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Spectral analysis on infinite Sierpiński fractafolds

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Abstract

A fractafold, a space that is locally modeled on a specified fractal, is the fractal equivalent of a manifold. For compact fractafolds based on the Sierpiński gasket, it was shown by the first author how to compute the discrete spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian. A similar problem was solved by the second author for the case of infinite blowups of a Sierpiński gasket, where spectrum is pure point of infinite multiplicity. Both works used the method of spectral decimations to obtain explicit description of the eigenvalues and eigenfunctions. In this paper we combine the ideas from these earlier works to obtain a description of the spectral resolution of the Laplacian for noncompact fractafolds. Our main abstract results enable us to obtain a completely explicit description of the spectral resolution of the fractafold Laplacian. For some specific examples, we turn the spectral resolution into a “Plancherel formula”. We also present such a formula for the graph Laplacian on the 3-regular tree, which appears to be a new result of independent interest. At the end, we discuss periodic fractafolds and fractal fields.

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References

  1. N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst, and A. Teplyaev, Vibration modes of 3n-gaskets and other fractals, J. Phys. A: Math Theor. 41 (2008) 015101; Vibration spectra of finitely ramified, symmetric fractals, Fractals 16 (2008), 243–258.

    Article  MathSciNet  Google Scholar 

  2. M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields 79 (1988), 543–623.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Béllissard, Renormalization group analysis and quasicrystals, Ideas and Methods in Quantum and Statistical Physics (Oslo, 1988), Cambridge Univ. Press, Cambridge, 1992, pp. 118–148.

    Google Scholar 

  4. P. Cartier, Harmonic analysis on trees, Harmonic Analysis on Homogeneous Spaces, Amer. Math. Soc., Providence, RI, 1973, pp. 419–424.

    Google Scholar 

  5. J. DeGrado, L. G. Rogers and R. S. Strichartz, Gradients of Laplacian eigenfunctions on the Sierpinski gasket, Proc. Amer. Math. Soc. 137 (2009), 531–540.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Derfel, P. Grabner, and F. Vogl, The zeta function of the Laplacian on certain fractals, Trans. Amer. Math. Soc. 360 (2008), 881–897.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Derfel, P. Grabner, and F. Vogl, Complex asymptotics of Poincaré functions and properties of Julia sets, Math. Proc. Cambridge Philos. Soc. 145 (2008), 699–718.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Drenning and R. Strichartz, Spectral decimation on Hambly’s homogeneous hierarchical gaskets, Illinois J. Math. 53 (2009), 915–937.

    MathSciNet  MATH  Google Scholar 

  9. A. Figa-Talamanca and C. Nebbia, Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees, Cambridge University Press, Cambridge, 1991.

    Book  MATH  Google Scholar 

  10. D. Ford and B. Steinhurst, Vibration spectra of the m-tree fractal, Fractals 18 (2010), 157–169.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal. 1 (1992), 1–35.

    Article  MathSciNet  MATH  Google Scholar 

  12. B. M. Hambly and T. Kumagai, Diffusion processes on fractal fields: heat kernel estimates and large deviations, Probab. Theory Related Fields 127 (2003), 305–352.

    Article  MathSciNet  MATH  Google Scholar 

  13. K. Hare, B. Steinhurst, A. Teplyaev, and D. Zhou, Disconnected Julia sets and gaps in the spectrum of Laplacians on symmetric finitely ramified regular fractals, Math. Res. Let. (2012), to appear.

  14. A. Ionescu, On the Poisson transform on symmetric spaces of real rank one, J. Funct. Anal. 174 (2000), 513–523.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Ionescu, Erin P. J. Pearse, L. G. Rogers, Huo-Jun Ruan, and R. S. Strichartz, The resolvent kernel for PCF self-similar fractals, Trans. Amer. Math. Soc. 362 (2010), 4451–4479.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Jordan, Comb graphs and spectral decimation, Glasg. Math. J. 51 (2009), 71–81.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Kigami, A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math. 6 (1989), 259–290.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Kigami, Analysis on Fractals, Cambridge University Press, Cambridge, 2001.

    Book  MATH  Google Scholar 

  19. J. Kigami, Harmonic analysis for resistance forms, J. Funct. Anal. 204 (2003), 399–444.

    Article  MathSciNet  MATH  Google Scholar 

  20. B. Krön and E. Teufl, Asymptotics of the transition probabilities of the simple random walk on self-similar graph, Trans. Amer. Math. Soc. 356 (2003), 393–414.

    Article  Google Scholar 

  21. P. Kuchment, On the Floquet theory of periodic difference equations, Geometrical and Algebraical Aspects in Several Complex Variables (Cetraro, 1989), EditEl, Rende, 1991, pp. 201–209.

