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 $M.$