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I saw this:

https://www.youtube.com/watch?v=rQ3kK58VvsI

In many parts of the video, you can see what looks like a fractal set similar to the Mandelbrot set, changing as the parameters to the underlying complex map are changed. Black, I presume, indicates the points within the set, with bounded orbit, and the colorful region is outside or escaping points, as is usual for this type of plot. But in some cases, you can see that there are areas of color within the fractalized "boundary curve" like a filling, which implies that part of that boundary curve actually lies outside the black set and is in the escaping region. Yet what then does this boundary curve mean, mathematically and dynamically, where it is separating two parts of the escaping region, as opposed to separating the escaping region from the bounded-orbit region? What kind of structure is being seen here?

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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. :)

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  • $\begingroup$ So what does the fractalized curve crossing the colored region mean? It doesn't separate black from color, but color from color. $\endgroup$ – The_Sympathizer Jan 17 '17 at 2:50
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    $\begingroup$ Is it a superposition of two fractals? Is that when one of $z_+$ and $z_-$ is bounded but the other isn't? $\endgroup$ – The_Sympathizer Jan 17 '17 at 2:52
  • $\begingroup$ 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. $\endgroup$ – Mark McClure Jan 17 '17 at 3:00

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