Mathematical Research Letters

Volume 5 (1998)

Number 2

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

Pages: 149 – 163



M. Guysinsky

A. Katok


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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