Julia Sets and the Mandelbrot Set

Universality of the Mandelbrot set

Are there more failures more serious than starting exactly on a basin boundary?

Are there large sets of starting points from which Newton's method does not converge to a root?

For example, can Newton's method ever have attracting cycles?

Curry, Garnett, and Sullivan experimented with the cubic polynomials f_c(z)

f_c(z) = z³ + (c - 1)z - c

Classical results show if there is an attracting cycle, it would have to attract z₀ = 0.

So the basic experiment was to apply Newton's method for each point c in the complex plane, always starting with z₀ = 0, and see if the sequence converges to a root.

Return to Universality of the Mandelbrot set.