## Return to Question

4 edit title, tags, capitalize abbreviations

# Istheresomeknownwaytocreatethe Mandelbrot asifsset(theboundary),withaniteratedfunctionsystem?

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 ifsIFS. 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 $M.$

3 changed some words

Is there some known way to create the Mandelbrot set (the boundary), with an iterated function system?

Julia sets can be created 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 create 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 $M.$

2 better definition

Is there some known way to create the Mandelbrot set (the boundary), with an iterated function system?

Julia sets can be created by iterating the two functions $z \mapsto \pm \sqrt{z-c},$ and this quickly converges to the julia 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 create 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 $M.$

1