Rational maps with symmetries

The group of automorphisms Aut(f) of a rational map f:P1->P1 of degree d>1 is defined to be the group of Moebius transformations that commute with that rational map. When Aut(f) is not trivial, we say that f is a rational map with symmetries. The group Aut(f) is a finite subgroup of PSL(2,C). Such groups are classified; up to conjugacy by an automorphism of P1, Aut(f) is either:

Any finite subgroup G of PSL(2,C) can be realized as the group of automorphisms of some rational map (see [Doyle-McMullen, Solving the quintic by iteration]). There are lots of rational maps f for which Aut(f)=G but the set of such rational maps has very few isolated points. We say that such isolated points are rigid rational maps with symmetries. The following pictures show the Julia sets of those hyperbolic and post-critically finite rigid rational maps with symmetries, radially projected from the Riemann sphere onto the Platonic solid whose group of symmetries coincide with Aut(f). By clicking on the picture, you will get a picture of the Julia set drawn on the sphere and an enlargement of the projection onto the Platonic solid. You may then get a pdf-file of a pattern that you can cut and paste.
You may also download directly all the patterns in a pdf-file patterns.pdf or in a compressed postscript file patterns.ps.gz (the size of the patterns is suitable for printing on A4 paper).