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Bulletin of the London Mathematical Society Advance Access originally published online on August 7, 2009
Bulletin of the London Mathematical Society 2009 41(5):885-897; doi:10.1112/blms/bdp065
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© 2009 London Mathematical Society

Reversibility in the group of homeomorphisms of the circle

Nick Gill

Department of Mathematics
University Walk
Bristol
BS8 1TW
United Kingdom
n.gill@bristol.ac.uk

Anthony G. O’Farrell

Department of Mathematics
National University of Ireland Maynooth
County Kildare
Ireland
anthonyg.ofarrell@gmail.com

Ian Short

Department of Pure Mathematics and Mathematical Statistics
Centre for Mathematical Sciences
Wilberforce Road
Cambridge
CB3 0WB
United Kingdom

Received 18 October 2007. Revision received 9 April 2009.

The group of orientation-preserving homeomorphisms of the circle is simple, and, because there are non-trivial involutions in this group, it must be generated by its involutions. We show that, in this group of homeomorphisms, each element can be expressed as a product of three involutions. We also characterise those elements of the group that can be expressed as a composite of two involutions, and perform a similar characterisation in the full group of homeomorphisms of the circle.

2000 Mathematics Subject Classification 37E10 (primary), 37B05, 57S25 (secondary).

The second and third authors were supported by Science Foundation Ireland, grant 05/RFP/MAT0003. The second author was also supported by the ESF Network HCAA.


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