= Attracting petal
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,
$$
f(z)=z+az^{k+1}+O(z^{k+2}),\qquad a\ne0,
$$
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 $az^k$ 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>.<Dynamics in one complex variable: introductory lectures by John Milnor>[Section 7, petal definition, Theorem 7.2 and Corollary 7.4]{ref}
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