arXiv:2610.33228v1 Announce Type: new 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.
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