Laplacians on a family of quadratic Julia sets I

Abstract:

We describe families of Laplacians on Julia Sets $\mathcal {J}_c$ for quadratic polynomials $P(z)=z^2+c$ in the spirit of Kigami’s construction of Laplacians on p.c.f. self-similar fractals. We consider an infinite family of Julia sets for $c$ in the interior of a bulb in the Mandelbrot set that includes the basilica and the Douady rabbit. We use the external ray parametrization of the Julia set which represents the Julia set as a circle with some points identified. There is a one-dimensional space of $P$-invariant energies that arises from the standard energy on the circle, but we show surprisingly that there are higher dimensional spaces of energies invariant under iterates of $P$. There are two natural measures associated with the dynamics of $P$ on $\mathcal {J}$, the equilibrium measure $\mu$, which is $P$-invariant but ignores the geometric aspects of the $P$ action, and the conformal measure $\nu$, which is not $P$-invariant but does transform according to a power of the Jacobian of the mapping. The $P$-invariant Laplacian $\Delta _\mu$ is built from the $P$-invariant energy and the measure $\mu$. This Laplacian will depend only on the topological type of $\mathcal {J}$ (so for quasicircles, it just gives the usual Laplacian on the circle). The conformal Laplacian $\Delta _\nu$ is built from the $P$-invariant energy and the measure $\nu$.

We describe numerical procedures to approximate the eigenvalues and eigenfunctions of the Laplacians $\Delta _\mu$ and $\Delta _\nu$ and present the computational results. For $\Delta _\mu$ we identify a 4-element ($\mathbf {Z}_2\oplus \mathbf {Z}_2$) group of symmetries. In the case of the basilica the symmetries are generated by horizontal and vertical reflections, but in the case of the rabbit and other Julia sets the symmetries are more hidden (only $z\to -z$ is obvious). Based on these symmetries we are able to classify eigenfunctions and explain the computational data.