Well the idea I have for local iteration, is that it is only possible if the function being iterated is conjugate to:
\[
\begin{align}
f &: \mathbb{D} \to \mathbb{D}\\
f(0) &= 0\\
0 < |f'(0)| &< 1\\
\end{align}
\]
Which I've yet to see a counter example for (so long as we do away with meromorphic functions), which all the evidence further points to.
This would mean, the only time you have a local iteration, is when it is the Schroder iteration about an attracting fixed point (or if you flip it on its head, A schroder iteration about a repelling fixed point). I've yet to see any counter evidence. And additionally, every Schroder iteration is conjugate to this scenario. So I don't share your point of view that this isn't in the real world, I think it very much is.
Also, it's much more important to the stuff MphLee and I talked about. Where we are talking about conjugating between Semi-group, and semi-group actions. I agree it doesn't help with iterating random functions--but foundationally I think it's very important. It would additionally show we can have a Canonical map between two function \(f,g\) which are locally iterable, and additionally share the same multiplier, that being \(\phi(z) = \Psi_f^{-1}(\Psi_g(z))\) which satisfies \(\phi g = f \phi\). Where \(\Psi\) is the Schroder map of the respective function.
But, then again, many of our mathematical preferences diverge, bo. So to each their own, lol.
\[
\begin{align}
f &: \mathbb{D} \to \mathbb{D}\\
f(0) &= 0\\
0 < |f'(0)| &< 1\\
\end{align}
\]
Which I've yet to see a counter example for (so long as we do away with meromorphic functions), which all the evidence further points to.
This would mean, the only time you have a local iteration, is when it is the Schroder iteration about an attracting fixed point (or if you flip it on its head, A schroder iteration about a repelling fixed point). I've yet to see any counter evidence. And additionally, every Schroder iteration is conjugate to this scenario. So I don't share your point of view that this isn't in the real world, I think it very much is.
Also, it's much more important to the stuff MphLee and I talked about. Where we are talking about conjugating between Semi-group, and semi-group actions. I agree it doesn't help with iterating random functions--but foundationally I think it's very important. It would additionally show we can have a Canonical map between two function \(f,g\) which are locally iterable, and additionally share the same multiplier, that being \(\phi(z) = \Psi_f^{-1}(\Psi_g(z))\) which satisfies \(\phi g = f \phi\). Where \(\Psi\) is the Schroder map of the respective function.
But, then again, many of our mathematical preferences diverge, bo. So to each their own, lol.
