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

TL;DR

This paper studies groups of real analytic circle diffeomorphisms with finite rotation number images, showing that nondiscrete groups must have finite orbits and classifying subgroups without finite orbits.

Contribution

It establishes a connection between the finiteness of the rotation number image and the dynamical properties of the group, including orbit structure and subgroup classification.

Findings

Nondiscrete groups with finite rotation number image have finite orbits.

Subgroups without finite orbits contain either a finite index cyclic group or a nonabelian free subgroup.

Provides a classification of such groups based on their orbit and subgroup structure.

Abstract

We consider groups of orientation-preserving real analytic diffeomorphisms of the circle which have a finite image under the rotation number function. We show that if such a group is nondiscrete with respect to the $C^1$-topology then it has a finite orbit. As a corollary, we show that if such a group has no finite orbit then each of its subgroups contains either a cyclic group of finite index or a nonabelian free subgroup.