= Parabolic fixed point
A <fixed point> $z_0$ of a <holomorphic map> is parabolic when its multiplier $f'(z_0)$ 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 $f'(z_0)=1$. 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.<Dynamics in one complex variable: introductory lectures by John Milnor>[Section 7]{ref}
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