Actually, Henryk, if you graph even a rough mockup of the regular slog developed for the fixed point z=2, the corrugation effect as we wrap around the fixed point at 4 appears, and furthermore, it looks like it should be there. In other words, it makes sense that they should be there, due to a switch from rings around the fixed point at 2 to cyclic wavy curves.

Start with a bunch of concentric rings around z=2, with a relatively small radius. This is just a mockup, so accuracy isn't required. Just make sure to get enough rings, so that when you take the logarithm of each, you don't get any major gaps.

Now, continue to take more and more logarithms of the set of concentric rings. The rings will become ovals, stretching out to the left and getting squished to the right. As the rings pass the origin, the next iteration swing out to minus infinity, and then become curves stretching from between the lines with imaginary parts , and of course, when we include all branches, they become infinitely long wave curves, cyclic at intervals of .

Further iterations will be show the corrugation, due to the change from rings to wavy lines, a switch that occurred when the first ring touched the origin.

Start with a bunch of concentric rings around z=2, with a relatively small radius. This is just a mockup, so accuracy isn't required. Just make sure to get enough rings, so that when you take the logarithm of each, you don't get any major gaps.

Now, continue to take more and more logarithms of the set of concentric rings. The rings will become ovals, stretching out to the left and getting squished to the right. As the rings pass the origin, the next iteration swing out to minus infinity, and then become curves stretching from between the lines with imaginary parts , and of course, when we include all branches, they become infinitely long wave curves, cyclic at intervals of .

Further iterations will be show the corrugation, due to the change from rings to wavy lines, a switch that occurred when the first ring touched the origin.

~ Jay Daniel Fox