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

[Gottfried]

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.

For me, when I played around with this, I got the impression, that it might be best expressed, that the (attracting) fixed point has a gradient-field around it. In polar notation, (where the fixed point is translated to the origin), the lengthes and the angle from one iteration to the next converged to a certain value for each starting point (and also for each fixed point). So the inverse process, starting at the fixed point to arrive at the starting point means to select a direction and a length-change - inexpressible for me in terms of the infinitesimal changes... But I remember the term "gradient field" from my "Hütte - mathematische Tabellen" and it seems to fit here.

Gottfried