Global rigidity of higher rank abelian Anosov algebraic actions. (English) Zbl 1312.37028
The following main theorem is proved in the paper:
Suppose \(\alpha:\mathbb{Z}^r\curvearrowright M\) is a \(C^\infty\) action on a compact infra-nilmanifold \(M\), where \(r\geq 2\), and let \(\rho\) be its linearization.
Assume that there exists \(n_0\in\mathbb R^n\) such that \(\alpha(n_0)\) is an Anosov diffeomorphism, and that \(\rho\) has no rank-one factor. Then \(\alpha\) is conjugate to \(\rho\) by a \(C^\infty\) diffeomorphism.
Suppose \(\alpha:\mathbb{Z}^r\curvearrowright M\) is a \(C^\infty\) action on a compact infra-nilmanifold \(M\), where \(r\geq 2\), and let \(\rho\) be its linearization.
Assume that there exists \(n_0\in\mathbb R^n\) such that \(\alpha(n_0)\) is an Anosov diffeomorphism, and that \(\rho\) has no rank-one factor. Then \(\alpha\) is conjugate to \(\rho\) by a \(C^\infty\) diffeomorphism.
Reviewer: Jan Andres (Olomouc)
MSC:
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
| 32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
| 37C15 | Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems |
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