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 D₀, D₁ are mapped by f_c 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 f_cⁿ(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