# Dynamical Systems/The Mandelbrot set

**Definition (Mandelbrot set)**:

For , define . The **Mandelbrot set** is the subset of the complex plane of points which are such that the set

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 of the complex plane such that

is a bounded subset of , where is a function holomorphic in two variables, and such that

will be unbounded once the modulus of any of its elements passes a certain threshold .

**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 be a Mandelbrot-type set. Then is simply connected.

**Proof:** Suppose not. Then the complement of would have a bounded component . Take any point . Then define inductively , with ; this is the iterated function, and it is holomorphic. By assumption, there exists such that . But by the maximum principle, for some , so that . But was closed.