Checkpoint 3.7.2
Locate the points from output of the previous code in the Mandelbrot set and sketch the corresponding Julia sets.
Subsection 3.7.1 Quadratics with super-attractive orbits
We've already seen that \(f_0(z)=z^2\) has a super-attractive fixed point at the origin and that the points \(z_0=0\) and \(z_1=-1\) form a super-attractive orbit of period 2 for \(f_{-1}(z)=z^2-1\text{.}\) We also know, from our work with real iteration, that there is a \(c\) value around \(-1.75\) so that \(f_c\) has a super-attractive orbit of period 3. It turns out that there are two more complex super-attractive period 3 parameters. We can find them with the following Sage code:
Checkpoint 3.7.3
Find the complex \(c\)values such that \(f_c\) has a super-attractive orbit of period 4 or 5. Locate these points in the Mandelbrot set and sketch the corresponding Julia sets.