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Reducing Conjugacy in the full diffeomorphism group of R to conjugacy in the subgroup of orientation-preserving maps

O'Farrell, Anthony G. and Roginskaya, Maria (2009) Reducing Conjugacy in the full diffeomorphism group of R to conjugacy in the subgroup of orientation-preserving maps. Journal of Mathematical Sciences, 158 (6). pp. 895-898. ISSN 1072-3374

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Abstract

Let Diffeo = Diffeo(R) denote the group of infinitely-differentiable diffeomorphisms of the real line R, under the operation of composition, and let Diffeo+ be the subgroup of diffeomorphisms of degree +1, i.e. orientation-preserving diffeomorphisms. We show how to reduce the problem of determining whether or not two given elements f, g ∈ Diffeo are conjugate in Diffeo to associated conjugacy problems in the subgroup Diffeo+. The main result concerns the case when f and g have degree −1, and specifies (in an explicit and verifiable way) precisely what must be added to the assumption that their (compositional) squares are conjugate in Diffeo+, in order to ensure that f is conjugated to g by an element of Diffeo+. The methods involve formal power series, and results of Kopell on centralisers in the diffeomorphism group of a half-open interval.

Item Type: Article
Additional Information: Preprint version of article. The original publication is available at www.springerlink.com DOI: 0.1007/s10958-009-9419-x. Supported by SFI under grant RFP05/MAT0003. The authors are grateful to Ian Short for useful comments.
Keywords: Diffeomorphism group; conjugacy; real line; orientation;
Subjects: Science & Engineering > Mathematics
Item ID: 2786
Identification Number: DOI: 10.1007/s10958-009-9419-x
Depositing User: Prof. Anthony O'Farrell
Date Deposited: 24 Oct 2011 14:29
Journal or Publication Title: Journal of Mathematical Sciences
Publisher: Springer
Refereed: Yes
URI:

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