 Fatou set

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]