Iteration with two analytic fixed points

[JmsNxn]

(but you can apply it to \(\sqrt{2}\)).

Ok, I think that is where we stopped with your proof attempts ... for base \(\sqrt{2}\) one has as domain whole \(\mathbb{C}\).
And we know already by several means (one from Karlin&McGregor, but also because the multipliers are not reciprocal) that it can not have the same regular iteration at both fixed points, but maybe its still interesting to have another proof.
So here you have met all your assumptions, how would *your* proof go in this scanario (function is entire, domain is \(\mathbb{C}\))?

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

\[
\begin{align}
f^{\circ n}(z) &\to 2\,\,\text{as}\,\,n\to\infty\\\\
f(z) &= \sqrt{2}^z\\
\end{align}
\]

And \(A\) is defined as the maximal connected domain which satisfies this--where \(2 \in A\) and \(f : A \to A\).

The Schroder function is holomorphic here; By which \(\Psi(A) \to \mathbb{C}\). This function is nonsingular, by which:

\[
\frac{d}{dz} \Psi(z) \neq 0\\
\]

So that locally, near \(z = 2\) we have:

\[
f^{\circ t}(z) = \Psi^{-1}\left(\log(2)^t \Psi(z)\right)\\
\]

For \(|\log(2)^t| < 1\). This function is analytically continuable to:

\[
f^{\circ t}(z) : \mathbb{C}_{\Re(t) > 0} \times A   \to A\\
\]

This can be constructed in a manner of ways. The manner I find myself most comfortable with is with Mellin transforms. We can always express these \(\sqrt{2}^z\) iterates using integrals.

The second way, is the manner that I code in my solutions. These would essentially just be: program in the inverse Schroder, program in the Schroder. Add in a recursive protocol, which pulls us near \(z=2\) for high accuracy in Taylor data, then pull back using the inverse. And this will run smoothly.

From here, we can note that \(A\) is actually simply connected--which is written in Devaney's Chaotic dynamical systems: Attracting basins of geometric fixed points are simply connected. So even on its maximal domain \(f^{\circ t}(z) : A \to A\) is mappable to \(f^{\circ t}(z) : \mathbb{D} \to \mathbb{D}\)--So long as \(f\) is euclidean (transcendental entire), we are okay.

Again, all the crazy behaviour starts happening when \(t \to -\infty\), where the neighbourhoods aren't as nice. And there's no iteration for \(t \in \mathbb{C}\). But there is a local iteration if we fix \(\Re(t) > 0\). Where upon, we are somewhat locally around the fixed point. As you limit \(t \to -\infty\) everything turns to chaos! And yes it can still be holomorphic there. But it may not be a local iteration, in the sense that it's writable as \(f^{\circ t}(z) : \mathcal{H} \times A \to A\).

I don't want to press this too far Bo, I just hope you can understand that a local iteration is a stricter requirement than Regular iteration; but the two ideas pretty much coincide everywhere.

EDIT:

I'd also like to add, that a lot of your uses of regular iterations, especially lately, are Group actions. These are essentially \(\{\mathbb{R}, +\}\) group actions on a function. I am focused on \(\{\mathcal{H}, +\}\) semi-group actions on a function. And they have their disagreements, despite being expandable in the same manner and ultimately just being regular iteration in a different disguise.

Don't get me wrong, everything I'm doing is regular iteration, just with further restrictions