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.