3. Julia Sets and the Mandelbrot Set

3. H. Universality of the Mandelbrot set

Curry, Garnett, and Sullivan studied ways in which Newton's method fails to converge to a root, and found a surprise. Let us see what they found.
Newton's method can fail if the starting point lies exactly on the boundary of the basin of attraction of a root.
Can Newton's method have an attracting cycle? If it does, how would we test this?
Here is the experiment of Curry, Garnett, and Sullivan.
We learn the Mandelbrot set is almost everywhere.

Return to the Mandelbrot set and Julia sets.