# The Fundamental Dichotomy for Julia sets

## Disconnected Julia sets

Let orbit of the critical point *z*_{c} = 0
(and therefore the *z = f(0) = c* point) goes to infinity.
Consider the Julia set *J(4)* as an example.
We take a circle *l'*, which goes through *z = c = 4* .
Each point of *l'* has two preimages *±(z - c)*^{½}
with the exeption of *z = c*, which has the only preimage
*z*_{c}. Therefore preimage of *l'* is the figure
eight curve *l*. The two disks *D*_{0 }, D_{1}
are mapped by *f*_{c} in one-to-one fashion
onto the *V* disk (it contains both *D*_{0} , D_{1}).
The Julia set *J(4)* is contained inside *D*_{0} ∪ D_{1}
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 a figure eight curve. In a similar way any closed curve
outside the Julia set never is a figure eight curve and can not
break the Julia set.
## The Fundamental Dichotomy

1. If *f*_{c}^{n}(0)→∞ ,
the filled Julia set of *f*_{c} is a Cantor set.

2. Otherwise, the filled Julia set of *f*_{c} is a connected
set.
[1] *John W. Milnor* "Dynamics in One Complex Variable" § 9

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*updated* 26 Nov 2013