Skip to main content
SpringerLink
Log in

Wave Equation on One-Dimensional Fractals with Spectral Decimation and the Complex Dynamics of Polynomials

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

We study the wave equation on one-dimensional self-similar fractal structures that can be analyzed by the spectral decimation method. We develop efficient numerical approximation techniques and also provide uniform estimates obtained by analytical methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Akkermans, E., Dunne, G.V., Teplyaev, A.: Thermodynamics of photons on fractals. Phys. Rev. Lett. 105, 230407 (2010)

    Article  Google Scholar 

  2. Akkermans, E., Dunne, G.V., Teplyaev, A.: Physical consequences of complex dimensions of fractals. Europhys. Lett. 88, 40007 (2009)

    Article  Google Scholar 

  3. Ambjørn, J., Jurkiewicz, J., Loll, R.: Quantum gravity as sum over spacetimes. Lect. Notes Phys. 807, 59 (2010)

    Article  MATH  Google Scholar 

  4. Ambjørn, J., Jurkiewicz, J., Loll, R.: Spectral dimension of the universe. Phys. Rev. Lett. 95, 171301 (2005)

    Article  Google Scholar 

  5. Arzano, M., Calcagni, G., Oriti, D., Scalisi, M.: Fractional and noncommutative spacetimes, preprint arXiv:1107.5308

  6. ben-Avraham, D., Havlin, S.: Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  7. Alexander, S., Orbach, R.: Density of states on fractals: “fractions”. J. Phys. Lett. 43, L625 (1982)

    Article  Google Scholar 

  8. Adams, B., Smith, S.A., Strichartz, R., Teplyaev, A.: The spectrum of the Laplacian on the pentagasket. Fractals in Graz 2001 Analysis Dynamics Geometry Stochastics, Trends in Mathematics, Birkhäuser Basel, pp. 1–24 (2003)

  9. Bajorin, N., Chen, T., Dagan, A., Emmons, C., Hussein, M., Khalil, M., Mody, P., Steinhurst, B., Teplyaev, A.: Vibration modes of \(3n\)-gaskets and other fractals. J. Phys. A 41, 015101 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bajorin, N., Chen, T., Dagan, A., Emmons, C., Hussein, M., Khalil, M., Mody, P., Steinhurst, B., Teplyaev, A.: Vibration spectra of finitely ramified. Symmetric Fractals Fractals 16, 243–258 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Barlow, M.T.: Diffusions on fractals. Lectures on Probability Theory and Statistics (Saint-Flour, 1995). Lecture Notes in Mathematics, vol. 1690, pp. 1–121. Springer, Berlin (1998)

    Book  Google Scholar 

  12. Barlow, M.T., Bass, R.F., Kumagai, T., Teplyaev, A.: Uniqueness of Brownian motion on Sierpinski carpets. J. Eur. Math. Soc. 12, 655–701 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Barnsley, M.F., Geronimo, J.S., Harrington, A.N.: Condensed Julia sets, with an application to a fractal lattice model Hamiltonian. Trans. Am. Math. Soc. 288, 537–561 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ben-Bassat, O., Strichartz, R.S., Teplyaev, A.: What is not in the Domain of the Laplacian on Sierpiński gasket Type Fractals. J. Funct. Anal. 166, 197–217 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Benedetti, D.: Fractal properties of quantum spacetime. Phys. Rev. Lett. 102, 111303 (2009)

    Article  MathSciNet  Google Scholar 

  16. Bird, E.J., Ngai, S.-M., Teplyaev, A.: Fractal Laplacians on the Unit Interval. Ann. Sci. Math. Qué. 27, 135–168 (2003)

    MathSciNet  MATH  Google Scholar 

  17. Caravelli, F., Modesto, L.: Fractal Dimension in 3D Spin-Foams, preprint arXiv:0905.2170

  18. Carlip, S.: Spontaneous Dimensional Reduction in Short-Distance Quantum Gravity, preprint arXiv:0909.3329

  19. Carlip, S.: The Small Scale Structure of Spacetime, preprint arXiv:1009.1136

  20. Chan, J., Ngai, S.-M., Teplyaev, A.: One-dimensional wave equations defined by fractal Laplacians. J. Anal. Math. 127, 219–246 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, J.P., Teplyaev, A.: Singularly continuous spectrum of a self-similar Laplacian on the half-line. J. Math. Phys. 57, 052104 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, J.P., Molchanov, S., Teplyaev, A.: Spectral dimension and Bohr’s formula for Schrödinger operators on unbounded fractal spaces. J. Phys. A 48, 395203 (2016)

