Non-locally discrete actions on the circle with at most N fixed points

A subgroup of Homeo_+(𝕊^1) is Möbius-like if every element is conjugate to an element of PSL(2,ℝ) . In general, a Möbius-like subgroup of Homeo_+(𝕊^1) is not necessarily (semi-)conjugate to a subgroup of PSL(2,ℝ) , as discovered by Kovačević (Trans Am Math Soc 351:4823–4835, 1999). Here we determine simple dynamical criteria for the existence of such a (semi-)conjugacy. We show that Möbius-like subgroups of Homeo_+(𝕊^1) which are elementary (namely, preserving a Borel probability measure), are semi-conjugate to subgroups of PSL(2,ℝ) . On the other hand, we provide an example of elementary subgroup of Diff^∞ _+(𝕊^1) satisfying that every non-trivial element fixes at most 2 points, which is not isomorphic to any subgroup of PSL(2,ℝ) . Finally, we show that non-elementary, non-locally discrete subgroups acting with at most N fixed points are conjugate to a dense subgroup of some finite central extension of PSL(2,ℝ) .