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

[jaydfox]

What I find strange is that it seems that the iterated logarithm of every point in the upper complex half plane tends to the one fixed point in the upper plane, though there are many other attracting fixed points of \( \ln \) in the upper half plane. Similarly for the lower half plane.

I could be wrong, but I think there's only one true fixed point per branch. The others are "images" of it from other branches of the natural logarithm.

For example, if you always use the branch with imaginary part in the range \( (\pi, 3\pi] \), then your fixed point is 2.06227773+7.588631178i.

Now, if you take the natural logarithm of that point with the principle branch, you get 2.06227773+1.305445871i. So that point would appear to be a fixed point of exponentiation. If you exponentiate it, you get back to 2.06227773+7.588631178i, and the next exponentiation stays there, using that point as a fixed point.

Therefore, there are many fixed points of exponentiation, but only one fixed point per branch of the natural logarithm (two if you count the conjugate as a separate value).