    Article  MATH  Google Scholar 

  23. Codello, A., Percacci, R., Rahmede, C.: Investigating the ultraviolet properties of gravity with a Wilsonian renormalization group equation. Ann. Phys. 324, 414 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Coletta, K., Dias, K., Strichartz, R.: Numerical analysis on the Sierpinski gasket with applications to Schr\({\ddot{\rm d}}\)inger equations, wave equation, and Gibbs’ phenomenon. Fractals 12, 413–449 (2004)

  25. Constantin, S., Strichartz, R., Wheeler, M.: Analysis of the Laplacian and spectral operators on the Vicsek commun. Pure Appl. Anal. 10, 1–44 (2011)

    MATH  Google Scholar 

  26. Dalrymple, K., Strichartz, R.S., Vinson, J.P.: Fractal differential equations on the Sierpinski gasket. J. Fourier Anal. Appl. 5, 203–284 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Domany, E., Alexander, S., Bensimon, D., Kadanoff, L.: Solutions to the Schrödinger equation on some fractal lattices. Phys. Rev. B (3) 28, 3110–3123 (1984)

    Article  Google Scholar 

  28. DeGrado, J., Rogers, L., Strichartz, R.: Gradients of Laplacian eigenfunctions on the Sierpinski gasket. Proc. Am. Math. Soc. 137(2), 531–540 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Dunne, G.V.: Heat kernels and zeta functions on fractals. J. Phys. A 45(37), 374016 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Chen, Z.-Q., Fukushima, M.: Symmetric Markov Processes, Time Change, and Boundary Theory. London Mathematical Society Monographs Series, vol. 35. Princeton University Press, Princeton (2012)

    MATH  Google Scholar 

  31. Englert, F., Frere, J.-M., Rooman, M., Spindel, Ph: Metric space-time as fixed point of the renormalization group equations on fractal structures. Nucl. Phys. B 280, 147–180 (1987)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Gibbons, M., Raj, A., Strichartz, R.: The finite element method on the Sierpinski gasket. Constr. Approx. 17(4), 561–588 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Goldstein, S.: Random walks and diffusions on fractals. In: Kesten, H. (ed.) Percolation Theory and Ergodic Theory of Infinite Particle Systems. Institute for Mathematics and Its Applications, vol. 8, pp. 121–129. Springer, New York (1987)

    Chapter  Google Scholar 

  35. Ionescu, M., Pearse, E., Rogers, L., Ruan, H., Strichartz, R.: The resolvent rernel for p.c.f. self-similar fractals. Trans. Am. Math. Soc. 362, 4451–4479 (2010)

    Article  MATH  Google Scholar 

  36. Kigami, J.: Harmonic calculus on p.c.f. self-similar sets. Trans. Am. Math. Soc. 335, 721–755 (1993)

    MathSciNet  MATH  Google Scholar 

  37. Kigami, J.: Analysis on Fractals. Cambridge Tracts in Mathematics, vol. 143. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  38. Kigami, J.: Local Nash inequality and inhomogeneity of heat kernels. Proc. Lond. Math. Soc. (3) 89, 525–544 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kigami, J., Lapidus, M.L.: Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals. Commun. Math. Phys. 158, 93–125 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kigami, J., Lapidus, M.L.: Self-similarity of volume measures for Laplacians on p.c.f. self-similar fractals. Commun. Math. Phys. 217, 165–180 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Lal, N., Lapidus, M.L.: Hyperfunctions and spectral zeta functions of Laplacians on self-similar fractals. J. Phys. A 45(36), 365205 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lauscher, O., Reuter, M.: Fractal spacetime structure in asymptotically safe gravity. JHEP 10, 050 (2005)

