Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


Short proofs of theorems of Malyutin and Margulis

Author: Eli Glasner
Journal: Proc. Amer. Math. Soc. 145 (2017), 5463-5467
MSC (2010): Primary 54H20, 37B05, 20B07
Published electronically: July 10, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Ghys-Margulis alternative asserts that a subgroup $ G$ of homeomorphisms of the circle which does not contain a free subgroup on two generators must admit an invariant probability measure. Malyutin's theorem classifies minimal actions of $ G$. We present a short proof of Malyutin's theorem and then deduce Margulis' theorem which confirms the G-M alternative. The basic ideas are borrowed from the original work of Malyutin, but our proof is considerably shorter.

References [Enhancements On Off] (What's this?)

  • [Au-66] Joseph Auslander, Regular minimal sets. I, Trans. Amer. Math. Soc. 123 (1966), 469-479. MR 0193629,
  • [Bek-04] L. A. Beklaryan, Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants, Uspekhi Mat. Nauk 59 (2004), no. 4(358), 3-68 (Russian, with Russian summary); English transl., Russian Math. Surveys 59 (2004), no. 4, 599-660. MR 2106645,
  • [Ell-69] Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
  • [Gh-01] Étienne Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), no. 3-4, 329-407. MR 1876932
  • [Gl-76] Shmuel Glasner, Proximal flows, Lecture Notes in Mathematics, Vol. 517, Springer-Verlag, Berlin-New York, 1976. MR 0474243
  • [Mal-07] A. V. Malyutin, Classification of group actions on the line and the circle, Algebra i Analiz 19 (2007), no. 2, 156-182 (Russian); English transl., St. Petersburg Math. J. 19 (2008), no. 2, 279-296. MR 2333902,
  • [Mar-00] Gregory Margulis, Free subgroups of the homeomorphism group of the circle, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 9, 669-674 (English, with English and French summaries). MR 1797749,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 54H20, 37B05, 20B07

Retrieve articles in all journals with MSC (2010): 54H20, 37B05, 20B07

Additional Information

Eli Glasner
Affiliation: Department of Mathematics, Tel Aviv University, 69978 Tel Aviv, Israel

Received by editor(s): June 28, 2016
Received by editor(s) in revised form: January 11, 2017, January 12, 2017, and January 17, 2017
Published electronically: July 10, 2017
Additional Notes: This research was supported by a grant of the Israel Science Foundation (ISF 668/13)
Communicated by: Nimish Shah
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society