Normal forms on contracting foliations: smoothness and homogeneous structure

Abstract

In this paper we consider a diffeomorphism f of a compact manifold \(\mathcal {M}\) which contracts an invariant foliation W with smooth leaves. If the differential of f on TW has narrow band spectrum, there exist coordinates \({\mathcal {H}}_x:W_x\rightarrow T_xW\) in which \(f|_W\) has polynomial form. We present a modified approach that allows us to construct maps \({\mathcal {H}}_x\) 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.