Pictures of the Chi-Star

[sheldonison]

The part that was miraculous was mapping the region to the unit disk so that the transfer map \( z \mapsto z+1 \) gets sent to \( z \mapsto \tau(z) \) where \( \tau \) is an automorphism of the unit disk without fixed points. That to me was the genius of the method. I'm still a little unclear on how he does it, but I'm starting to see the general picture.

I'm going to describe the Riemann mapping as best as I understand it even though my understanding of how Kneser uses the RiemannMapping region is not complete.  You take the \( \alpha \) Abel function of the real number line to get the repeating yellow contour from the previous post.
\( \alpha(\Re);\;\;\;\alpha(z)=\frac{\ln(\Psi(z))}{\lambda} \)

And then to get Kneser's RiemannMapping region which is mapped to the unit circle, you need to multiply by \( 2\pi i \) so now its 2pi i periodic instead of unit periodic.  Finally, you take the exponent so all of the unit sub-segments are now mapped on top of each other.  This encloses an infinite region which includes the z=0 center corresponding to \( \Im(\infty) \).   And then you take the RiemannMapping of that region ...  If you accept all of that, then you have:

\( U(z)=\text{RiemannMapping}(e^{2\pi i \alpha(\Re)})\;\;\; \) It is valid to work with the line segment from 0 to 1 instead of the Real number line.  

I'm not sure if that is the correct Riemann mapping terminology, but \( U(z) \) represents the RiemannMapping unit circle function, with the requirement that \( U(0)=0 \), and that \( U(1) \) is the singularity; the rest of the unit circle is analytic.  My next step is to generate the \( z+\theta(z) \) function from the RiemannMapping.  You have to take the ln and divide by \( 2\pi i \) to get the repeating yellow region mapped to the real axis.

\( z+\theta(z)=\frac{\ln(U(e^{2\pi i z}))}{2\pi i } \)
\( \lim_{z \to i\infty}\theta(z)=k\;\; \) where k is a constant, and theta goes to k as imag(z) goes to infinity
\( \alpha^{-1}(z+\theta(z))\;\;\; \) Kneser's Tetration function which is real valued at the real axis...

Finally, here is the picture of the Riemann mapping region: \( e^{2\pi i \alpha(\Re)} \)