= Leau-Fatou flower theorem
{c}
Suppose a <holomorphic function> near zero has expansion
$$
f(z)=z+az^{k+1}+O(z^{k+2}),\qquad a\ne0,\quad k\ge1.
$$
The <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.<Dynamics in one complex variable: introductory lectures by John Milnor>[Section 7, parabolic fixed points and petals]{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 $z_n^k\sim-1/(ka n)$ 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.
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