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.