 Antipodal fixed-point classification of Möbius transformations

For a nonidentity Möbius transformation whose two fixed points are antipodal under stereographic projection, conjugate by a sphere rotation sending them to $0,\infty$. The conjugated map fixes both, hence is $z\mapsto az$ with $a\ne 0$. If $|a|=1$ it is an axial rotation; if $|a|<1$ every finite orbit tends to zero; if $|a|>1$ every nonzero orbit tends to infinity. Undoing the rotation gives either a sphere rotation or one attracting fixed point with the other as the sole exceptional initial point.