Subsection4.4.1Parabolic points

A parabolic fixed point is a fixed point with the property that the multiplier \(\lambda\) has the form \(\lambda = e^{2\pi\alpha i}\) where \(\alpha\) is rational number.

Theorem4.4.4

Suppose that \(f\) has the form

\begin{equation*} f(z) = z + a z^{n+1} + O(z^{n+2}). \end{equation*}

Then zero is a fixed, parabolic point in the Fatou set of \(f\text{.}\) Suppose also that \(v\) is a solution of

\begin{equation*} nav^n=1. \end{equation*}

Then, \(f\) is repulsive in the direction \(v\text{.}\) Suppose, on the other hand that \(v\) is a solution of

\begin{equation*} nav^n=-1. \end{equation*}

Then, \(f\) is attractive in the direction \(v\text{.}\)