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5 The fundamental dichotomy

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 x² + 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 x² --- 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 x² + 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 x² + 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 x² + 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

• J_c is connected, or, equivalently
• The orbit of 0 under x² + c does not tend to infinity.

It is amazing that the orbit of 0 "knows" the shape of the filled Julia set for x² + c. The reason that 0 is so special stems from the fact that 0 is the critical point of x² + c. That is, the derivative of x² + c is 2x, and this derivative only vanishes at x=0.

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