Iteration with two analytic fixed points

[JmsNxn]

There exists an immediate attracting basin \(\mathcal{A}\) about the fixed point \(z=2\). This domain is connected and open.

I am not seeing how this proves that both fixed points can not have the same iteration. Your approach looks a bit like:
We take the attracting basin continue the iteration to this domain and then show that it can not have a repelling fixed point.

\(\mathcal{A}\) is a maximal domain of this iteration (it equates to a boundary of a function whose maximal domain is \(f : \mathbb{D} \to \mathbb{D}\)). So in \(\sqrt{2}\)'s case, the value \(4\) is on the boundary of \(\mathcal{A}\)--and this function is not holomorphic anywhere on the boundary of \(A\). Thereby we cannot have that \(f^{\circ t}(z)\) is holomorphic near \(4\). Because this is the maximal domain. The only draw back to this statement, is that we are not using LFT's; which are meromorphic except for \(\lambda z + c\)--and these are mappable to \(\lambda z\). And then, this iteration has the entirety of \(\mathbb{C}\) as their \(A\). This is the only example of such happening. Where it is the only example of additionally having a fixed point at \(\infty\) which is repelling.



Trying to say that we can iterate about two fixed points in the manner I am talking is just flat out wrong. And you continue to doubt it. You can iterate about two fixed points, but never in a semi-group manner. And especially! you don't even come close for entire transcendental functions (mappings of \(\mathbb{C} \to \mathbb{C}\) with an essential singularity at \(z = \infty\)).



So for you to ask that a function: \(f : \mathbb{C} \to \mathbb{C}\) has a "local iteration" representation about two fixed points is nonsense. And, frankly, I'm a little tired of having to continuously argue this point. You have iterated LFTs and Meromorphic functions--which are of an entirely different mathematical class. For entire functions; even polynomial ones--you can't have a SEMI-GROUP ACTION holomorphic at two fixed points.

We can find half roots, and third roots, which are holomorphic on larger domains than \(A\); but the thing is, they are not holomorphic for \(\Re(t) > 0\).

I hope to not burn bridges. But I'll leave it alone now. I don't want to talk about this anymore.



Every function \(b^z\) for \(1 < b < \eta\) follows this rule. The main repelling fixed point is on the boundary of \(A\), and within \(A\) is the left half plane, and a good amount of the right half plane.



And the function \(f^{\circ t}(z)\) has its MAXIMAL DOMAIN on \(A\) for \(\Re(t) > 0\). By which--even asking that it can be reconcilable with another fixed point is nonsense. \(4 \not\in A\).



ADDITIONALLY: \(A\) is always simply connected; and therefore mappable to \(\mathbb{D}\). So that all of your regular iterations, are actually actions of the semigroup \(\mathbb{C}_{\Re(t) > 0}\) (or some equivalent halfplane) on \(f: \mathbb{D} \to \mathbb{D}\) with \(f(0) = 0\). The only time your regular iterations stray from this is with parabolic iteration. Whereby you are mapping \(f: \mathbb{D} \to \mathbb{D}\) but \(f(1) = 1\)--so the fixed point is on the boundary.



I'm sorry bo, but for you to say I am the one repeating myself isn't correct. You are making similar claims over and over; and I'm trying to reexplain something that is taught heavily in complex dynamics. We choose the dynamics of our functions like we choose our fixed points. There's no having your cake and eating it too.

I have nothing but respect for you, but I'm just very frustrated right now. Everyone of your holomorphic solutions satisfies the rule \(\mathbb{C}_{\Re(t) > 0}\) acts on \(f\) near the first fixed point; and \(\mathbb{C}_{\Re(t) < 0}\) acts on the other point--and that is absolutely no coincidence. \(\mathbb{C}_{\Re(t) > 0}\) cannot act on \(f\) about both fixed points.

I'm fine being an asshole here--But much of what you've been arguing has been wrong since my original statements; in the sense that you've found contradicitons. I am trying to classify the types of regular iterations. And one of the most important classes is: SEMIGROUP ACTION. It's pretty much all everyone cares about.