 Leau-Fatou flower theorem

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.