Digital expansions with negative real bases

Similarly to Parry's characterization of $β$-expansions of real numbers in real bases $β> 1$, Ito and Sadahiro characterized digital expansions in negative bases, by the expansions of the endpoints of the fundamental interval. Parry also described the possible expansions of 1 in base $β> 1$. In the same vein, we characterize the sequences that occur as $(-β)$-expansion of $\frac{-β}{β+1}$ for some $β> 1$. These sequences also describe the itineraries of 1 by linear mod one transformations with negative slope.

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