Parametrization of unstable manifolds and Fatou disks for parabolic skew-products

Abstract

Given a parabolic map in one dimension f (z) = z +O(z 2 ), f = Id, it is known that there exists the analogous of stable and unstable domains. That is, domains in which every point is attracted by f (or by the inverse f −1 ) towards the fixed point. In this paper we prove that there exists a natural parametrization for the unstable manifold in terms of iterates for some subset of parabolic maps. Furthermore, we prove that this parametrization is valid also in the case of skew-product maps that satisfy certain conditions. Finally, we give an application of this fact to construct Fatou disks for skew-product maps, in a similar way than in the paper [5].

Local dynamics of parabolic skew-products

“…In section 3 we recall results in one dimension. In section 4 we recall results from [26] where we found a complete description of the dynamics of a more particular class of parabolic maps on a whole neighborhood of the origin. Finally in the last section we prove the main theorem.…”