On the connectedness of self-affine tiles

Ibrahim Kirat*, Ka Sing Lau

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

66 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)291-304
Number of pages14
JournalJournal of the London Mathematical Society
Issue number1
Publication statusPublished - Aug 2000
Externally publishedYes


Acknowledgements. The authors would like to thank Professor Y. Wang for introducing this topic while he was visiting the Institute of Mathematical Sciences at the Chinese University of Hong Kong. They would also like to thank the referee for many helpful suggestions and comments about revising the paper. Thanks are also extended to Professor K. W. Leung for clarifying some algebraic facts, and to Dr S. M. Ngai and Dr H. Rao for modifying the connectedness criterion. The first author was supported by a grant from Sakarya University in Turkey and the Institute of Mathematical Sciences at the Chinese University of Hong Kong. The second author was supported by an RGC grant from Hong Kong.

FundersFunder number
Sakarya University in Turkey
Institute of Mathematical Sciences
Chinese University of Hong Kong


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