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

Matsuda, Yoshifumi

doi:10.5802/aif.2477

Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function
[Des groupes de difféomorphismes analytiques réels du cercle qui ont une image finie sous l’application du nombre de rotation]

Matsuda, Yoshifumi ¹

¹ University of Tokyo Graduate School of Mathematical Sciences 3-8-1 Komaba Meguro Tokyo 153-8914 (Japan)

Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 1819-1845.

Résumé
Abstract

Nous considérons des groupes de difféomorphismes directs et analytiques réels du cercle qui ont une image finie sous l’application du nombre de rotation. Nous montrons que si un tel groupe est non-discret pour la topologie $C^{1}$ alors il a une orbite finie. Comme corollaire, nous montrons que si un tel groupe n’a aucune orbite finie alors chacun de ses sous-groupes contient soit un sous-groupe cyclique d’indice fini, soit un sous-groupe libre non-abélien.

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 subgroup of finite index or a nonabelian free subgroup.

MR Zbl

DOI : 10.5802/aif.2477

Classification : 37E45, 37E10, 57S05, 37B05, 20F67
Keywords: Rotation number, circle diffeomorphisms, groups, local vector fields.
Mot clés : nombre de rotation, difféomorphisms du cercle, groupes, champs du vecteur locaux

Affiliations des auteurs :

Matsuda, Yoshifumi ¹

¹ University of Tokyo Graduate School of Mathematical Sciences 3-8-1 Komaba Meguro Tokyo 153-8914 (Japan)

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     title = {Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function},
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Matsuda, Yoshifumi. Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function. Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 1819-1845. doi : 10.5802/aif.2477. http://www.numdam.org/articles/10.5802/aif.2477/

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