Confluent Parry numbers, their spectra, and integers in positive- and negative-base number systems

In this paper we study the expansions of real numbers in positive and negative real base as introduced by R'enyi, and Ito & Sadahiro, respectively. In particular, we compare the sets ${\mathbb{Z}}_{\beta }^{+}$ and ${\mathbb{Z}}_{-\beta }$ of nonnegative $\beta$-integers and $(-\beta )$-integers. We describe all bases $(\pm \beta )$ for which ${\mathbb{Z}}_{\beta }^{+}$ and ${\mathbb{Z}}_{-\beta }$ can be coded by infinite words which are fixed points of conjugated morphisms, and consequently have the same language. Moreover, we prove that this happens precisely for $\beta$ with another interesting property, namely that any integer linear combination of non-negative powers of the base $-\beta$ with coefficients in $\{0,1,\dots ,\lfloor \beta \rfloor \}$ is a $(-\beta )$-integer, although the corresponding sequence of digits is forbidden as a $(-\beta )$-integer.