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Topological structure of fractal squares

Published online by Cambridge University Press:  01 March 2013

KA–SING LAU ,
JUN JASON LUO  and
HUI RAO
KA–SING LAU
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, P.R. China. e-mail: kslau@math.cuhk.edu.hk
JUN JASON LUO
Affiliation:
Department of Mathematics, Shantou University, Shantou 515063, Guangdong, P.R. China. e-mail: luojun2011@yahoo.com.cn
HUI RAO
Affiliation:
Department of Mathematics, Hua Zhong Normal University, Wuhan 430079, P.R. China. e-mail: hrao@mail.ccnu.edu.cn
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Abstract

Given an integer n ≥ 2 and a digit set ⊊ {0,1,. . .,n − 1}2, there is a self-similar set F ⊂ ℝ2 satisfying the set equation: F=(F+)/n. We call such F a fractal square. By studying a periodic extension H= F + ℤ2, we classify F into three types according to their topological properties. We also provide some simple criteria for such classification.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 1 , July 2013 , pp. 73 - 86
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Bandt, C. and Gelbrich, G.Classification of self-affine lattice tilings. J. London Math. Soc. 50 (1994), 581–593.CrossRefGoogle Scholar
[2]Bandt, C. and Wang, Y.Disklike self-affine tiles in ℝ2. Discrete Comput. Geom. 26 (2001), no.4, 591–601.CrossRefGoogle Scholar
[3]Deng, Q. R. and Lau, K. S.Connectedness of a class of planar self-affine tiles. J. Math. Anal. Appl. 380 (2011), 493–500.CrossRefGoogle Scholar
[4]Devaney, R.An Introduction to Chaotic Dynamical Systems (Addison–Wesley, 1989).Google Scholar
[5]Falconer, K.Fractal Geometry: Mathematical Foundations and Applications (John Wiley, 2003).CrossRefGoogle Scholar
[6]Hata, M.On the structure of self-similar sets. Japan J. Appl. Math. 2 (1985), 381–414.CrossRefGoogle Scholar
[7]Kirat, I. and Lau, K. S.On the connectedness of self-affine tiles. J. London Math. Soc. 62 (2000), 291–304.CrossRefGoogle Scholar
[8]Luo, J. J. and Lau, K. S.Lipschitz equivalence of self-similar sets and hyperbolic boundaries. Adv Math. 235 (2013), 555–579.CrossRefGoogle Scholar
[9]Leung, K. S. and Lau, K. S.Disklikeness of planar self-affine tiles. Trans. Amer. Math. Soc. 359 (2007), 3337–3355.CrossRefGoogle Scholar
[10]Leung, K. S. and Luo, J. J.Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets. J. Math. Anal. Appl. 395 (2012), 208–217.CrossRefGoogle Scholar
[11]Leung, K. S. and Luo, J. J. Connectedness of planar self-affine sets associated with non-collinear digit sets. Preprint.Google Scholar
[12]Luo, J. J. Topological structure and Lipschitz equivalence of fractal sets. Ph.D. thesis. The Chinese University of Hong Kong (2012).Google Scholar
[13]Luo, J., Rao, H. and Tan, B.Topological structure of self-similar sets. Fractals 10 (2002), 223–227.CrossRefGoogle Scholar
[14]Ngai, S. M. and Tang, T. M.A technique in the topology of connected self-similar tiles. Fractals 12 (2004), 389–403.CrossRefGoogle Scholar
[15]Ngai, S. M. and Tang, T. M.Topology of connected self-similar tiles in the plane with disconnected interiors. Topology Appl. 150 (2005), 139–155.CrossRefGoogle Scholar
[16]Roinestad, K. Geometry of fractal squares. Ph.D. thesis. The Virginia Polytechnic Institute and State University (2010).Google Scholar
[17]Taylor, T. D.Connectivity properties of Sierpinski relatives. Fractals 19 (2011), 481–506.CrossRefGoogle Scholar
[18]Taylor, T. D., Hudson, C. and Anderson, A.Examples of using binary Cantor sets to study the Connectivity of Sierpinski relatives. Fractals 20 (2012), 61–75.CrossRefGoogle Scholar
[19]Xi, L.-F. and Xiong, Y.Self-similar sets with initial cubic patterns. CR Acad. Sci. Paris, Ser. I 348 (2010), 15–20.CrossRefGoogle Scholar