Pictures of near-parabolic renormalizations are here.

The figures here are the attracting Fatou coordinates of Q(z-b) (or its 1st renormalization) for various b, where Q(z)=z(1+1/z)^6/(1-1/z)^4 (restricted outside E={x+iy| (x/1.24)^2+(y/1.04)^2<=1}). They are drawn by this program (binary for Mac OS X (PPC). Run from command prompt (e.g., Terminal.app)).

Black regions are divided into two cases:- Forward orbits eventually lie in E.
- The number of iteration we need to determine what happens there is greater than what we expected. (We usually stop iteration after 280,000 times.)