Laplacians on a family of quadratic Julia sets I

Authors:
Taryn C. Flock and Robert S. Strichartz

Journal:
Trans. Amer. Math. Soc. **364** (2012), 3915-3965

MSC (2010):
Primary 28A80

DOI:
https://doi.org/10.1090/S0002-9947-2012-05398-0

Published electronically:
March 21, 2012

MathSciNet review:
2912440

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Abstract: We describe families of Laplacians on Julia Sets for quadratic polynomials 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 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 -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 . There are two natural measures associated with the dynamics of on , the equilibrium measure , which is -invariant but ignores the geometric aspects of the action, and the conformal measure , which is not -invariant but does transform according to a power of the Jacobian of the mapping. The -invariant Laplacian is built from the -invariant energy and the measure . This Laplacian will depend only on the topological type of (so for quasicircles, it just gives the usual Laplacian on the circle). The conformal Laplacian is built from the -invariant energy and the measure .

We describe numerical procedures to approximate the eigenvalues and

eigenfunctions of the Laplacians and and present the computational results. For we identify a 4-element ( ) 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 is obvious). Based on these symmetries we are able to classify eigenfunctions and explain the computational data.

**[ADS10]**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.**[ASST03]**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)****[Bar98]**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)****[Bra89]**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****[CSW09]**Sarah Constantin, Robert S. Strichartz, and Miles Wheeler,*Analysis of the Laplacian and spectral operators on the Vicsek set*, Comm. Pure Appl. Anal.**10**(2011), no. 1, 1-44. MR**2746525****[DH84]**Adrien Douady and John H. Hubbard,*Etude dynamique des polynomes complexes i*, Publ. Math. d'Orsay (1984).**[Flo08]**Taryn C. Flock,*Laplacians on a family of quadratic julia sets*,`http://www.math.cornell.edu/~taryn/data.html`, September 2008.**[FS92]**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)****[Kig01]**Jun Kigami,*Analysis on fractals*, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR**1840042 (2002c:28015)****[KL93]**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)****[RT09]**Luke G. Rogers and Alexander Teplyaev,*Laplacians on the Basilica julia sets*, Comm. Pure Appl. Anal.**9**(2010), no. 1, 211-231. MR**2556753 (2011c:28024)****[Str06]**Robert S. Strichartz,*Differential equations on fractals*, Princeton University Press, Princeton, NJ, 2006, A tutorial. MR**2246975 (2007f:35003)****[Zho09]**Denglin Zhou,*Spectral analysis of laplacians on viscek sets*, Pac. J. Math.**241**(2009), 369-398.**[Zho10]**-,*Criteria for spectral gaps of Laplacians on fractals*, J. Fourier Anal. Appl.**16**(2010), no. 1, 76-96. MR**2587582 (2011b:28026)**

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Additional Information

**Taryn C. Flock**

Affiliation:
Department of Mathematics, University of California Berkeley, Berkeley, California 94709

Email:
tflock@math.berkeley.edu

**Robert S. Strichartz**

Affiliation:
Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853

Email:
str@math.cornell.edu

DOI:
https://doi.org/10.1090/S0002-9947-2012-05398-0

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

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.