Möbius-like groups of homeomorphisms of the circle

An orientation preserving homeomorphism of S¹ is Möbius-like if it is conjugate in Homeo(S¹1) to a Möbius transformation. Our main result is: given a (noncyclic) group G Homeo₊(S¹) whose every element is Möbius-like, if G has at least one global fixed point, then the whole group G is conjugate in Homeo(S¹1) to a Möbius group if and only if the limit set of G is all of S¹. Moreover, we prove that if the limit set of G is not all of S¹, then after identifying some closed subintervals of S¹ to points, the induced action of G is conjugate to an action of a Möbius group. Said differently, G 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 G is isomorphic, as a group, to a Möbius group. This result has another interpretation. Namely, we prove that a group G of orientation preserving homeomorphisms of R whose every element can be conjugated to an affine map (i.e., a map of the form x ↦ ax + b) is just the conjugate of a group of affine maps, up to a certain insertion of intervals. In any case, the group structure of G is the one of an affine group.