Consider the function $f:=ghg^{-1}$ on $\widetilde{\mathbb{C}}$ where $g$ is a homeomorphism and $h$ is a rational map. Why is it true that branch points of $h$ are transformed by $g$ to removeable singularities and thus to branch points of $f$.
Here's my thinking, but I'm not really confident in it:
If $p\in\widetilde{\mathbb{C}}$ is such that $h(p)$ were a branch point, then does $g$ send $h(p)$ to a removeable singularity of $h$? If so, then since $(f\circ g)(p) = (g \circ h)(p)$, then $p$ is a removeable singularity of $f \circ g$. Then, does $g$ send $p$ to a branch point of $f$?
This method might be wrong, but if not, any help explaining why these steps are true would be greatly appreciated.
For context, I am working on Sullivan's No Wandering Domain Theorem where the issue I'm having can be found on page 412 (https://www.math.stonybrook.edu/~bishop/classes/math627.S13/Sullivan-1985-Nonwandering.pdf).