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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an expanding matrix, a digit set and the associated self-affine set. It has been asked by Gröchenig and Haas (1994) that given any expanding matrix , whether there exists a digit set such that 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 is a connected tile. Moreover, for such digit sets, we give necessary and sufficient conditions for 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.