Different Coloring of Julia Sets

Question

I have known about Julia Sets for a while now, and today I had an idea about the coloring of Julia and Mandelbrot Sets. What if someone were to color them not only by how quickly z diverges, but also how quickly it converges. How would I go about doing this in a program? Has it already been done?

Answer 1

In fact, just is exactly how WolframAlpha colors its Julia sets. Here's the result of the query "julia set -1", after the image type is set to escape time:

Furthermore, this is really the only way to go when drawing Julia sets for rational functions. Here's the Julia set of $z^{-2}-1$:

Finally, Newton's method pictures, with the different basins of attractions shaded in different colors are generated this way.

Answer 2

First because "... a single algorithm for computing all quadratic Julia sets does not exist." (Mark Braverman, Michael Yampolsky) one have to find what type of dynamics ( inside Julia set) has.

Then one can use :

• attraction time, Koenigs or Boettcher coordinate
• Siegel disc coloring
• Cauchy Convergence Algorithm (CCA)
• ....