Suppose that $z_0$ in $\mathbb{C}$ is a fixed point of an analytic function of $F$. Then $z_0$ is:
- Super-attracting if $f'(z_0)=0$
- Attracting if $0<|f'(z_0)|<1$
- Repelling if $|f'(z_0)|>1$
- Rationally indifferent if $f'(z_0)$ is a root of unity
- irrationally indifferent if $|f'(z_0)|=1$, but $f'(z_0)$ is not a root of unity
These are the two questions:
(a) Show that $z \to z^2$ has no indifferent fixed points.
(b) Show that $z\to z+z^2$ has no repelling fixed points.
I understand how to determine what fixed points these functions have, but I'm not sure how one would go about showing that a particular function does NOT have a certain type of fixed point. My idea is below:
(a) Suppose $f(z)=z^2$ had a rationally indifferent fixed point. Then there exists a fixed point $z_0$ such that $z_0^2-z_0=0$ and $2z_0=e^{2\pi i k}$ or $z_0=\frac{e^{2pi i k}}{2}$. I believe this shows that the fixed point would have to be $\frac{1}{2}$, which does not work.
(b) Suppose it had a repelling fixed point. Then there exists a $z_0$ such that $z_0^2+z_0=z_0$ and $|2z_0+1|>1$. Eventually this shows that it has a large set fixed point, none of which actually work.
Am I on the right track with these problems?