Acta Mathematica Hungarica

, Volume 139, Issue 1–2, pp 106–119 | Cite as

Digital expansions with negative real bases

Article

Abstract

Similarly to Parry’s characterization of β-expansions of real numbers in real bases β>1, Ito and Sadahiro characterized digital expansions in negative bases, by the expansions of the endpoints of the fundamental interval. Parry also described the possible expansions of 1 in base β>1. In the same way, we characterize the sequences that occur as (−β)-expansion of \(\frac{-\beta}{\beta+1}\) for some β>1. These sequences also describe the itineraries of 1 by linear mod one transformations with negative slope.

Key words and phrases

digital expansion β-expansion β-transformation negative basis 

Mathematics Subject Classification

11A63 37B10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Z. Daróczy and I. Kátai, On the structure of univoque numbers, Publ. Math. Debrecen, 46 (1995), 385–408. MathSciNetMATHGoogle Scholar
  2. [2]
    D. Dombek, Z. Masáková and E. Pelantová, Number representation using generalized (−β)-transformation, Theoret. Comput. Sci., 412 (2011), 6653–6665. MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    B. Faller, Contribution to the Ergodic Theory of Piecewise Monotone Continuous Maps, Ph.D. Thesis, École Polytechnique Fédérale de Lausanne, 2008. Google Scholar
  4. [4]
    C. Frougny and A. C. Lai, Negative bases and automata, Discrete Math. Theor. Comput. Sci., 13 (2011), 75–94. MathSciNetGoogle Scholar
  5. [5]
    P. Góra, Invariant densities for generalized β-maps, Ergodic Theory Dynam. Systems, 27 (2007), 1583–1598. MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    S. Ito and T. Sadahiro, Beta-expansions with negative bases, Integers, 9 (2009), 239–259, A22. MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    V. Komornik and P. Loreti, On the topological structure of univoque sets, J. Number Theory, 122 (2007), 157–183. MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    L. Liao and W. Steiner, Dynamical properties of the negative beta-transformation, Ergodic Theory Dynam. Systems, to appear. Google Scholar
  9. [9]
    W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401–416. MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477–493. MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2012

Authors and Affiliations

  1. 1.LIAFA, CNRS UMR 7089Université Paris Diderot – Paris 7Paris Cedex 13France

Personalised recommendations