In that paper, we constructed a branched covering on the moduli space of topological polynomials with the same post-critical dynamics as the rabbit polynomial. The movies illustrate the Julia set of these topological polynomials as a point moves in moduli space; they should be thought of as an exploration of a fractal in C^2, which is a bundle over a moduli space fractal (in grey) with fibres the Julia sets of the corresponding topological polynomial.
I wrote in PDF this brief explanation of the movies, with active hyperlinks to them.
These movies are best understood in conjunction with Nekrashevych's preprint An uncountable family of 3-generated groups acting on the binary tree.
The movies are produced by the following C++ code, run on PC/Linux: movie.C, Makefile.