Let $f:\mathbb{CP}^1\rightarrow \mathbb{CP}^1$ be defined by $$f\left(\left[x:y\right]\right)=\left[x\left(cx+dy\right):y\left(ax+by\right)\right]$$ or in affine form $$f\left(x\right)=\frac{x\left(a+bx\right)}{\left(c+dx\right)}$$ where $a,b,c,d\in \mathbb{C}$ satisfy $ad-bc\neq 0$.
When do the iterates $f^k\left(x\right)$ converge? There are generally 3 fixed points: for a given fixed point $x_\ast\in \mathbb{CP}^1$ which initial $x\in \mathbb{CP}^1$ converge to $x_\ast$?
The fixed points are $\left[0:1\right],\left[1:0\right],\left[d-b:a-c\right]$.
I know that the superficially similar problem of iterates of Mobius transformations is easily solved.
Bonus: what changes if we move to $\mathbb{RP}^1$?