Alternate base; Normalization; Spectrum; Pisot number; Parry alternate base
Abstract :
[en] The first aim of this article is to give information about the algebraic properties of alternate bases determining sofic systems. We show that a necessary condition is that the product d of the bases is an algebraic integer and all of the bases belong to the algebraic field Q(d). On the other hand, we also give a sufficient condition: if the product d of the bases is a Pisot number and all the bases belong to Q(d), then the system associated with the alternate base is sofic. The second aim of this paper is to provide an analogy of Frougny's result concerning normalization of real bases representations. We show that given an alternate base such that the product d of the bases is a Pisot number and all the bases belong to Q(d), the normalization function is computable by a finite Büchi automaton, and furthermore, we effectively construct such an automaton. An important tool in our study is the spectrum of numeration systems associated with alternate bases. The spectrum of a real number d>1 and an alphabet A of integers was introduced by Erdös et al. For our purposes, we use a generalized concept for a complex number d and a complex alphabet A and study its topological properties.
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