Question: Find fixed points of involution g: $P_1(C) -> P_1(C)$, $g^2$ = Id, if g(2/3) = 3 and g(-2/3)= 1/4
My ideas: to use cross-ratio, maybe we can say g(3) = 2/3 and g(1/4) = -2/3 so we can count cross ratio [3,1/4,2/3,-2/3], but I don't know it can help with finding fixed points
An isomorphism of $P^1_\mathbb{C}$ is a 2x2 invertible matrix modulo $\mathbb {C}^*$. The constraints you have give 4 equations : \begin{align} g(3)=2/3\\ g(2/3)=3\\ g(-2/3)=1/4\\ g(1/4)=-2/3\end{align} These should be what you need to determine a representative of $g$ as a matrix. Then, you will have to find the eigenvectors of your matrix.