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Simple Infinitely Ramified Self-Similar Sets

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Recent Developments in Fractals and Related Fields

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Summary

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.

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References

  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. 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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bandt, C., Hung, N.V.: Fractal n-gons and their Mandelbrot sets. Nonlinearity 21, 2653–2670 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bandt, C., Mesing, M.: Self-affine fractals of finite type. Banach Center Publications, Vol. 84, 131–148 (2009)

    Article  MathSciNet  Google Scholar 

  6. Barlow, M.T.: Diffusion on fractals. Lecture Notes Math., Vol. 1690, Springer, Berlin (1998)

    Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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. Broomhead, D., Montaldi, J., Sidorov, N.: Golden gaskets: variations on the Sierpiński sieve. Nonlinearity 17, 1455–1480 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Falconer, K.J.: Fractal Geometry. Wiley, New York (1990)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Kigami, J.: Analysis on Fractals. Cambridge University Press, Cambridge, UK (2001)

    Book  MATH  Google Scholar 

  14. Kusuoka, S., Zhou, X.Y.: Dirichlet forms on fractals: Poincaré constant and resistance. Probab. Theory Related Fields 93, 169–196 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mesing, M.: Fraktale endlichen Typs. Ph.D. thesis, University of Greifswald, Germany (2007)

    Google Scholar 

  16. Ngai, S.-M., Wang, Y.: Hausdorff dimension of self-similar sets with overlaps. J. Lond. Math. Soc. 63, 655–672 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Schief, A.: Separation properties for self-similar sets. Proc. Am. Math. Soc. 122, 111–115 (1994)

    MATH  MathSciNet  Google Scholar 

  18. Strichartz, R.S.: Differential equations on fractals. Princeton University Press, Princeton, NJ (2006)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Greifswald, Greifswald, Germany

    Christoph Bandt

Authors
  1. Christoph Bandt
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Correspondence to Christoph Bandt .

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Editors and Affiliations

  1. Département de Mathématiques, LAGA (UMR CNRS 7539), Institut Galilée, Université Paris 13, Villetaneuse, 93430, France

    Julien Barral

  2. Université Paris-Est, Laboratoire d'Analyse et de Mathématique, Bâtiment P3 4ème étage Bureau 441, CRETEIL Cedex, 94 010, France

    Stéphane Seuret

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Bandt, C. (2010). Simple Infinitely Ramified Self-Similar Sets. In: Barral, J., Seuret, S. (eds) Recent Developments in Fractals and Related Fields. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4888-6_15

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