# What is going on in this fractal video?

I saw this:

I haven't have time to generate the images on my own, but I believe that the video presents an investigation of the complex dynamics of the family of functions $$f_{c,k}(z) = c\,z(z+k)(z+1).$$ Note that this family has two critical points at $$z_{\pm} = \frac{1}{3} \left(\pm\sqrt{k^2-k+1}-k-1\right).$$ That is, $f_{c,k}'(z_{\pm})=0$ independent of $c$. The Julia set of $f_{c,k}$ will be connected precisely when the orbits of both these points remain bounded under iteration of $f_{c,k}$. If we fix a value of $k$ and let $c$ vary in the complex plane, we get a slice in the complex $c$-plane. The first minute of the video plots these slices, I believe. The black region should correspond to values of $c$ for which the orbits of both $z_{+}$ and $z_{-}$ remain bounded. The animation illustrates how the pictures change as $k$ ranges from $k=-1$ through $k=1$. The remainder of the video illustrates the corresponding Julia sets - I think. :)
• Is it a superposition of two fractals? Is that when one of $z_+$ and $z_-$ is bounded but the other isn't? – The_Sympathizer Jan 17 '17 at 2:52
• Yes, there are four regions of interest: both critical points bounded, $z_{+}$ bounded but not $z_{-}$, $z_{-}$ bounded but not $z_{+}$, and both unbounded. If you Google "cubic connected locus", you will find lots of information on this topic, including one paper that I wrote illustrating an exploration with Mathematica. – Mark McClure Jan 17 '17 at 3:00