Mobius-like groups of homeomorphisms of the circle

An orientation preserving homeomorphism of S-1 is Mobius-like if it is conj ugate in Homeo(S-1 1) to a Mobius transformation. Our main result is: given a (noncyclic) group G hooked right arrow Homeo(+)(S-1) whose every element is Mobius-like, if G has at least one global fixed point, then the whole g roup G is conjugate in Homeo(S-1 1) to a Mobius group if and only if the li mit set of G is all of S-1. Moreover, we prove that if the limit set of G i s not all of S-1, then after identifying some closed subintervals of S-1 to points, the induced action of G is conjugate to an action of a Mobius grou p. Said differently, G is obtained from a group which is conjugate to a Mob ius group, by a sort of generalized Denjoy's insertion of intervals. In thi s case G is isomorphic, as a group, to a Mobius group. This result has another interpretation. Namely, we prove that a group G of orientation preserving homeomorphisms of R whose every element can be conju gated to an affine map (i.e., a map of the form x bar right arrow ax + b) i s just the conjugate of a group of affine maps, up to a certain insertion o f intervals. In any case, the group structure of G is the one of an affine group.