Möbius-like groups of homeomorphisms of the circle

An orientation preserving homeomorphism of is Möbius-like if it is conjugate in to a Möbius transformation. Our main result is: given a (noncyclic) group whose every element is Möbius-like, if has at least one global fixed point, then the whole group is conjugate in to a Möbius group if and only if the limit set of is all of . Moreover, we prove that if the limit set of is not all of , then after identifying some closed subintervals of to points, the induced action of is conjugate to an action of a Möbius group. Said differently, is obtained from a group which is conjugate to a Möbius group, by a sort of generalized Denjoy's insertion of intervals. In this case is isomorphic, as a group, to a Möbius group.

This result has another interpretation. Namely, we prove that a group of orientation preserving homeomorphisms of whose every element can be conjugated to an affine map (i.e., a map of the form ) is just the conjugate of a group of affine maps, up to a certain insertion of intervals. In any case, the group structure of is the one of an affine group.