NIL-ANOSOV ACTIONS

Thierry Barbot (1) , Carlos Maquera (2)

1 LMA - EA2151 Laboratoire de Mathématiques d'Avignon
2 Instituto de Ciências Matemáticas e de Computação

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.

Keywords: Anosov action; Nilpotent Lie group

Domains: Dynamical Systems [math.DS]

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