The way it is stated in that Wkipedia article, an iterated function system is a set of functions that map the space you work in (for instance

) to a smaller region of itself. I feel like that's a bit inaccurate in that, in this definition, there isn't even a mention of iteration.

As it is written it sounds like an IFS is simply a (finite) set of (contractive) functions.

Such sets provedly have a unique, nonempty, compact fixed Set.

I.e. there is a specially finite set of points, associated with a given IFS which, if you apply any function of the IFS to that set, the resulting point will again land inside the same set.

For the Buddhabrot, if I understand correctly, the corresponding associated set simply is all the points that are still visible in the anti-Buddhabrot as the number of iterations

which, if I'm not mistaken, simply is the MSet.

However that set isn't the IFS itself. It's just the unique associated set.

The IFS simply is the collection of function.

For the Buddhabrot said set simply is:

f1: x->x^2-y^2+c1

f2: y->2xy+c2

This is actually a Julia Set with fixed constants.

So if we are being super technical, the MSet's generating functions are not an IFS because the corresponding set is not finite nor even countable: they define a family of two functions in one real parameter each. Furthermore one of the two involved functions is nowhere contractive. (Though the article mentions that this condition can and in practice will be relaxed.)

Also, the family of functions in the MSet do not interact with each other so more accurately one might say that the generating function of the MSet represents an entire family of IFS with two real parameters.

None of this is actually related to how the IFS looks like or is constructed. - the IFS itself, as stated, does not look one way or another. if you want to construct the associated set, though, one way to do it is in the classic way of iterating (randomly or combinatorially to a finite order) across the set of functions. - that's in fact not how to get a Julia Set either. The two functions are always applied in parallel rather than in arbitrary order (btw how would that look like?)

One way out of this is to take the two functions and interpret them together as a single function in

but then it's hardly a function

**system** (though granted sets with one element are still sets), just a function, but the associated sets would simply be the given Julia Sets corresponding to c=c1+c2 i.

Those same sets should be constructable in an escape time manner (the set is only the region inside the usually black blobs, not the colorful thing around it) or an orbit map (anti-style) in both cases taking the iteration count to infinity.

So given that, based on the articles's definition (which, again, somehow doesn't even include iteration), no, neither the MSet nor Buddhabrot are IFS. They are particular renderings of a family of sets that are associated with an IFS with a single function over

as element.

Finally if you, for instance, take your space to be that of collections of connected line segments and your functions to be ones that map one such collection to another, you essentially recover L-Systems.