Neighbours of Self-affine Tiles in Lattice Tilings

Scheicher, Klaus; Thuswaldner, Jörg Μ.

doi:10.1007/978-3-0348-8014-5_10

Cited by 25 publications

(51 citation statements)

References 17 publications

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“…Each of our proofs uses a different matrix (or matrices) and mostly a different class (or classes) of digit sets. The implementation of those algorithms in [20,21] for each proof and each digit set here does not look much easier than our calculations as can be seen from [5]. On the other hand, the ad hoc methods used in this paper to determine the neighbors of the tiles directly give explicit infinite digit expansions of a few boundary points for each tile.…”

Section: Theorem 23 ([3]) Let T Be a Z 2 -Tile With Not More Than Smentioning

confidence: 99%

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Disk-like tiles and self-affine curves with noncollinear digits

Kirat

2009

Math. Comp.

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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.

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Section: Theorem 23 ([3]) Let T Be a Z 2 -Tile With Not More Than Smentioning

confidence: 99%

“…In order to use these theorems, it is crucial to obtain the neighbors. There are also graph-theoretic algorithms to determine the neighbors of T [20,21]. It is mentioned in [20] that the algorithm there is more suitable for a large class of tiles and faster than the one in [21].…”

Section: Theorem 23 ([3]) Let T Be a Z 2 -Tile With Not More Than Smentioning

confidence: 99%

Disk-like tiles and self-affine curves with noncollinear digits

Kirat

2009

Math. Comp.

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“…Then we describe the criterion of Gröchenig and Haas [7] that will permit us to conclude that stair-like tiles are Z 2 -tiles. After that an algorithm of Scheicher and Thuswaldner [21] for finding the neighbors of a Z m -tile is presented.…”

Section: Preliminariesmentioning

confidence: 99%

“…Bandt and Wang [4] gave a criterion for the disk-likeness of a Z 2 -tile T in terms of its "neighbors". Finding the neighbors of a Z 2 -tile was discussed in Strichartz and Wang [23] as well as in Scheicher and Thuswaldner [21]. Classes of Z 2 -tiles related to number systems are studied by Akiyama et al in [1,2].…”

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confidence: 99%

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On Disk-like Self-affine Tiles Arising from Polyominoes

Gmainer¹,

Thuswaldner²

2006

Methods and Applications of Analysis

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Abstract. In this paper we study a class of plane self-affine lattice tiles that are defined using polyominoes. In particular, we characterize which of these tiles are homeomorphic to a closed disk. It turns out that their topological structure depends very sensitively on their defining parameters.In order to achieve our results we use an algorithm of Scheicher and the second author which allows to determine neighbors of tiles in a systematic way as well as a criterion of Bandt and Wang, with that we can check disk-likeness of a self-affine tile by analyzing the set of its neighbors.

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Elementary Fractal Geometry. 3. Complex Pisot Factors Imply Finite Type

Bandt

2024

Discrete Comput Geom

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Self-similar sets require a separation condition to admit a nice mathematical structure. The classical open set condition (OSC) is difficult to verify. Zerner proved that there is a positive and finite Hausdorff measure for a weaker separation property which is always fulfilled for crystallographic data. Ngai and Wang gave more specific results for a finite type property (FT), and for algebraic data with a real Pisot expansion factor. We show how the algorithmic FT concept of Bandt and Mesing relates to the property of Ngai and Wang. Merits and limitations of the FT algorithm are discussed. Our main result says that FT is always true in the complex plane if the similarity mappings are given by a complex Pisot expansion factor $$\lambda $$ λ and algebraic integers in the number field generated by $$\lambda .$$ λ . This extends the previous results and opens the door to huge classes of separated self-similar sets, with large complexity and an appearance of natural textures. Numerous examples are provided.

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Neighbours of Self-affine Tiles in Lattice Tilings

Cited by 25 publications

References 17 publications

Disk-like tiles and self-affine curves with noncollinear digits

Disk-like tiles and self-affine curves with noncollinear digits

On Disk-like Self-affine Tiles Arising from Polyominoes

Elementary Fractal Geometry. 3. Complex Pisot Factors Imply Finite Type

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