Möbius-like groups of homeomorphisms

of the circle

Author:
Natasa Kovacevic

Journal:
Trans. Amer. Math. Soc. **351** (1999), 4791-4822

MSC (1991):
Primary 57S05

DOI:
https://doi.org/10.1090/S0002-9947-99-02189-3

Published electronically:
August 23, 1999

MathSciNet review:
1473447

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Abstract | References | Similar Articles | Additional Information

Abstract: An orientation preserving homeomorphism of is Möbius-like if it is conjugate in to a Möbius transformation. Our main result is: given a (noncyclic) group whose every element is Möbius-like, if has at least one global fixed point, then the whole group is conjugate in to a Möbius group if and only if the limit set of is all of . Moreover, we prove that if the limit set of is not all of , then after identifying some closed subintervals of to points, the induced action of is conjugate to an action of a Möbius group. Said differently, 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 is isomorphic, as a group, to a Möbius group.

This result has another interpretation. Namely, we prove that a group of orientation preserving homeomorphisms of whose every element can be conjugated to an affine map (i.e., a map of the form ) is just the conjugate of a group of affine maps, up to a certain insertion of intervals. In any case, the group structure of 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:
https://doi.org/10.1090/S0002-9947-99-02189-3

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