Normal forms for non-uniform contractions

Let $f$ be a measure-preserving transformation of a Lebesgue space $(X,\mu )$ and let $\text{f}$ be its extension to a bundle $\text{E}=X\times \text{Rm}$ by smooth fiber maps ${\text{f}}_x:{\text{E}}_x\to {\text{E}}_{fx}$ so that the derivative of $\text{f}$ at the zero section has negative Lyapunov exponents. We construct a measurable system of smooth coordinate changes ${\text{h}}_x$ on ${\text{E}}_x$ for $\mu$-a.e. $x$ so that the maps ${\text{p}}_x={\text{h}}_{fx}\circ {\text{f}}_x\circ {\text{h}}_x^{-1}$ are sub-resonance polynomials in a finite dimensional Lie group. Our construction shows that such ${\text{h}}_x$ and ${\text{p}}_x$ are unique up to a sub-resonance polynomial. As a consequence, we obtain the centralizer theorem that the coordinate change $\text{h}$ 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 ${\text{h}}_x:W_x\to T_{x}W$ such that the maps $\h_{fx} \circ f \circ \h_x ^{-1}$ 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.