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

Abstract

A subgroup of ${\mathrm{Homeo}}_{+}({\mathbb{S}}^1)$ is Möbius-like if every element is conjugate to an element of $\mathrm{PSL}(2,\mathbb{R})$. In general, a Möbius-like subgroup of ${\mathrm{Homeo}}_{+}({\mathbb{S}}^1)$ is not necessarily (semi-)conjugate to a subgroup of $\mathrm{PSL}(2,\mathbb{R})$, as discovered by N. Kovačević [Trans. Amer. Math. Soc. 351 (1999), 4823-4835]. Here we determine simple dynamical criteria for the existence of such a (semi-)conjugacy. We show that Möbius-like subgroups of ${\mathrm{Homeo}}_{+}({\mathbb{S}}^1)$ which are elementary (namely, preserving a Borel probability measure), are semi-conjugate to subgroups of $\mathrm{PSL}(2,\mathbb{R})$. On the other hand, we provide an example of elementary subgroup of ${\mathrm{Diff}}_{+}^{\infty }({\mathbb{S}}^1)$ satisfying that every non-trivial element fixes at most 2 points, which is not isomorphic to any subgroup of $\mathrm{PSL}(2,\mathbb{R})$. 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 $\mathrm{PSL}(2,\mathbb{R})$.

16 pages. This work is part of the second author's PhD thesis at Université de Bourgogne; to appear in Math. Z