Fatou set ($F(f)$)

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]

A Fatou component is a connected component of the Fatou set of a rational map. It is open. A rational map sends each Fatou component onto another one: complete invariance of the Fatou set puts it inside the inverse image of the target component; its maximal connectedness makes it one full inverse-image component. Restricting the globally proper sphere map to such a component is proper, since the inverse image of a compact subset intersected with the component is closed in a compact set. Its image is both open and closed in the connected target component, hence the whole target. A component is periodic if some iterate returns it to itself.

A Herman ring is a doubly connected periodic Fatou component on which the first-return map is holomorphically conjugate to an irrational rotation of an annulus $r<|w|<1$. Compactness of the circle of rotation factors proves normality of the return iterates, and the finitely many intervening iterates preserve normality. It is the annular rotation-domain counterpart of a Siegel disc.

A Siegel disc is a simply connected periodic Fatou component on which the first-return map is holomorphically conjugate to an irrational rotation of the unit disc: $h\circ R^p\circ h^{-1}(w)=e^{2\pi i\theta }w$, with $\theta \notin \mathbb{Q}$. Its return iterates are normal because any sequence of rotation factors has a convergent subsequence in the unit circle, giving locally uniform limits after conjugation. The iterates generally do not converge to a constant. This is a rotation domain, illustrating stability without attraction.

Suppose a Fatou component $U$ satisfies $R(U)\subseteq U$, and some $a\in U$ has $R^{-1}(a)\subseteq U$. Every connected component of $R^{-1}(U)$ maps properly and surjectively to $U$: properness follows by intersecting compact preimages with the relatively closed component; openness and closedness then give surjectivity. Such a component must meet the entire fiber over $a$, which lies in $U$. Since $U$ is itself one inverse-image component, there can be no other. Hence $R^{-1}(U)=U$ and $R(U)=U$, proving complete invariance.

A periodic Fatou component of a rational map is an immediate attracting basin, an immediate parabolic basin, a Siegel disc, or a Herman ring. Attracting and parabolic basins capture critical orbits, while the boundaries of rotation domains lie in the closure of the postcritical set.

Every immediate basin of an attracting fixed point of a rational map contains a critical point. Otherwise the map on the basin would be an unbranched covering; lifting it to the unit disc would give an automorphism fixing a point, contradicting strict contraction at the attracting fixed point.

With the closed-set convention, the postcritical set of a rational map $f$ iswhere $\mathrm{Crit}(f)$ is the set of critical points of a rational map. Thus one starts at the critical values, takes all their forward orbits, and then takes the closure. Some authors omit the closure; stating the convention matters when a critical orbit accumulates without becoming periodic.

For $f(z)=z^2-1$, the critical points of a rational map are zero and infinity. Their forward images give $P(f)=\{-1,0,\infty \}$, because $0\mapsto -1\mapsto 0$ and infinity is fixed. The postcritical set combines all critical orbits, rather than recording only one chosen starting point.

A parabolic basin consists of points that eventually enter an attracting direction of a parabolic cycle. For a suitable return iterate, an attracting petal is a forward-invariant open region with the periodic point on its boundary, and the return iterates tend to that point there. The union of its inverse images is an open parabolic basin contained in the Fatou set; the Fatou components meeting the attracting petals form the immediate basin. Its boundary lies in the Julia set.

The parabolic cycle itself and points that land exactly on it belong to the Julia set, so mere pointwise convergence to that cycle is not the definition of the open parabolic basin. For $f(z)=z+z^2$, zero is a fixed point with periodic-orbit multiplier one. Small negative real starting values approach zero without reaching it, whereas zero itself is a Julia set point. The reflected example parabolic disk for z minus z squared gives an explicit forward-invariant region and a quantitative convergence bound.^[ref]

For $P(z)=z-z^2$, the disc $\Delta =\{|z-1/2|<1/2\}$ corresponds under $w=1/z$ to $\Re w>1$. The transformed map is $T(w)=w+1+1/(w-1)$, so $\Re T(w)>\Re w+1$. Thus $P(\Delta )\subseteq \Delta$ and $|P^n(z)|<1/(n+1)$ there. The Fatou component $U$ containing the disc has $P^n\to 0$ locally uniformly by normality and the identity theorem. It is forward invariant. The point $1/4$ in the disc has the entire fiber $\{1/2\}$, so the Fatou component containing an entire fiber is completely invariant criterion gives $P^{-1}(U)=U$. Zero itself is a Julia point: a normal subsequential limit near zero would be zero by the disc limit, but every iterate has derivative one at zero, contradicting derivative convergence.

Every Fatou component of a rational map of degree at least two is eventually periodic. Thus the periodic-component classification accounts for the entire Fatou set.