Normal forms for non-uniform contractions
Boris Kalinin - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States (email) Abstract: Let $f$ be a measure-preserving transformation of a Lebesgue space $(X,\mu)$ and let $\mathfrak{F}$ be its extension to a bundle $\mathfrak{E} = X \times\mathbb{R}^m$ by smooth fiber maps $\mathfrak{F}_x : \mathfrak{E}_x \to \mathfrak{E}_{fx}$ so that the derivative of $\mathfrak{F}$ at the zero section has negative Lyapunov exponents. We construct a measurable system of smooth coordinate changes $\mathfrak{H}_x$ on $\mathfrak{E}_x$ for $\mu$-a.e. $x$ so that the maps $\mathfrak{P}_x =\mathfrak{H}_{fx} \circ \mathfrak{F}_x \circ \mathfrak{H}_x^{-1}$ are sub-resonance polynomials in a finite dimensional Lie group. Our construction shows that such $\mathfrak{H}_x$ and $\mathfrak{P}_x$ are unique up to a sub-resonance polynomial. As a consequence, we obtain the centralizer theorem that the coordinate change $\mathfrak{H}$ 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 $\mathfrak{H}_x: W_x \to T_xW$ such that the maps $\mathfrak{H}_{fx} \circ f \circ \mathfrak{H}_x^{-1}$ 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.
Keywords: Normal form, contracting foliation, non-uniform hyperbolicity, Lyapunov
exponents, polynomial map, homogeneous structure.
Received: May 2016; Revised: March 2017; Available Online: April 2017. |