I have to
Prove that a loxodromic transformation has an attractor and a repeller as fixed points.
I have no idea how to start the proof, or what i need to do, formally. Basically, I only know the geometric picture of what an attractor and a repeller look like. Right now, it is beyond my scope of understanding, to figure out on my own what I have to do. So some help would be very appreciated, if there exists any.
$Def (loxodromic): \overline{T}$ lives in $PSL(2, \mathbb{C}).$ It is such that it fixes exactly 2 points in $\mathbb{\hat{C}}.$ $\overline{T}$ is conjugate in $PSL(2, \mathbb{C})$ to $\overline{S}(z) = \alpha z.$ If $|\alpha| \neq 1 $ and $\alpha \in \mathbb{R^+ }, \overline{T}$ is called loxodromic.
Using the additional information given in your comment, here's what to do.
First, $\overline S$ has an attractor and repeller as fixed points. To see this, use the equation $\overline S^n(z) = a^n z$. If $\alpha > 1$ then $0$ and $\infty$ are the only fixed points, and for any other value of $z$ in the closed upper half plane one has $\lim_{n \to \infty} a^n = \infty$ and $\lim_{n \to -\infty} a^n = 0$, so $\infty$ is an attractor and $0$ is a repeller. Similarly, of $0<\alpha<1$ then $0$ is an attractor and $\infty$ is a repeller.
Second, if $\overline T = \overline R \overline S \overline R^{-1}$ and if $\alpha > 1$ then $\overline R(\infty)$ is the attractor for $T$ and $\overline R(0)$ is the repeller for $T$; and if $0 < \alpha < 1$ then $\overline R(0)$ is the attractor for $T$ and $\overline R(\infty)$ is the repeller for $T$.