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

Yoshifumi Matsuda[1]

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

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 5, page 1819-1845
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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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 59.5 (2009): 1819-1845. <http://eudml.org/doc/10441>.

@article{Matsuda2009,
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 subgroup of finite index or a nonabelian free subgroup.},
affiliation = {University of Tokyo Graduate School of Mathematical Sciences 3-8-1 Komaba Meguro Tokyo 153-8914 (Japan)},
author = {Matsuda, Yoshifumi},
journal = {Annales de l’institut Fourier},
keywords = {Rotation number; circle diffeomorphisms; groups; local vector fields; Poincaré rotation number},
language = {eng},
number = {5},
pages = {1819-1845},
publisher = {Association des Annales de l’institut Fourier},
title = {Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function},
url = {http://eudml.org/doc/10441},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Matsuda, Yoshifumi
TI - Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 5
SP - 1819
EP - 1845
AB - 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.
LA - eng
KW - Rotation number; circle diffeomorphisms; groups; local vector fields; Poincaré rotation number
UR - http://eudml.org/doc/10441
ER -

References

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  9. A. Navas, On uniformly quasisymmetric groups of circle diffeomorphisms, Ann. Acad. Sci. Fenn. Math. 31 (2006), 437-462 Zbl1098.22011MR2248825
  10. Julio C. Rebelo, Ergodicity and rigidity for certain subgroups of Diff ω ( S 1 ) , Ann. Sci. École Norm. Sup. (4) 32 (1999), 433-453 Zbl0968.37002MR1693579
  11. A. Selberg, On discontinuous groups in higher-dimmensional symmetric spaces, Contributions to function theories (1960), 147-164, Tata Institute of Fundamental Research, Bombay Zbl0201.36603MR130324
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