Geometry fo Julia sets:
In this section, we return to the quadratic map
fc(z) = z2 + c
and discuss the geometry of typical Julia sets for some chosen c values. A classification on the geometry of the Julia sets in the c-plane will be covered in the next section.

First of all, Julia sets Jc have the following geometric symmetries:

1. For any value of c, Jc is symetric about the origin, i.e.
  if z0 is in Jc , so does -z0.

Proof: z0 is in Jc implies figure 50 + c is in Jc. But figure 44.
(this proof is based on the invariance property of Julia sets.)
2. If c is real, then Jc is symmetric about the real (z) axis, i.e.
  If z0 is in Jc , so does figure 45

Proof: z0 is in kc implies figure 46 is bounded implies figure 47 is bounded.
implies figure 48 is bounded implies z* is in kk.
Thus kc is symmetric about the real (z) axis.
It follows that Jc = δkc is symmetric about real (z) axis.
3. If c is real, then Jc is symmetric about the Im(z) axis, i.e.
  If z0 is in Jc , so does figure 49

Proof: z0 is in Jc according to the first and second symmetries,
-z0 is in Jc implies figure 49 is in Jc.
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