Spectrum, algebraicity and normalization in alternate bases

Abstract: The first aim of this work is to give information about the algebraic properties of alternate bases determining sofic systems. We exhibit two conditions: one necessary and one sufficient. Comparing the setting of alternate bases to that of one real base, these conditions exhibit a new phenomenon: the bases should be expressible as rational functions of their product. The second aim is to provide an analogue of Frougny's result concerning normalization of real bases representations. Under some suitable condition (i.e., our previous sufficient condition for being a sofic system), we prove that 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. For our purposes, we use a generalized concept of spectrum associated with a complex base and complex digits, and we study its topological properties.

This is joint work with Célia Cisternino, Zuzana Masáková and Edita Pelantová.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper )

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