File:Newton-Fractal 2z3-2z+2.png

English: Newton fractal for the polynomial ${\displaystyle f:z\mapsto z^{3}-2z+2}$ in the range [-2,2] × [-1.5,1.5]${\displaystyle i}$.

Numbers in the orange, green and blue basins converge to zeros of ${\displaystyle f}$, numbers in the red basins are attracted by the attractive cycle ${\displaystyle \{0,1\}}$ (i.e. they do not converge to a zero of ${\displaystyle f}$).

The white structure is the Newton fractal for ${\displaystyle f}$, i.e. it is the Julia set ${\displaystyle J}$ for the meromorphic function ${\displaystyle g:z\mapsto z-{\frac {f(z)}{f'(z)}}=2{\frac {z^{3}-1}{3z^{2}-2}}}$

Zeros of ${\displaystyle f}$ are attractive fixpoints of ${\displaystyle g}$. Moreover, 1 and 0 are fixpoints of ${\displaystyle g\circ g}$ but not of ${\displaystyle g}$.

The set ${\displaystyle J}$ has the following properties:

• its interior is empty
• it is closed
• ${\displaystyle \#J=\#\mathbb {R} }$
• ${\displaystyle J}$ is connected
• ${\displaystyle J=\partial B_{\mathrm {red} }=\partial B_{\mathrm {orange} }=\partial B_{\mathrm {blue} }=\partial B_{\mathrm {green} }}$

${\displaystyle \partial B}$

${\displaystyle J}$