On Disk-like Self-affine Tiles Arising from Polyominoes

Gmainer, Johannes; Thuswaldner, Jörg Μ.

doi:10.4310/maa.2006.v13.n4.a3

Cited by 3 publications

(2 citation statements)

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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: 98%

“…They assumed that D is a consecutive collinear digit set in Less is known about the disklikeness when the digits are noncollinear. In regard to this problem, Gmainer and Thuswaldner [5] study the disklikeness of a class of tiles with noncollinear digits arising from polyominoes. The study of such digits is more delicate than collinear digits.…”

Section: Introductionmentioning

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: 98%

Section: Introductionmentioning

confidence: 99%

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

Kirat

2009

Math. Comp.

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Boundaries of Disk-Like Self-affine Tiles

Leung

Luo

2013

Discrete Comput Geom

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Let T := T (A, D) be a disk-like self-affine tile generated by an integral expanding matrix A and a consecutive collinear digit set D, and let f (x) = x 2 + px + q be the characteristic polynomial of A. In the paper, we identify the boundary ∂T with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair (A, D) to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm ω, we find the generalized Hausdorff dimension dim ω H (∂T ) = 2 log ρ(M )/ log |q| where ρ(M ) is the spectral radius of certain contact matrix M . Especially, when A is a similarity, we obtain the standard Hausdorff dimension dim H (∂T ) = 2 log ρ/ log |q| where ρ is the largest positive zero of the cubic polynomial x 3 −(|p|−1)x 2 −(|q|−|p|)x−|q|, which is simpler than the known result.

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