  22. P. Kuchment, Floquet Theory for Partial Differential Equations, Birkhäuser Verlag, Basel, 1993.

    Book  MATH  Google Scholar 

  23. P. Kuchment, Quantum graphs II. Some spectral properties of quantum and combinatorial graphs, J. Phys. A. 38 (2005), 4887–4900.

    Article  MathSciNet  MATH  Google Scholar 

  24. P. Kuchment and O. Post, On the spectra of carbon nano-structures, Comm. Math. Phys. 275 (2007), 805–826.

    Article  MathSciNet  MATH  Google Scholar 

  25. P. Kuchment and B. Vainberg, On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators, Comm. Math. Phys. 268 (2006), 673–686.

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Malozemov and A. Teplyaev, Self-similarity, operators and dynamics, Math. Phys. Anal. Geom. 6 (2003), 201–218.

    Article  MathSciNet  MATH  Google Scholar 

  27. R. Oberlin, B. Street, and R. S. Strichartz, Sampling on the Sierpinski gasket, Experiment. Math. 12 (2003), 403–418.

    MathSciNet  MATH  Google Scholar 

  28. K. Okoudjou, L. G. Rogers, and R. S. Strichartz, Generalized eigenfunctions and a Borel theorem on the Sierpinski gasket. Canad. Math. Bull. 52 (2009), 105–116.

    Article  MathSciNet  MATH  Google Scholar 

  29. K. Okoudjou, L. G. Rogers, and R. S. Strichartz, Szegö limit theorems on the Sierpiński gasket, J. Fourier Anal. Appl. 16 (2010), 434–447.

    Article  MathSciNet  MATH  Google Scholar 

  30. O. Post, Equilateral quantum graphs and boundary triples, Analysis on Graphs and its Applications, Amer. Math. Soc., Providence, RI, 2008, pp. 469–490.

    Google Scholar 

  31. J.-F. Quint, Harmonic analysis on the Pascal graph, J. Funct. Anal. 256 (2009), 3409–3460.

    Article  MathSciNet  MATH  Google Scholar 

  32. M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, 1980.

  33. T. Shima, On eigenvalue problems for Laplacians on p.c.f. self-similar sets, Japan J. Indust. Appl. Math. 13 (1996), 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  34. T. Shirai, The spectrum of infinite regular line graphs, Trans. Amer. Math. Soc. 352 (2000), 115–132.

    Article  MathSciNet  MATH  Google Scholar 

  35. R. S. Strichartz, Harmonic analysis as spectral theory of Laplacians. J. Funct. Anal. 87 (1989) 51–148, Corrigendum, J. Funct. Anal. 109 (1992), 457–460.

    Article  MathSciNet  MATH  Google Scholar 

  36. R. S. Strichartz, Fractals in the large, Canad. J. Math. 50 (1998), 638–657.

    Article  MathSciNet  MATH  Google Scholar 

  37. R. S. Strichartz, Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc. 355 (2003), 4019–4043.

    Article  MathSciNet  MATH  Google Scholar 

  38. R. S. Strichartz, Laplacians on fractals with spectral gaps have nicer Fourier series. Math. Res. Lett. 12 (2005), 269–274.

    MathSciNet  MATH  Google Scholar 

  39. R. S. Strichartz, Differential Equations on Fractals: a Tutorial, Princeton University Press, Princeton, NJ, 2006.

    MATH  Google Scholar 

  40. R. S. Strichartz, Transformation of spectra of graph Laplacians, Rocky Mountain J. Math. 40 (2010), 2037–2062.

    Article  MathSciNet  MATH  Google Scholar 

  41. A. Teplyaev, Spectral analysis on infinite Sierpiński gaskets J. Funct. Anal. 159 (1998), 537–567.

    Article  MathSciNet  MATH  Google Scholar 

  42. D. Zhou, Spectral analysis of Laplacians on the Vicsek set, Pacific J. Math. 241 (2009), 369–398.

    Article  MathSciNet  MATH  Google Scholar 

  43. D. Zhou, Criteria for spectral gaps of Laplacians on fractals, J. Fourier Anal. Appl. 16 (2010), 76–96.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department Of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Robert S. Strichartz

  2. Department Of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA

    Alexander Teplyaev

Authors
  1. Robert S. Strichartz
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  2. Alexander Teplyaev
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Corresponding author

Correspondence to Robert S. Strichartz.

Additional information

Research supported in part by the National Science Foundation, grant DMS-0652440.

Research supported in part by the National Science Foundation, grant DMS-0505622.

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Strichartz, R.S., Teplyaev, A. Spectral analysis on infinite Sierpiński fractafolds. JAMA 116, 255–297 (2012). https://doi.org/10.1007/s11854-012-0007-5

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  • DOI: https://doi.org/10.1007/s11854-012-0007-5

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