I wanted visual confirmation that the webbing is everywhere. Here is a deep zoom into the nearly empty area in the top right of yesterday’s image. The color has been changed from previous image to enhance the features. The lines get thin very quickly. You see a horizontal line, intersected by many vertical lines, and each of them connected by almost invisible horizontal lines. Because the lines in this third set are thinner than the pixels in the image, Moiré patterns appear.
To save typing and simplify some calculations, define . Note that .
With this notation, . Set and expand the iteration .
After the first step, we always have . With a little algebra, it is seen that if . So the real part of the iteration keeps getting larger, and we have many orbits escaping to infinity.
A similar analysis fails with . On each iteration step,
The real part is always bounded, , so the only hope of escape lies with the imaginary part. But if x is a multiple of a*pi, then u(x) = 0, and y has to start over.