To be honest this is a bit too much to digest for me, and unfortunately in the next time I will not have time to further participate at the forum. Anyways, the stuff sounds really intriguing.
Particularly about the period dial. So you can change the regular iteration at fixed point 2 (base \(\sqrt{2}\)) into the regular solution at fixed point 4, just by turning the period dial from \(2\pi i/\log(\log(2))\) to \(2\pi i/\log(\log(4))\) ?! Very interesting!
And then turning it to \(\infty\) you would get the base-continued crescent/Paulsen iteration (i.e. something that is not real on the real line but nearly (supersmall imaginary part))?
Then I have a question about this assertion:
Just want to make some remarks about the "Kneser iteration" - for me it is not interchangeable with the crescent iteration, but is a very special case. It means to finding a real analytic iteration if there is no real fixed point by using a conjugated fixed point pair. In this special case one can make use of the Riemann mapping theorem. If however we are not on the real line anymore, but just have two arbitrary fixed points with a fundamental region, Kneser's construction does not work anymore (or at least Paulsen and me don't see a way to generalize Kneser's construction) in that case Riemann's mapping theorem is not sufficient anymore, but one needs the *measurable* Riemann mapping theorem)
I also wonder why you only consider \(\eta_-\) to be the Suez canal, should it not behave like all the other parabolic fixed points which don't explode (in terms of creating more fixed points when perturbed). Did you try the beta method on non-real parabolic fixed points?
I have already the conjecture that the fixed point of base \(\eta\) is the only indifferent fixed point where the regular iteration is not analytic. Some computations are quite supporting that the regular iteration powerseries at parabolic fixed points have convergence radius (while it is quite a difficult proof that the regular iteration powerseries of \(e^x-1\) and hence of \(\eta^x\) has zero convergence radius). I am not completely sure, but does this mean that all the Fatou-Coordinates in the petals are just one function? I also guess this has to do with the \(\eta\)-fixed point being the only "exploding" one.
And maybe this in turn would lead to a proof that the Shell-Thron boundary is permeable except at \(\eta\).
Do we btw have a strict proof that any iteration of a non-trivial function can never be holomorphic at two fixed points (except for integer iterates)? I think I saw you reasoning about that topic, but I was not sure how strict it was ... this would be surely worth a dedicated thread.
Öhm, I don't know how the Kneser iteration (in what sense ever) can be the same as the regular iteration .... please explain!
Particularly about the period dial. So you can change the regular iteration at fixed point 2 (base \(\sqrt{2}\)) into the regular solution at fixed point 4, just by turning the period dial from \(2\pi i/\log(\log(2))\) to \(2\pi i/\log(\log(4))\) ?! Very interesting!
And then turning it to \(\infty\) you would get the base-continued crescent/Paulsen iteration (i.e. something that is not real on the real line but nearly (supersmall imaginary part))?
Then I have a question about this assertion:
Quote:Iterating about periodic points is pretty standard for the repelling case, (a bit more complicated with neutral). The thing is, that the local iterations can never contain the periodic points.What exactly do you mean by iterating about the periodic points?
Just want to make some remarks about the "Kneser iteration" - for me it is not interchangeable with the crescent iteration, but is a very special case. It means to finding a real analytic iteration if there is no real fixed point by using a conjugated fixed point pair. In this special case one can make use of the Riemann mapping theorem. If however we are not on the real line anymore, but just have two arbitrary fixed points with a fundamental region, Kneser's construction does not work anymore (or at least Paulsen and me don't see a way to generalize Kneser's construction) in that case Riemann's mapping theorem is not sufficient anymore, but one needs the *measurable* Riemann mapping theorem)
Quote:the branch cut at \(\eta\) happens exactly along \((\eta,\infty)\)Dunno what you mean "happens" - isn't the branch cut something you set instead of something that happens?
I also wonder why you only consider \(\eta_-\) to be the Suez canal, should it not behave like all the other parabolic fixed points which don't explode (in terms of creating more fixed points when perturbed). Did you try the beta method on non-real parabolic fixed points?
I have already the conjecture that the fixed point of base \(\eta\) is the only indifferent fixed point where the regular iteration is not analytic. Some computations are quite supporting that the regular iteration powerseries at parabolic fixed points have convergence radius (while it is quite a difficult proof that the regular iteration powerseries of \(e^x-1\) and hence of \(\eta^x\) has zero convergence radius). I am not completely sure, but does this mean that all the Fatou-Coordinates in the petals are just one function? I also guess this has to do with the \(\eta\)-fixed point being the only "exploding" one.
And maybe this in turn would lead to a proof that the Shell-Thron boundary is permeable except at \(\eta\).
Do we btw have a strict proof that any iteration of a non-trivial function can never be holomorphic at two fixed points (except for integer iterates)? I think I saw you reasoning about that topic, but I was not sure how strict it was ... this would be surely worth a dedicated thread.
(07/26/2022, 02:42 AM)JmsNxn Wrote: To construct the regular iteration (which is kneser's iteration) about \(b = \eta_- - 0.5\), we can think ...
Öhm, I don't know how the Kneser iteration (in what sense ever) can be the same as the regular iteration .... please explain!
