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Dynamics for β-shifts and Diophantine approximation

Published online by Cambridge University Press:  01 December 2007

BORIS ADAMCZEWSKI  and
YANN BUGEAUD
BORIS ADAMCZEWSKI
Affiliation:
CNRS, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 21 avenue Claude Bernard, 69622 Villeurbanne Cedex, France (email: Boris.Adamczewski@math.univ-lyon1.fr)
YANN BUGEAUD
Affiliation:
Université Louis Pasteur, U. F. R. de mathématiques, 7, rue René Descartes, 67084 Strasbourg Cedex, France (email: bugeaud@math.u-strasbg.fr)
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Abstract

We investigate the β-expansion of an algebraic number in an algebraic base β. Using tools from Diophantine approximation, we prove several results that may suggest a strong difference between the asymptotic behaviour of eventually periodic expansions and that of non-eventually periodic expansions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 6 , December 2007 , pp. 1695 - 1711
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Adamczewski, B. and Bugeaud, Y.. On the complexity of algebraic numbers II. Continued fractions. Acta Math. 195 (2005), 1–20.Google Scholar
[2]Adamczewski, B. and Bugeaud, Y.. On the complexity of algebraic numbers I. Expansions in integer bases. Ann. of Math. (2) 165 (2007), 547–565.CrossRefGoogle Scholar
[3]Adamczewski, B., Bugeaud, Y. and Luca, F.. Sur la complexité des nombres algébriques. C. R. Acad. Sci. Paris 339 (2004), 11–14.CrossRefGoogle Scholar
[4]Adamczewski, B. and Delaunay, C.. Some computations on the β-expansion of algebraic numbers in an algebraic basis β, in progress.Google Scholar
[5]Allouche, J.-P. and Cosnard, M.. The Komornik–Loreti constant is transcendental. Amer. Math. Monthly 107 (2000), 448–449.CrossRefGoogle Scholar
[6]Allouche, J.-P. and Shallit, J. O.. Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge, 2003.CrossRefGoogle Scholar
[7]Bailey, D. and Crandall, R. E.. On the random character of fundamental constant expansions. Experiment. Math. 10 (2001), 175–190.CrossRefGoogle Scholar
[8]Bernat, J.. Propriétés arithmétiques de la β-numération. PhD Thesis, Université de la Méditerranée, 2005.Google Scholar
[9]Berthé, V. and Siegel, A.. Tilings associated with β-numeration and substitutions. Integers 5(3) (2005), A2 (46pp).Google Scholar
[10]Bertrand-Mathis, A.. Développements en base θ, répartition modulo un de la suite (xθ n)n≥0, langages codés et θ-shift. Bull. Soc. Math. France 114 (1986), 271–323.CrossRefGoogle Scholar
[11]Bertrand-Mathis, A.. Points génériques de Champernowne sur certains systèmes codés; application aux θ-shifts. Ergod. Th. & Dynam. Sys. 8 (1988), 35–51.CrossRefGoogle Scholar
[12]Blanchard, F.. β-expansions and symbolic dynamics. Theoret. Comput. Sci. 65 (1989), 131–141.CrossRefGoogle Scholar
[13]Borel, É.. Sur les chiffres décimaux de et divers problèmes de probabilités en chaǐne. C. R. Acad. Sci. Paris 230 (1950), 591–593.Google Scholar
[14]Boyd, D.. Salem numbers of degree four have periodic expansions. Théorie des nombres (Quebec, PQ, 1987). de Gruyter, Berlin, 1989, pp. 57–64.Google Scholar
[15]Boyd, D.. On the beta expansion for Salem numbers of degree 6. Math. Comp. 65 (1996), 861–875.CrossRefGoogle Scholar
