Growth rate for beta-expansions

Growth Rate for Beta-expansions

Let β > 1 and let m > β be an integer. Each x ∈ Iβ := [0, m−1 β−1 ] can be represented in the form x = ∞ ∑ k=1 εkβ −k, where εk ∈ {0, 1, . . . , m − 1} for all k (a β-expansion of x). It is known that a.e. x ∈ Iβ has a continuum of distinct β-expansions. In this paper we prove that if β is a Pisot number, then for a.e. x this continuum has one and the same growth rate. We also link this rate to...

On the Average Growth Exponent for Beta-expansions

Let β ∈ (1, 2). Each x ∈ Iβ := [0, 1 β−1 ] can be represented in the form

beta-EXPANSIONS WITH DELETED DIGITS FOR PISOT NUMBERS beta

On beta expansions for Pisot numbers

Given a number β > 1, the beta-transformation T = Tβ is defined for x ∈ [0, 1] by Tx := βx (mod 1). The number β is said to be a betanumber if the orbit {Tn(1)} is finite, hence eventually periodic. In this case β is the root of a monic polynomial R(x) with integer coefficients called the characteristic polynomial of β. If P (x) is the minimal polynomial of β, then R(x) = P (x)Q(x) for some pol...

Beta Expansions for Regular Pisot Numbers

A beta expansion is the analogue of the base 10 representation of a real number, where the base may be a non-integer. Although the greedy beta expansion of 1 using a non-integer base is, in general, infinitely long and non-repeating, it is known that if the base is a Pisot number, then this expansion will always be finite or periodic. Some work has been done to learn more about these expansions...