TITLE: Is there some known way to create the Mandelbrot set (the boundary), with an iterated function system?
QUESTION [18 upvotes]: Is there some known way to create the Mandelbrot set (the boundary),
with an iterated function system (IFS)?
Julia sets can be formed by iterating the two functions $z \mapsto \pm \sqrt{z-c},$
and this quickly converges to the Julia set. Formally, the two functions form a Hutchinson operator,
and the Julia set is the unique compact fixed set. 
That is, $J = f_1(J) \cup f_2(J)$ where $f_1,f_2$ are the two functions, and $J$ is the Julia set.
However, Mandelbrot is the iteration of many functions, with start value 0,
so there is no obvious way to define an IFS. Is there some non-obvious way?
That is, find maps $g_1,\dots,g_n$ s.t. the Mandelbrot $M$ can be expressed as
$M = \bigcup_i g_i(M),$ where each $g_i$ maps $M$ to some subset of $M,$ but not the entire $M.$

REPLY [1 votes]: You may take a look at Barnsley, Fractals everywhere, pp. 318f, Figures viii.237 and viii.238 (http://books.google.de/books?id=oh7NoePgmOIC&lpg=PP1&hl=de&pg=PA319#v=onepage&q&f=false). It looks not exactly like the Mandelbrot set, but there is a similarity. Unfortunately Im not able to reproduce these images (using algorithm 2.1 from page 302).
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