The Mandelbrot Set and Julia Sets

The Role of Critical Points

In the Dichotomy Theorem for Julia sets obtianed by iterating a polynomial function f(z) we observed
    the Julia set is connected if the orbits of all critical points remain bounded, and
    the Julia set is a Cantor set if the orbits of all critical points escape to infinity.
For polynomials having a single critical point, f(z) = zn + c for example, Julia sets must be either connected or Cantor sets, and the Mandelbrot set is the collection of all c for which the Julia set is connected.
On the other hand, the function f(z) = z2/2 + z3/3 + c has two critical points, z = 0 and z = -1, so we have several possible definitions for the analog of the Mandelbrot set.
To illustrate the greater complexity that can arise for functions having several critical points, consider the polynomial f(z) and look at the Mandelbrot set consisting of those c for which the iterates of z = 0 remain bounded.
Zoom in on the red boxClick the picture for an additional zoom, revealing that the black spot is a (quadratic!) Mandelbrot set.
Here is the Julia set for the point selected in the above right picture. Here is a magnification of a portion of the Julia set.
This Julia set is not connected, yet neither is it a Cantor set: substantially increasing the maximum number of iterations Does not alter the filled-in balck regions of the Julia set.

Return to the Mandelbrot set definition.