The number of pieces of a Julia set is severely limited: |
Dichotomy Theorem The Julia set Jc of z2 + c
is either connected (one piece) or totally disconnected (infinitely many pieces, a
distorted Cantor set). |
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This is a consequence of a theorem proved in
1918-19 by Pierre Fatou and Gaston Julia. |
This theorem contains a cautionary lesson about interpreting computer
graphics. The middle and right pictures are magnifications of the picture to
their left. |
|
In the third picture we see the filled-in Julia set contains a solid piece, so the Julia set
is not a Cantor set. |
Consequently, it must be connected. That is, despite appearances, the colorful filaments
contain very small black dots attaching all parts of the Julia set togther. |
Yet to the eye, the black regions appear to be isolated. |
In the next section,
we shall see the Dichotomy Theorem is the
foundation of the definition of the Mandelbrot set. |