(09/01/2022, 05:50 AM)JmsNxn Wrote: Super fascinating! I really love these explanations by you, because they are reinforcing many things I've read, but never quite "clicked with". I've always focused too much on geometric cases; and I always thought I got the parabolic case, but I wasn't confident enough to speak on it. These last few weeks of your comments and visual descriptions have really made me see it clearer.
I am glad and that's super motivating too.
Also it cleared up things for me. For example I always thought that the parabolic case is (in a quite direct manner) the merge of hyperbolic fixed points. But this is surely not true when considering the number of petals which is 2*(n-1) while the number of perturbed fixed points is n. (Only for the case n=2 the numbers would equal.) Also depending on the perturbation the fixed points would come from different directions and have different multipliers. I made a picture of the perturbed Leau-Fatou flowers for the case n=5, \(f(x)=x+x^5 - c^5\) where I moved c around 0, \(|c|=0.4\) or \(|c|=0.2\):
The 5 fixed points are equally spaced on the circle with radius 0.4 or 0.2.
Red fixed point means repelling, green means attracting.
PS: the left picture is like a symbol of the outer chaotic world, while the right picture looks like a state of meditation ... perhaps we just have to find the parabolic fixed point inside ourselves!