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Classification of the group actions on the real line and circle
Author(s):
A.
V.
Malyutin
Translated by:
the author
Original publication:
Algebra i Analiz,
tom 19
(2007),
nomer 2.
Journal:
St. Petersburg Math. J.
19
(2008),
279-296.
MSC (2000):
Primary 54H15;
Secondary 57S25, 57M60, 54H20, 37E05, 37E10
Posted:
February 7, 2008
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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.
References:
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- 1.
- L. A. Beklaryan, Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants, Uspekhi Mat. Nauk 59 (2004), no. 4, 3-68; English transl., Russian Math. Surveys 59 (2004), no. 4, 599-660. MR 2106645 (2005i:37044)
- 2.
- É. Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), 329-407. MR 1876932 (2003a:37032)
- 3.
- H. Furstenberg, Boundary theory and stochastic processes on homogeneous spaces, Harmonic Analysis on Homogeneous Spaces (Williams College, Williamstown, MA, 1972), Proc. Sympos. Pure Math., vol. 26, Amer. Math. Soc., Providence, RI, 1973, pp. 193-229. MR 0352328 (50:4815)
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Additional Information:
A.
V.
Malyutin
Affiliation:
St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email:
malyutin@pdmi.ras.ru
DOI:
10.1090/S1061-0022-08-00999-0
PII:
S 1061-0022(08)00999-0
Keywords:
Circle,
line,
group of homeomorphisms,
action,
proximal,
distal,
semiconjugacy
Received by editor(s):
16/JUN/2006
Posted:
February 7, 2008
Additional Notes:
The author was partially supported by grant NSh-4329.2006.1 and by RFBR grant no.~05-01-00899
Copyright of article:
Copyright
2008,
American Mathematical Society
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