(07/25/2022, 08:19 PM)JmsNxn Wrote: 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.OMG, this is really a very long (and cumbersome to read, because it was not clear to me what "local iteration" means - you mean regular iteration, right? - also you spoke about the case \(b=\eta_-\) while I spoke about perturbed \(b\) and the resulting fixed points coming into existence - which in the end makes no difference though, just making it more confusing) explanation for:
If you define an Abel function \(\alpha\), then \(\alpha(f^{\circ 2}(z_0)) = \alpha(z_0) = \alpha(z_0) + 2\) so that we must have \(z_0\) is
...
For a regular iteration at \(z_0\) we have \(g^{\circ t}(z_0)=z_0\) and for the crescent iteration we have at least \(\lim_{z\to z_i}g^{\circ t}(z)=z_i, i=1,2\), therefore iterating \(g=f^{\circ 2}\) at non-fixed/periodic point(s) of \(f\) would would give \(z_0=g^{\frac{1}{2}}(z_0)\neq f(z_0)\).
I never considered periodic points, so I was not aware of this drawback.
Btw. its not *a* "crescent iteration" its *the* "crescent iteration" as in perturbed Fatou coordinates. Not some method that works somehow on some fundamental region, but a clearly defined (not numerically though) and uniqueness proven iteration.
If I understood that correctly your beta method is also approaching the crescent iteration (like Sheldons, Kouznetsovs and Paulsens)? Its just a different way of computing it?
Quote:Well the periodic points can actually be seen in my graph!
Several issues trouble me here. I thought there would be no real half-iterate of \(\eta_-^z\). So how can it be that the super function is real valued?
And second: You mean not an exact periodic point and hence not an exact period, but just influences of periodic points on your graph? (Because there are no real periodic points of \(\eta_-^z\)).
