The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!]

[jaydfox]

I wonder, however, if there are "mixed" fixed points. You know, alternate between two branches for every other logarithm, or three branches for every third, etc. What type of values do we settle on?

I dont think so. We have in each rectangle of values \( x+iy \) with \( x>0 \), \( 2\pi k < y < 2\pi k + \frac{1}{2}\pi \), \( k\ge 0 \) exactly one fixed point and these together with its conjugates are all fixed points of \( e^x \).
I think the fixed point with imaginary part in \( 2\pi k ... 2\pi k + \frac{1}{2}\pi \) is the limit of the iterated \( \ln(x)+2\pi i k \) for a starting value with positive imaginary part.

Sorry, I think I wasn't clear. I was thinking that if we alternated logarithms between two branches, we should settle on two points (one for every other iteration) that are "fixed", such that if you exponentiated, you would go back and forth between two points, until you'd moved sufficiently far to escape. Similarly, three points for three-cycle iterated logarithms, etc.

At any rate, while fascinating (and something I may study further in the future), I think I've drifted far enough away from my original line of inquiry that its time to start steering back on course. All of this study is to get a better idea of how iterated exponentiation "works", so that I can figure out how to derive a continuous solution based on principles of exponentiation, rather than from Andrew's sneaky trick of solving a linear system of equations based on the inverse of iterated exponentiation. Unforunately, the best I've found so far is a relation between the primary fixed point of natural logarithms and the coefficients of the power series of Andrew's slog for base e.

It is my hope to find a direct solution for base e, have it be reasonably defensible, and hopefully have it match either Andrew's solution or mine (preferably Andrew's).