Images of Green's Functions.

Here are images of the zero set of some Green's functions for self maps of the Riemann Sphere. This set lies in complex two dimensional space, but it is fixed under multiplication by any complex number on the unit circle. We can picture the zero set of a Green's function in three dimensional real space by plotting only those points of it with a real second coordinate (so we are plotting the intersection with a real hyperplane). This gives a complete picture in the sense that we could reconstruct the original set in complex two space from its intersection with the above hyperplane.

The maps used were are all quadratic. All but the second to last one are polynomial. For these the value "c" below an image is the value of the parameter used (i.e. the map used was z squared plus c). The points in red are above a simple approximate Julia set for the given map (they are above the preimages of the origin under the eleventh or so iterate of the map).
c=1 c=-3
c=-.75 c=-.125+.65i
c=-1.40155 A quadratic rational map.
This last one is in white because it represents cauliflower greens (what a terrible pun, but it reminds me a little of a section of cauliflower stem).