# Dynamical Systems/The Mandelbrot set

Definition (Mandelbrot set):

For ${\displaystyle c\in \mathbb {C} }$, define ${\displaystyle f_{c}(z):=z^{2}+c}$. The Mandelbrot set is the subset ${\displaystyle M}$ of the complex plane ${\displaystyle \mathbb {C} }$ of points ${\displaystyle c}$ which are such that the set

${\displaystyle \{f_{c}(0),f_{c}(f_{c}(0)),f_{c}(f_{c}(f_{c}(0))),\ldots \}\subset \mathbb {C} }$

is bounded.

Here is a picture of the Mandelbrot set:

This set motivates the following more precise definition:

Definition (Mandelbrot-type set):

A Mandelbrot-type set we define to be as the set of points ${\displaystyle c}$ of the complex plane such that

${\displaystyle \{f(c,0),f(c,f(c,0)),f(c,f(c,f(c,0))),\ldots \}}$

is a bounded subset of ${\displaystyle \mathbb {C} }$, where ${\displaystyle f:\mathbb {C} \times \mathbb {C} \to \mathbb {C} }$ is a function holomorphic in two variables, and such that

${\displaystyle \{f(c,0),f(c,f(c,0)),f(c,f(c,f(c,0))),\ldots \}}$

will be unbounded once the modulus of any of its elements passes a certain threshold ${\displaystyle C>0}$.

Proposition (mandelbrot-type set is bounded):

Proposition (mandelbrot-type set is closed):

Proposition (mandelbrot-type set is compact):

Proposition (mandelbrot-type set is simply connected):

Let ${\displaystyle M}$ be a Mandelbrot-type set. Then ${\displaystyle M}$ is simply connected.

Proof: Suppose not. Then the complement of ${\displaystyle M}$ would have a bounded component ${\displaystyle A\subset \mathbb {C} }$. Take any point ${\displaystyle c_{0}\in {\overset {\circ }{A}}}$. Then define inductively ${\displaystyle \phi _{n}(c):=f(c,\phi _{n-1}(c))}$, with ${\displaystyle \phi _{0}(c)=0}$; this is the iterated function, and it is holomorphic. By assumption, there exists ${\displaystyle n}$ such that ${\displaystyle |\phi _{n}(c_{0})|>C}$. But by the maximum principle, ${\displaystyle |\phi _{n}(c_{1})|>C}$ for some ${\displaystyle c_{1}\in \partial A}$, so that ${\displaystyle c_{1}\in \partial A\cap M^{c}}$. But ${\displaystyle M}$ was closed. ${\displaystyle \Box }$