Anosov actions: minimality of foliations or suspension action

TL;DR

This paper establishes a dichotomy for Anosov actions of b^k on compact manifolds, showing that stable and unstable leaves are either dense or the action is a suspension of a b^k-Anosov action, advancing Verjovsky's conjecture.

Contribution

It proves a key dichotomy for Anosov b^k actions, linking leaf density to suspension structures, thus making progress on Verjovsky's extended conjecture.

Findings

Stable and unstable leaves are either dense or form suspension actions.

Progress toward Verjovsky's extended conjecture for Anosov actions.

Characterization of Anosov actions via topological conjugacy.

Abstract

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.