It is known that the automorphisms of the unit disk in the complex plane $\mathbb{D}$ = $\{z \in \mathbb{C}:|z| < 1\}$ all take the form: $$\psi_\alpha(z)= e^{i\theta}\frac{z-\alpha}{1-\overline\alpha z}$$ where $\theta \in S^1$, and $\alpha \in \mathbb{D}$. The former factor $e^{i\theta}$ evidently depicts a rotation, which aligns with some geometric intuition regarding one family of conformal maps from the disk to itself.
However, I was wondering whether there was some intuitive way to make sense of the right-hand mapping - I've shown its properties of conformality and idempotence, but I still feel like it comes "out of nowhere", and I was interested to know if there is a way to understand these Blaschke factors and how they act on the disk beyond the purely symbolic proofs that they are automorphisms. Thank you so much.
