Normal forms and invariant geometric structures for dynamical systems with invariant contracting follations

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∞) contractions of ℝⁿ. 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 Poincaré and developed in the modern form for C∞ 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 semi-simple Lie groups.