Laplacians on a family of quadratic Julia sets II

Tarik Aougab; Stella Chuyue Dong; Robert S. Strichartz

doi:10.3934/cpaa.2013.12.1

Article Contents

2013, Volume 12, Issue 1: 1-58. Doi: 10.3934/cpaa.2013.12.1

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Laplacians on a family of quadratic Julia sets II

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

Received: June 2011

Revised: May 2012

Published: January 2013

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Abstract

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:

Julia sets,
Laplacians.

Mathematics Subject Classification: Primary: 28A80.

Citation:

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References

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