exp(a/z)*z^2+c

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Offline gerrit

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« on: June 01, 2019, 07:57:04 AM »
with a a number and c the pixel seems like it should look like the Mandelbrot set for small a and it does, for small a you get a M-set but you start to see distortions at zoom 1/a. There are 2 critical points (and orbits) at a/2 and 0. The latter may seem suspicious, what is exp(1/0), but it's at least an essential zero coming from the negative side so maybe that's enough to count as critical point. I've read mathematicians doing this without even bothering to mention it, so maybe it's obvious.

It seems there are many minis in the shapes you get, but they are not all the same like in the M-set but come in infinite varieties. That's good but there are no nontrivial embedded Julia sets as far as I can tell by some exploring. There are actual M-set minis and they have the usual embedded Julia sets but nothing new.

Here's an image for a=1/2, combining both images (from 2 critical orbits which are quite different) using Ultra Fractal "layers" which I understand at the level of select this and I get that. I'm sure there's a better way.

Offline gerrit

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« Reply #1 on: June 02, 2019, 01:03:12 AM »
Two images based on orbit of a/2 critical point, which seems usually more interesting, with a=0.84524.
First is full parameter space, second close up of the structure in between the two tilted minibrots.
Third image full parameter space for the other critical orbit.

Offline gerrit

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« Reply #2 on: June 13, 2019, 02:50:21 AM »
a=-0.30952, a/2 orbit. This is inside the main cardioblob.