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

[jaydfox]

Argh, that's embarrassing... I was right in some respects, but I made a key error in my thought experiment. Large values will tend towards 0 with high probability (essentially 50% once the imaginary part gets large), and iterating values close to zero will tend to resemble iteration with real numbers (i.e., integer tetrations of e). Eventually, an infinitesimal imaginary part will get large enough to cause a repeat. However, with very large numbers, the next time through the cycle, we'll be even closer to 0, so that the iterates further resemble real numbers, and we have to exponentiate even further before the imaginary part becomes large enough to send us back even closer to 0, and so on. If taken to its logical conclusion, we should be able to eventually recover any number of consecutive integer tetrations of e with arbitrary precision, by iteratively exponentiating a random complex number.

In short, if you exponentiate a complex number enough times, it should eventually reach a point where it's essentially a real number. Of course, exponentiate it enough times further, and it will get "sent" back to an infinitesimal value near 0, where the process repeats (and goes a little further the time through).

And on the other hand, if you take iterated logarithms of any complex number, you should eventually converge on a fixed point. Assuming you use the principal branch, you should settle on the primary fixed point (or its conjugate). As far as I can tell, the other fixed points require use of different branches of the logarithm.

And, having been wrong already, I reserve the right to be wrong again...