Normal forms for non-uniform contractions

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Abstract

Let f be a measure-preserving transformation of a Lebesgue space (X,µ) and let ℱ be its extension to a bundle ℰ = X × ℝm by smooth fiber maps ℱx: ℰx → ℰfx so that the derivative of ℱ at the zero section has negative Lyapunov exponents. We construct a measurable system of smooth coordinate changes ℋx on ℰx for µ-a.e. x so that the maps Px = ℋfx°ℱx°ℋ-1x are sub-resonance polynomials in a finite dimensional Lie group. Our construction shows that such ℋx and Px are unique up to a sub-resonance polynomial. As a consequence, we obtain the centralizer theorem that the coordinate change H also conjugates any commuting extension to a polynomial extension of the same type. We apply our results to a measure-preserving diffeomorphism f with a non-uniformly contracting invariant foliation W. We construct a measurable system of smooth coordinate changes ℋx Wx → Tx W such that the maps ℋfx° f °ℋ-1x are polynomials of sub-resonance type. Moreover, we show that for almost every leaf the coordinate changes exist at each point on the leaf and give a coherent atlas with transition maps in a finite dimensional Lie group.

Original languageEnglish (US)
Pages (from-to)341-368
Number of pages28
JournalJournal of Modern Dynamics
Volume11
DOIs
StatePublished - Jan 1 2017

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

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