Monatshefte für Mathematik

, Volume 188, Issue 4, pp 717–751 | Cite as

Zeta function and negative beta-shifts

  • Florent Nguema NdongEmail author


Given a real number \( \beta > 1\), we study the associated \( (-\,\beta )\)-shift introduced by Ito and Sadahiro. We compare some aspects of the \((-\,\beta )\)-shift to the \(\beta \)-shift. When the expansion in base \( -\,\beta \) of \( -\,\frac{\beta }{\beta +1} \) is periodic with odd period or when \( \beta \) is less than the golden ratio, the \( (-\,\beta )\)-shift cannot be coded because its language is not transitive. This intransitivity of words explains the existence of gaps in the interval \([-\,\frac{\beta }{\beta +1}, \frac{1}{\beta +1})\). We observe that an intransitive word appears in the \((-\,\beta )\)-expansion of a real number taken in the gap. Furthermore, we determine the Zeta function \(\zeta _{-\,\beta }\) of the \((-\,\beta )\)-transformation and the associated lap-counting function \(L_{T_{-\,\beta }}\). These two functions are related by \(\zeta _{-\,\beta }=(1-z^2)L_{T_{-\,\beta }}\). We observe some similarities with the zeta function of the \(\beta \)-transformation. The function \(\zeta _{-\,\beta }\) has a simple pole at \( \frac{1}{\beta }\) and no other singularities z such that \(\vert z \vert =\frac{1}{\beta }\). We also note an influence of gaps (\(\beta \) less than the golden ratio) on the zeta function.


Negative bases \(\beta \)-Expansions Coded system Zeta function 

Mathematics Subject Classification

11A63 11B05 11M06 37E05 



  1. 1.
    Artin, M., Mazur, B.: On periodic points. Ann. Math. 82, 82–99 (1965)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bertrand-Mathis, A.: Points génériques de Champernowne sur certains systèmes codes; application aux \(\theta \)-shifts. Erg.Theory Dyn. Syst. 8(1), 35–51 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertrand-Mathis, A.: Comment écrire les nombres relatifs dans une base qui n’est pas entière. Unif. Distrib. Theory 9(2), 135–156 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Blanchard, F., Hansel, G.: Systèmes codés. Theor. Comput. Sci. 44(1), 17–49 (1986)CrossRefzbMATHGoogle Scholar
  5. 5.
    Flatto, L., Lagarias, J.C., Poonen, B.: The zeta function of the beta transformation. Erg. Theory Dyn. Syst. 14(2), 237–266 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ito, S., Sadahiro, T.: Beta-expansions with negative bases. Integers 9(A22), 239–259 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Lagarias, J.C.: Number theory zeta functions and dynamical zeta functions. In: Spectral Problems in Geometry and Arithmetic (Iowa City, IA, 1997), Contemporary Mathematics., vol. 237, pp. 45–86. American Mathematical Society, Providence, RI, (1999)Google Scholar
  8. 8.
    Liao, L., Steiner, W.: Dynamical properties of the negative beta-transformation. Erg. Theory Dyn. Syst. 32(5), 1673–1690 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ndong, F.N.: On the lyndon dynamical system. Adv. Appl. Math. 78, 1–26 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rényi, A.: Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar 8, 477–493 (1957)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Université des Sciences et Techniques de MasukuFrancevilleGabon

Personalised recommendations