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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?

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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:

enter image description here

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

enter image description here

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

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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 :

like here - parabolic dynamics

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