On the Connectedness of Self-Affine Tiles |
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| Authors: | Kirat Ibrahim; Lau Ka-Sing |
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| Institution: | Department of Mathematics, University of Pittsburgh Pittsburgh, PA 15260, USA, ibkst+{at}pitt.edu
Department of Mathematics, Chinese University of Hong Kong Hong Kong, kslau{at}math.cuhk.edu.hk |
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| Abstract: | Let T be a self-affine tile in Rn defined by an integral expandingmatrix A and a digit set D. The paper gives a necessary andsufficient condition for the connectedness of T. The conditioncan be checked algebraically via the characteristic polynomialof A. Through the use of this, it is shown that in R2, for anyintegral expanding matrix A, there exists a digit set D suchthat the corresponding tile T is connected. This answers a questionof Bandt and Gelbrich. Some partial results for the higher-dimensionalcases are also given. |
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