TITLE: Mandelbrot-like sets for functions other than $f(z)=z^2+c$?
QUESTION [33 upvotes]: Are there any well-studied analogs to the Mandelbrot set using functions other than $f(z)= z^2+c$ in $\mathbb{C}$?

REPLY [3 votes]: I couldn't say anything about "well-studied". What I do know is this:

You can grab any random function you like and iterate it. Generally, the results aren't particularly interesting.
You seem to get the most "natural" looking images (i.e., the ones most like the usual Mandelbrot and Julia sets) if you stick to nice, well-behaved complex-valued functions.
If you want to be picky about it, the Mandelbrot set really ought to be the set of parameter values with connected Julia sets. There's some theorem about every basin of attraction containing on critical point or something like that, but I don't really understand the details. The important point is, the parameter-space image tends to "look nicest" when you use critical points of the interation.
It's quite possible for the function to have more than one parameter. This doesn't change the Julia sets much (except that there are more of them), but it makes the Mandelbrot set have more than 2 dimensions.

My personal favourite is the cubic function $f(z) = z^3 - 3a^2z + b$. This has two critical points, $+a$ and $-a$. The Julia sets have 3-fold symmetry, and aren't especially stunning. The Mandelbrot set, however, is strictly speaking the intersection of two sets, one iterating with $z_0 = +a$ and the other $z_0 = -a$. If you combine the colourings of these two iterations, you get a strange "shadowy" effect. On top of that, the Mandelbrot set is 4D, and plotting various 3D slices of it looks interesting.