Newton fractal

Example

Let $p(z) = z^3 - 1$, with roots $1$, $\omega = e^{2\pi i/3}$ and $\omega^2$. Newton's method for $p$ iterates the rational function $$N(z) = z - \frac{p(z)}{p'(z)} = \frac{2z^3 + 1}{3z^2}.$$ The Newton fractal of $p$ is the Julia set $J(N)$. It is usually drawn with each point of the Fatou set coloured by the root its orbit tends to.

Check of the main features:

1. Each root $r$ is a superattracting fixed point: $N(r) = r$ since $p(r) = 0$, and $N'(z) = p(z) p''(z) / p'(z)^2$ vanishes at $r$.
2. So each root has an open basin of attraction, contained in the Fatou set.
3. The map $N$ commutes with $z \mapsto \omega z$, so the picture has threefold rotational symmetry.
4. The Julia set is the common boundary of the three basins: every point of $J(N)$ lies in the closure of each basin.