(07/18/2022, 06:12 PM)bo198214 Wrote: And as I am already at making motion pictures:
I always wanted to know what happens if we don't go through the base \(\eta\) but circling around it.
So I took a circle around \(\eta\) so that the lower real base is \(\sqrt{2}\)
It turns out that there is no bifurcation happening.
However the fixed points move through the image of the Shell-Thron boundary, i.e. they changing from attracting to repelling and vice versa (red attracting, blue repelling).
Does this mean, that the regular iteration at one of the fixed points has a singularity (as a function of the base) at all points of the Shell-Thron region?
I find it interesting how the fixed points swap place at the first round of the base and then change back at the second round
Can one conclude that the regular iteration at the lower fixed point is just a branch of the regular iteration at the upper fixed point (in case there is no singularity at the Shell-Thron region, except \(\eta\))?
There is only this little time frame where we have two repelling fixed points, perhaps these are the only bases where one can construct the Kneser/perturbed Fatou Abel function?
Too bad that Sheldon is not around, I think he tested his algorithm on different complex bases.
So many questions.
Perhaps at next I could investigate how the regular iteration behaves when going through the Shell-Thron boundary - guys, I currently in my extra vacation, maybe the only time in the year where I have time to dedicate to Tetration...
I'm excited to see bo back!
Have you read Paulsen's paper on Complex tetration?
He claims you can construct holomorphic tetration \(\text{tet}_b(z)\) in \(b\) everywhere except a branching problem at \(\eta\) and \(0\). His paper is very enlightening, as the majority of it is analysing how the fixed points behave within and outside the Shell-Thron region.
https://www.researchgate.net/profile/Wil...-bases.pdf
It is essentially a paper on extending Kneser's construction for \(b > \eta\), to everywhere in the complex plane.
EDIT: Also, are you using pari to create these animations, if so you have to tell me how, I've wanted to animate things for so long...
