The Mandelbrot Set and Julia Sets

Julia Sets - Dichotomy Theorem

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).
connected Cantor set
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.

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