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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Differentiable conjugacy for groups of area-preserving circle diffeomorphisms


Author: Daniel Monclair
Journal: Trans. Amer. Math. Soc. 370 (2018), 6357-6390
MSC (2010): Primary 37A05, 37C05, 37C15, 37D20, 37E10; Secondary 53B30, 53C50
DOI: https://doi.org/10.1090/tran/7124
Published electronically: May 17, 2018
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Abstract: We study groups of circle diffeomorphisms whose action on the cylinder $ \mathcal C=\mathbb{S}^1\times \mathbb{S}^1\setminus \Delta $ preserves a volume form. We first show that such a group is topologically conjugate to a subgroup of $ \rm {PSL}(2,\mathbb{R})$, then discuss the existence of a differentiable conjugacy.

For some groups, we find that this conjugacy is automatically differentiable. These rigidity results can be seen as particular cases of theorems of Herman (for circle diffeomorphisms conjugate to rotations) and Ghys (for actions of surface groups), with much simpler proofs.

For other groups (typically deformations in $ \mathrm {Diff}(\mathbb{S}^1)$ of Schottky groups in $ \rm {PSL}(2,\mathbb{R})$), we show that there is much more flexibility and that a differentiable conjugacy does not always exist.


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Additional Information

Daniel Monclair
Affiliation: Université du Luxembourg, Campus Kirchberg, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg
Email: daniel.monclair@uni.lu

DOI: https://doi.org/10.1090/tran/7124
Received by editor(s): September 1, 2016
Received by editor(s) in revised form: November 14, 2016
Published electronically: May 17, 2018
Additional Notes: Partially supported by ANR project GR-Analysis-Geometry (ANR-2011-BS01-003-02)
Article copyright: © Copyright 2018 American Mathematical Society

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