Iteration with two analytic fixed points

[JmsNxn]

Well the idea I have for local iteration, is that it is only possible if the function being iterated is conjugate to:

\[
\begin{align}
f &: \mathbb{D} \to \mathbb{D}\\
f(0) &= 0\\
0 < |f'(0)| &< 1\\
\end{align}
\]

This, was never the problematic point. Problematic was this domain is also the domain for all (integer!) iterates. Which excludes all functions that have a singularity in the attractive basin around the fixed point.
Also Schröder iteration does not have this assumption, so it would not lead to local iteration in most cases. The convergence radius of the iterates can depend on t.

No no no no.

You've seemed to misunderstand.

There is always a half plane \(\mathcal{H}\) which is closed under additions of elements, where the Schroder iteration is conjugate to:

\[
f^{\circ t}(z) : \mathcal{H} \times \mathbb{D} \to \mathbb{D}\\
\]

I'm aware the domain depends on \(t\) as we try to grow it indefinitely. But if we stick to the half plane where \(\mathcal{H} = \{t \in \mathbb{C}\,|\,|\lambda^t| < \delta\}\) for \(\lambda\) the multiplier, then we are stuck in a neighborhood of the fixed point (hence local iteration). Which is then conjugate to this scenario.

To be completely explicit. Consider:

\[
f^{\circ t}(z) = \Psi^{-1}(\lambda^t \Psi(z))\\
\]

Choose \(|z| < \epsilon\) such that \(|\Psi(z)| < \rho\) and then \(|\Psi^{-1}(q)| < \epsilon\) when \(|q| < \rho\). Which this construction is always possible, so long as we shrink \(\epsilon, \rho\), and our multiplier \(|\lambda| < 1\). Then we have that: For \(\mathcal{H} = \{t \in \mathbb{C}\,|\,|\lambda^t| < 1\}\):

\[
f^{\circ t}(z) : \mathcal{H} \times \mathcal{N} \to \mathcal{N}\\
\]

Where \(\mathcal{N} = \{z \in \mathbb{C}\,|\, |z| < \epsilon\}\).

This is where the "local iteration" terminology is inspired from. Which, as you can see, is biholomorphic to the above uses of the Unit disk.

What I want to say, is that this completely describes every possible local iteration of \(f : \mathbb{C} \to \mathbb{C}\). Which, iconically avoids the LFT conundrums. Such that we have an equivalence of statements: The Schroder iteration of a euclidean function \(f\) = The local iteration of a euclidean function \(f\). So that the equivalence is reflexive. Whereby, if we can identify that this is a local iteration of some kind, that it must be a Schroder iteration.

Again, this may seem stupid to you Bo, but it is related to Semi-group actions in particular. By which we can consider the semi-group of a half plane ACTING on the function \(f : \mathbb{D} \to \mathbb{D}\). This obviously has no effect of how we iterate, or iteration through numerical evidence, so on and so forth. But it does help a lot with the work Mphlee and I have been trying to put together. Which is far more algebraic stuff; nothing to do with plugging in numbers. I'll shut up now