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Mathematische Zeitschrift

, Volume 287, Issue 3–4, pp 1279–1305 | Cite as

Nil-Anosov actions

  • Thierry BarbotEmail author
  • Carlos Maquera
Article
  • 97 Downloads

Abstract

We consider Anosov actions of a Lie group G of dimension k on a closed manifold of dimension \(k+n\). We introduce the notion of Nil-Anosov action of G (which includes the case where G is nilpotent) and establishes the invariance by the entire group G of the associated stable and unstable foliations. We then prove a spectral decomposition Theorem for such an action when the group G is nilpotent. Finally, we focus on the case where G is nilpotent and the unstable bundle has codimension one. We prove that in this case the action is a Nil-extension over an Anosov action of an abelian Lie group. In particular:
  • if \(n \ge 3,\) then the action is topologically transitive,

  • if \(n=2,\) then the action is a Nil-extension over an Anosov flow.

Keywords

Anosov action Compact orbit Closing lemma Transitivity Anosov flow 

Mathematics Subject Classification

37C85 

Notes

Acknowledgements

This paper was partly written while the second author stayed at Laboratoire de de Mathématiques, Université d’Avignon et des Pays de Vaucluse. It has been concluded while the second author was visiting the University of Campinas (UNICAMP) and supported by the french-brazilian program “Chaires franco-brésiliennes dans l’état de São Paulo”. Pr. J.R. Varão Filho contributed to the conclusion of the paper, mainly in the elaboration of Theorem 8.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.LMAAvignon UniversityAvignon CedexFrance
  2. 2.Instituto de ciências Matemáticas e de ComputaçãoUniversidade de São Paulo - São CarlosSão CarlosBrazil

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