TY - JOUR

T1 - Classification of the group actions on the real line and circle

AU - Malyutin, A. V.

PY - 2008/1/1

Y1 - 2008/1/1

N2 - 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.

AB - 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.

KW - Action

KW - Circle

KW - Distal

KW - Group of homeomorphisms

KW - Line

KW - Proximal

KW - Semiconjugacy

UR - http://www.scopus.com/inward/record.url?scp=85009776486&partnerID=8YFLogxK

U2 - 10.1090/S1061-0022-08-00999-0

DO - 10.1090/S1061-0022-08-00999-0

M3 - статья

AN - SCOPUS:85009776486

VL - 19

SP - 279

EP - 296

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 2

ER -