Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T04:05:09.995Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
YANN BUGEAUD
Affiliation:
Université Louis Pasteur, U. F. R. de mathématiques, 7, rue René Descartes, 67084 Strasbourg Cedex, France (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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