The Julia set symmetry
The J-set is centrally symmetric as since
fc(z) = z 2 + c is an even function.
The Julia set J(c) is made of all points zj ,
which do not go to an attractor (it may be at infinity too)
under iterations. As since iterations of the points
fc(zj ) do not go to an attractor too,
therefore the Julia sets are invariant under fc .
The J(0) set
For z = r e iφ
fo(z) = z 2 = r 2
e 2iφ (*).
Therefore the J(0) set is the circle with the unit radius
r = 1.
The map fo wraps twice the circle onto itself
and is similar to the Sawtooth map.
Therefore unstable orbits are everywhere dense on the circle.
Preimages of an unstable orbit are everywhere dense too.
The Julia set self-similarity
In accordance with (*) fc maps one half
of the Julia set onto the whole set. Moreover the whole Julia set
can be obtained from its any small piece by the map
fcon for finite n.
Therefore the Julia set is self-similar.
You can trace quadratic map dynamics here.
The white square is mapped in the region with faded colors.
You see that Julia set is similar in both regions.
You can test by hand that any merging point of three bulbs is
a preimage of the unstable fixed point z1
and these preimages are dense in the set.
Controls: Drag the white square to move it
(its coordinates are shown).
You can see self-similarity of "midget" Julia sets.
The "cauliflower" J(0.35) set is self-similar too.
Previous: Iterations of inverse maps
Next: Critical points and Fatou theorem
updated 4 Jan 2014