Parabolic fixed point

A fixed point $z_0$ of a holomorphic map is parabolic when its multiplier $f^{\prime }(z_0)$ is a root of unity and the corresponding return iterate is not locally the identity. In the usual rational map setting of degree of a rational map of the Riemann sphere at least two, the latter exclusion is automatic. If the multiplier has order $q$, then $f^q$ has derivative one at $z_0$ and a nonzero higher-order term.

For $f(z)=z^2+1/4$, the point $z_0=1/2$ is fixed and $f^{\prime }(z_0)=1$. Writing $z=1/2+u$ gives exactly $u\mapsto u+u^2$. Small negative real $u$ approach zero, whereas small positive real $u$ move away. The Leau-Fatou flower theorem describes the corresponding sectors in the complex plane. This behavior differs from uniform local attraction with multiplier of modulus less than one, and from an irrational rotation multiplier of modulus one.^[ref]