While investigating the iteration dynamics of different Julia sets I found two interesting points in all of them which followed a certain recurring pattern. They are pretty obvious once you know what to look for, and in this case the question to ask is: what points in a Julia set are the most stable? Or in other words, what points in a given Julia set move only little or not at all when iterated with the Mandelbrot function. The answer can be found by solving the equation z²+c=z.

Sometime after figuring this out I had the – perhaps in hindsight obvious – idea of using these special points in my z₀ sampler implementation. As a result the random c values fed to the iterated Mandelbrot function are always coupled with a z₀ value that results in a maximally stable orbit. Although this scheme cannot make the eventually diverging orbits eternally stable due to the inherent imprecision of the underlying floating-point arithmetic it does succeed at least in producing this striking image.