Laplacians on the basilica Julia set

Details Laplacians on the basilica Julia set Laplacians on the basilica Julia set

2008-02-22

arxiv.org/abs/0802.3248v3

Mathematics

Classical Analysis and ODEs

24 pages, one figure in separate eps file. Replaced the theorem that was removed in the second version, with a corrected proof

Scientific paper

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 spectral dimension 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 spectral dimension 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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