    Article  MathSciNet  Google Scholar 

  43. Lee, Y.-T.: Infinite Propagation Speed For Wave Solutions on Some P.C.F. Fractals, Submitted. arXiv:1111.2938

  44. Magliaro, E., Perini, C., Modesto, L.: Fractal Space-Time from Spin-Foams, preprint arXiv:0911.0437

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Milnor, J.: Dynamics in One Complex Variable, 3rd edn. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  47. Rammal, R.: Spectrum of harmonic excitations on fractals. J. Phys. 45, 191–206 (1984)

    Article  MathSciNet  Google Scholar 

  48. Rammal, R., Toulouse, G.: Random walks on fractal structures and percolation clusters. J. Phys. Lett. 44, L13–L22 (1983)

    Article  Google Scholar 

  49. Reuter, Martin, Saueressig, Frank: Fractal space-times under the microscope: a renormalization group view on Monte Carlo Data. J. High Energy Phys. 12, 1–31 (2011)

    MATH  Google Scholar 

  50. Reuter, M., Schwindt, J.M.: Scale-dependent metric and causal structures in Quantum Einstein Gravity. JHEP 01, 049 (2007)

    Article  MathSciNet  Google Scholar 

  51. Rogers, L.: Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups. Trans. Am. Math. Soc. 2012, 1633–1685 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  52. Rogers, L., Strichartz, R.: Distribution theory on P.C.F. fractals. J. Anal. Math. 112, 137–191 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  53. Sabot, C.: Electrical networks, symplectic reductions, and application to the renormalization map of self-similar lattices. J. Phys. Lett. 44, L13–L22 (1983). Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 1. Proceedings of Symposia in Pure Mathematics 72, Amer. Math. Soc., (2004), 155–205

    Article  Google Scholar 

  54. Steinhurst, B., Teplyaev, A.: Existence of a meromorphic extension of spectral zeta functions on fractals. Lett. Math. Phys. 103, 1377–1388 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  55. Strichartz, R.: A priori estimates for the wave equation and some applications. J. Funct. Anal. 5, 218–235 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  56. Strichartz, R.: A guide to distribution theory and Fourier transforms. Reprint of the 1994 original (CRC, Boca Raton). World Scientific Publishing Co., River Edge (2003)

    Google Scholar 

  57. Strichartz, R.S.: Function spaces on fractals. J. Funct. Anal. 198, 43–83 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  59. Strichartz, R.: Waves are recurrent on noncompact fractals. J. Fourier Anal. Appl. 16, 148–154 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  60. Shima, T.: On eigenvalue problems for Laplacians on p.c.f. self-similar sets. Jpn. J. Ind. Appl. Math. 13, 1–23 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  62. Teplyaev, A.: Spectral zeta function of symmetric Sierpiński gasket type fractals. Fractal Geometry and Stochastics III, Progress in Probability, vol. 57, pp. 245–262. Birkhäuser (2004)

  63. Teplyaev, A.: Spectral zeta functions of fractals and the complex dynamics of polynomials. Trans. Am. Math. Soc. 359, 4339–4358 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Research supported in part by the NSF Grants DMS-1106982 and DMS-1262929. The authors are very grateful to Daniel Kelleher, Hugo Panzo and Antoni Brzoska for many helpful discussions, and to Luke Rogers for explaining the eigenfunction estimates based on his paper [51].  A.T. also thanks Sze-Man Ngai and Alexander Grigor’yan for very valuable advice. The authors thank anonymous referees for corrections and a substantial list of constructive suggestions leading to improvements in the first version of our paper, and for the suggestion to include the infinite wave propagation speed Remark 1.1.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Ulysses Andrews, Grigory Bonik & Alexander Teplyaev

  2. Mathematics Department, Colgate University, Hamilton, NY, 13346, USA

    Joe P. Chen

  3. Mathematics Department, SUNY Broome Community College, Binghamton, NY, 13902, USA

    Richard W. Martin

Authors
  1. Ulysses Andrews
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Grigory Bonik
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Joe P. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Richard W. Martin
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Alexander Teplyaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Teplyaev.

Additional information

Communicated by Robert S. Strichartz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Andrews, U., Bonik, G., Chen, J.P. et al. Wave Equation on One-Dimensional Fractals with Spectral Decimation and the Complex Dynamics of Polynomials. J Fourier Anal Appl 23, 994–1027 (2017). https://doi.org/10.1007/s00041-016-9494-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-016-9494-6

Keywords

Mathematics Subject Classification

Search

Navigation