P. and Cosnard, M., The Komornik-Loreti constant is transcendental. Amer. Math. Monthly  107 (2000) 448–449. CrossRef
Allouche, J.-P. and Cosnard, M., Non-integer bases, iteration of continuous real maps, and an arithmetic self-similar set. Acta Math. Hungar. 91 (2001) 325–332. CrossRef
Allouche, J.-P., Frougny, C. and Hare, K.G., On univoque Pisot numbers. Math. Comput. 76 (2007) 1639–1660. CrossRef
J.-P. Allouche and A. Glen, Extremal properties of (epi)sturmian sequences and distribution modulo 1, Preprint (2007).
Bugeaud, Y. and Dubickas, A., Fractional parts of powers and Sturmian words. C. R. Math. Acad. Sci. Paris  341 (2005) 69–74. CrossRef
Bullett, S. and Sentenac, P., Ordered orbits of the shift, square roots, and the devil's staircase. Math. Proc. Cambridge 115 (1994) 451–481. CrossRef
Chi, D.P. and Kwon, D., Sturmian words, β-shifts, and transcendence. Theor. Comput. Sci. 321 (2004) 395–404. CrossRef
Cosnard, M., Étude de la classification topologique des fonctions unimodales. Ann. Inst. Fourier  35 (1985) 59–77. CrossRef
Erdős, P., Joó, I. and Komornik, V., Characterization of the unique expansions 1 = ∑ q-ni and related problems. Bull. Soc. Math. France  118 (1990) 377–390. CrossRef
Komornik, V. and Loreti, P., Unique developments in non-integer bases. Amer. Math. Monthly  105 (1998) 636–639. CrossRef
M. Lothaire, Algebraic Combinatorics On Words, Encyclopedia of Mathematics and its Applications, Vol. 90. Cambridge University Press (2002).
Pirillo, G., Inequalities characterizing standard Sturmian words. Pure Math. Appl. 14 (2003) 141–144.
Veerman, P., Symbolic dynamics and rotation numbers. Physica A  134 (1986) 543–576. CrossRef
Veerman, P., Symbolic dynamics of order-preserving orbits. Physica D  29 (1987) 191–201. CrossRef