On the stability of rank-one factors for Anosov ${\mathbb{Z}}^k$-actions

Abstract

We prove $C^1$-stability of $C^{\infty }$ rank-one factors for a large class of higher rank Anosov ${\mathbb{Z}}^k$-actions. If the factor is a projection of a smooth fiber bundle and the action contains an element which acts as the identity on the rank-one factor and is Anosov on every fiber, then any sufficiently $C^1$-small $C^{\infty }$ perturbation of the action will also have a $C^{\infty }$ rank-one factor. In particular, we prove the $C^1$-stability of the product structure for Anosov ${\mathbb{Z}}^2$-actions which are obtained as a (diagonal) product of two $\mathbb{Z}$-Anosov actions.

arXiv comment13 pages