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An attracting petal is a forward-invariant region approaching a parabolic fixed point of a holomorphic map. For a local map fixing zero with multiplier one,one can choose a small connected open region $U$, with zero on its boundary, such that $f(\overline{U})\subseteq U\cup \{0\}$ and the iterated images of $\overline{U}$ shrink to zero. In particular, every orbit starting in $U$ tends to the fixed point.

The Leau-Fatou flower theorem gives $k$ attracting petals, alternating with $k$ repelling petals. The leading directions are determined by $a{z}^k$ being negative real. For $f(z)=z+z^2$, negative real points sufficiently close to zero move monotonically toward zero; the attracting petal surrounds that direction. This attraction comes from the nonlinear term, since the derivative at the fixed point has modulus one.

For a parabolic cycle whose multiplier is another root of unity, the petal construction is applied to a suitable return iterate with multiplier one. Points whose orbits eventually enter an attracting petal form its parabolic basin.^[ref]

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]

In the quadratic polynomial family $f_c(z)=z^2+c$, a hyperbolic component is a connected component of the set of parameters for which $f_c$ has an attracting periodic point. The finite critical orbit is then attracted to that cycle. Its exact period is constant throughout the component. The component is open because an attracting periodic point and its strict multiplier inequality persist under small parameter changes.

The main cardioid has an attracting periodic point of period one; the period-two bulb has an attracting exact two-cycle. Their periodic-orbit multipliers give the explicit coordinates $c=\lambda /2-{\lambda }^2/4$ and $\lambda =4(c+1)$, respectively, with $|\lambda |<1$. More generally the attracting-cycle multiplier gives a conformal coordinate on each quadratic polynomial hyperbolic component.

This is a definition within the specified parameter family. For general rational maps, the existence of one attracting cycle alone does not require every critical orbit to approach an attracting cycle, so it does not by itself establish global hyperbolicity.^[ref]

John Milnor works across topology, geometry and dynamical systems. His expositions connect local calculations with global structure: Morse Theory by John Milnor develops the topology of a manifold through critical points, while his complex-dynamics lectures study iteration of holomorphic maps.^[ref][ref]

Near a nondegenerate critical point, the Morse lemma reduces a smooth function exactly to a signed quadratic form in suitable coordinates. Its index controls the topological change in sublevel sets. With James D. Stasheff, Milnor also wrote Characteristic Classes by John Milnor and James D. Stasheff, developing invariants of vector bundles.

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]

Let $f$ be a rational map of degree of a rational map of the Riemann sphere at least two. Its nonempty Julia set $J$ is infinite: a finite completely invariant closed set of a rational map lies in the Fatou set. Since $J$ is compact, its set $J^{\prime }$ of accumulation points is nonempty. It is closed, contained in $J$, and completely invariant; local openness of a nonconstant holomorphic map gives the inverse-image direction.

Apply the completely invariant closed set of a rational map classification to $J^{\prime }$. Its alternative with at most two points would put the nonempty set $J^{\prime }$ in the Fatou set, contradicting $J^{\prime }\subseteq J$. Hence $J\subseteq J^{\prime }$, so $J=J^{\prime }$ and the Julia set has no isolated points. This supplies the perfect property without assuming it in advance.^[ref]

Suppose a holomorphic function near zero has expansionThe Leau-Fatou flower theorem gives $k$ attracting petals for forward iteration and $k$ repelling petals for the local inverse, alternating around the parabolic fixed point. They can be chosen so that together with zero they cover a neighborhood. Within an attracting petal, every forward orbit tends to zero along its attracting direction.^[ref]

The change of variable $w=-1/(ka{z}^k)$ explains the geometry: substitution gives $w(f(z))=w(z)+1+o(1)$. Iteration therefore behaves like translation far out in a half-plane, and the $k$ inverse branches produce the $k$ petals. In particular $z_n^k\sim -1/(kan)$ along an attracting orbit. For $f(z)=z+z^2$, small negative real points approach zero like $-1/n$, while small positive real points move away. A derivative equal to one hides dynamics that the first nonlinear term reveals.

A family $\mathcal{F}$ of meromorphic functions on a domain is a normal family if every sequence from $\mathcal{F}$ has a subsequence converging uniformly on every compact subset, measured in the chordal metric on the Riemann sphere. The limit is a meromorphic function or the constant infinity. For holomorphic functions, a finite limit is holomorphic; a sequence may also tend locally uniformly to infinity.

The constant functions $f_n(z)=n$ form a normal family, because any sequence has either a constant subsequence or a subsequence tending uniformly to infinity in the chordal metric. In contrast $g_n(z)=nz$ is not a normal family on any neighbourhood of zero. Every subsequence tends to infinity at every nonzero point but remains zero at zero; this pointwise pattern cannot be a continuous Riemann sphere valued limit.

For a rational map $f$, the Fatou set is precisely where its iterations of a map form a normal family. Normality asks for convergent subsequences, so it allows stable rotations as well as convergence to an attracting periodic point.^[ref]

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 a rational map of degree of a rational map of the Riemann sphere at least two, a parabolic cycle is a cycle of a holomorphic map whose periodic-orbit multiplier is a root of unity. If the exact period is $p$ and the multiplier has order $q$, the return iterate $f^{pq}$ has multiplier one. In a local coordinate $u$ centred at a point of the cycle it has the formThe nonzero nonlinear term gives attracting petals and repelling petals, with the former underlying a parabolic basin. A multiplier of modulus one with irrational argument is not parabolic.

For $f_c(z)=z^2+c$ at $c=1/4$, the fixed point $z=1/2$ has multiplier one. The translation $z=1/2+u$ gives the exact local map $u\mapsto u+u^2$, exposing the nonlinear term responsible for attraction from one direction and repulsion from another. The parabolic cycle belongs to the Julia set even though nearby points in its parabolic basin approach it.^[ref]

A fixed point $z_0$ of a holomorphic map is parabolic when its multiplier $f^{\prime }(z_0)$ is a root of unity and the corresponding return iterate is not locally the identity. In the usual rational map setting of degree of a rational map of the Riemann sphere at least two, the latter exclusion is automatic. If the multiplier has order $q$, then $f^q$ has derivative one at $z_0$ and a nonzero higher-order term.

For $f(z)=z^2+1/4$, the point $z_0=1/2$ is fixed and $f^{\prime }(z_0)=1$. Writing $z=1/2+u$ gives exactly $u\mapsto u+u^2$. Small negative real $u$ approach zero, whereas small positive real $u$ move away. The Leau-Fatou flower theorem describes the corresponding sectors in the complex plane. This behavior differs from uniform local attraction with multiplier of modulus less than one, and from an irrational rotation multiplier of modulus one.^[ref]

A repelling petal for a holomorphic map at a parabolic fixed point is an attracting petal for its local inverse. The inverse exists near the fixed point because the multiplier is nonzero. Backward iterates remain in the petal and approach the fixed point, whereas a nonzero point followed forward eventually leaves the small repelling region.

For $f(z)=z+a{z}^{k+1}+O(z^{k+2})$, its inverse begins $f^{-1}(z)=z-a{z}^{k+1}+O(z^{k+2})$. Repelling directions for $f$ therefore have $a{z}^k$ positive real, alternating with the attracting directions. The Leau-Fatou flower theorem provides $k$ petals of each kind.

For $f(z)=z+z^2$, small positive real points increase under forward iteration, while the inverse branch $f^{-1}(z)=(-1+\sqrt{1+4z})/2$ moves them toward zero. This verifies the direction of repulsion without treating multiplier one as an expanding linear map.^[ref]