[16]Chi, D. P. and Kwon, D. Y.. Sturmian words, β-shifts, and transcendence. Theoret. Comput. Sci. 321 (2004), 395–404.CrossRefGoogle Scholar
[17]Corvaja, P. and Zannier, U.. Some new applications of the subspace theorem. Compos. Math. 131 (2002), 319–340.CrossRefGoogle Scholar
[18]Dajani, K. and Kraaikamp, C.. Ergodic Theory of Numbers (Carus Mathematical Monographs, 29). Mathematical Association of America, Washington, 2002.CrossRefGoogle Scholar
[19]Evertse, J.-H.. An improvement of the quantitative Subspace theorem. Compos. Math. 101 (1996), 225–311.Google Scholar
[20]Gazeau, J.-P. and Verger-Gaugry, J.-L.. Geometric study of the beta-integers for a Perron number and mathematical quasicrystals. J. Théor. Nombres Bordeaux 16 (2004), 125–149.CrossRefGoogle Scholar
[21]Gheorghiciuc, I.. The subword complexity of a class of infinite binary words. Adv. Appl. Math 39 (2007), 237–259.CrossRefGoogle Scholar
[22]Hofbauer, F.. β-shifts have unique maximal measure. Monatsh. Math. 85 (1978), 189–198.CrossRefGoogle Scholar
[23]Ito, S. and Shiokawa, I.. A construction of β-normal sequences. J. Math. Soc. Japan 27 (1975), 20–23.CrossRefGoogle Scholar
[24]Lagarias, J. C.. On the normality of arithmetical constants. Experiment. Math. 10 (2001), 355–368.Google Scholar
[25]Lang, S.. Introduction to Diophantine Approximations. Springer, New York, 1995.CrossRefGoogle Scholar
[26]Lothaire, M.. Algebraic Combinatorics on Words (Encyclopedia of Mathematics and its Applications, 90). Cambridge University Press, Cambridge, 2002.CrossRefGoogle Scholar
[27]Mahler, K.. Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen. Math. Ann. 101 (1929), 342–366. Corrigendum 103 (1930), 532.CrossRefGoogle Scholar
[28]Meyer, Y.. Quasicrystals, Diophantine approximation and algebraic number. Beyond Quasicrystals (Les Houches, 1994). Springer, New York, 1995, pp. 3–16.CrossRefGoogle Scholar
[29]Moody, R. V.. Meyer sets and their duals. The Mathematics of Long-Range Aperiodic Order (Waterloo, ON, 1995) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 489). Kluwer, Dordrecht, 1997, pp. 403–441.CrossRefGoogle Scholar
[30]Parry, W.. On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.CrossRefGoogle Scholar
[31]Pytheas Fogg, N.. Substitutions in Dynamics, Arithmetics and Combinatorics (Lecture Notes in Mathematics, 1794). Springer, New York, 2002.CrossRefGoogle Scholar
[32]Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493.CrossRefGoogle Scholar
[33]Schmeling, J.. Symbolic dynamics for β-shifts and self-normal numbers. Ergod. Th. & Dynam. Sys. 17 (1997), 675–694.CrossRefGoogle Scholar
[34]Schmidt, K.. On periodic expansions of Pisot and Salem numbers. Bull. London Math. Soc. 12 (1980), 269–278.CrossRefGoogle Scholar
[35]Schmidt, W. M.. Diophantine Approximation (Lecture Notes in Mathematics, 785). Springer, New York, 1980.Google Scholar
[36]Thurston, W. P.. Groups, tilings and finite state automata. Summer 1989 AMS Colloquium Lectures, 1989.Google Scholar
[37]Verger-Gaugry, J.-L.. On gaps in Rényi β-expansions of unity and β-integers for β>1 an algebraic number. Ann. Inst. Fourier (Grenoble) 56 (2006), 2565–2579.Google Scholar
[38]Waldschmidt, M.. Diophantine Approximation on Linear Algebraic Groups (Grundlehren der Mathematischen Wissenschaften, 326). Springer, New York, 2000.CrossRefGoogle Scholar