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

Laplacians on a family of quadratic Julia sets II

Tarik Aougab 1, , Stella Chuyue Dong 2, and  Robert S. Strichartz 3,

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.
Keywords: Laplacians., Julia sets.
Mathematics Subject Classification: Primary: 28A8.
Citation: Tarik Aougab, Stella Chuyue Dong, Robert S. Strichartz. Laplacians on a family of quadratic Julia sets II. Communications on Pure and 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), 1-44.

[2]

Stella C. Dong, Laplacians on a family of quadratic Julia sets II, http://www.math.cornell.edu/ cdong01/, March 2010.

[3]

A. Douady, Descriptions of compact sets in $\mathbb{C}$, in "Topological Methods in Modern Mathematics'' (eds. L. Goldberg and A. Phillips), Publish or Perish, Houston (1993), 429-465.

[4]

T. Flock and R. Strichartz, Laplacians on a family of Julia sets I, Trans. Amer. Math. Soc., to appear.

[5]

Jun Kigami, "Analysis on Fractals," volume 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001.

[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), 93-125.

[7]

J. Milnor, Periodic orbits, externals rays and the Mandelbrot set: an expository account, Geometrie Complexe et Systemes Dynamiques, 261 (2000), 277-333.

[8]

Luke G. Rogers and Alexander Teplyaev, Laplacians on the basilica Julia set, Comm. Pure Appl. Anal., 9 (2010), 211-231.

[9]

R. Strichartz, Fractals in the large, Can. J. Math., 50 (1998), 638-657.

[10]

Robert S. Strichartz, "Differential Equations on Fractals, A Tutorial,'' Princeton University Press, Princeton, NJ, 2006.

[11]

A. Teplyaev, Spectral analysis on infinite Sierpinski gaskets, J. Functional Anal., 159 (1998), 537-567.

show all references

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), 1-44.

[2]

Stella C. Dong, Laplacians on a family of quadratic Julia sets II, http://www.math.cornell.edu/ cdong01/, March 2010.

[3]

A. Douady, Descriptions of compact sets in $\mathbb{C}$, in "Topological Methods in Modern Mathematics'' (eds. L. Goldberg and A. Phillips), Publish or Perish, Houston (1993), 429-465.

[4]

T. Flock and R. Strichartz, Laplacians on a family of Julia sets I, Trans. Amer. Math. Soc., to appear.

[5]

Jun Kigami, "Analysis on Fractals," volume 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001.

[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), 93-125.

[7]

J. Milnor, Periodic orbits, externals rays and the Mandelbrot set: an expository account, Geometrie Complexe et Systemes Dynamiques, 261 (2000), 277-333.

[8]

Luke G. Rogers and Alexander Teplyaev, Laplacians on the basilica Julia set, Comm. Pure Appl. Anal., 9 (2010), 211-231.

[9]

R. Strichartz, Fractals in the large, Can. J. Math., 50 (1998), 638-657.

[10]

Robert S. Strichartz, "Differential Equations on Fractals, A Tutorial,'' Princeton University Press, Princeton, NJ, 2006.

[11]

A. Teplyaev, Spectral analysis on infinite Sierpinski gaskets, J. Functional Anal., 159 (1998), 537-567.

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