Möbius-like groups of homeomorphisms of the circle

An orientation preserving homeomorphism of $S^1$ is Möbius-like if it is conjugate in $Homeo(S^1{1})$ to a Möbius transformation. Our main result is: given a (noncyclic) group $G\hookrightarrow Homeo_{+}(S^{1})$ whose every element is Möbius-like, if $G$ has at least one global fixed point, then the whole group $G$ is conjugate in $Homeo(S^1{1})$ to a Möbius group if and only if the limit set of $G$ is all of $S^1$. Moreover, we prove that if the limit set of $G$ is 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 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 $\boldsymbol{R}$ whose every element can be conjugated to an affine map (i.e., a map of the form $x \mapsto ax + b$) 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.