Reducing conjugacy in the full diffeomorphism group of $\mathbb R$ to conjugacy in the subgroup of orientation-preserving maps
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamics systems, combinatorial methods. Part XVI, Tome 360 (2008), pp. 231-237
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Let $\operatorname{Diffeo}=\operatorname{Diffeo}(\mathbb R)$ denote the group of infinitely-differentiable diffeomorphisms of the real line $\mathbb R$, under the operation of composition, and let $\operatorname{Diffeo}^+$ 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 $f,g\in\operatorname{Diffeo}$ are conjugate in $\operatorname{Diffeo}$ to associated conjugacy problems in the subgroup $\operatorname{Diffeo}^+$. 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 $\operatorname{Diffeo}^+$, in order to ensure that $f$ is conjugated to $g$ by an element of $\operatorname{Diffeo}^+$. The methods involve formal power series, and results of Kopell on centralisers in the diffeomorphism group of a half-open interval. Bibl. – 4 titles.
@article{ZNSL_2008_360_a9,
author = {A. G. O'Farrell and M. Roginskaya},
title = {Reducing conjugacy in the full diffeomorphism group of~$\mathbb R$ to conjugacy in the subgroup of orientation-preserving maps},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {231--237},
year = {2008},
volume = {360},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_360_a9/}
}
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A. G. O'Farrell; M. Roginskaya. Reducing conjugacy in the full diffeomorphism group of $\mathbb R$ to conjugacy in the subgroup of orientation-preserving maps. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamics systems, combinatorial methods. Part XVI, Tome 360 (2008), pp. 231-237. http://geodesic.mathdoc.fr/item/ZNSL_2008_360_a9/
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