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.
| Original language | English |
|---|---|
| Pages (from-to) | 291-304 |
| Number of pages | 14 |
| Journal | Journal of the London Mathematical Society |
| Volume | 62 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - Aug 2000 |
| Externally published | Yes |
Funding
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.
| Funders | Funder number |
|---|---|
| RGC | |
| Sakarya University in Turkey | |
| Institute of Mathematical Sciences | |
| Chinese University of Hong Kong |