Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-08T19:32:39.416Z Has data issue: false hasContentIssue false
  • English
  • Français

Sur les rapprochements par conjugaison en dimension 1 et classe $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}C^1$

Part of: Low-dimensional dynamical systems Smooth dynamical systems: general theory Representation theory of groups

Published online by Cambridge University Press:  19 June 2014

Andrés Navas
Andrés Navas*
Affiliation:
Dpto. de Matemática y C.C., Universidad de Santiago de Chile (USACH), Alameda 3363, Estación Central, 71783-5 Santiago, Chile email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the space of actions of every finitely generated, nilpotent group by $C^1$ orientation-preserving diffeomorphisms of the circle is path-connected. This is done via a general result that allows any given action on the interval to be connected to the trivial one by a continuous path of topological conjugates.

Keywords

nilpotent groupconnectednessactions on the interval

MSC classification

Secondary: 20D15: Nilpotent groups, $p$-groups 37C85: Dynamics of group actions other than ${bf Z}$ and ${bf R}$, and foliations 37E99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 7 , July 2014 , pp. 1183 - 1195
Copyright
© The Author 2014 

References

Bonatti, C. and Eynard, H., Connectedness of the space of smooth actions of ${\mathbb{Z}}^n$ on the interval. Prépublication (2012).Google Scholar
Bonatti, C. and Guelman, N., Smooth conjugacy classes of circle diffeomorphisms with irrational rotation number. Prépublication (2012).Google Scholar
Breuillard, E. and Green, B., Approximate groups. I: The torsion-free nilpotent case, J. Inst. Math. Jussieu 10 (2011), 37–57.CrossRefGoogle Scholar
Cantwell, J. and Conlon, L., An interesting class of C 1 foliations, Topology Appl. 126 (2002), 281–297.Google Scholar
Castro, G., Jorquera, E. and Navas, A., Sharp regularity for certain nilpotent group actions on the interval. À paraître dans, Math. Ann.Google Scholar
Deroin, B., The group of almost-periodic homeomorphisms of the real line, Enseign. Math. 59 (2013), 183–194; doi:10.4171/LEM/59-1-7.Google Scholar
Deroin, B., Kleptsyn, V. and Navas, A., Sur la dynamique unidimensionnelle en régularité intermédiaire, Acta Math. 199 (2007), 199–262.Google Scholar
Eynard, H., Sur deux questions connexes de connexité concernant les feuilletages et leurs holonomies. Thèse de Doctorat, École Normale Supérieure de Lyon (2009). Disponible sur, http://tel.archives-ouvertes.fr/tel-00436304/fr/.Google Scholar
Eynard, H., A connectedness result for commuting diffeomorphisms of the interval, Ergodic Theory Dynam. Systems 31 (2011), 1183–1191.Google Scholar
Farb, B. and Franks, J., Groups of homeomorphisms of one-manifolds. III. Nilpotent subgroups, Ergodic Theory Dynam. Systems 23 (2003), 1467–1484.Google Scholar
Farinelli, É., Classes de conjugaison des difféomorphismes de l’intervalle en régularité $C^1$. Prépublication (2012).Google Scholar
Ghys, É., Langevin, R. and Walczak, P., Entropie géométrique des feuilletages, Acta Math. 160 (1988), 105–142.Google Scholar
Guelman, N. and Liousse, I., C 1-actions of Baumslag–Solitar groups on S 1, Algebr. Geom. Topol. 11 (2011), 1701–1707.CrossRefGoogle Scholar
Harrison, J., Unsmoothable diffeomorphisms, Ann. of Math. (2) 102 (1975), 85–94.Google Scholar
Herman, M., Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5–233.Google Scholar
Hurder, S., Entropy and dynamics of $C^{1}$ foliations. Prépublication (2000).Google Scholar
Jorquera, E., A universal nilpotent group of C 1-diffeomorphisms of the interval, Topol. Appl. 159 (2012), 2115–2126.Google Scholar
Navas, A., Growth of groups and diffeomorphisms of the interval, Geom. Funct. Anal. 18 (2008), 988–1028.CrossRefGoogle Scholar
Navas, A., Groups of Circle Diffeomorphisms, Chicago Lectures in Mathematics (University of Chicago Press, 2011).Google Scholar
Navas, A., On centralizers of interval diffeomorphisms in critical (intermediate) regularity, J. Anal. Math. 121 (2013), 1–30.Google Scholar
Ollagnier, J. M. and Pinchon, D., Systèmes dynamiques topologiques I. Étude des limites de cobords, Bull. Soc. Math. France 105 (1977), 405–414.Google Scholar
Rivas, C., On spaces of Conradian group orderings, J. Group Theory 13 (2010), 337–353.Google Scholar
Tsuboi, T., Γ 1-structures avec une seule feuille, Asterisque 116 (1984), 222–234.Google Scholar
Volný, D. and Weiss, B., Coboundaries in L 0∞, Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004), 771–778.Google Scholar