Qs on extension of continuous iterations from analytic functs to non-analytic

[Leo.W]

I guess you can do the same but instead of rotating you applying scaling by a non-negative scalar \(k\in\mathbb R\).
Given an arbitrary function \(k:{}S^1 \to \mathbb R^+\) that assigns to every point on the unitary circle a real number we can define \(f_k:\mathbb C\to\mathbb C\) defined as

...
Since the pasting of the functions has no information about how the various classes/fibers \(X_i\) paste together into the topological/analytical structure of the total space \(\mathbb C\) we can end up with non-analytic \(f:\mathbb C\to \mathbb C\) that can be continuously iterated.

Yes you're perfectly right about this!

And thanks for your post I now know how to type formula using \+( and \+) in texts, helped a lot.

I'd thought about the partitions, and your partitions worked out very well, em, I don't think the functions have to distinguish each self in \(\mathcal{C}^0,\mathcal{C}^1\) or etc.,  maybe globally \(\mathcal{C}^0\) is what I meant mostly, (thus it won't allow someone to approximate any iteration by fixed point or other analytic methods)
This inspires me more,
For \(\mathcal{C}^0\), one can cut the \(\mathbb{C}\) with parellel lines and the function can be for each slice, move it in the direction of these parellel lines, which is an action of addition, or \(f_k^t(z)=z+tk_{a*Im(z)+b*Re(z)}\)
however I don't quite understand all the terms (spectral decomposition of linear operators to fiber bundles especially) in topology, but I'll learn them as soon as possible.

But partition won't work out for more cases, like abs(z), sgn(z), arg(z), log(abs(arg(z)+sgn(z))) and more wierder functions,
I'm still stuck about the conj(z), as its transformation cannot be described by countably infinitely many partitions \(X_i\) and each has an \(f:X_i\to X_i\).
I wonder how one can get a continuous iteration without endomorphism, since it breaks the symmetry \(f:X_i\to X_i\), or even a more complex question, like their iterations to complex heights, since then you can't use eigen decomposition or linear operators, like the cases \(k\in \mathbb{C},f_k^t(z)=k_{\arg(z)}z\)

Anyway thank u