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We consider groups of orientation-preserving real analytic diffeomorphisms ofthe circle which have a finite image under the rotation number function. Weshow that if such a group is nondiscrete with respect to the $C^1$-topologythen it has a finite orbit. As a corollary, we show that if such a group has nofinite orbit then each of its subgroups contains either a cyclic group offinite index or a nonabelian free subgroup.

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