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Connectedness of the space of smooth actions of $\mathbb{Z}^{n}$ on the interval

Published online by Cambridge University Press:  20 April 2015

CHRISTIAN BONATTI  and
HÉLÈNE EYNARD-BONTEMPS
CHRISTIAN BONATTI
Affiliation:
Institut de Mathématiques de Bourgogne, CNRS - UMR 5584, Université de Bourgogne, 9 av. A. Savary, 21000 Dijon, France email bonatti@u-bourgogne.fr
HÉLÈNE EYNARD-BONTEMPS
Affiliation:
IMJ - PRG, CNRS - UMR 7586, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris cedex, France email helene.eynard-bontemps@imj-prg.fr
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Abstract

We prove that the spaces of ${\mathcal{C}}^{\infty }$ orientation preserving actions of $\mathbb{Z}^{n}$ on $[0,1]$ and non-free actions of $\mathbb{Z}^{2}$ on the circle are connected.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 7 , October 2016 , pp. 2076 - 2106
Copyright
© Cambridge University Press, 2015 

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References

Benhenda, M.. Circle diffomorphisms: quasi-reducibility and commuting diffeomorphisms. Nonlinearity 25(7) (2012), 1981–1995.CrossRefGoogle Scholar
Bonatti, C. and Firmo, S.. Feuilles compactes d’un feuilletage générique en codimension 1. Ann. Sci. Éc. Norm. Supér. (4) 427(4) (1994), 407–462.CrossRefGoogle Scholar
Brunella, M.. Remarks on structurally stable proper foliations. Math. Proc. Cambridge Philos. Soc. 115(1) (1994), 111–120.CrossRefGoogle Scholar
Eynard-Bontemps, H.. Sur deux questions connexes de connexité concernant les feuilletages et leurs holonomies. PhD Dissertation, http://tel.archives-ouvertes.fr/tel-00436304/fr/.Google Scholar
Eynard, H.. A connectedness result for commuting diffeomorphisms of the interval. Ergod. Th & Dynam. Sys. 31(4) (2011), 1183–1191.CrossRefGoogle Scholar
Eynard, H.. On the centralizer of diffeomorphisms of the half-line. Comment. Math. Helv. 86(2) (2011), 415–435.CrossRefGoogle Scholar
Eynard-Bontemps, H.. On the connectedness of the space of codimension one foliations on a closed 3-manifold. Preprint, 2014, arXiv:1404.5884.CrossRefGoogle Scholar
Fayad, B. and Khanin, K.. Smooth linearization of commuting circle diffeomorphisms. Ann. of Math. (2) 170(2) (2009), 961–980.CrossRefGoogle Scholar
Kopell, N.. Commuting diffeomorphisms. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV) . American Mathematical Society, Providence, RI, 1968, pp. 165–184.Google Scholar
Larcanché, A.. Topologie locale des espaces de feuilletages en surfaces des variétés fermées de dimension 3. Comment. Math. Helv. 82 (2007), 385–411.CrossRefGoogle Scholar
Navas, A.. Groups of Circle Diffeomorphisms (Chicago Lectures in Mathematics) . University of Chicago Press, Chicago, 2011, Ch. 4, translation of the 2007 Spanish edition.CrossRefGoogle Scholar
Navas, A.. Sur les rapprochements par conjugaison en dimension 1 et classe ${\mathcal{C}}^{1}$ . Preprint, 2012, arXiv:1208.4815.Google Scholar
Rosenberg, H. and Roussarie, R.. Some remarks on stability of foliations. J. Differential Geom. 10 (1975), 207–219.CrossRefGoogle Scholar
Sergeraert, F.. Feuilletages et difféomorphismes infiniment tangents à l’identité. Invent. Math. 39 (1977), 253–275.CrossRefGoogle Scholar
Szekeres, G.. Regular iteration of real and complex functions. Acta Math. 100 (1958), 203–258.CrossRefGoogle Scholar
Takens, F.. Normal forms for certain singularities of vector fields. Ann. Inst. Fourier 23 (1973), 163–195.CrossRefGoogle Scholar
Tsuboi, T.. Hyperbolic compact leaves are not C 1 -stable. Geometric Study of Foliations (Tokyo, 1993). World Scientific, River Edge, NJ, 1994, pp. 437–455.Google Scholar
Wood, J.. Foliations on 3-manifolds. Ann. of Math. (2) 89 (1969), 336–358.CrossRefGoogle Scholar
Yoccoz, J-C.. Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Astérisque 231 (1995), 125–160.Google Scholar