$\beta$-expansions for cubic Pisot numbers

Frédérique Bassino (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge

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.

Domaines: Algorithme et structure de données [cs.DS]

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