Normal forms for non-uniform contractions

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.