Mathematical Research Letters

Volume 5, Issue 2, March 1998 pp. 149-163.

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

Authors: M. Guysinsky and A. Katok
Author institution: Pennsylvania State University

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 $\Bbb R^n$. The central concept is a notion of a {\it 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 ${C^{\infty}}$ maps by S. Sternberg and K.-T. Chen. Applicability of these concepts depends on the {\it 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.

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