One of the most beautiful results in all of complex dynamics is the following theorem. This result dates back to 1919 and was proved by both G. Julia and P. Fatou. The fundamental dichotomy for x^{2} + c is: For each c-value, the filled Julia set is either a connected set or a Cantor set. A connected set is a set that consists of just one piece. It may be a simple object, like the unit disk---the filled Julia set for x^{2} --- or it may be any of the more complicated figures depicted in Figure 7. All of these filled Julia sets are connected.
On the other hand, a Cantor set consists of infinitely many pieces --- in fact, uncountably many pieces. Moreover, each piece is a point and every point in the Cantor set is a limit point of other points in this set. So a Cantor set should be visualized as a "cloud" of points---no two points touching, but infinitely many points scattered in any region around a given point.
So the fundamental dichotomy says that filled Julia sets for x^{2} + c come in one of two varieties: connected sets (one piece) or Cantor sets (infinitely many pieces). There is no in-between: there are no c-values for which J_{c} consists of 10 or 20 or 756 pieces.
How do we decide what shape a given J_{c} assumes? Amazingly, it is the orbit of 0 that determines this. For if the orbit of 0 tends to infinity under iteration of x^{2} + c, then the fact is that J_{c} is a Cantor set. On the other hand, if the orbit of 0 does not tend to infinity, then J_{c} is a connected set.
A visual way to view this dichotomy is given by the Mandelbrot set. If c lies in M, then we know that the orbit of 0 does not escape to infinity under iteration of x^{2} + c, so J_{c} must be connected. If c does not lie in M then J_{c} is a Cantor set. This dichotomy thus gives us a second interpretation of the Mandelbrot set.
The Mandelbrot set consists of all c-values for which
It is amazing that the orbit of 0 "knows" the shape of the filled Julia set for x^{2} + c. The reason that 0 is so special stems from the fact that 0 is the critical point of x^{2} + c. That is, the derivative of x^{2} + c is 2x, and this derivative only vanishes at x=0.