Some Facts about Julia Sets

For the family Qc(z), the orbit of 0 is extremely important.

If the orbit of 0 is bounded, then the Filled Julia Set is one piece.

But if the orbit of 0 escapes to infinity, then the Filled Julia Set is topologically the same as the Cantor Ternary Set, i.e. all real numbers between 0 and 1 which have at least one ternary expansion which contains only 0s and 2s.

Some facts about the Cantor Ternary Set are:

The Cantor Ternary Set is perfect, i.e. a point is in the Cantor Ternary Set if and only if it is a limit point of the Cantor Ternary Set.
The Cantor Ternary Set is nowhere dense, i.e. it contains no intervals.
By property 2, the Cantor Ternary Set has an empty interior.
The Cantor Ternary Set is of Lebesgue outer measure zero, i.e. it can be contained in a countable collection of open intervals of arbitrarily small total length.
The Cantor Ternary Set is uncountable with cardinality the same as the reals.

So either the Julia Set is one peice or infinitely many totally disconnected peices.

This is known as the Fundamental Dichotomy.

The reason that 0 is so important is because for any c, the only critical point of Qc(z) is z = 0. Recall that a critical point of a function is one at which the first derivative vanishes.

The definition I have given for the Julia Set is specific to Qc(z). For entire transcendental functions, points whose orbits escape to infinity are in the Julia Set.

Examples of entire transcendental functions are the complex exponential, lambda e^z, the complex sine, lambda sin(z), and the complex cosine, lambda cos(z).

For these families, there exists the phenomenon of exploding Julia Sets. For each of these functions, there are certain numbers such that if the magnitude of lambda is larger than this number, the Julia Set explodes.