Generalized radix representations and dynamical systems II

Abstract

Case I: z = (n, −m) with n, m ∈ N. Then τ (z) = (−m, s) with s = m − ⌊(n − m)x⌋. Case I.1: s ≤ 0. Then τ (z) ∈ (−N) 2 and we are done by (iii). Case I.2: s > 0. Case I.2.1: s ≥ m. Then τ (z) ∈ M and we are done by (ii). Case I.2.2: s < m. Then n > m, τ (z) ∈ L and τ (z) 1 = m + s < z 1 , hence we are done by induction hypothesis. Case II: z = (−n, m) with n, m ∈ N. Case II.1: m ≥ n. Then z ∈ M and we are done by (ii). Case II.2: m < n. Case III: z = (n, m) with n, m ∈ N. Then τ (z) = (m, −p) with p ≥ m and τ 2 (z) = (−p, q) with q ≥ p. Thus τ 2 (z) ∈ M and our assertion follows from (ii).(v) By (iii) and (iv) we finally see Z 2 ⊆ S(x, x + 1), thereby completing the proof of the lemma.Proof. If m > n > 0 then τ (x,−x−1) (n, m) = (m, p) with p > m. Thus the sequence (τ k (x,−x−1) (1, 2)) k∈N is strictly increasing with respect to the norm · 1 .Proof of Theorem 2.1. (i) By the Schur-Cohn criterion ( 6 ) and ([4, Lemmas 4.1 and 4.2]) we know that E 2 ⊆ D 2 ⊆ E 2 .(ii) Let (x, y) ∈ L. We are going to show that (x, y) belongs to D 2 .Case I: x < 1. Case I.1: |y| < 1 + x. Then we are done by (i). Case I.2: |y| = 1 + x. Case I.2.1: y < 0. Then −y = 1 + x, x ≤ 0, and we are done by Lemma 2.2. Case I.2.2: y ≥ 0. Thus y = 1 + x. Case I.2.2.1: x ≤ 0. We are done by Lemma 2.2. Case I.2.2.2: x > 0. Our assertion drops out of Lemma 2.3. Case II: x = 1. Then y ∈ {−1, 0, 1} and the assertion can easily be checked.

Periodicity of Certain Piecewise Affine Planar Maps

“…The problem currently seems hopeless for cases (4) and (5). However, a nice observation on rational values of λ with prime-power denominator p n is exhibited in [9].…”

“…This conjecture originated on the one hand from a discretization process in a rounding-off scheme occurring in computer simulation of dynamical systems (we refer the reader to [19,27] and the literature quoted there), and on the other hand in the study of shift radix systems (see [4,2] for details). Extensive numerical evidence on the periodicity of integer sequences satisfying (1.1) was first observed in [26].…”

“…The present paper deals with a conjecture on the periodicity of a certain kind of these maps: Conjecture 1.1. [4,27] For every real λ with |λ| < 2, all integer sequences (a k ) k∈Z satisfying (1.1) 0 ≤ a k−1 + λa k + a k+1 < 1 for all k ∈ Z are periodic.…”

Remarks on a conjecture on certain integer sequences

“…But even in dimension d = 2 this easier problem is not completely settled. It is believed [1] that the line segment {(1, λ) ∈ R 2 ||λ| < 2} belongs to D 2 . Going back to the definitions we realize that we are exactly left with the problem stated in Conjecture 1.1.…”

“…In this note we will analyze the following conjecture (see [1]): Conjecture 1.1. Let λ ∈ R and assume that the sequence of integers (a n ) n∈Z satisfies the inequalities (1.1) 0 ≤ a n−1 + λa n + a n+1 < 1 (n ∈ Z).…”