Let T be a self-affine tile in n defined by an integral expanding matrix A and a digit set D. The paper gives a necessary and sufficient condition for the connectedness of T. The condition can be checked algebraically via the characteristic polynomial of A. Through the use of this, it is shown that in #, for any integral expanding matrix A, there exists a digit set D such that the corresponding tile T is connected. This answers a question of Bandt and Gelbrich. Some partial results for the higher-dimensional cases are also given.
Abstract.Little is known about the connectedness of self-affine tiles in R n . In this note we consider this property on the self-affine tiles that are generated by consecutive collinear digit sets. By using an algebraic criterion, we call it the height reducing property, on expanding polynomials (i.e., all the roots have moduli > 1), we show that all such tiles in R n , n ≤ 3, are connected. The problem is still unsolved for higher dimensions. For this we make another investigation on this algebraic criterion. We improve a result of Garsia concerning the heights of expanding polynomials. The new result has its own interest from an algebraic point of view and also gives further insight to the connectedness problem.
Abstract. Let A ∈ M n (Z) be an expanding matrix, D ⊂ Z n a digit set and T = T (A, D) the associated self-affine set. It has been asked by Gröchenig and Haas (1994) that given any expanding matrix A ∈ M 2 (Z), whether there exists a digit set such that T is a connected or disk-like (i.e., homeomorphic to the closed unit disk) tile. With regard to this question, collinear digit sets have been studied in the literature. In this paper, we consider noncollinear digit sets and show the existence of a noncollinear digit set corresponding to each expanding matrix such that T is a connected tile. Moreover, for such digit sets, we give necessary and sufficient conditions for T to be a disk-like tile.
Abstract. Let T be a self-affine tile that is generated by an expanding integral matrix A and a digit set D. It is known that many properties of T are invariant under the Zsimilarity of the matrix A. In [LW1] Lagarias and Wang showed that if A is a 2 × 2 expanding matrix with |det(A)| = 2, then the Z-similar class is uniquely determined by the characteristic polynomial of A. This is not true if |det(A)| > 2. In this paper we give complete classifications of the Z-similar classes for the cases |det(A)| = 3, 4, 5. We then make use of the classification for |det(A)| = 3 to consider the digit set D of the tile and show that µ(T ) > 0 if and only if D is a standard digit set. This reinforces the conjecture in [LW3] on this.
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