Pictures of parabolic renormalizations

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.,

Black regions are divided into two cases:
  1. Forward orbits eventually lie in E.
  2. 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.)
The second case corresponds to strips which tends to +infinity*i and their backward images.

Pictures for Q(z)

Pictures for Q(z-2)

Pictures for Q(z+2)

Pictures for Q(z-2i)

Pictures for Q(z+2i)