Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2015; 52(4): 1047-1057

Printed July 31, 2015

https://doi.org/10.4134/BKMS.2015.52.4.1047

Copyright © The Korean Mathematical Society.

On the period of $\beta$-expansion of Pisot or Salem series over $\mathbb{F}_{q}((x^{-1}))$

Ghorbel Rim and Zouari Sourour

Facult\'e des Sciences de Sfax, Facult\'e des Sciences de Sfax

Abstract

In \cite{rhs1}, it is proved that the lengths of periods occurring in the $\beta$-expansion of a rational series $r$ noted by $Per_{\beta}(r)$ depend only on the denominator of the reduced form of $r$ for quadratic Pisot unit series. In this paper, we will show first that every rational $r$ in the unit disk has strictly periodic $\beta$-expansion for Pisot or Salem unit basis under some condition. Second, for this basis, if $r =\frac{P}{Q} $ is written in reduced form with $|P| < |Q|$, we will generalize the curious property ``$Per_{\beta}(\frac{P}{Q})=Per_{\beta}(\frac{1}{Q})$".

Keywords: formal power series, $\beta$-expansion, Pisot series, Salem series

MSC numbers: 11R06, 37B50