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

A subgroup of \(\textrm{Homeo}_+(\mathbb {S}^1)\) ${\text{Homeo}}_{+}({\mathbb{S}}^1)$ is Möbius-like if every element is conjugate to an element of \(\textrm{PSL}(2,\mathbb {R})\) $\text{PSL}(2,\mathbb{R})$ . In general, a Möbius-like subgroup of \(\textrm{Homeo}_+(\mathbb {S}^1)\) ${\text{Homeo}}_{+}({\mathbb{S}}^1)$ is not necessarily (semi-)conjugate to a subgroup of \(\textrm{PSL}(2,\mathbb {R})\) $\text{PSL}(2,\mathbb{R})$ , 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 \(\textrm{Homeo}_+(\mathbb {S}^1)\) ${\text{Homeo}}_{+}({\mathbb{S}}^1)$ which are elementary (namely, preserving a Borel probability measure), are semi-conjugate to subgroups of \(\textrm{PSL}(2,\mathbb {R})\) $\text{PSL}(2,\mathbb{R})$ . On the other hand, we provide an example of elementary subgroup of \(\textrm{Diff}^\infty _+(\mathbb {S}^1)\) ${\text{Diff}}_{+}^{\infty }({\mathbb{S}}^1)$ satisfying that every non-trivial element fixes at most 2 points, which is not isomorphic to any subgroup of \(\textrm{PSL}(2,\mathbb {R})\) $\text{PSL}(2,\mathbb{R})$ . 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 \(\textrm{PSL}(2,\mathbb {R})\) $\text{PSL}(2,\mathbb{R})$ .