Abstract
We study a class of planar self-affine sets \(T(A,{\mathcal {D}})\) generated by the integer expanding matrices \(A\) with \(|\det A|=3\) and the non-collinear digit sets \({\mathcal {D}}=\{0, v, kAv\}\) where \(k\in {\mathbb {Z}}\setminus \{0\}\) and \(v\in {\mathbb {R}}^{2}\) such that \(\{v, Av\}\) is linearly independent. By examining the characteristic polynomials of \(A\) carefully, we prove that \(T(A,{\mathcal {D}})\) is connected if and only if the parameter \(k=\pm 1\).
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Acknowledgments
The authors would like to thank Professor Ka-Sing Lau for suggesting a related question and advice on the work; we also thank the referee for pointing out an error in Sect. 4.
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The research is partially supported by National Natural Science Fundation of China (No. 11301322), Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20134402120007), and the Foundation for Distinguished Young Talents in Higher Education of Guangdong (2013LYM_0028).
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Leung, KS., Luo, J.J. Connectedness of planar self-affine sets associated with non-collinear digit sets. Geom Dedicata 175, 145–157 (2015). https://doi.org/10.1007/s10711-014-0033-9
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DOI: https://doi.org/10.1007/s10711-014-0033-9