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A subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is M\"obius-like if everyelement is conjugate to an element of $\mathrm{PSL}(2,\mathbb{R})$. In general,a M\"obius-like subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is not necessarily(semi-)conjugate to a subgroup of $\mathrm{PSL}(2,\mathbb{R})$, as discoveredby N. Kova\v{c}evi\'{c} [Trans. Amer. Math. Soc. 351 (1999), 4823-4835]. Herewe determine simple dynamical criteria for the existence of such a(semi-)conjugacy. We show that M\"obius-like subgroups of$\mathrm{Homeo}_+(\mathbb{S}^1)$ which are elementary (namely, preserving aBorel probability measure), are semi-conjugate to subgroups of$\mathrm{PSL}(2,\mathbb{R})$. On the other hand, we provide an example ofelementary subgroup of $\mathrm{Diff}^\infty_+(\mathbb{S}^1)$ satisfying thatevery non-trivial element fixes at most 2 points, which is not isomorphic toany subgroup of $\mathrm{PSL}(2,\mathbb{R})$. Finally, we show thatnon-elementary, non-locally discrete subgroups acting with at most $N$ fixedpoints are conjugate to a dense subgroup of some finite central extension of$\mathrm{PSL}(2,\mathbb{R})$.

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