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
\(f_k(z)=k_{{\rm arg}(z)}\cdot z\).
Obviously we can extend it as \(f_k^t(z)=k^t_{{\rm arg}(z)}\cdot z\). If we define \(k^t:{}S^1\to \mathbb R^+\) as the composition \((k^t)_\theta:=(k_\theta)^t\) we have
\[f^t_k=f_{k^t}\]
If you look carefully on the construction what we have done is consider a partition of \(\bigcup _{i\in I}X_i=\mathbb C\) were we define on each class \(X_i\) a function \(f_i:X_i\to X_i\) that we know how to continuously iterate, i.e. how to extend it to a group action of \(\mathbb R\) over it. We then just paste all the functions/actions \(f_i\) into an endofunction of \(\mathbb C\).
\[f=\coprod_{i\in I}f_i:\bigcup _{i\in I}X_i\to \bigcup _{i\in I}X_i \]
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.
Addendum: note that the construction I have given provides an infinite amount of examples and how to build them.
In my and your case we use the polar decomposition of \(\mathbb C\setminus\{0\}\simeq S^1\times \mathbb R^+\), i.e. as modulus and argument, to obtain a partition of \(\mathbb C\).
The complex can be presented as a bundle \({\rm arg}:\mathbb C\to S^1 \), i.e. \(\bigcup_{\theta\in S^1}\mathbb R^+e^{i\theta}\to S^1\), or as the bundle \(|\cdot|:\mathbb C\to\mathbb R^+\), i.e. \(\bigcup_{r\in \mathbb R^+}rS^1\to \mathbb R^+\).
All of this is very tied to decomposition techniques one sees in the spectral decomposition of linear operators to fiber bundles. The resulting map \(f\) is, in fact, a bundle endomorphism. In fact, I suspect that to make sure that your pasting is \(\mathcal C^0\) ,\(\mathcal C^1\),\(\mathcal C^\infty\) or \(\mathcal C^\omega\) you have to ask not only for a bundle that decompose the domain, but an additional structure on it (parallel transport? a connection?)
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
\(f_k(z)=k_{{\rm arg}(z)}\cdot z\).
Obviously we can extend it as \(f_k^t(z)=k^t_{{\rm arg}(z)}\cdot z\). If we define \(k^t:{}S^1\to \mathbb R^+\) as the composition \((k^t)_\theta:=(k_\theta)^t\) we have
\[f^t_k=f_{k^t}\]
If you look carefully on the construction what we have done is consider a partition of \(\bigcup _{i\in I}X_i=\mathbb C\) were we define on each class \(X_i\) a function \(f_i:X_i\to X_i\) that we know how to continuously iterate, i.e. how to extend it to a group action of \(\mathbb R\) over it. We then just paste all the functions/actions \(f_i\) into an endofunction of \(\mathbb C\).
\[f=\coprod_{i\in I}f_i:\bigcup _{i\in I}X_i\to \bigcup _{i\in I}X_i \]
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.
Addendum: note that the construction I have given provides an infinite amount of examples and how to build them.
In my and your case we use the polar decomposition of \(\mathbb C\setminus\{0\}\simeq S^1\times \mathbb R^+\), i.e. as modulus and argument, to obtain a partition of \(\mathbb C\).
The complex can be presented as a bundle \({\rm arg}:\mathbb C\to S^1 \), i.e. \(\bigcup_{\theta\in S^1}\mathbb R^+e^{i\theta}\to S^1\), or as the bundle \(|\cdot|:\mathbb C\to\mathbb R^+\), i.e. \(\bigcup_{r\in \mathbb R^+}rS^1\to \mathbb R^+\).
All of this is very tied to decomposition techniques one sees in the spectral decomposition of linear operators to fiber bundles. The resulting map \(f\) is, in fact, a bundle endomorphism. In fact, I suspect that to make sure that your pasting is \(\mathcal C^0\) ,\(\mathcal C^1\),\(\mathcal C^\infty\) or \(\mathcal C^\omega\) you have to ask not only for a bundle that decompose the domain, but an additional structure on it (parallel transport? a connection?)
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)
