Properties of Julia Sets:
In this section, we discuss some general dynamical properties of Julia sets for polynomial maps. Of course, these properties apply to the quadratic map ƒc as a special case.

Suppose P(z) is a polynomial function, then its Julia set J(P) has the following properties:

1. Let P be a polynomial of degree ≥ 2. Then J(P) ≠ empty set, i.e. nonempty.
2. J(P) subset K(P) where K(P) is the filled in Julia set of P.
3. J(P) = J(P n), where n is a positive integer.
4. J(P) is completely invariant, i.e. if z is in J(P), then P(z) and P-1(z) are also in J(P).
5. Let z0 is in J(P), then
figure 42
i.e. J(P) is the closure of all the preimages z0
6. J(P) is the closure of the set of repelling periodic points of P.
7. J(P) has empty interior.
8. P is chaotic J(P).

Note: Property 5 is often used to plot the Julia sets graphically, and property 6 is often usedas an alternative definition for the Julia set.

Most of the above properties on Julia sets can be extended to rational maps. Refer to Devaney (1986) and Devaney (1994) for more details.

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