(09/08/2022, 12:19 PM)bo198214 Wrote: No, James, this has nothing to do with real-valued iteration, though the curves look similar to the thread "Real Multivalued/Parametrized Iteration Groups".
This is plain old regular (hyperbolic) iteration approaching the parabolic case in the limit.
What I remember from Milnor he does not really consider the proper parabolic case, rather says just take the k-fold iterate then you have multiplier 1. Can you perhaps refer to the section where he talks about parabolic case != 1?!
I realize this isn't a thread on real valued iterations, but real valued iterations are iterations which map a curve to a curve. And that's more so what I meant. You are right though, I'm being incomplete.
Perhaps here, I have a later edition of the book. But Milnor specifically explains that if \(e^{2\pi i \varphi}\) is the multiplier, and \(\varphi\) is a \(q\)'th root of unity, then the different surfaces are modulo each other upto \(q\); exactly like the action \(z \mapsto e^{2\pi i/q}z\). So yes, he only talks about \(=1\), but he describes how to construct abel functions on every parabolic multiplier, and gives pretty clear language on this. Where there are \(q\) solutions, just like \(j = k \,\,\text{mod}\,\,q\). And then he has a whole section on siegel disks--(Section 11)---which requires a discussion of irrational numbers which are of the above form as I wrote. In this case, we don't solve for the Abel function, we solve for the inverse Schroder function as \(\lambda = e^{2 \pi i \varphi}\) and \(f(h(w)) = h(\lambda w)\)--which does implicitly solve an Abel function.
I'm not sure what you mean by no discussion of the general parabolic case, because Milnor gives a large treatment on how to conjugate every parabolic case to the case \(\lambda = 1\), and for non-parabolic, but "indifferent", he has a lot to say about different ways of describing it. It's very Riemann surface heavy, but he essentially describes the problem, though most of it is hidden in subtle descriptions.
You are correct, he reduces it to \(\lambda = 1\), but he does give a treatment describing what the surface looks like for every parabolic fixed point; and does so a bit off hand, and not to full depth. But there is the above statement, that parabolic fixed points \(e^{2 \pi i j/q}z + o(z)\), where \(j\) is coprime to \(q\), behave like the action \(z \mapsto e^{2\pi i j/q} z\) locally. Additionally, we can construct an abel function about here, and not only that. Expect the swirling I discussed. This then becomes a "permutation of the petals"--or something like that. This is hard to explain, but I hope you can visualize it. It rotates the petals a small amount if you only affect the original multiplier.
Try writing:
\[
e^{z}-1 -2z = - z + \sum_{k=2}^\infty \frac{z^k}{k!}\\
\]
And observe how things get permuted in comparison to:
\[
e^{z} -1\\
\]
It is very much the difference between \(z \mapsto -z\) and \(z \mapsto z\). The negative z's send to positive, and the positive to negatives, rather than the clean behaviour of \(z \mapsto z\). This is the "swirling" I'm referring to. It's literally just the action of the unit of the circle \(-1\).
I don't have much time lately, but I'll try to keep this conversation going
