(08/21/2022, 01:29 AM)JmsNxn Wrote: 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.
