I need to find the fixed points of the following:
dilation on $\mathbb{C}_{\infty}$
translation on $\mathbb{C}_{\infty}$
inversion on $\mathbb{C}_{\infty}$
I'm thinking using Möbius transformations, I know $S(z) = z$ would be a fixed point for a Möbius transformation $S$, and $S(z) = za$ would be a dilation on it, with $z,a \in \mathbb{C}$.
We know $S(z) = z$ gives a fixed point.
Suppose $S(z) = az$ is a dilation with $a \neq 1$, then we can see only $z = 0$ and $z = \infty$ are points mapped to themselves.
Suppose $S(z) = z + a$ is a translation with $a \neq 0$, then we can see only $z = \infty$ is mapped to itself.
Suppose $S(z) = \frac{1}{z}$ is an inversion, then we can see only $z = 1$ is mapped to itself.
Am I on the right track?