Kneser method question

[tommy1729]

It has been a while since I considered kneser’s method.

So I will try to describe it and then ask my question.
Correct me if I make mistake.

Step one :

We use the first upper fixpoint of exp : L.

From L we compute the solution to 

Exp(f(z))  = f(L z)

By using the koenigs function.

Step 2 :

We find the value 1. 
From this 1 we trace the positive reals.

We notice the reals make a corner at every rotation L or equivalently at 1,e,e^e,....

First question : why at these values and not at say pi,exp(pi),exp(exp(pi)),... ?

Second question : this indicates there are singularities at 1,e,e^e,....
What kind of singularities are they ?? 
The corners seem to suggest log or sqrt or such, but I am not sure.

Third question : do we pick branches ? I thought we did not. 

I can imagine that statistically such singularities or corners are expected because we need to respect the rotation. But why it is necessary, I have no formal explaination.

So I guess that makes it question 4.

We continue.

First we take a log base L to solve 

Exp(g(z)) = g(z) + 1

Notice we added a log on top of the singularities , right ?

So we arrive at question 5 and 6 :

Question 5 : did the log remove or simplify the singularities ?
This is only possible with powers like log x^a = a log x I think.

Question 6 : all singularities are still there right ??

Ok, so now we have a kind of Abel function g(z) but it does not map the positive reals to the positive reals.

Notice also the positive reals values are now arranged 1-periodic within the function g(x).

But often stacked on top of each other, hence unfortunately not describable by a Fourier series.

So we need to map the positive reals to the positive reals kinda.

We know that is possible with a somewhat unique analytic function from riemann’s mapping theorem.

However the riemann mapping is mysterious to many.

Apart from how it is done and closed form issues , error terms etc there is also this :

Question 7 :

How does the riemann mapping not destroy the functional equation ?

Question 8 :
The big question :

How does the riemann mapping remove those singularities at 1,e,e^e,... ??

Question 9 : what happened to the other singularities ?

I have the feeling I’m not the only wondering about these things.

Regards

Tommy1729