Normal forms for non-uniform contractions

Abstract

Let $f$ be a measure-preserving transformation of a Lebesgue space $(X,\mu)$ and let $\f$ be its extension to a bundle $\E = X \times\Rm$ by smooth fiber maps $\f_x : \E_x \to \E_{fx}$ 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 $\mu$-a.e. $x$ so that the maps $\p_x =\h_{fx} \circ \f_x \circ \h_x ^{-1}$ 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 $\h_x: W_x \to T_xW$ 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.