Are dilation and rotation the same thing in Mobius transformation? Speaking of Möbius transformation, I think a rotation counter-clockwise by $\theta$ is actually a dilation by a magnitude of $e^{-i\theta}$. Is this correct?

1. Usually you use the words "dilation" and "rotation" to speak about the transformation from a geometric point of view (they are different geometric transformations).
2. Actualy multiplying by $e^{i\theta}$ produces a counter-clockwise rotation by $\theta$ radiants.