Monatshefte für Mathematik

, Volume 188, Issue 4, pp 717–751

# Zeta function and negative beta-shifts

• Florent Nguema Ndong
Article

## 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

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