The Mandelbrot Set and Julia Sets

Julia Sets - Computational Issues

Accuracy vs. time

However, suppose all of z₀, ..., z₁₀₀ lie within a distance of 2 from the origin.

Can we conclude the sequence never runs away to infinity?

Unfortunately not: maybe z₂₀₀ will be farther than 2 from the origin.

Now we must make a choice. Select some maximum number, N, of iterations we are willing to try.

If all of z₀, ..., z_N lie within a distance of 2 from the origin, we assert the sequence will never run away to infinity and so z₀ belongs to K_c.

This leaves open the possibility that we incorrectly conclude some points belong to K_c.

For example, here are two renderings of K_c (the points painted black) for the same c. On the left, N = 10; on the right, N = 50.

Generally, the larger N, the fewer mistakes we make. On the other hand, the larger N, the more computer time needed to generate the picture.

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