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Approximation of Fractals by Discrete Graphs: Norm Resolvent and Spectral Convergence

  • Olaf Post
  • Jan Simmer
Article
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Abstract

We show a norm convergence result for the Laplacian on a class of pcf self-similar fractals with arbitrary Borel regular probability measure which can be approximated by a sequence of finite-dimensional weighted graph Laplacians. As a consequence other functions of the Laplacians (heat operator, spectral projections etc.) converge as well in operator norm. One also deduces convergence of the spectrum and the eigenfunctions in energy norm.

Keywords

Post-critically finite fractals Spectral convergence Convergence of operators Discrete approximation 

References

  1. 1.
    Adams, B., Smith, S.A., Strichartz, R.S., Teplyaev, A.: The spectrum of the Laplacian on the pentagasket. In: Grabner, P., Woess, W. (eds.) Fractals in Graz 2001. Trends in Mathematics, pp. 1–24. Birkhäuser, Basel (2003)Google Scholar
  2. 2.
    Ben-Bassat, O., Strichartz, R.S., Teplyaev, A.: What is not in the domain of the Laplacian on Sierpinski gasket type fractals. J. Funct. Anal. 166, 197–217 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brzoska, A., Coffey, A., Hansalik, M., Loew, S., Rogers, L.G.: Spectra of magnetic operators on the diamond lattice fractal. arXiv:1704.01609 (2017)
  4. 4.
    Berry, T., Heilman, S.M., Strichartz, R.S.: Outer approximation of the spectrum of a fractal Laplacian. Exp. Math. 18, 449–480 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Blasiak, A., Strichartz, R.S., Uğurcan, B.E.: Spectra of self-similar Laplacians on the Sierpinski gasket with twists. Fractals 16, 43–68 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dondl, P., Cherednichenko, K., Rösler, F.: Norm-resolvent convergence in perforated domains. Asymptot. Anal. arXiv:1706.05859 (2018) (to appear)
  7. 7.
    Fukushima, M., Shima, T.: On a spectral analysis for the Sierpiński gasket. Potential Anal. 1, 1–35 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gibbons, M., Raj, A., Strichartz, R.S.: The finite element method on the Sierpinski gasket. Constr. Approx. 17, 561–588 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hyde, J., Kelleher, D., Moeller, J., Rogers, L., Seda, L.: Magnetic Laplacians of locally exact forms on the Sierpinski Gasket. Commun. Pure Appl. Anal. 16, 2299–2319 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hinz, M., Rogers, L.: Magnetic fields on resistance spaces. J. Fractal Geom. 3, 75–93 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hinz, M., Teplyaev, A.: Vector analysis on fractals and applications. In: Carfì, D., Lapidus, M.L., Pearse, E.P.J., van Frankenhuijsen, M. (eds.) Fractal geometry and Dynamical Systems in Pure and Applied Mathematics. II: Fractals in Applied. Contemporary Mathematics, vol. 601, pp. 147–163. American Mathematical Society, Providence (2013)Google Scholar
  12. 12.
    Hinz, M., Teplyaev, A.: Closability, regularity, and approximation by graphs for separable bilinear forms. Zap. Nauchn. Sem. S. Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 441, 299–317 (2015)zbMATHGoogle Scholar
  13. 13.
    Ionescu, M., Pearse, E.P.J., Rogers, L.G., Ruan, H.-J., Strichartz, R.S.: The resolvent kernel for PCF self-similar fractals. Trans. Am. Math. Soc. 362, 4451–4479 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)zbMATHGoogle Scholar
  15. 15.
    Kigami, J.: Harmonic calculus on p.c.f. self-similar sets. Trans. Am. Math. Soc. 335, 721–755 (1993)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kigami, J.: Harmonic metric and Dirichlet form on the Sierpiński gasket. In: Elworthy, K.D., Ikeda, N. (eds.) Asymptotic Problems in Probability Theory: Stochastic Mand Diffusions on Fractals (Sanda, Kyoto, 1990). Pitman Research Notes in Mathematics Series, vol. 283, pp. 201–218. Longman Scientific and Technical, Harlow (1993)Google Scholar
  17. 17.
    Kigami, J.: Analysis on Fractals. Cambridge Tracts in Mathematics, vol. 143. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  18. 18.
    Khrabustovskyi, A., Post, O.: Operator estimates for the crushed ice problem. Asymptot. Anal. arXiv:1710.03080 (2018) (to appear)
  19. 19.
    Kuwae, K., Shioya, T.: Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry. Commun. Anal. Geom. 11, 599–673 (2003)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mosco, U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123, 368–421 (1994)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Post, O.: Spectral convergence of quasi-one-dimensional spaces. Ann. Henri Poincaré 7, 933–973 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Post, O.: Spectral Analysis on Graph-Like Spaces. Lecture Notes in Mathematics, vol. 2039. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  23. 23.
    Post, O.: Boundary pairs associated with quadratic forms. Math. Nachr. 289, 1052–1099 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Post, O., Simmer, J.: Approximation of fractals by manifolds and other graph-like spaces. arXiv:1802.02998 (2018)
  25. 25.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York (1980)zbMATHGoogle Scholar
  26. 26.
    Shima, T.: On eigenvalue problems for Laplacians on p.c.f. self-similar sets. Jpn. J. Ind. Appl. Math. 13, 1–23 (1996)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Strichartz, R.S.: Fractafolds based on the Sierpiński gasket and their spectra. Trans. Am. Math. Soc. 355, 4019–4043 (2003)CrossRefGoogle Scholar
  28. 28.
    Strichartz, R.S.: Differential Equations on Fractals. Princeton University Press, Princeton (2006)zbMATHGoogle Scholar
  29. 29.
    Strichartz, R.S., Usher, M.: Splines on fractals. Math. Proc. Camb. Philos. Soc. 129, 331–360 (2000)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Teplyaev, A.: Harmonic coordinates on fractals with finitely ramified cell structure. Can. J. Math. 60, 457–480 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Fachbereich 4 – MathematikUniversität TrierTrierGermany

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