Definition (Mandelbrot set):
For , define . The Mandelbrot set is the subset of the complex plane of points which are such that the set
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.