An archived instance of discourse for discussion in undergraduate Complex Dynamics.

An alternate viewpoint for Newton’s method

mark

The images below show Julia sets and basins of attraction related to the function $f(z)=z^5-1$. I emphasize related to; the basins of attraction of $f$ are not shown.

The first image shows the basins of attraction of $N(z)$, where $N$ is the Newton's method function for $f$.
Problem 1: Write down the function $N$ whose Julia set is shown in figure 1.

The second and third images are obtained by conjugating $N$ so that zero maps to $-1$ and $\infty$ maps to 1.
Problem 2: Indicate a strategy to find the funciton that generated those figures.

RedCrayon

$$N(z)\equiv z-\frac{f(z)}{f'(z)}=z-\frac{z^5-1}{5z^4}=\frac{1}{5z^4}+\frac{4z}{5}$$ A function that maps zero to $-1$ and $\infty$ to $1$ is $$\phi(z)=\frac{z-1}{z+1}$$ The inverse is $$\phi^{-1}(z)=-\frac{z+1}{z-1}$$ Conjugating with $N$ gives $$\phi\circ N\circ\phi^{-1}(z)=\frac{20z^2+20z^3+20z^4+4z^5}{5+15z+30z^2+10z^3+5z^4-z^5}$$ The image below is of the basins of $\phi\circ N\circ\phi^{-1}(z)$. The second pair of axes is a bug.

Basins of $\phi\circ N\circ\phi^{-1}(z)$

mark

@RedCrayon Looks great! I wonder what's with the weird, double real-axis?

RedCrayon

If you begin a scan with "axes" true, and then begin another scan with different "z min" and "z max," this bug occurs. You get the updated picture, but the updated axes are overlaid with the old axes.