Digital expansions with negative real bases

Abstract

Abstract. Similarly to Parry's characterization of β-expansions of real numbers in real bases β > 1, Ito and Sadahiro characterized digital expansions in negative bases, by the expansions of the endpoints of the fundamental interval. Parry also described the possible expansions of 1 in base β > 1. In the same vein, we characterize the sequences that occur as (−β)-expansion of −β β+1 for some β > 1. These sequences also describe the itineraries of 1 by linear mod one transformations with negative slope.

Permutations and Negative Beta-Shifts

“…We have already seen in Lemma 13 that W −α ∩ W −β = ∅. If α = 1, then we have s ≤ u < w. If α > 1, then we have d −α (1) < d −β (1) by Theorem 3 in [11]. As d −α (1) ∈ W −α , d −β (1) ∈ W −β , and the elements of W −α and W −β respectively are contiguous by Lemmas 14 and 15, we obtain that s < w. We also obtain that…”

“…By Theorem 2 of [11], we have d −b(w) (1) = w for all w ∈ (n−2)(n−3) · · · 10, n − 2, (n−2)(n−3) · · · 11, (n−2)(n−3) · · · 210 , except for w = 210, as b(210) = 2. For w = (n−2)(n−3) · · · 10, we have ⌊b(w)⌋ = n− 2, w > n − 2, w > (n−2)(n−3) · · · 210 if n is even, w > (n−2)(n−3) · · · 11 if n is odd.…”

“…As v > u, we have s = 0 and thus v = s k−1 s ′ , contradicting Theorem 2 of [11]. If |v| is odd, then we have v ′ = lim x→1 d −β (x) by…”

Critical itineraries of maps with constant slope and one discontinuity

“…For the cases corresponding to Ito-Sadahiro's (−β)-transformations, we obtain the following characterization from [25]. Let d = 100111001001001110011 · · · ∈ Ω be the word starting with ϕ n (1) for all n ≥ 0, where ϕ denotes the morphism on {0, 1} * defined by ϕ(1) = 100, ϕ(0) = 1.…”

“…The critical itineraries of f −2,1/3 are (0010; 0010), which is a pair satisfying the conditions above. However, the word 10 does not satisfy condition (1.8) in [25] because 10 ∈ {2, 10} ω .…”