Let $T$ be a Möbius transformation, with fixed points $p$ and $q$. Hence the normal form of $T$ is $$ \frac{T(z) - p}{T(z) - q} = \lambda \frac{z - p}{z - q} $$
I want to show that the normal form of $T^n$ is $$ \frac{T^n(z) - p}{T^n(z) - q} = \lambda^n \frac{z - p}{z - q} $$
It is easily seen that $p$ and $q$ must also be fixed points for $T^n$, but I am confused as to how to obtain $\lambda^n$ as the coefficient.