β-shift, systèmes de numération et automates
Journal de théorie des nombres de Bordeaux, Tome 7 (1995) no. 2, pp. 473-498.

In this note we prove that the language of a numeration system is the language of a β -shift under some assumptions on the basis. We deduce from this result a partial answer to the question when the language of a numeration system is regular. Moreover, we give a characterization of the arithmetico-geometric sequences and the mixed radix sequences that are basis of a numeration system for which the language is regular. Finally, we study the Ostrowski systems of numeration and give another proof of the result of J. Shallit : the Ostrowski systems having a regular langage are exactly the ones associated to a quadratic number.

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Nathalie Loraud. $\beta $-shift, systèmes de numération et automates. Journal de théorie des nombres de Bordeaux, Tome 7 (1995) no. 2, pp. 473-498. https://jtnb.centre-mersenne.org/item/JTNB_1995__7_2_473_0/

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