The Fundamental Dichotomy for Julia sets
Disconnected Julia sets
Let orbit of the critical point zc = 0
(and therefore the z = c point) goes to infinity.
Concider the Julia set J(4) as an example.
We take a circle l', which goes through c .
Each point of l' has two preimages ±(z - c)½
with the exeption of z = c, which has the only preimage
zc. Therefore preimage of l' is the figure
eight curve l. Two discks D0 , D1
are mapped by fc in one-to-one fashion
onto the V disk (it containes both D0 , D1).
The Julia set J(4) is contained inside D0 ∪ D1
and is divided by the l curve in two disconnected parts.
As you can see in these pictures, one can proceed this process ad infinitum
therefore the Julia set J(4) has infinitely many components.
Connected Julia sets
If the critical orbit does not escape to infinity, then J(c)
is a connected set. You can see to the left, that f -1 maps
l' curve into l one. l' never contains z = c value,
therefore each point of l' has two different preimages
±(z - c)½ and l
can not be the figure eight curve. In a similar way any closed curve
outside the Julia set never is the figure eight curve and can not
break the Julia set.
The Fundamental Dichotomy
1. If fcn(0)→∞ ,
the filled Julia set of fc is a Cantor set.
2. Otherwise, the filled Julia set of fc is a connected
 John W. Milnor "Dynamics in One Complex Variable" § 9
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updated 26 Nov 2013