3.1: The Julia set symmetry

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The J-set is centrally symmetric as since f_c(z) = z² + c is an even function.

The Julia set J(c) is made of all points z_j, which do not go to an attractor (it may be at infinity too) under iterations. As since iterations of the points f_c(z_j) do not go to an attractor too, therefore the Julia sets are invariant under f_c.

The J(0) set

For z = r e^iφ

f_o(z) = z² = r² e^2iφ (*).

Therefore the J(0) set is the circle with the unit radius r = 1. The map f_o 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 (*) f_c 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 f_c^on 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.