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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Laplacians on a family of quadratic Julia sets I
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by Taryn C. Flock and Robert S. Strichartz PDF
Trans. Amer. Math. Soc. 364 (2012), 3915-3965 Request permission


We describe families of Laplacians on Julia Sets $\mathcal {J}_c$ for quadratic polynomials $P(z)=z^2+c$ in the spirit of Kigami’s construction of Laplacians on p.c.f. self-similar fractals. We consider an infinite family of Julia sets for $c$ in the interior of a bulb in the Mandelbrot set that includes the basilica and the Douady rabbit. We use the external ray parametrization of the Julia set which represents the Julia set as a circle with some points identified. There is a one-dimensional space of $P$-invariant energies that arises from the standard energy on the circle, but we show surprisingly that there are higher dimensional spaces of energies invariant under iterates of $P$. There are two natural measures associated with the dynamics of $P$ on $\mathcal {J}$, the equilibrium measure $\mu$, which is $P$-invariant but ignores the geometric aspects of the $P$ action, and the conformal measure $\nu$, which is not $P$-invariant but does transform according to a power of the Jacobian of the mapping. The $P$-invariant Laplacian $\Delta _\mu$ is built from the $P$-invariant energy and the measure $\mu$. This Laplacian will depend only on the topological type of $\mathcal {J}$ (so for quasicircles, it just gives the usual Laplacian on the circle). The conformal Laplacian $\Delta _\nu$ is built from the $P$-invariant energy and the measure $\nu$.

We describe numerical procedures to approximate the eigenvalues and eigenfunctions of the Laplacians $\Delta _\mu$ and $\Delta _\nu$ and present the computational results. For $\Delta _\mu$ we identify a 4-element ($\mathbf {Z}_2\oplus \mathbf {Z}_2$) group of symmetries. In the case of the basilica the symmetries are generated by horizontal and vertical reflections, but in the case of the rabbit and other Julia sets the symmetries are more hidden (only $z\to -z$ is obvious). Based on these symmetries we are able to classify eigenfunctions and explain the computational data.

  • Tarik Aougab, Chu Yue (Stella) Dong, and Robert S. Strichartz, Laplacians of a family of quadratic julia sets II, Comm. Pure Appl. Math., to appear.
  • Bryant Adams, S. Alex Smith, Robert S. Strichartz, and Alexander Teplyaev, The spectrum of the Laplacian on the pentagasket, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, pp. 1–24. MR 2091699 (2006g:28017)
  • Martin T. Barlow, Diffusions on fractals, Lectures on probability theory and statistics (Saint-Flour, 1995), Lecture Notes in Math., vol. 1690, Springer, Berlin, 1998, pp. 1–121. MR 1668115 (2000a:60148)
  • Bodil Branner, The Mandelbrot set, Chaos and fractals (Providence, RI, 1988), Proc. Sympos. Appl. Math., vol. 39, Amer. Math. Soc., Providence, RI, 1989, pp. 75–105. MR 1010237
  • Sarah Constantin, Robert S. Strichartz, and Miles Wheeler, Analysis of the Laplacian and spectral operators on the Vicsek set, Commun. Pure Appl. Anal. 10 (2011), no. 1, 1–44. MR 2746525, DOI 10.3934/cpaa.2011.10.1
  • Adrien Douady and John H. Hubbard, Etude dynamique des polynomes complexes i, Publ. Math. d’Orsay (1984).
  • Taryn C. Flock, Laplacians on a family of quadratic julia sets,, September 2008.
  • M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal. 1 (1992), no. 1, 1–35. MR 1245223 (95b:31009)
  • Jun Kigami, Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR 1840042 (2002c:28015)
  • Jun Kigami and Michel L. Lapidus, Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys. 158 (1993), no. 1, 93–125. MR 1243717 (94m:58225)
  • Luke G. Rogers and Alexander Teplyaev, Laplacians on the basilica Julia sets, Commun. Pure Appl. Anal. 9 (2010), no. 1, 211–231. MR 2556753, DOI 10.3934/cpaa.2010.9.211
  • Robert S. Strichartz, Differential equations on fractals, Princeton University Press, Princeton, NJ, 2006, A tutorial. MR 2246975 (2007f:35003)
  • Denglin Zhou, Spectral analysis of laplacians on viscek sets, Pac. J. Math. 241 (2009), 369–398.
  • Denglin Zhou, Criteria for spectral gaps of Laplacians on fractals, J. Fourier Anal. Appl. 16 (2010), no. 1, 76–96. MR 2587582, DOI 10.1007/s00041-009-9087-8
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Additional Information
  • Taryn C. Flock
  • Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94709
  • MR Author ID: 976421
  • Email:
  • Robert S. Strichartz
  • Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853
  • Email:
  • Received by editor(s): June 30, 2009
  • Received by editor(s) in revised form: March 29, 2010
  • Published electronically: March 21, 2012
  • Additional Notes: The research of the first author was supported by the National Science Foundation through the Research Experiences for Undergraduates Program at Cornell
    The research of the second author was supported in part by the National Science Foundation, grant DMS-0652440
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 3915-3965
  • MSC (2010): Primary 28A80
  • DOI:
  • MathSciNet review: 2912440