Julia Set - Formal Definition

Formal Definition

Let be a complex rational function from the plane into itself, that is, where and are complex polynomials. Then there are a finite number of open sets, that are left invariant by and are such that:

1. the union of the 's is dense in the plane and
2. behaves in a regular and equal way on each of the sets .

The last statement means that the termini of the sequences of iterations generated by the points of are either precisely the same set, which is then a finite cycle, or they are finite cycles of circular or annular shaped sets that are lying concentrically. In the first case the cycle is attracting, in the second it is neutral.

These sets are the Fatou domains of, and their union is the Fatou set of . Each of the Fatou domains contains at least one critical point of, that is, a (finite) point z satisfying, or z = ∞, if the degree of the numerator is at least two larger than the degree of the denominator, or if for some c and a rational function satisfying this condition.

The complement of is the Julia set of . is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers). Like, is left invariant by, and on this set the iteration is repelling, meaning that for all w in a neighbourhood of z (within ). This means that behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and they make up an infinitely small part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.

There has been extensive research on the Fatou set and Julia set of iterated rational functions, known as rational maps. For example, it is known that the Fatou set of a rational map has either 0,1,2 or infinitely many components. Each component of the Fatou set of a rational map can be classified into one of four different classes.