I'm currently trying to create a fast OpenGL application that can render Julia Sets in real time. I was able to do this with the Mandelbrot Set, thanks to this handy trick to quickly identify period 1 and 2 regions. But when I moved over to Julia Sets, I realized quickly that I can't use the same formulas for optimization, as the final picture looked like a normal Julia set but only with a black Mandelbrot smack dab covering it in the middle. How can I do the same kind of simple optimization with any given Julia Set, with finding the boundaries of the first couple periodic regions?
Example: $c=-.13+-.8i$
https://www.marksmath.org/visualization/julia_sets/