I got stuck on the following statement:
Let $g$ be an element of $\mathrm{Homeo}(\mathbb{S}^1)$ such that $g^m = e$ for some integer $m>1$. Pick any point $x \in \mathbb{S}^1$. Then $A_x = \{g^k(x): k \in \mathbb{Z}\}$ is finite, and $g$ permutes the components of $\mathbb{S}^1 \backslash A_x$. Therefore, $g$ is conjugate to an elliptic Mobius transformation.
$\textbf{My question is}$: Why do these conditions imply that $g$ is conjugate to an elliptic Möbius transform?
Any hints are welcome! Thanks in advance!