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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.

A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique such expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter finite right transducer. Every real number has at least one such expansion and a countable infinite set of them have more than one. We explain how these expansions can be approximated and characterize the expansions of reals that have two expansions.These results are not only developed for their own sake but also as they relate to other problems in combinatorics and number theory.

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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.

A standard way to parametrize the boundary of a connected fractal tile T is proposed. The parametrization is Hölder continuous from R/Z to ∂T and fixed points of ∂T have algebraic preimages. A class of planar tiles is studied in detail as sample cases and a relation with the recurrent set method by Dekking is discussed. When the tile T is a topological disk, this parametrization is a bi-Hölder homeomorphism.

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.

In the present paper we give an overview of topological properties of self-affine tiles. After reviewing some basic results on self-affine tiles and their boundary we give criteria for their local connectivity and connectivity. Furthermore, we study the connectivity of the interior of a family of tiles associated to quadratic number systems and give results on their fundamental group. If a self-affine tile tessellates the space the structure of the set of its 'neighbors' is discussed.