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January  2010, 9(1): 211-231. doi: 10.3934/cpaa.2010.9.211

Laplacians on the basilica Julia set

Luke G. Rogers 1, and  Alexander Teplyaev 1,

1. 

Department of Mathematics, University of Connecticut, Storrs CT 06269-3009, United States, United States

Received  January 2009 Revised  May 2009 Published  October 2009

We consider the basilica Julia set of the polynomial $P(z)=z^{2}-1$ and construct all possible resistance (Dirichlet) forms, and the corresponding Laplacians, for which the topology in the effective resistance metric coincides with the usual topology. Then we concentrate on two particular cases. One is a self-similar harmonic structure, for which the energy renormalization factor is $2$, the exponent in the Weyl law is $\log9/\log6$, and we can compute all the eigenvalues and eigenfunctions by a spectral decimation method. The other is graph-directed self-similar under the map $z\mapsto P(z)$; it has energy renormalization factor $\sqrt2$ and Weyl exponent $4/3$, but the exact computation of the spectrum is difficult. The latter Dirichlet form and Laplacian are in a sense conformally invariant on the basilica Julia set.
Keywords: Dirichlet form, spectral decimation., Fractal, Resistance form, eigenvalues, self-similarity, Julia set, Laplacian, eigenfunctions.
Mathematics Subject Classification: Primary: 28A80; Secondary: 37F50, 31C2.
Citation: Luke G. Rogers, Alexander Teplyaev. Laplacians on the basilica Julia set. Communications on Pure & Applied Analysis, 2010, 9 (1) : 211-231. doi: 10.3934/cpaa.2010.9.211
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