| Fatou's theorem states that if a rational function f(z) has an attracting cycle, then a critical point of f(z) must be attracted to the cycle. |
| For the Newton function |
| Nf(z) = z - f(z)/f '(z) |
| a straightforward calculation gives |
| N 'f(z) = f(z)f ''(z)/(f '(z)2) |
| So the critical points of Nf(z) are the points z for which f(z) = 0 (the roots of f(z)) and those where f ''(z) = 0. The roots of f(z) are attracting fixed points, so each belongs to its own basin of attraction. Consequently, the only critical points of Nf(z) that could be attracted to something other than a root of f(z) are the points z for which f ''(z) = 0. |
| For the Curry, Garnett, Sullivan polynomials f(z) = z3 + (c - 1)z - c, we see f ''(z) = 6z. Consequently, only z = 0 gives f ''(z) = 0. |
Return to universality of the Mandelbrot set.