Exercises

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Section 5.6 Exercises

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

$\begin{equation*} g(z) = z^m + a_{m+1}' z^{m+1}+\cdots \end{equation*}$