Laplacians on the basilica Julia set

TL;DR

This paper constructs and analyzes Laplacians on the basilica Julia set of a quadratic polynomial, exploring self-similar and graph-directed structures, and computing spectra in specific cases.

Contribution

It introduces all resistance forms on the basilica Julia set where the resistance metric matches the usual topology, and explicitly computes spectra for certain self-similar Laplacians.

Findings

Spectral dimension is log9/log6 for the self-similar case.

Eigenvalues and eigenfunctions are computed via spectral decimation.

A conformally invariant Laplacian is identified for the basilica Julia set.

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 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.