Möbius transformations with n-cycles geometric viewpoint

Let MM be the group of Möbius transformations on C∞=C∪{∞}C∞=C∪{∞} and 〈f〉M={fn;n∈Z} the cyclic subgroup of MM generated by f  , for f∈Mf∈M. If 〈f〉M〈f〉M is finite of order n, f is called an n-cycle. We prove in the first part that if f is an n  -cycle, then for any α∈C∞α∈C∞, the set {fn(α);n∈Z}=Of(α) lies on a circle. Furthermore we characterize with geometric arguments the circles which are invariant under this kind of transformations.