In this note we prove that the language of a numeration system is the language of a -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.
@article{JTNB_1995__7_2_473_0, author = {Loraud, Nathalie}, title = {$\beta $-shift, syst\`emes de num\'eration et automates}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {473--498}, publisher = {Universit\'e Bordeaux I}, volume = {7}, number = {2}, year = {1995}, mrnumber = {1378592}, zbl = {0843.11013}, language = {fr}, url = {http://archive.numdam.org/item/JTNB_1995__7_2_473_0/} }
TY - JOUR AU - Loraud, Nathalie TI - $\beta $-shift, systèmes de numération et automates JO - Journal de théorie des nombres de Bordeaux PY - 1995 SP - 473 EP - 498 VL - 7 IS - 2 PB - Université Bordeaux I UR - http://archive.numdam.org/item/JTNB_1995__7_2_473_0/ LA - fr ID - JTNB_1995__7_2_473_0 ER -
Loraud, Nathalie. $\beta $-shift, systèmes de numération et automates. Journal de théorie des nombres de Bordeaux, Volume 7 (1995) no. 2, pp. 473-498. http://archive.numdam.org/item/JTNB_1995__7_2_473_0/
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