the features of the action of $\Gamma = \langle \gamma \rangle$ on $\mathbb{C}\mathbb{P}^1$ Let $\gamma$ be an Elliptic element of ${\rm PSL}(2,\mathbb{C})$ representing an Irrational rotation.
Let $\Gamma$ be the subgroup of ${\rm PSL}(2,\mathbb{C})$ generated by $\gamma$ (i.e., $\Gamma = \langle \gamma \rangle$).
Let $\mathbb{C}\mathbb{P}^1$ be the Complex Projective Line.
Suppose the group $\Gamma$ acting on $\mathbb{C}\mathbb{P}^1$.
What are the main features (properties) of the action of the group $\Gamma$ on $\mathbb{C}\mathbb{P}^1$?
 A: I'll elaborate on my comment, to turn it into an answer.
First, the action of $\gamma$ on $\mathbb CP^1$ has two fixed points $P,Q \in \mathbb CP^1$, which one can easily seen by solving the equation $Mz=z$ where the matrix $M \in \text{SL}(2,\mathbb C)$ represents $\gamma \in \text{PSL}(2,\mathbb C)$.
Next, there exists an element $\delta \in \text{PSL}(2,\mathbb C)$ taking those two points $P$ and $Q$ to $0$ and $\infty$. When you conjugate $\gamma$ by that element, the result is an element $\gamma_1 = \delta\gamma\delta^{-1}$ of the form $\gamma_1(z) = e^{2 \pi i r} z$ or $e^{-2 \pi i r} z$.
What else is there to say? ... The map $\gamma'$ preserves the family of  circles centered on the origin; those circles limit down to $0$ in the inward direction and $\infty$ in the outward direction, acting as an irrational rotation on each such circle. It follows that $\gamma$ preserves a family of circles limiting down to $P$ on one direction and $Q$ in the other direction, acting by a map which is conjugate to the irrational rotation.
