Beta-Expansions with Negative Bases

Abstract

Abstract This paper investigates representations of real numbers with an arbitrary negative base – β < –1, which we call the (– β )-expansions. They arise from the orbits of the (– β )-transformation which is a natural modification of the β -transformation. We show some fundamental properties of (– β )-expansions, each of which corresponds to a well-known fact of ordinary β -expansions. In particular, we characterize the admissible sequences of (– β )-expansions, give a necessary and sufficient condition for the (– β )- shift to be sofic, and explicitly determine the invariant measure of the (– β )-transformations.