Expansions in Cantor real bases

We introduce and study series expansions of real numbers with an arbitrary Cantor real base β=(β _n)_n∈ℕ , which we call β -representations. In doing so, we generalize both representations of real numbers in real bases and through Cantor series. We show fundamental properties of β -representations, each of which extends existing results on representations in a real base. In particular, we prove a generalization of Parry’s theorem characterizing sequences of nonnegative integers that are the greedy β -representations of some real number in the interval [0, 1). We pay special attention to periodic Cantor real bases, which we call alternate bases. In this case, we show that the β -shift is sofic if and only if all quasi-greedy β^(i) -expansions of 1 are ultimately periodic, where β^(i) is the i -th shift of the Cantor real base β .