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.

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Introduction. By a famous theorem of Siegel [S], the number of integral points on an elliptic curve E over an algebraic number field K is finite. A conjecture of Lang and Demyanenko (see [L3], p. 140) states that, for a quasiminimal model of E over K, this number is bounded by a constant depending only on the rank of E over K and on K (see also [HSi], [Zi4] [GSch]). The Siegel-Baker method (see [L3]) for the calculation of integer points on elliptic curves over K = Q requires some detailed information about certain quartic number fields. Computing these fields often represents a hard problem and, moreover, this approach does not seem to be appropriate. That is why in general all the results mentioned above cannot be used for the actual calculation of all integral points on an elliptic curve E over Q.However, there is another method suggested by Lang [L1], [L3] and further developed by Zagier [Za]. We shall work out the Lang-Zagier method and turn it into an algorithm for determining all integral points on elliptic curves E over Q using elliptic logarithms. The algorithm requires the knowledge of a basis of the Mordell-Weil group E(Q) and of an explicit lower bound for linear forms in elliptic logarithms. Compared to the Siegel-Baker

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.

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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.

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