Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-14T18:26:29.005Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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