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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]

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]

The parabolic basin $A$ is open and lies in $F(g)$. Suppose $z\in \partial A\cap F(g)$ and let $U$ be the Fatou component containing $z$. A small neighborhood of $z$ in $U$ meets $A$. On one point of $U$, the iterates eventually enter the chosen attracting petal and converge to zero along its attracting vector. Normality and the identity theorem for the limiting iterates make the same true throughout the connected component $U$. Hence $U\subseteq A$, which is impossible for a boundary point. Therefore the parabolic basin satisfies