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Monatshefte für Mathematik

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

Zeta function and negative beta-shifts

  • Florent Nguema NdongEmail author
Article
  • 51 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

11A63 11B05 11M06 37E05 

Notes

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Copyright information

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

Authors and Affiliations

  1. 1.Université des Sciences et Techniques de MasukuFrancevilleGabon

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