Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function (Q1041267)

The paper is concerned with groups of orientation-preserving real analytic diffeomorphisms of the circle which have a finite image under the rotation number function. It is shown that if such a group is nondiscrete with respect to the \(C^1\)-topology then it has a finite orbit. The main paper result is the following Theorem. Let \(\Gamma\) be a subgroup of \(\text{Diff}^{\omega}_{+}(S^1)\) which is nondiscrete with respect to the \(C^1\)-topology. Then \(\Gamma\) has a finite image under the rotation number function if and only if it has a finite orbit. As a consequence, it is shown that if such a group has no finite orbit then each of its subgroups contains either a cyclic subgroup of finite index or a nonabelian free subgroup.