Abstract. Let β be a real number with 1 < β < 2. We prove that the language of the β-shift is ∆ 0 n iff β is a ∆n-real. The special case where n is 1 is the independently interesting result that the language of the β-shift is decidable iff β is a computable real. The “if ” part of the proof is non-constructive; we show that for Walters ’ version of the β-shift, no constructive proof exists. 1

Abstract not found

This chapter focuses on multiple tilings associated with substitutive dynamical systems. We recall that a substitutive dynamical system (Xσ, S) is a symbolic dynamical system where the shift S acts on the set Xσ of infinite words having the same language as a given infinite word which is

ABSTRACT. Given β ∈ (1, 2), a β-expansion of a real x is a series in decreasing powers of β with coefficients 0 and 1 whose sum equals x. The aim of this note is to study certain problems related to the universality and combinatorics of β-expansions. Our main result is as follows: for each β ∈ (1, 2) and a.e. x ∈ (0, 1) there always exists a universal β-expansion of x in the sense of Erdős and Komornik, i.e., a β-expansion whose complexity function is 2 n. Besides, we study some properties of points having a “small ” set of β-expansions and finish the paper by considering normal β-expansions. 1. FORMULATION OF MAIN RESULTS Let β ∈ (1, 2) be our parameter and put Σ = � ∞ 1 {0, 1}. Fix x ≥ 0; we call a sequence ε ∈ Σ a β-expansion of x, if it satisfies (1.1) x = πβ(ε): = εnβ −n. Remark 1.1. Note that traditionally this term implies the greedy β-expansion of x (see below) but we prefer to use it in the above sense, because we will be interested in all β-expansions of a given x. We hope this will not cause any confusion. It is clear that since ε = (εn) ∞ 1 ∈ Σ, each x representable in the form of the series (1.1), must belong to the interval Iβ: = [0, 1/(β − 1)]. On the other hand, each x ∈ Iβ does have at least one β-expansion, namely, the greedy β-expansion mentioned above. It is defined as follows: if x ∈ [0, 1), let Tβ(x) = βx mod 1, and put εn = ⌊βT n−1 β x⌋, n ≥ 1 (here the power stands for the corresponding iteration and ⌊· ⌋ denotes the integral part of a number). If x ∈ [1, (β − 1)), then we put n=1 ℓ = min {k ≥ 1: x − β −1 − · · · − β −k ∈ (0, 1)} and apply the greedy algorithm to x − β −1 − · · · − β ℓ to obtain the digits εℓ+1, εℓ+2, etc. Finally, if x = 1/(β − 1), then inevitably εn ≡ 1. An important property of the greedy β-expansions is their monotonicity. Namely, if x < y, then the greedy β-expansion ε of x is lexicographically less than the greedy β-expansion ε ′ of y, i.e., εn < ε ′ n for the smallest n ≥ 1 such that εn � = ε ′ n. A detailed description of all possible greedy β-expansions was given by Parry [10] and is briefly described in Section 2.

Noname manuscript No. (will be inserted by the editor) Numeration and discrete dynamical systems