All automorphisms of the complex unit disk $\Delta = \{z \in \mathbb{C} : |z| < 1\}$ to itself, can be written in the form $f_a(z) = e^{i \theta} \frac{z-a}{1-\overline{a}z}$ where $a \in \Delta$ and $\theta \in S^1$

This map sends $a$ to $0$ $1/\overline{a}$ to $\infty$ and the unit circle to the unit circle.