Geodesic
Classification of the group actions on the real line and circle
Algebra i analiz, Tome 19 (2007) no. 2, pp. 156-182.

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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. 
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     author = {A. V. Malyutin},
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A. V. Malyutin. Classification of the group actions on the real line and circle. Algebra i analiz, Tome 19 (2007) no. 2, pp. 156-182. http://geodesic.mathdoc.fr/item/AA_2007_19_2_a8/

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