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Conjugacy of real diffeomorphisms. A survey

Authors: A. G. O’Farrell and M. Roginskaya
Original publication: Algebra i Analiz, tom 22 (2010), nomer 1.
Journal: St. Petersburg Math. J. 22 (2011), 1-40
MSC (2010): Primary 37E05, 20E45
Published electronically: November 16, 2010
MathSciNet review: 2641079
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Abstract: Given a group $ G$, the conjugacy problem in $ G$ is the problem of giving an effective procedure for determining whether or not two given elements $ f,g\in G$ are conjugate, i. e., whether there exists $ h\in G$ with $ fh=hg$. This paper is about the conjugacy problem in the group $ \operatorname{Diffeo}(I)$ of all diffeomorphisms of an interval $ I\subset\mathbb{R}$.

There is much classical work on the subject, solving the conjugacy problem for special classes of maps. Unfortunately, it is also true that many results and arguments known to the experts are difficult to find in the literature, or simply absent. We try to repair these lacunae, by giving a systematic review, and we also include new results about the conjugacy classification in the general case.

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Additional Information

A. G. O’Farrell
Affiliation: Mathematics Department, NUI, Maynooth, Co. Kildare, Ireland

M. Roginskaya
Affiliation: Mathematics Department, Chalmers University of Technology, and Göteborg University, SE-412 96 Göteborg, Sweden

Keywords: Diffeomorphism group, conjugacy, real line, orientation
Received by editor(s): May 1, 2009
Published electronically: November 16, 2010
Additional Notes: Supported by Grant SFI RFP05/MAT0003 and the ESF Network HCAA
Dedicated: Dedicated to V. P. Havin on the occasion of his 75th birthday
Article copyright: © Copyright 2010 American Mathematical Society