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 $H _x:W_x\to T_xW$ 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.