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
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
To see this , let F0 , F1 denote the two
branches of fc-1 defined on
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.
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
and has infinitely many connected components.
Now let s = s0s1...
be an infinite sequence of 0's and 1's. We may form the infinite composition
limn®¥ Fs0 o
Fs1 o ... o Fsn(z)
for any z in D0 È D1 . This limit
point must lie in J(c) and
fc(Fs0s1...(z)) = Fs1s2...(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
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
 Robert L. Devaney
Saddle-Node Bifurcations § 2
 John W. Milnor "Dynamics in One Complex Variable" § 9
Previous: Critical points and Fatou theorem
Next: The fixed points and periodic orbits
updated 24 November 2002