# American Institute of Mathematical Sciences

January  2013, 12(1): 1-58. doi: 10.3934/cpaa.2013.12.1

## Laplacians on a family of quadratic Julia sets II

 1 Mathematics Department, Yale University, New Haven, CT 06510, United States 2 Mathematics Department, New York University, New York, NY 10012, United States 3 Mathematics Department, Malott Hall, Cornell Univeristy, Ithaca, NY 14853, United States

Received  June 2011 Revised  May 2012 Published  September 2012

This paper continues the work started in [4] to construct $P$-invariant Laplacians on the Julia sets of $P(z) = z^2 + c$ for $c$ in the interior of the Mandelbrot set, and to study the spectra of these Laplacians numerically. We are able to deal with a larger class of Julia sets and give a systematic method that reduces the construction of a $P$-invariant energy to the solution of nonlinear finite dimensional eigenvalue problem. We give the complete details for three examples, a dendrite, the airplane, and the Basilica-in-Rabbit. We also study the spectra of Laplacians on covering spaces and infinite blowups of the Julia sets. In particular, for a generic infinite blowups there is pure point spectrum, while for periodic covering spaces the spectrum is a mixture of discrete and continuous parts.
Citation: Tarik Aougab, Stella Chuyue Dong, Robert S. Strichartz. Laplacians on a family of quadratic Julia sets II. Communications on Pure & Applied Analysis, 2013, 12 (1) : 1-58. doi: 10.3934/cpaa.2013.12.1
##### References:
 [1] S. Constantin, R. Strichartz and M. Wheeler, Analysis of the Laplacian and spectral operators on the Vicsek set,, Comm. Pure Appl. Anal, 10 (2011), 1.   Google Scholar [2] Stella C. Dong, Laplacians on a family of quadratic Julia sets II,, \url{http://www.math.cornell.edu/ cdong01/}, (2010).   Google Scholar [3] A. Douady, Descriptions of compact sets in $\mathbbC$,, in, (1993), 429.   Google Scholar [4] T. Flock and R. Strichartz, Laplacians on a family of Julia sets I,, Trans. Amer. Math. Soc., ().   Google Scholar [5] Jun Kigami, "Analysis on Fractals,", volume 143 of Cambridge Tracts in Mathematics, (2001).   Google Scholar [6] 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), 93.   Google Scholar [7] J. Milnor, Periodic orbits, externals rays and the Mandelbrot set: an expository account,, Geometrie Complexe et Systemes Dynamiques, 261 (2000), 277.   Google Scholar [8] Luke G. Rogers and Alexander Teplyaev, Laplacians on the basilica Julia set,, Comm. Pure Appl. Anal., 9 (2010), 211.   Google Scholar [9] R. Strichartz, Fractals in the large,, Can. J. Math., 50 (1998), 638.   Google Scholar [10] Robert S. Strichartz, "Differential Equations on Fractals, A Tutorial,'', Princeton University Press, (2006).   Google Scholar [11] A. Teplyaev, Spectral analysis on infinite Sierpinski gaskets,, J. Functional Anal., 159 (1998), 537.   Google Scholar

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##### References:
 [1] S. Constantin, R. Strichartz and M. Wheeler, Analysis of the Laplacian and spectral operators on the Vicsek set,, Comm. Pure Appl. Anal, 10 (2011), 1.   Google Scholar [2] Stella C. Dong, Laplacians on a family of quadratic Julia sets II,, \url{http://www.math.cornell.edu/ cdong01/}, (2010).   Google Scholar [3] A. Douady, Descriptions of compact sets in $\mathbbC$,, in, (1993), 429.   Google Scholar [4] T. Flock and R. Strichartz, Laplacians on a family of Julia sets I,, Trans. Amer. Math. Soc., ().   Google Scholar [5] Jun Kigami, "Analysis on Fractals,", volume 143 of Cambridge Tracts in Mathematics, (2001).   Google Scholar [6] 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), 93.   Google Scholar [7] J. Milnor, Periodic orbits, externals rays and the Mandelbrot set: an expository account,, Geometrie Complexe et Systemes Dynamiques, 261 (2000), 277.   Google Scholar [8] Luke G. Rogers and Alexander Teplyaev, Laplacians on the basilica Julia set,, Comm. Pure Appl. Anal., 9 (2010), 211.   Google Scholar [9] R. Strichartz, Fractals in the large,, Can. J. Math., 50 (1998), 638.   Google Scholar [10] Robert S. Strichartz, "Differential Equations on Fractals, A Tutorial,'', Princeton University Press, (2006).   Google Scholar [11] A. Teplyaev, Spectral analysis on infinite Sierpinski gaskets,, J. Functional Anal., 159 (1998), 537.   Google Scholar
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