Borel summation, Mellin Transforms, Parabolic iteration

[JmsNxn]

So to begin, we'll get some concepts clear. A parabolic fixed point \(f(0) = 0\) and \(f'(0) = e^{2 \pi i m/q}\) for \(q,m \in \mathbb{Z}\). And the multiplier of this fixed point, as Milnor states it, or the \(\ell = \text{valit}\) as Bo, refers to it, is given in the expansion:

I think you mean "multiplicity", perhaps edit that in your original post - because it is called "multiplier" in the hyperbolic case for the first coefficient, right?
EDIT: After reading Milnor again: the multiplicity is valit+1, they are not the same. The multiplicity is the index of the first non-zero coefficient after the first coefficient. Valit is the multiplicity - 1. The number of petals is 2*valit.

Oh yes, sorry. I misplaced my terms. What I meant is still the same as you though, I just thought the multiplicity = valit. Either way I meant that:

\[
f(z) = z(1+az^n + o(z^n))\\
\]

And that \(n = \) valit. I guess I just changed the variables to fast, and forgot the nomenclature that the multiplicity (not multiplier--I'm bad with language, bo) is \(n+1\), you are right.

But I would like to remark here that asymptotics is for all petals the same, only the branch(cut) of the logarithm is what needs to be adjusted from petal to petal.

Yes, you are absolutely correct, I had not noticed that entirely yet. I thought there was something more subtle going on, but yes. You are correct. I was also just trying to be reserved in my estimations of the result.

Don't you need more terms \(\frac{1}{u^k}\) and also having coefficients, like:

\[ \alpha(u) = \left(\sum_{k=1}^{\ell} \frac{c_k}{u^k}\right) - A \log(u) + \sum_{k=0}^\infty b_k u^k \]

OKAY! So you are right here to a point--but not exactly. I've been working a lot lately, so I haven't had time to explain these constructions perfectly. I think to draw out the mellin transform isomorphism, is the right move here--and further describes what I am getting at.


Here is where things get spicy! Bo, I just want you to remember we are doing all of these variable changes, "under the integral".

So When I write:

\[
g(w) \sim w + \log(w) + \sum_{k=0}^\infty b_k w^{-k}\\\
\]

And we know that:

\[
\int_0^\infty g(w)w^{s-1}\,dw\\
\]

Converges, we can invert, as such, changing \(u = 1/w\) and \(du = -\frac{dw}{w^2}\) creates an interchange under the integral:

\[
\int_0^\infty g(1/u)u^{-s-1}\,du\\
\]

And now the asymptotic series is at \(0\)--rather than at infinity. And we can do the above transformation/asymptotic expansion.

I believe your mistake is thinking we have to conjugate the Abel function--which we don't do. We conjugate the function the abel function is associated with--within the Abel function. By which, I mean:

\[
\alpha(u) = \alpha(1/w)\\
\]

We do not perform a flip on the series itself, just on the inner terms. So I believe you are mistaken here. At least, as I know of it. I'm going to draw this out a bit, just to paint a more vivid picture. Assume that:

\[
\alpha(f(w)) = \alpha(w) + 1\\
\]

And now let's define a function \(F\) such that \(F(w) = \frac{1}{f(1/w)}\). Then, the function \(A(w) = \alpha(1/w)\), satisfies:

\[
A(F(w)) = A(w) + 1\\
\]

Which, I'm sure you can check.

Where do \(\delta\) and N come from?

Oh, I apologize here, Bo. These are terms to do with Mellin transforms--and are essentially variables which act as limiters.

The value \(N\) is a natural number greater than \(0\), and each of these expansions are valid for all \(N\)--and \(\delta>0\), is my way of saying a small value \(\delta\). Where such a value always exists, you may just need to choose small enough.

I think I need to write a much more in depth construction of these concepts, I can kind of just blaze through it, and forget you guys aren't as familiar with Mellin transform/analytic number theory kind of stuff. I will definitely try my best to explain everything though, bo. Apologize if this is confusing.

I really want us to shift from "Borel Sums" into, this exists in a Mellin transformed space and we transform back. Which is the "sophisticated" way of doing borel sums, lol. D. Zagier's paper is sort of my bible right now. I'm working on writing a much better write up, I'm just not sure if I have the time to write a full paper on this. But Parabolic iteration is making so much fucking sense that it never made before--precisely because I can do this Mellin shit now. Which, I guess, is just how my mathematics works. I'll work harder at explaining the details 

Regards, James