Anosov actions: minimality of foliations or suspension action

We prove that an Anosov action of ${\mathbb{R}}^k$ over a compact manifold $M$ transitive on regular sub-cones satisfies the dichotomy: each stable and unstable leaf is dense or the Anosov action is topologically conjugated to a suspension of a ${\mathbb{Z}}^k$-Anosov action. This represents an important progress toward addressing Verjovsky’s extended conjecture for Anosov actions, as developed by Barbot and Maquera.