On n-periodic points of the exp() - A discussion with pictures and methods

[Gottfried]

On n-periodic points of the exp() - A discussion with pictures and methods                          

Initially triggered by a completely unrelated (seemingly on a first glance) other question in MSE I worked on the problem of periodic points of the exponential function and got a -I think: marvelous- result which I like to share here:

MSE - on periodic points of the exp()-function


Perhaps I'll transfer the full copy of the text and the images here later, but for the moment I'm bit lazy after that intense researching, computing & documenting.

update: another question which asks for generalization to arbitrary (real) bases and their 2-periodic points
see  MSE - on 2-periodic points for iterated b^z    (Because this discussion has evolved much and is much worthful, I post another statement pointing at it)

update2: a short exposé of my idea in mathoverflow.net and the relevant question: "is my method for finding n-periodic points of exhaustive?"


update3 (7'20): A compilation into a draft article see here
  periodic points compact.pdf (Size: 309.84 KB / Downloads: 711)  

Gottfried

[Gottfried]

A very nice discussion I've been involved, but which is now too long to be copied here, is Jun,20 in Math Stack Exchange .     
"how to compute the 2-periodic points of \( w=z^{z^w} \)  .  I apply my newly found method to the example bases   \( z \) with protocols of errors and progresses in the iteration and method of finding.

This has also developed into a (re-) discussion on Yiannis' generalization of the Lambert-W function, called "HyperW" of "HW()" which he has presented in his article in 2005 (available here in tetration-forum-library Galidakis2005).    

See Dominic, Yiannis Galidakis and me in discussion...

Gottfried