Number representation using generalized $(-\beta )$-transformation

We study non-standard number systems with negative base $-\beta$. Instead of the Ito-Sadahiro definition, based on the transformation $T_{-\beta }$ of the interval $[-\frac{\beta }{\beta +1},\frac{1}{\beta +1})$ into itself, we suggest a generalization using an interval $[l,l+1)$ with $l\in (-1,0]$. Such generalization may eliminate certain disadvantages of the Ito-Sadahiro system. We focus on the description of admissible digit strings and their periodicity.