Julia Sets and the Mandelbrot Set

Universality of the Mandelbrot set

We have already mentioned that when starting from a point on the bounday of the basins of the roots, Newton's method does not converge, but oscillates chaotically.

In one way this is not such an interesting failure, because the basin boundary is very sparse in the sense that small changes in the starting point typically give a point in one of the basins of attraction.
So this type of failure is unlikely to happen.

Return to Universality of the Mandelbrot set.