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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]

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 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]