Connectedness of planar self-affine sets associated with non-collinear digit sets

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• 作者：King-Shun Leung (1)
  Jun Jason Luo (2)
  
  1. Department of Mathematics and Information Technology ; The Hong Kong Institute of Education ; Tai Po ; Hong Kong
  2. College of Mathematics and Statistics ; Chongqing University ; Chongqing ; 401331 ; People鈥檚 Republic of China
  
• 关键词：Connectedness ; Self ; affine set ; Digit set ; $${\mathcal {E}}$$ E ; Connected ; Primary 28A80 ; Secondary 52C20 ; 52C45
• 刊名：Geometriae Dedicata
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：175
• 期：1
• 页码：145-157
• 全文大小：5,462 KB
• 参考文献：1. Akiyama, S, Gjini, N (2005) Connectedness of number-theoretic tilings. Discret. Math. Theoret. Comput. Sci. 7: pp. 269-312
  2. Bandt, C, Gelbrich, G (1994) Classification of self-affine lattice tilings. J. Lond. Math. Soc. 50: pp. 581-593 CrossRef
  3. Bandt, C, Wang, Y (2001) Disk-like self-affine tiles in $${\mathbb{R}}^2$$ R 2. Discret. Comput. Geom. 26: pp. 591-601 CrossRef
  4. Deng, QR, Lau, KS (2011) Connectedness of a class of planar self-affine tiles. J. Math. Anal. Appl. 380: pp. 493-500 CrossRef
  5. Fu, XY, Gabardo, JP (2015) Self-affine scaling sets in $${\mathbb{R}}^2$$ R 2. Mem. Am. Math. Soc. 233: pp. 1097
  6. Gr枚chenig, K, Haas, A (1994) Self-similar lattice tilings. J. Fourier Anal. Appl. 1: pp. 131-170 CrossRef
  7. Hacon, D, Saldanha, NC, Veerman, JJP (1994) Remarks on self-affine tilings. Exp. Math. 3: pp. 317-327 CrossRef
  8. Hata, M (1985) On the structure of self-similar sets. Jpn. J. Appl. Math. 2: pp. 381-414 CrossRef
  9. He, XG, Kirat, I, Lau, KS (2011) Height reducing property of polynomials and self-affine tiles. Geom. Dedic. 152: pp. 153-164 CrossRef
  10. Kirat, I (2010) Disk-like tiles and self-affine curves with non-collinear digits. Math. Comp. 79: pp. 1019-1045 CrossRef
  11. Kirat, I, Lau, KS (2000) On the connectedness of self-affine tiles. J. Lond. Math. Soc. 62: pp. 291-304 CrossRef
  12. Kirat, I, Lau, KS, Rao, H (2004) Expanding polynomials and connectedness of self-affine tiles. Discret. Comput. Geom. 31: pp. 275-286 CrossRef
  13. Lagarias, JC, Wang, Y (1996) Self-affine tiles in $${\mathbb{R}}^n$$ R n. Adv. Math. 121: pp. 21-49 CrossRef
  14. Leung, KS, Lau, KS (2007) Disk-likeness of planar self-affine tiles. Trans. Am. Math. Soc. 359: pp. 3337-3355 CrossRef
  15. Leung, KS, Luo, JJ (2012) Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets. J. Math. Anal. Appl. 395: pp. 208-217 CrossRef
  16. Leung, KS, Luo, JJ (2013) Boundaries of disk-like self-affine tiles. Discret. Comput. Geom. 50: pp. 194-218 CrossRef
  17. Liu, JC, Luo, JJ, Xie, HW (2014) On the connectedness of planar self-affine sets. Chaos Solitons Fractals 69: pp. 107-116 CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1572-9168

文摘

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 \(k=\pm 1\) .