all I did was keep expanding the polynomial and looking out for patterns.
Iterations:
0:
1:
2:
3:
4:
5:
etc.
Look at the first n positions for the nth step:
0:
1:
2:
3:
4:
5:
I tried this up to iteration 14 or so and it kept holding true.
The rest I explained in my previous post.
Note, however, that this series explicitly gives me just that one thing
, not the other one
.
The only connection between those two is that they are both inverses of the inverse function to the first one:
.
I can't say whether that second one actually has anything to do with it.
What I /can/ say is that it does give rise to nice fractal orbits.
Note, however, that none of those orbits resemble an entire Mandelbrot set like you'd hope for the limit Metabrot. (some are pretty close, however, like the bulb-tracing orbits towards the antenna or the green inverse Mandelbrot inside the main circle)
Sadly, beyond being sheer unfeasible to do with high precision, this rendering method has another problem. I tried sampling more points. The whole thing just becomes too cluttered to see anything and it looks like I'm filling the entire plane.
Part of that is because I do not stay within the actual closed bounds of the fractal. I'm tracing points of dust-regions too. Not /quite/ sure how to fix that in this case. Something like "if any point in this sub-sequence has a distance from the point (1/4,0) greater than 2, do not render anything in this sequence or iterate it deeper" would, I take it, be the equivalent of the normal Mandelbrot set bailout. I'm unsure whether that would work for this particular method.
Also I'm not sure how I'd implement that into Mathematica.
Perhaps rendering this buddhabrot style would be preferable, giving all the points the same color, but have them accumulate to give a density map instead of a map of each individual orbit. That way one could probably remove some of the clutter.
Oh and
is a fixed point in that that is
and
.
will keep shrinking if you multiply such values repeatedly, while things like
will grow indefinitely. As such, 1 is an unstable multiplicative fixed point. Unstable because any small deviation from 1 (along the real axis) will quickly cause accumulated errors.
I don't think a similar situations arises in a square-root, and that kind of fixed point probably works differently from other fixed points.
also is a fixed point of derivation with
, but that's an entirely different situation.
And yeah I have seen those multibranch renders. Still really nice to look at