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 Rényi, and Ito & Sadahiro, respectively. In particular, we compare the sets $\mathbb{Z}_β^+$ and $\mathbb{Z}_{-β}$ of nonnegative $β$-integers and $(-β)$-integers. We describe all bases $(\pmβ)$ for which $\mathbb{Z}_β^+$ and $\mathbb{Z}_{-β}$ 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 $β$ with another interesting property, namely that any integer linear combination of non-negative powers of the base $-β$ with coefficients in $\{0,1,\dots,\lfloorβ\rfloor\}$ is a $(-β)$-integer, although the corresponding sequence of digits is forbidden as a $(-β)$-integer.

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