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Algebra i Analiz, 2007, Volume 19, Issue 2, Pages 156–182 (Mi aa118)  

This article is cited in 8 scientific papers (total in 8 papers)

Research Papers

Classification of the group actions on the real line and circle

A. V. Malyutin

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: The group actions on the real line and circle are classified. It is proved that each minimal continuous action of a group on the circle is either a conjugate of an isometric action, or a finite cover of a proximal action. It is also shown that each minimal continuous action of a group on the real line either is conjugate to an isometric action, or is a proximal action, or is a cover of a proximal action on the circle. As a corollary, it is proved that a continuous action of a group on the circle either has a finite orbit, or is semiconjugate to a minimal action on the circle that is either isometric or proximal. As a consequence, a new proof of the Ghys-Margulis alternative is obtained.

Keywords: Circle, line, group of homeomorphisms, action, proximal, distal, semiconjugacy.

Full text: PDF file (263 kB)
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English version:
St. Petersburg Mathematical Journal, 2008, 19:2, 279–296

Bibliographic databases:

MSC: Primary 54H15; Secondary 57S25, 57M60, 54H20, 37E05, 37E10
Received: 16.06.2006

Citation: A. V. Malyutin, “Classification of the group actions on the real line and circle”, Algebra i Analiz, 19:2 (2007), 156–182; St. Petersburg Math. J., 19:2 (2008), 279–296

Citation in format AMSBIB
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\paper Classification of the group actions on the real line and circle
\jour Algebra i Analiz
\yr 2007
\vol 19
\issue 2
\pages 156--182
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\zmath{https://zbmath.org/?q=an:1209.37009}
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\transl
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 2
\pages 279--296
\crossref{https://doi.org/10.1090/S1061-0022-08-00999-0}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Hattab H., “Pointwise recurrent one-dimensional flows”, Dyn. Syst., 26:1 (2011), 77–83  crossref  mathscinet  zmath  isi
    2. Deroin B., Kleptsyn V., Navas A., Parwani K., “Symmetric Random Walks on Homeo(+)(R)”, Ann. Probab., 41:3B (2013), 2066–2089  crossref  mathscinet  zmath  isi
    3. A. V. Malyutin, “Groups acting on dendrons”, J. Math. Sci. (N. Y.), 212:5 (2016), 558–565  mathnet  crossref
    4. Hattab H., “Flows of Locally Finite Graphs”, Boll. Unione Mat. Ital., 10:4 (2017), 671–679  crossref  mathscinet  zmath  isi
    5. Glasner E., Megrelishvili M., “Circularly Ordered Dynamical Systems”, Mon.heft. Math., 185:3 (2018), 415–441  crossref  mathscinet  zmath  isi
    6. Salem A.H., Hattab H., “Group Action on Local Dendrites”, Topology Appl., 247 (2018), 91–99  crossref  mathscinet  zmath  isi  scopus
    7. Shi E., Zhou L., “Topological Transitivity and Wandering Intervals For Group Actions on the Line R”, Group. Geom. Dyn., 13:1 (2019), 293–307  crossref  mathscinet  zmath  isi  scopus
    8. A. V. Malyutin, “The Rotation Number Integer Quantization Effect in Braid Groups”, Proc. Steklov Inst. Math., 305 (2019), 182–194  mathnet  crossref  crossref  isi  elib
  • Алгебра и анализ St. Petersburg Mathematical Journal
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