  We continue the study of constructing invariant Laplacians on Julia sets, and
studying properties of their spectra. In this paper we focus on two types of
examples: 1) Julia sets of cubic polynomials $z^3 + c$ with a single critical
point; 2) formal matings of quadratic Julia sets. The general scheme introduced
in earlier papers in this series involves realizing the Julia set as a circle
with identifications, and attempting to obtain the Laplacian as a renormalized
limit of graph Laplacians on graphs derived form the circle with
identifications model. In the case of cubic Julia sets the details follows the
pattern established for quadratic Julia sets, but for matings the details are
quite challenging, and we have only been completely successful for one example.
Once we have constructed the Laplacian, we are able to use numerical methods to
approximate the eigenvalues and eigenfunctions. One striking observation from
the data is that for the cubic Julia sets the multiplicities of all eigenspaces
(except for the trivial eigenspace of constants) are even numbers. Nothing like
this is valid for the quadratic julia sets studied earlier. We are able to
explain this, based on the fact that three is an odd number, and more precisely
because the dihedral-3 symmetry group has only two distinct one-dimensional
irreducible representations.
