### 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 × ℝ^{m} by smooth fiber maps ℱ_{x}: ℰ_{x} → ℰ_{fx} so that the derivative of ℱ at the zero section has negative Lyapunov exponents. We construct a measurable system of smooth coordinate changes ℋ_{x} on ℰ_{x} for µ-a.e. x so that the maps P_{x} = ℋ_{fx}°ℱ_{x}°ℋ^{-1}_{x} are sub-resonance polynomials in a finite dimensional Lie group. Our construction shows that such ℋ_{x} and P_{x} 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 ℋ_{x} W_{x} → T_{x} W such that the maps ℋ_{fx}° f °ℋ^{-1}_{x} 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 language | English (US) |
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Pages (from-to) | 341-368 |

Number of pages | 28 |

Journal | Journal of Modern Dynamics |

Volume | 11 |

DOIs | |

State | Published - Jan 1 2017 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Applied Mathematics