NORMAL FORMS AND INVARIANT GEOMETRIC STRUCTURES FOR DYNAMICAL-SYSTEMSWITH INVARIANT CONTRACTING FOLIATIONS

We present a certain version of the ''non-stationary'' normal forms th eory for extensions of topological dynamical systems (homeomorphisms o f compact metrizable spaces) by smooth (C-infinity) contractions of R- n. The central concept is a notion of a sub-resonance relation which i s an appropriate generalization of the notion of resonance between the eigenvalues of a matrix which plays a similar role in the local norma l forms theory going back to Poincare and developed in the modern form for C-infinity maps by S. Sternberg and K.-T. Chen. Applicability of these concepts depends on the narrow band condition, a certain collect ion of inequalities between the relative rates of contraction in the f ibers. One of the ways to formulate our first conclusion (the sub-reso nance normal form theorem) is to say that there is a continuous invari ant family of geometric structures in the fibers whose automorphism gr oups 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 r ather close to the normal forms in a neighborhood of an invariant mani fold, studied in the literature, the centralizer theorem seems to be n ew even in the classical local case. The principal situation where our analysis applies is a smooth system on a compact manifold with an inv ariant contracting foliation. In this case we also establish smoothnes s of the sub-resonance normal form along the fibers. The principal app lications so far are to local differentiable rigidity of algebraic Ano sov actions of higher-rank abelian groups and algebraic Anosov and par tially hyperbolic actions of lattices in higher-rank semi-simple Lie g roups.