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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Möbius-like groups of homeomorphisms
of the circle


Author: Natasa Kovacevic
Journal: Trans. Amer. Math. Soc. 351 (1999), 4791-4822
MSC (1991): Primary 57S05
Published electronically: August 23, 1999
MathSciNet review: 1473447
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Abstract: An orientation preserving homeomorphism of $S^1$ is Möbius-like if it is conjugate in $Homeo(S^1{1})$ to a Möbius transformation. Our main result is: given a (noncyclic) group $G\hookrightarrow Homeo_{+}(S^{1})$ whose every element is Möbius-like, if $G$ has at least one global fixed point, then the whole group $G$ is conjugate in $Homeo(S^1{1})$ to a Möbius group if and only if the limit set of $G$ is all of $S^1$. Moreover, we prove that if the limit set of $G$ is not all of $S^1$, then after identifying some closed subintervals of $S^1$ to points, the induced action of $G$ is conjugate to an action of a Möbius group. Said differently, $G$ is obtained from a group which is conjugate to a Möbius group, by a sort of generalized Denjoy's insertion of intervals. In this case $G$ is isomorphic, as a group, to a Möbius group.

This result has another interpretation. Namely, we prove that a group $G$ of orientation preserving homeomorphisms of $\boldsymbol{R}$ whose every element can be conjugated to an affine map (i.e., a map of the form $x \mapsto ax + b$) is just the conjugate of a group of affine maps, up to a certain insertion of intervals. In any case, the group structure of $G$ is the one of an affine group.


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Additional Information

Natasa Kovacevic
Affiliation: Department of Mathematics, University of Toronto, 100 St. George Street, Room 4072, Toronto, Ontario M5S 1A1, Canada
Email: natasak@home.com

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02189-3
PII: S 0002-9947(99)02189-3
Received by editor(s): March 7, 1995
Received by editor(s) in revised form: July 31, 1997
Published electronically: August 23, 1999
Article copyright: © Copyright 1999 American Mathematical Society