Multiplicative Calculus

[fivex]

So I was reading about multiplicative calculus. And I thought that there might be an analog of the m-set in it.
After some trial and error, I came across the following formula:
Which does not have a additive critical point, but has three multiplicative critical points(i.e. it's geometric derivative equals 1 at three points): +/- 0.606530659712633 and 0.
The formula does have some issues, though. The first, obviously, is that it has multiple critical points. The second is that something is raised to z, which means that the fractal is bailout sensitive.
That being said, it does work:

A close up of the left side:

I'm going to try to find a function that has neither of those problems; however such a function is probably quite complicated!

[hsmyers]

Interesting! Do you mean n to be the current iteration count? And could you explain the bailout sensitivity?

--hsm

[fivex]

Interesting! Do you mean n to be the current iteration count? And could you explain the bailout sensitivity?

--hsm

Yes.
The bailout sensitivity comes from the fact that raising something to z effectively forms an infinite power tower, thus turning it into a hybrid with the highly bailout sensitive tetration fractal
This is the fractal with a bailout of 10³⁰:

And here is a close up of a region:

Can you spot the minibrot?
A much cleaner fractal with the same effect is :

If you pick the right region it can look quite nice:

[hsmyers]

Excellent! Much thanks and damn good work BTW 

--hsm

[Tglad]

Yes e^x mandelbrot set is cool isn't it, I'd like to see more. Do you know why it has discontinuities?

[hsmyers]

Not enough of a mathematician to tell you. That said, check your code for any division where the potential for zero exists in the denominator and make sure your basic equation is continuous in the given domain. Since div by zero generally causes an interrupt, dis-continuous functions would be a guess. Likewise there might be a problem with the machine accuracy for particularly if these show up at depth so to speak. Just blue sky thoughts on possible answersmaybe a more knowledgeable fractalier will chime in?

--hsm

[fivex]

Yes e^x mandelbrot set is cool isn't it, I'd like to see more. Do you know why it has discontinuities?

You get that even in slightly modified tetration fractals, too. This is what happens if you iterate :


Also, I just noticed that in the standard tetration fractal, if you turn the bailout low enough, you get structures that heavily resemble the m-set. With bailout 7:


I think that there is a deeper connection between the two fractals than we realize.

[Tglad]

I believe the connection is that the mandelbrot is universal (http://www.math.harvard.edu/~ctm/papers/home/text/papers/muniv/muniv.pdf)
It explains why it crops up inside several other fractals, including the newton fractal I think.