Abstract
Let A be an expanding matrix with characteristic polynomial \(f(x)=x^2+px+3\) and \(\mathcal {D}=\{0,v,\ell v+kAv\}\) be a digit set where \(\ell ,k\in {\mathbb {Z}},\ v\in {\mathbb {R}}^2\) so that \(\{v,Av\}\) is linearly independent. It is well known that there exists a unique self-affine fractal T satisfying \(AT=T+\mathcal {D}\). In this paper, we give a complete characterization for the connectedness of T by using radix expansion.
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Acknowledgements
We would like to express our gratitude to the referee whose comments and suggestions improved the paper. We would also like to thank Professor Jun Jason Luo of Chongqing University for his valuable advice on a draft of this paper. The first author was supported by the Fundamental Research Funds for the Central Universities of China (2019CDXYST0015). The second author was supported by two projects (FLASS/DRF/IRS-9 and MIT/SGA02/19-20) from The Education University of Hong Kong.
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Wang, L., Leung, KS. Radix expansions and connectedness of planar self-affine fractals. Monatsh Math 193, 705–724 (2020). https://doi.org/10.1007/s00605-020-01461-0
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DOI: https://doi.org/10.1007/s00605-020-01461-0