If I have given four different points in a disc : $a,b,c,d$ Does always exist a Möbius transformation $w$ which PRESERVES UNIT DISC , so that $w(a)=c$ and $w(b)=d$? If yes, is it unique?
If given 6 points: $a,b,c,d,e,f$, then I look for a transformation(which preserves unit disc) that maps: $w(a)=d , w(b)=e , w(c)=f$. If this transformation (also) exists, is it unique?
Thank you in advance!
To answer the first question: no, $w$ need not exist, although when it does exist it is unique. The reason is that the Poincare distance $d(x,y)$ between two points $x$ and $y$ in the unit disc is invariant under all Möbius transformations, so a necessary condition for the existence of $w$ is that $d(a,b)=d(c,d)$. That equation is also a sufficient condition for the existence of $w$, and moreover if $w$ exists then it is unique.
This pretty obviously addresses your second question as well: no, $w$ need not exist.