# NIL-ANOSOV ACTIONS

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: i) if $n \geq 3,$ then the action is topologically transitive, ii) if $n=2,$ then the action is a Nil-extension over an Anosov flow.
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Preprints, Working Papers, ...

https://hal-univ-avignon.archives-ouvertes.fr/hal-01197491
Contributor : Thierry Barbot <>
Submitted on : Sunday, June 5, 2016 - 9:47:50 AM
Last modification on : Tuesday, January 14, 2020 - 10:38:02 AM
Long-term archiving on: : Tuesday, September 6, 2016 - 10:12:15 AM

### Files

Nilp_transitive.pdf
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### Identifiers

• HAL Id : hal-01197491, version 2
• ARXIV : 1509.03816

### Citation

Thierry Barbot, Carlos Maquera. NIL-ANOSOV ACTIONS. 2015. ⟨hal-01197491v2⟩

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