In this paper, we study representations of real numbers in the positional numeration system with negative basis, as introduced by Ito and Sadahiro. We focus on the set $\Z_{ -\beta}$ of numbers whose representation uses only non-negative powers of $-\beta$, the so-called $(-\beta)$-integers. We describe the distances between consecutive elements of $\Z_{ -\beta}$. In case that this set is non-trivial we associate to $\beta$ an infinite word $\boldsymbol{ v}_{ -\beta}$ over an (in general infinite) alphabet. The self-similarity of $\Z_{ -\beta}$, i.e., the property $-\beta \Z_{ -\beta}\subset \Z_{ -\beta}$, allows us to find a morphism under which $\boldsymbol{ v}_{ -\beta}$ is invariant. On the example of two cubic irrational bases $\beta$ we demonstrate the difference between Rauzy fractals generated by $(-\beta)$-integers and by $\beta$-integers.
Categories: math.NT