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$\beta$-expansions for cubic Pisot numbers

Abstract : Real numbers can be represented in an arbitrary base > 1 using the transformation T : x ! x (mod 1) of the unit interval; any real number x 2 [0; 1℄ is then expanded into d (x) = (xi)i 1 where xi = b T i1 (x) . The losure of the set of the expansions of real numbers of [0; 1[ is a subshift of fa 2 N j a < g N , alled the beta-shift. This dynami al system is hara terized by the beta-expansion of 1; in parti ular, it is of nite type if and only if d (1) is nite; is then alled a simple beta-number. We rst ompute the beta-expansion of 1 for any ubi Pisot number. Next we show that ubi simple beta-numbers are Pisot numbers.
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Contributor : Frédérique Bassino Connect in order to contact the contributor
Submitted on : Thursday, October 6, 2011 - 1:32:14 PM
Last modification on : Wednesday, January 19, 2022 - 4:42:04 PM
Long-term archiving on: : Saturday, January 7, 2012 - 2:21:38 AM


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  • HAL Id : hal-00619858, version 1



Frédérique Bassino. $\beta$-expansions for cubic Pisot numbers. 5th Latin American Theoretical INformatics (LATIN'2002), 2002, United States. pp.141-152. ⟨hal-00619858⟩



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