# Inverse-Iteration Method

Robert P. Munafo, 1993 Dec 23.

A method of drawing Julia Sets, which exploits the fact that the iteration function can be performed in reverse, yielding two possible values of Z(n) when a value of Z(n+1) is known.

This method of iterating causes the boundary of the Julia Set to be attractive rather than repelling as it normally is. This in turn means that after a sufficient number of iterations (about 100), it is pretty much guaranteed that every subsequent iteration will give a point on the boundary.

The method does not work too well unless the iterating algorithm is very careful when picking which of the two values of z(n) to persue. If you just pick a value at random each time, you end up with a plot that emphasizes points whose external angle has a small power of 2 in the denominator. The Modified IIM algorithm overcomes this.

The method is described in The Science Of Fractal Images, pages 173-178 (Peitgen and Saupe, editors).

See also Inverse Mandelbrot Iteration.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018. Mu-ency index

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