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ON THE PERIOD OF Ξ²-EXPANSION OF PISOT OR SALEM SERIES OVER 𝔽q((x-1))

  • RIM, GHORBEL (FACULTE DES SCIENCES DE SFAX DEPARTEMENT DE MATHEMATIQUES) ;
  • SOUROUR, ZOUARI (FACULTE DES SCIENCES DE SFAX DEPARTEMENT DE MATHEMATIQUES)
  • Received : 2013.03.15
  • Published : 2015.07.31
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Abstract

In [6], 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 "$Per_{\beta}(\frac{P}{Q})=Per_{\beta}(\frac{1}{Q})$".

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References

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