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Iterates for the Mandelbrot Set

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The Mandelbrot set is the set of numbers in the complex plane for which the map remains finite for all , where . If the sequence does not converge. In the graph, the path taken by the iterations is traced until Abs . The Mandelbrot set is displayed in the background with the colors indicating the first for which .

Contributed by: Felipe Dimer de Oliveira (March 2011)
Open content licensed under CC BY-NC-SA

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The background figure was generated by a density plot of the number of iterations it takes for the map to satisfy with a maximum number of iterations set to 100. In the figures above, from top to bottom: is chosen so the map satisfies for all , for (light blue region), and for .

Reference: T. W. Gamelin, Complex Analysis, New York: Springer–Verlag, 2001.

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