Laplacians  on the basilica Julia set

Luke G. Rogers; Alexander Teplyaev

doi:10.3934/cpaa.2010.9.211

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2010, Volume 9, Issue 1: 211-231. Doi: 10.3934/cpaa.2010.9.211

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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

Received: December 31, 2008

Revised: April 30, 2009

Published: December 2009

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Abstract

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.

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Mathematics Subject Classification: Primary: 28A80; Secondary: 37F50, 31C25.

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