Dynamical Systems/The Mandelbrot set

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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.