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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.

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]