Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-16T23:41:25.278Z Has data issue: false hasContentIssue false
  • English
  • Français

A connectedness result for commuting diffeomorphisms of the interval

Published online by Cambridge University Press:  04 June 2010

HÉLÈNE EYNARD
HÉLÈNE EYNARD*
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan (email: heynardb@umpa.ens-lyon.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let 𝒟r+[0,1], r≥1, denote the group of orientation-preserving 𝒞r diffeomorphisms of [0,1]. We show that any two representations of ℤ2 in 𝒟r+[0,1], r≥2, are connected by a continuous path of representations of ℤ2 in 𝒟1+[0,1] . We derive this result from the classical works by G. Szekeres and N. Kopell on the 𝒞1 centralizers of the diffeomorphisms of [0,1) that are at least 𝒞2 and fix only 0 .

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 4 , August 2011 , pp. 1183 - 1191
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[Ey1]Eynard, H.. On the centralizer of diffeomorphisms of the half-line. Comment. Math. Helv., to appear. Preprint, 2008, arXiv:0811.1173v1.Google Scholar
[Ey2]Eynard, H.. Sur deux questions connexes de connexité concernant les feuilletages et leurs holonomies. PhD Dissertation, 2009, ENS Lyon, France, available athttp://tel.archives-ouvertes.fr/tel-00436304/fr/.Google Scholar
[Ko]Kopell, N.. Commuting diffeomorphisms. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV). American Mathematical Society, Providence, RI, 1968, pp. 165–184.Google Scholar
[Na]Navas, A.. Groups of Diffeomorphisms of the Circle (Mathematical Surveys, 13). Sociedade Brasileira de Matemática, Rio de Janeiro, 2007.Google Scholar
[Se]Sergeraert, F.. Feuilletages et difféomorphismes infiniment tangents à l’identité. Invent. Math. 39 (1977), 253–275.Google Scholar
[Sz]Szekeres, G.. Regular iteration of real and complex functions. Acta Math. 100 (1958), 203–258.CrossRefGoogle Scholar
[Ta]Takens, F.. Normal forms for certain singularities of vector fields. Ann. Inst. Fourier 23 (1973), 163–195.CrossRefGoogle Scholar
[Yo]Yoccoz, J.-C.. Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Astérisque 231 (1995), 89–242.Google Scholar