Julia set

Julia sets, described by Gaston Julia, are fractal shapes defined on the complex number plane. Given an iterated map of the complex plane to itself, or a collection of such maps, the Julia set for this system can be defined (informally) as the set of points for which nearby points do not exhibit similar behaviour under repeated iterations of the map(s).

Julia set using reversed formula

It is also possible to generate Julia sets using a method derived from the iterated function system (IFS) "random game" method. Instead of using an "escape time" method to find the points that do not belong to the Julia set, the "random game" method uses the reverse formula to track the convergence of a single point towards the edge of the Julia set.

Related Topics:
Iterated function system - Reverse - Convergence - Point - Edge

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For example, for the Mandelbrot set, the reverse formula is

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:z_{n+1} = sqrt{z_n - c}

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At each iteration one of the two roots is selected at random. After a sufficiently large number of iterations the current position of z is plotted. The process is then repeated with a different random sequence of roots.

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The process of "reversing" the mapping essentially exchanges the attractors and repellers; the Julia set is now an attractor, with infinity and possibly the usual periodic cycle now being repellers. A Julia set which has a fractal structure is a strange attractor for the reversed mapping.

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Note that certain parts of the Julia set - notably the "waists" - are quite hard to reach with the reverse Julia algorithm, since the "points" or "tips" of the Julia set are more strongly attractive. But this can be avoided by using a consistent choice of root a random number of times in a row (maximum 4-6 are good) making the center attractors stronger.

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