Slog(Exponential Factorial(x))

[JmsNxn]

Paulsen and Cowgill's paper is a mathematical paper, which constructs Kneser, and gives a uniqueness condition for Kneser. They don't really make an algorithm, per se, just explain how to grab Taylor series and calculate them. Which, I guess, is an algorithm in and of itself.

http://myweb.astate.edu/wpaulsen/tetration2.pdf

It's a fantastic paper. Paulsen has another paper exploring complex bases, but I think it lacks a good amount of the umph! of this paper, lol. It's not too hard to read, and in my mind, is a very cogent explanation of Kneser.


Also, I just have an irrational hatred of matrices. Not even that it's particularly hard, I can still use matrices if pressed--I just hate them. Lol I'm matrix phobic, lol.

Im still left with many many questions.

anyways,
I had the vague idea of mapping a boundary R to the real line iteratively.

that is to say by some kind of averaging.

by analogue when you want to map f(x) to the zero function you do something like 

f(x)/2

f(x)/2 /2 

f(x) /2 /2 /2

and 

g1(f(x)) = f(x)/2

g2(g1(f(x))) = f(x)/4

etc

the sequence g_n are bounded functions that get weaker and weaker ( closer to id(z)) , so the sequence g_n(g_{n-1}(... )  is analytic.

maybe this is a garbage idea but it seems intuitive to me.

In fact , what do we know about fractional (iterations of) riemann mappings ?

In essense they are just fractional iterations of analytic functions but still ... I feel we might be missing something here.


regards

tommy1729

Fractional iterations of Riemann mappings are actually pretty simple to understand.

Let's say: \(f : S \to \mathbb{D}\), the only way to iterate this is, is to restrict \(S  = \mathbb{D}\) (the unit disk). So it reverts into iterating automorphisms of \(\mathbb{D}\). Every automorphism of \(\mathbb{D}\) is given by a Blashcke product, and therefore just looks like iterating a linear fractional transformation. By which, iterating any Riemann mapping looks like iterating linear fractional transformations upto conjugation.

What I think you're getting at, is trying to iterate a solution to Kneser's Riemann mapping. Which would equate to \(f^{\circ n}(y) \to K(y)\), where this is Kneser's Riemann mapping. That wouldn't be possible. But it would be possible to write \(f_n(y) \to K(y)\), this is discoverable with Taylor series, so it is perfectly possible there exists an iteration formula for this. Which could possibly look like \(g_n(g_{n-1}(...)) \to K(y)\). No idea how to do that though.

Fractional iterations of Riemann Mappings are just iterations of automorphisms though. It's the only way the statement makes sense... Unless you meant something else?

Regards