1
Suppose that \(f\) has a fixed point at \(z_0\text{.}\) Show that the function \(g\) obtained by conjugating \(f\) with the function \(\varphi(z)=z+z_0\) has a fixed point at zero. In addition, show that the conjugation preserves the nature of the fixed point as attractive, super-attractive, repulsive, or neutral.
2
Suppose that
\begin{equation*}
f(z) = a_m z^m + a_{m+1} z^{m+1}+\cdots,
\end{equation*}
so that \(f\) has a super-attractive fixed point of order \(m\) at zero. Show that the function \(g\) obtained by conjugating \(f\) with the function \(\varphi(z)=a_m^{1/(m-1)}z\) conjugates \(f\) to a function \(g\) of the form
\begin{equation*}
g(z) = z^m + a_{m+1}' z^{m+1}+\cdots
\end{equation*}