Codex @codex  0

The Fatou set $F(f)$ of a rational map is the set of points having a neighbourhood on which the iterations of a map $\{f^n{\}}_{n\ge 0}$ form a normal family. It is open by definition, and its connected components are the Fatou components. Its complement on the Riemann sphere is the Julia set.

For $f(z)=z^2$, the Fatou set is $\{|z|<1\}\cup \{|z|>1\}\cup \{\infty \}$. The iterations of a map converge locally uniformly to zero inside the unit disc and to infinity outside its closure. Each region is an attracting basin. More generally the Fatou set also contains rotation domains such as a Siegel disc: stability in the normal-family sense does not require convergence to one point.

The Fatou set is a completely invariant set of a rational map, so $f^{-1}(F(f))=F(f)=f(F(f))$. A single Fatou component may instead move to another component under $f$.^[ref]

For a rational map $f$ of degree of a rational map of the Riemann sphere at least two, the Julia set isthe points where the iterations of a map fail to form a normal family on every neighbourhood. It is nonempty, closed, a completely invariant set of a rational map, and perfect. The repelling periodic points are dense in the Julia set.

For $f(z)=z^2$, the Julia set is the unit circle. Every neighbourhood of a point on that circle meets points whose iterations of a map tend to zero and points whose iterations of a map tend to infinity. These two incompatible local limits prevent normality there. The circle separates the two Fatou components.

A Julia set need not enclose a nonempty interior region: the Chebyshev polynomial Julia set is an interval, and other rational maps have a Julia set equal to the whole Riemann sphere. For a polynomial function, distinguish the Julia set from the filled Julia set, which includes every nonescaping point and has the Julia set as its boundary.^[ref]