Editing Julia set (section)

==Formal definition==
Let <math>f(z)</math> be a non-constant [[meromorphic function]] from the [[Riemann sphere]] onto itself. Such functions <math>f(z)</math> are precisely the non-constant [[complex number|complex]] [[rational function]]s, that is, <math>f(z) = p(z)/q(z)</math> where <math>p(z)</math> and <math>q(z)</math> are [[complex polynomial]]s. Assume that ''p'' and ''q''  have no common [[root of a polynomial|roots]], and at least one has [[degree of a polynomial|degree]] larger than 1. Then there is a finite number of [[open set]]s <math>F_1, ..., F_r</math> that are left invariant by <math>f(z)</math> and are such that:
# The [[union (set theory)|union]] of the sets <math>F_i</math> is [[dense set|dense]] in the plane and
# <math>f(z)</math> behaves in a regular and equal way on each of the sets <math>F_i</math>.

The last statement means that the termini of the sequences of iterations generated by the points of <math>F_i</math> 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 case it is ''neutral''.

These sets <math>F_i</math> are the Fatou domains of <math>f(z)</math>, and their union is the Fatou set <math>\operatorname{F}(f)</math> of <math>f(z)</math>. Each of the Fatou domains contains at least one [[critical point (mathematics)|critical point]] of <math>f(z)</math>, that is, a (finite) point ''z'' satisfying <math>f'(z) = 0</math>, or <math>f(z) = \infty</math> if the degree of the numerator <math>p(z) </math> is at least two larger than the degree of the denominator <math>q(z)</math>, or if <math>f(z) = 1/g(z) + c</math> for some ''c'' and a rational function <math>g(z)</math> satisfying this condition.

The complement of <math>\operatorname{F}(f)</math> is the Julia set <math>\operatorname{J}(f)</math> of <math>f(z)</math>. If all the critical points are preperiodic, that is they are not periodic but eventually land on a periodic cycle, then <math>\operatorname{J}(f)</math> is all the sphere. Otherwise, <math>\operatorname{J}(f)</math> is a nowhere dense set (it is without interior points) and an [[uncountable]] set (of the same [[cardinality]] as the real numbers). Like <math>\operatorname{F}(f)</math>, <math>\operatorname{J}(f)</math> is left invariant by <math>f(z)</math>, and on this set the iteration is repelling, meaning that <math>|f(z) - f(w)| > |z - w| </math> for all ''w'' in a neighbourhood of ''z'' (within <math>\operatorname{J}(f)</math>). This means that <math>f(z)</math> 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 infinitesimal 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 [[Connected component (analysis)|components]].<ref>Beardon, ''Iteration of Rational Functions'', Theorem 5.6.2.</ref> Each component of the Fatou set of a rational map can be classified into one of [[Classification of Fatou components|four different classes]].<ref>Beardon, ''Iteration of Rational Functions'', Theorem 7.1.1.</ref>