The Fundamental Dichotomy for Julia sets

Connected Julia sets

If the critical orbit does not escape to infinity, then J(c) is a connected set. This occurs since closed curves outside the Julia set never contain the critical value, and therefore their preimage is never a figure eight.
It is evident that for large enough z any quadratic map fc(z) is conjugate ("very similar", see [1,2] for rigorous math :) to the squaring function fo(z) = z2. We have seen that fo(z) and its reverse function map circles into circles, therefore conjugate function fc-1(z) = +-(z - c)1/2 maps circle l' into oval l . One can extend this conjugacy up to the z = c point (which has only one preimage, namely 0 , whereas any other point has two distinct preimages). So if c lays within Julia set, it follows that we can continue this process indefinitely to the entire basin of attraction of infinity (and a simple "similar to a circle" loop is mapped by fc-1(z) into one more simple loop). This basin is simply connected, therefore its compliment is a closed, connected set. But this compliment is just the filled Julia set.

Disconnected Julia sets

If the critical point zc = 0 (and therefore the z = c point) goes to infinity, a very different topological picture emerges. We take a circle l', which goes through the c . Each point of l' has two preimages +-(z - c)1/2 (e.g. for c = 4 the point z = -4 has z0,1 = +-81/2i) with the exeption of z = c , which has the only zc preimage. So the 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 D0È D1 and has infinitely many connected components.
To see this [1], let F0 , F1 denote the two branches of fc-1 defined on D0È D1, with Fj taking values in Dj . Since fc|Dj is one-to-one, both F0 , F1 are analitic. For example F0 maps D0 in D00 and D1 in D01 and so on ad infinitum.

Now let s = s0s1... be an infinite sequence of 0's and 1's. We may form the infinite composition
    Fs0s1...(z) = limn®¥ Fs0 o Fs1 o ... o Fsn(z)
for any z in D0 È D1 . This limit point must lie in J(c) and
    Fs0s1...(z) Î Ds0
    fc(Fs0s1...(z)) = Fs1s2...(z) Î Ds1
    fcn(Fs0s1...(z)) = Fsn...(z) Î Dsn ,

As a consequence, if two sequences disagree somewhere, say in the nth spot, then the corresponding limit points lie in different components of J(c) (fcn of one of these limit points lies in D0 the other in D1). Thus the Julia set consists of uncountably many components. Moreover, each component is a single point, that is Fs0s1...(z) is independent of z . This follows from the fact that both F0 , F1 are strict contractions on D0 , D1 . So the distance between Fs0 o ... o Fsn(z1) and Fs0 o ... o Fsn(z2) decreases by a definite factor at each additional iteration. Consequently J(c) is totally disconnected. It is also closed and perfect. So J(c) is a Cantor set when fcn(0)®¥ . This then gives us one of the major results in quadratic dynamics:

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] Robert L. Devaney Complex Complex Saddle-Node Bifurcations § 2
[2] John W. Milnor "Dynamics in One Complex Variable" § 9

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updated 24 November 2002