On classification of resonance-free Anosov \(\mathbb{Z}^k\) actions (Q2470256)

The authors consider the problem of classification of higher-rank abelian Anosov actions, obtaining the following assertions. Let \(\alpha\) be an action of \(\mathbb{Z}^k\), \(k \geq 2\), by \(C^\infty\) diffeomorphisms of a compact connected smooth manifold \(M\). Then, if (1) all nontrivial elements of \(\alpha\) are Anosov and at least one is transitive; (2) \(\alpha\) is uniformly quasiconformal on each coarse Lyapunov distribution; (3) \(\alpha\) is totally nonsymplectic; and (4) for any Lyapunov functions \(\chi_i\), \(\chi_j\), and \(\chi_l\), the functional \((\chi_i - \chi_j)\) is not proportional to \(\chi_l\), then a finite cover of \(\alpha\) is \(C^\infty\) conjugate to a \(\mathbb{Z}^k\) action by affine automorphisms of a torus. Further, under the conditions (1)-(3), the Lyapunov functionals are the same for all invariant measures, and, moreover, there exists a Hölder continuous Riemannian metric on \(M\) such that, for any \(a \in \mathbb{Z}^k\) and any Lyapunov functional \(\chi\), \(\| Da(v) \| = e^{\chi(a)} \| v \|\) for any vector \(v\) in the corresponding Lyapunov distribution.