by Luke G Rogers and Alexander Teplyaev.

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.