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 y = fc(zj) do not go to an atractor too. Therefore the Julia sets are invariant under fc .
The J-set is centrally symmetric since fc(z) = z 2 + c is an even function.

For   z = r e if the squared value is   z 2 = r 2 e 2if. 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 = z 2.

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
    f(z1+ε) = f(z1) + f '(z1)ε + O(ε 2) ~ z2 + f '(z1) ε.
Sosmall 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.

In the applet below you can trace quadratic map dynamics. The white square is mapped in the region with inverted colors. You see thet Julia set is similar in both regions ("inverted" square is deformed due to ~ε 2 and higher terms in the Taylor's formula).

Controls: Drag the white square by mouse to move it. Z are coordinates the white square center, R is its size, dz is the scale of the visible region. As ususal press <Enter> to set new parameters values. E.g. set Cr=Ci=0 to test J(0) dynamics.

You can see below self-similarity of "dendrite" and "midgets" Julia sets.

More examples


 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'.

Moreover for c = -1.77289 the biggest J(-1) midget located at z = 0 (to the right below) in a similar way is mapped (twice) into a small one (in the white square to the left). Renormalization theory and the Julia midgets scaling will be discussed later.

It is easy to see, that the "cauliflower" J(0.35) set has the same "squaring" symmetry.

Two pictures below illustrate the squaring transformation for the Douady rabbit. The line R-R cuts the creature in two symmetric halfs and each part is mapped by fc into the whole J-set. Two points R pass into R' one. A "unit" square is used instead of a circle.

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updated 27 Apr 2006