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Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations. (English) Zbl 0988.37063
Summary: We present a certain version of the “non-stationary” normal forms theory for extensions of topological dynamical systems (homeomorphisms of compact metrizable spaces) by smooth \((C^\infty)\) contractions of \(\mathbb{R}^n\). The central concept is a notion of a sub-resonance relation which is an appropriate generalization of the notion of resonance between the eigenvalues of a matrix which plays a similar role in the local normal forms theory going back to PoincarĂ© and developed in the modern form for \(C^\infty\) maps by S. Sternberg and K.-T. Chen. Applicability of these concepts depends on the narrow band condition, a certain collection of inequalities between the relative rates of contraction in the fibers. One of the ways to formulate our first conclusion (the sub-resonance normal form theorem) is to say that there is a continuous invariant family of geometric structures in the fibers whose automorphism groups are certain finite-dimensional Lie groups.
Our central result is the joint normal form for the centralizer for an extension satisfying the narrow band condition. While our non-stationary normal forms are rather close to the normal forms in a neighborhood of an invariant manifold, studied in the literature, the centralizer theorem seems to be new even in the classical local case. The principal situation where our analysis applies is a smooth system on a compact manifold with an invariant contracting foliation. In this case we also establish smoothness of the sub-resonance normal form along the fibers. The principal applications so far are to local differentiable rigidity of algebraic Anosov actions of higher-rank abelian groups and algebraic Anosov and partially hyperbolic actions of lattices in higher-rank semisimple Lie groups.

MSC:
37G05 Normal forms for dynamical systems
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
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