(09/04/2022, 06:50 AM)bo198214 Wrote: Compared to \(f'(0)=1\) where the iterates of a point remain in the same petal, for any other \(f'(0)=e^{i\varphi}\), \(0<\varphi<2\pi\) the iterates need to traverse around the fixed point 0. And it seems some get entangled there
Hey, Bo. I don't have the fancy graphs you just posted. Not even sure how to conjure the graphs you are making using my own code. But there is a Cyclone, a tornado, a star shaped domain, at a neutral fixed point \(f'(0) = e^{i\varphi}\). If \(e^{i\varphi}\) is a unit of the circle, then there is a dimension of \(n\) on the amount of spirals about this \(0\). We usually cancel that out in discussions, where we call it parabolic--and depending on the Milnor multiplier/valit of the \(f(0)\), we achieve that many petals--but the "swirling rate" is based off of \(\varphi\). Nothing more than a rate like \(|\varphi - \frac{p}{q}| < \frac{1}{|q|^{1+\epsilon}}\) for \(p,q \in \mathbb{Z}\)--which is the basis of siegel disks. If your \(\varphi\) satisfies this construction, you can remap this domain much more favourably.
What you have described above, is a symptom of a domain of holomorphy that is a petal, and it is swirling. By such the domain seems much more insignificant.
So if you fix the petal as real to real, and you vary the swirling, its common that you will hit a flat line of \(y =0\) for all values involved.
You are trying to swirl what needs to be complex, and keeping it real.
You are trying to eat your cake and have it too.
Are we writing \(e^{\pi i t}z + o(z)\) or; are we writing a summation of these weird fibonacci things we've learned? By which, creating a holomorphic solution in \(t\), which is real valued, is asking for trouble. But it gets really complicated. And especially because we will have to add much more terms than LFT's--but they are related in principle.
Jesus, I hope this makes sense. I've been really busy lately, just trying to get out what I can.