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Mathematics of Computation

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


Author: Ibrahim Kirat
Journal: Math. Comp. 79 (2010), 1019-1045
MSC (2000): Primary 52C20, 05B45; Secondary 37C70
DOI: https://doi.org/10.1090/S0025-5718-09-02301-1
Published electronically: September 24, 2009
MathSciNet review: 2600554
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Abstract: Let $ A\in M_n(\mathbb{Z})$ be an expanding matrix, $ D\subset \mathbb{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\in M_2(\mathbb{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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Additional Information

Ibrahim Kirat
Affiliation: Department of Mathematics, Istanbul Technical University, Maslak 34469, Istanbul, Turkey
Email: ibkst@yahoo.com

DOI: https://doi.org/10.1090/S0025-5718-09-02301-1
Keywords: Self-affine tiles, disk-like tiles, connectedness
Received by editor(s): November 7, 2007
Received by editor(s) in revised form: July 25, 2008
Published electronically: September 24, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.