The automorphism group of the Poincare' disk has elements called elliptic, which have a single fixed point in the interior of the disk, and can be represented as a rotation around this fixed point.

It also has parabolic elements, which have a single fixed point in the boundary, and in a sense are a rotation around this ideal point.

If $x \in D \to p \in \partial D$, can we see a rotation around $x$ continuosly becoming a rotation around $p$, in some appropriate topology?