Zeta function and negative beta-shifts

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.