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.

Abstract. In this paper we are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot Number Systems as well as Canonical Number Systems.

Abstract. We determine periodic and aperiodic points of certain piecewise affine maps in the Euclidean plane. Using these maps, we prove for λ ∈ { ±1± √ 5 2 , ± √ 2, ± √ 3} that all integer sequences (a k ) k∈Z satisfying 0 ≤ a k−1 + λa k + a k+1 < 1 are periodic.

Abstract. The periodicity of sequences of integers (an) n∈Z satisfying the inequalities 0 ≤ a n−1 + λan + a n+1 < 1 (n ∈ Z)is studied for real λ with |λ| < 2. Periodicity is proved in case λ is the golden ratio; for other values of λ statements on possible period lengths are given. Further interesting results on the morphology of periods are illustrated.The problem is connected to the investigation of shift radix systems and of Salem numbers.

For r = (r 1 , . . . , r d ) ∈ R d the mapping τ r :where • denotes the floor function, is called a shift radix system if for each a ∈ Z d there exists an integer k > 0 with τ k r (a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like βexpansions and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).

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Recently, Akiyama et al. introduced so-called shift radix systems. These simple dynamical systems form a common generalization of several well-known notions of number systems like beta numeration and canonical number systems. In the present paper we generalize shift radix systems as follows: for (r1, . . . , r d ) ∈ C d we study mappingswhere for x ∈ C we set ⌊x⌋ = ⌊ℜx⌋ + i⌊ℑx⌋. We study basic dynamical properties of this class of mappings and relate them to known notions of number systems.