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

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. 1.Extensions of continuous maps by contractions 1.1. Linear extensions. Let X be a compact metric space, f : X → X a homeomorphism (continuous dynamical system), V a vector bundle over X with projection π : V → X, F : V → V a continuous invertible linear extension of f . Let us fix a continuous family of Riemannian metrics in the fibers. Extension F is called a contraction if ‖DF‖ < 1. Consider the induced operator F ∗ in the Banach space of continuous sections of V provided with the uniform norm for F ∗v(x) = F (v(f−1x). Under a very mild assumption that non–periodic points of f are dense the spectrum of the operator F ∗ is the union of finitely many closed annuli centered at the origin. Hence the characteristic set of F , χ(F ) = {λ ∈ R− : expλ ∈ spF ∗} is the union of finitely many intervals. Let us denote these intervals by ∆1, . ∆l. Let ∆i = [λi, μi] and assume that the intervals are ordered in increasing order so that λi+1 > μi. Let us assume that the space X is connected or that the map f is topologically transitive. Then the bundle V splits into the direct sum of F–invariant subbundles V1, . , Vl such that χ(F ) Vi = ∆i, i = 1, . , l [K1]. Let mi be the dimension of the sub-bundle Vi. Denote the given scalar product in the fiber Vx over the point x by < ·, · >x. Given > 0 we will call the standard –Lyapunov scalar product (or metric) the scalar product Normal forms and geometric structures 3 < ·, · >x, in Vx defined as follows (Cf. [K2,Section 1.1]): For u, v ∈ Vi(x), i = 1, . , l: