I start with an automorphism $f$ of the complex unit disc $S^1 = \{ z \in \mathbb{C} : |z| \leq 1\}$. I assume that such a map is given by a Mobius transform, namely
$$
f(z) = \frac{z - a}{-\overline{a}z + 1},
$$
for some $a \in S^1$ with $|a| < 1$. (I know the automorphisms of $S^1$ are slightly more general but for my purpose this is enough.) My question is how far does this map move points on the boundary? In other words, what is the most a point on the boundary can rotate by. Are there 'good' bounds on this by which I mean that these bounds should go to zero as $|a|$ does.



How far do conjugated Mobius transforms move points?
You can assume $a$ is real. Then, $f(z) - z = (z-1)/(-\overline{z} a + 1),$ for $z$ on the unit circle. Since the modulus of this thing is the chordal distance, which is related to the angle in the obvious way, this reduces to a simple calculus optimization problem.


