Ito-Sadahiro Numbers Vs. Parry Numbers
Acta Polytechnica Vol. 51 No. 4/2011 Ito-Sadahiro numbers vs. Parry numbers Z. Mas´akov´a, E. Pelantov´a Abstract We consider a positional numeration system with a negative base, as introduced by Ito and Sadahiro. In particular, we focus on the algebraic properties of negative bases −β for which the corresponding dynamical system is sofic, which β happens, according to Ito and Sadahiro, if and only if the (−β)-expansion of − is eventually periodic. We call β +1 such numbers β Ito-Sadahiro numbers, and we compare their properties with those of Parry numbers, which occur in thesamecontextfortheR´enyi positive base numeration system. Keywords: numeration systems, negative base, Pisot number, Parry number. ∈AN 1 Introduction According to Parry, the string x1x2x3 ... rep- resents the β-expansion of a number x ∈ [0, 1) if and The expansion of a real number in the positional only if number system with base β>1, as defined by R´enyi [12] is closely related to the transformation ∗ xixi+1xi+2 ...≺ d (1) (2) T :[0, 1) → [0, 1), given by the prescription T (x):= β βx −#βx$.Everyx ∈ [0, 1) is a sum of the infinite for every i =1, 2, 3,... series D { | ∞ Condition (2) ensures that the set β = dβ(x) xi − ∈ } D x = , where x = #βT i 1(x)$ (1) x [0, 1) is shift invariant, and so the closure of β βi i N i=1 in A , denoted by Sβ, is a subshift of the full shift N for i =1, 2, 3,... A . The notion of β-expansion can naturally be ex- Directly from the definition of the transformation T tended to all non-negative real numbers: The expres- we can derive that the ‘digits’ x take values in the set i sion of a positive real number y in the form {0, 1, 2,...,%β&−1} for i =1, 2, 3,...
[Show full text]