Simple Infinitely Ramified Self-Similar Sets

  • Christoph Bandt
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Self-similar sets form a mathematically tractable class of fractals. Any family of contractive mappings f 1,..., f m generates a unique corresponding fractal set A. It is more difficult to find conditions for the f k which ensure that A has a nice structure. We describe a technique which allows us to determine self-similar sets with a particularly simple structure. Some of the resulting examples are known, like the Sierpiński gasket and carpet, some others seem to be new. A simple structure is necessary when we want to do classical analysis on A; for instance, define harmonic functions and a Laplace operator. So far, much analysis has been realized on a very small class of fractal spaces–essentially the relatives of the Sierpiński gasket. In this chapter, we discuss two classes of infinitely ramified fractals which seem to be more realistic from the point of view of physical modeling, and we give examples for which fractal analysis seems to be possible. A property of the boundaries for these fractal classes is verified.


Brownian Motion Harmonic Function Heat Kernel Dirichlet Form Neighbor Type 
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  1. 1.
    Bandt, C.: Self-similar measures. In: Fiedler, B. (ed.) Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Springer, Berlin (2001)Google Scholar
  2. 2.
    Bandt, C., Graf, S.: Self-similar sets VII. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Am. Math. Soc. 114, 995–1001 (1992)MATHMathSciNetGoogle Scholar
  3. 3.
    Bandt, C., Hung, N.V.: Self-similar sets with an open set condition and great variety of overlaps. Proc. Am. Math. Soc. 136, 3895–3903 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bandt, C., Hung, N.V.: Fractal n-gons and their Mandelbrot sets. Nonlinearity 21, 2653–2670 (2008)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bandt, C., Mesing, M.: Self-affine fractals of finite type. Banach Center Publications, Vol. 84, 131–148 (2009)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Barlow, M.T.: Diffusion on fractals. Lecture Notes Math., Vol. 1690, Springer, Berlin (1998)Google Scholar
  7. 7.
    Barlow, M.T., Bass, R.F.: Construction of Brownian motion on the Sierpiński carpet. Ann. Inst. H. Poincaré Probab. Statist. 25, 225–257 (1989)MATHMathSciNetGoogle Scholar
  8. 8.
    Barlow, M.T., Bass, R.F., Kumagai, T., Teplyaev, A.: Uniqueness of Brownian motion on Sierpiński carpets, Preprint, arXiv:0812.1802v1 (2008)Google Scholar
  9. 9.
    Broomhead, D., Montaldi, J., Sidorov, N.: Golden gaskets: variations on the Sierpiński sieve. Nonlinearity 17, 1455–1480 (2004)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Falconer, K.J.: Fractal Geometry. Wiley, New York (1990)MATHGoogle Scholar
  11. 11.
    Grigor’yan, A., Hu, J.: Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces. Invent. Math. 174, 81–126 (2008)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Grigor’yan, A., Kumagai, T.: On the dichotomy in the heat kernel two-sided estimates. In: P. Exner et al. (eds.) Analysis on Graphs and Applications. Proc. Symposia Pure Math. 77, Amer. Math. Soc., Providence, RI (2008)Google Scholar
  13. 13.
    Kigami, J.: Analysis on Fractals. Cambridge University Press, Cambridge, UK (2001)MATHCrossRefGoogle Scholar
  14. 14.
    Kusuoka, S., Zhou, X.Y.: Dirichlet forms on fractals: Poincaré constant and resistance. Probab. Theory Related Fields 93, 169–196 (1992)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Mesing, M.: Fraktale endlichen Typs. Ph.D. thesis, University of Greifswald, Germany (2007)Google Scholar
  16. 16.
    Ngai, S.-M., Wang, Y.: Hausdorff dimension of self-similar sets with overlaps. J. Lond. Math. Soc. 63, 655–672 (2001)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Schief, A.: Separation properties for self-similar sets. Proc. Am. Math. Soc. 122, 111–115 (1994)MATHMathSciNetGoogle Scholar
  18. 18.
    Strichartz, R.S.: Differential equations on fractals. Princeton University Press, Princeton, NJ (2006)MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of GreifswaldGreifswaldGermany

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