Normal forms on contracting foliations: smoothness and homogeneous structure

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→ 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 C¹-small perturbations of algebraic systems.