Mandelbrot-like sets with normal mandelbrot minibrots

[TheRedshiftRider]

Recently Karl implemented a few functions which I suggested.
All of them are based on this function:

F(z)=z^2+z^n+c

Where n is any positive number.

The general shape of these sets is different but the shape of the minibrots is just like the mandebrot set (or higher power versions if the z^2 is replaced with a different power).

Does anyone know why this is?

[TheRedshiftRider]

https://imgur.com/a/Ah7u1

I tried some higher powers for the z^n. A new mandelbrot set grows out of it if you increase the power. And embedded juliasets can be found within the main set. These are also like normal mandelbrot sets.

[claude]

Does anyone know why this is?

Does Karl's implementation use the correct critical values?  The critical values for are which has 0 as an obvious solution but also n-2 other complex solutions .  I'm not sure what the correct way to render Mandelbrot sets with multiple critical values is - the decision criterion is either all escape for exterior or none escape for interior, not sure which is right.

For the 0 value, the periodic cycle of a minibrot will hit 0 again.  Close to 0, z^2 is much larger than z^n, so its folding-in-two effect will dominate - at other points in the periodic cycle the relevant region is small and far from 0 so the effect of the iteration is mostly linear with a small non-linear warping.  So I think the lowest power > 1 for any combination of powers decides what shape the minibrots are likely to have.