[split] Understanding Kneser Riemann method

[sheldonison]

@sheldonison

\( \text{tet}_b(z) =\;^z b= \exp_b^z(0) \)

just as a point of order, I think you meant 1 instead of 0.

I've tried to understand your Pari/GP scripts, but I think my fundamental issue is with the Kneser construction / Riemann mapping thing. So here is my understanding so far.....

The value of such a construction is that it allows us to compare regular iteration and intuitive iteration for \( b > \eta \). But there are too many unknowns for me: what are the properties of this Riemann mapping? how do we find it? what is the result? is it analytic? wouldn't this just be equivalent to

\( f^{-1}(x) = \text{tet}_b^{Reg}(\text{slog}_b^{Int}(x)) \)

and if this is the method for calculating the Riemann mapping, then we can't expect to learn anything about the two methods of iteration. Perhaps I should revisit this when I'm less confused.

Your equation is pretty close. I'm going to re-phrase it in terms that I prefer using, using a theta(z) mapping. I'm not surprised about the confusion. It would be nice to try to encapsulate the Kneser mappings into something as compact as possible. I think the equation linking your f(x) with my theta equation is: \( f(z)=z+\theta(z) \)

Lets say we have the Schroeder function, and its inverse \( S(z)\;\;S^{-1}(z) \) which have corresponding Abel and super functions, \( \alpha(z)=\log_L(S(z))\;\;\alpha^{-1}(z)=S^{-1}(L^z)\;\; \)

I think that's what you mean by regular iteration. This Abel function is complex valued for bases>eta. Also, there's actually two fixed points, which are complex conjugates of each other.

Now, here's the interesting thing. Start with a real valued slog(z) function, that meets the uniqueness criteria. We can generate that slog as a function of the \( \alpha(z) \) above as follows:
\( \theta(z)=\text{slog}(\alpha^{-1}(z))-z\;\;\;\theta(z) \) is a 1-cyclic function, theta(z+1)=theta(z)
\( \text{slog}(z) = \alpha(z) + \theta(\alpha(z))\;\; \) real valued slog(z) in terms of the Schroeder function and theta(z)

I'm writing this equations in terms of the slog, since my latest program, fatou.gp calculates the slog. The uniqueness criteria, equivalent to Kneser, is that the upper complex plane theta(z) has a very special property, that as \( \Im(z) \) approaches +imag infinity, theta(z) approaches a constant. Since theta(z) is a 1-cyclic function, this tells you that:
\( \theta(z) = \sum_{n=0}^{\infty} a_n \cdot \exp(2n\pi i z)\;
\; \) notice the absence of negative terms as compared with the general 1-cyclic: \( \sum_{n=-\infty}^{\infty} a_n \cdot \exp(2n\pi i z)\;
\; \)

So, what my latest fatou.gp program does, is find a way to compute a pair of \( \theta(z) \) mappings for the two fixed points, in the upper and lower halves of the complex plane, in addition to iterating and calculating an approximation for the real valued slog(z) Taylor series. This is equivalent to Kneser's construction, although Kneser never talked about 1-cyclic functions much, but his equations and his Riemann mapping can be equivalently expressed in terms of 1-cyclic mappings, like I'm doing here.

So, now this tells you that as \( \alpha(z) \) approaches +Im infinity, Kneser's slog approaches \( \alpha(z)+a_0 \) where \( a_0 \) is the constant term from the \( \theta(z) \) equation. Of course, the \( \alpha(z) \) approaches +Im infinity as z gets closer to the fixed point of L. Perhaps I will post more later; hope this helps.