File:Newton-lplane-Mandelbrot-smooth.jpg

${\displaystyle N_{p_{\lambda }}}$

${\displaystyle p_{\lambda }:z\mapsto z^{3}+(\lambda -1)z-\lambda }$

in the complex plane. ${\displaystyle N}$ denotes the Newton operator

${\displaystyle N_{f}:f\mapsto {\mathit {id}}-{\frac {f}{f'}}}$

For ${\displaystyle \lambda }$ in the black part of the plane, the critical point 0 of ${\displaystyle N_{p_{\lambda }}}$ does not converge to a zero of ${\displaystyle p_{\lambda }}$. This means that the set of start values for which Newton's method does not converge to a zero of ${\displaystyle p_{\lambda }}$ is a set of full measure. The black set is

${\displaystyle M=\{\lambda \in \mathbb {C} :N_{p_{\lambda }}^{k}(0)\not \to {\text{ root of }}p_{\lambda }{\text{ as }}k\to \infty \}}$

${\displaystyle \lambda =0.3+1.64i}$