A bit more numerical work, and some pretty pictures. First, the points in white are those u-values for which the orbit of u under f(z) = uz
diverges to infinity (or at least has absolute value great enough to overflow Matlab) at or before the 1000th iteration. The images go from -3.5 to 6.5 on the real axis, and -7.5 to 7.5 on the complex axis. This region shows most of the “interesting” behavior. Click to view full-size, where pixels are sampled every 0.01 units.
: white points numerically
calculated to diverge by 1000th iteration,
black points have not diverged
after 1000 iterations of exponential
map, with arctan-scaled shading
after 1000 iterations of exponential map, with arctan-polar coloration
after 1000 iterations
Black = Chaotic, or cycle-length greater than 10
Blue = 1-cycle (fixed-point attractor)
Green = 2-cycle (2-point attractor)
Yellow = 10-cycle
White = Numerically observed to diverge
Here are the same things, but drawing the result for z at the point log(z) (technically, drawing at point z the result for exp(z), so these are 2πi-periodic). That is, I have simply applied the map Log(z) to the complex plane, thereby warping the locations of points, but not changing any of the values in the iterated exponential map.
As you can see, it very strongly appears that the locus of fixed-point attractors, shown in blue in the last image, has natural log in the shape of a cardioid. The apparent cusp is at 1/e, and the opposite side is at -e. The top and bottom seem to be at c±2i for some real number c I haven’t tried to identify.
To find the cycle-lengths, I calculated the 1000th through 1010th iterations, and compared them pointwise to see when the orbit returned to its value from the 1000th iteration, within a tolerance of 0.001. To visualize how the cycles go, here are the 1000th through 1010th iterations in order, with arctan-polar coloration:
Edit: it also looks like the red area (period 3) in the log-plot has bounds that approach ±π/2 in the limit of large real component.