# The Fundamental Dichotomy for Julia sets

## Disconnected Julia sets

Let orbit of the critical point zc = 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 zc. Therefore preimage of l' is the figure eight curve l. The two disks D0 , D1 are mapped by fc in one-to-one fashion onto the V disk (it contains 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 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 fcn(0)→∞ , the filled Julia set of fc is a Cantor set.
2. Otherwise, the filled Julia set of fc is a connected set.

[1] John W. Milnor "Dynamics in One Complex Variable" § 9

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updated 26 Nov 2013