Normal forms on contracting foliations: smoothness and homogeneous structure

TL;DR

This paper studies the smoothness and structure of contracting foliations under certain spectral conditions, showing that the dynamics can be represented in polynomial form with smooth coordinate changes and a Lie group structure.

Contribution

It introduces a modified approach to construct smooth leaf-dependent coordinate maps that form a coherent atlas with Lie group transition maps, extending to small perturbations of algebraic systems.

Findings

Existence of polynomial form coordinates for contracting foliations with narrow band spectrum

Construction of smooth leaf-dependent coordinate maps

Transition maps form a finite dimensional Lie group

Abstract

In this paper we consider a diffeomorphism $f$ of a compact manifold $M$ which contracts an invariant foliation $W$ with smooth leaves. If the differential of $f$ on $TW$ has narrow band spectrum, there exist coordinates $H _x:W_x\to T_xW$ in which $f|_W$ has polynomial form. We present a modified approach that allows us to construct maps $H_x$ that depend smoothly on $x$ along the leaves of $W$. Moreover, we show that on each leaf they give a coherent atlas with transition maps in a finite dimensional Lie group. Our results apply, in particular, to $C^1$-small perturbations of algebraic systems.