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ON THE CONNECTEDNESS OF SELF-AFFINE TILES

Published online by Cambridge University Press:  30 October 2000

IBRAHIM KIRAT  and
KA-SING LAU
IBRAHIM KIRAT
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA; [email protected]
KA-SING LAU
Affiliation:
Department of Mathematics, Chinese University of Hong Kong, Hong Kong; [email protected]
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Abstract

Let T be a self-affine tile in ℝn 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 ℝ2, 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.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 62 , Issue 1 , August 2000 , pp. 291 - 304
Copyright
The London Mathematical Society 2000

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