 Iteration of a one-fixed-point Möbius transformation

A nonidentity Möbius transformation with only one fixed point is conjugate to a nonzero translation of the Riemann sphere. Its translation iterates are $z+nb$, tending to infinity in the topology of the Riemann sphere for finite $z$, with infinity itself fixed. Conjugating back proves that every point tends to the unique fixed point. This is convergence on the sphere; if the fixed point is finite, intermediate iterates can still pass through infinity.