Normal forms for non-uniform contractions

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Let f be a measure-preserving transformation of a Lebesgue space (X,μ) and let be its extension to a bundle = X × by smooth fiber maps ₓ: ₓ → so that the derivative of at the zero section has negative Lyapunov exponents. We construct a measurable system of smooth coordinate changes ₓ on ₓ for μ-a.e. x so that the maps ₓ = ∘ ₓ ∘ ₓ ⁻¹ are sub-resonance polynomials in a finite dimensional Lie group. Our construction shows that such ₓ and ₓ are unique up to a sub-resonance polynomial. As a consequence, we obtain the centralizer theorem that the coordinate change 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 ₓ: Wₓ → TₓW such that the maps ∘ f ∘ ₓ ⁻¹ 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.