The Julia sets symmetry
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.
It is evident, that iterations of the points fc(zj )
do not go to an atractor too.
Therefore the Julia sets are invariant under fc .
The J-set is centrally symmetric as since
fc(z) = z 2 + c is an even function.
For z = r e iφ
the squared value is z 2 = r 2
e 2iφ. Therefore the map
fc wraps twice the complex plane z onto itself
(with quadratic deformation of r and displacement by c).
This is the simplest Julia set for c = 0 + 0i . As since
for | zo | < 1, zn converges to
the fixed point z = 0,
for | zo | > 1, zn go to infinity and
for | zo | = 1, zn rotates and
stays on the same circle | z | = 1.
The circle is the Julia set J(0).
It is evident, that the circle is invariant under fo =
The Julia sets self-similarity
Let f maps a point z1 into z2 =
f(z1). For small enough
ε it follows from the Taylor's theorem that
= z2 + f '(z1)ε +
So small neighbourhood of z1 is mapped
linearly (by scaling and rotation) into the z2 one.
Therefore the Julia set is self-similar in these regions.
As iterated preimages f o(-n)(z1) are everywhere
dense in J therefore J is self-similalar in every point.
You can trace quadratic map dynamics here.
The white square is mapped in the region with faded colors.
You see thet Julia set is similar in both regions ("faded" square
is deformed due to ~ε 2 and higher terms in the
Controls: Drag the white square by mouse to move it
(its coordinates are shown).
You can see below self-similarity of "midget" Julia sets.
It is not difficult to imagine how f-1 maps points of the
J(-1) set from the Re(z) > 0 (or Im(z) > 0) half-plane
onto the whole J(-1).
Squaring "moves" J(-1) to the right (the lower picture) and after
addition of c = -1 the Julia set returns into its original position.
Note, that the two points a are mapped into one point a'.
It is easy to see, that the "cauliflower" J(0.35) set has the same
Previous: The Mandelbrot, Julia and
Next: Critical points and Fatou theorem
updated 11 Sep 2013