Parametrization of unstable manifolds for parabolic skew-products

TL;DR

This paper develops a natural parametrization for unstable manifolds in parabolic and skew-product maps, enabling the construction of Fatou disks and advancing understanding of their local dynamics.

Contribution

It introduces a new parametrization method for unstable manifolds in parabolic skew-product maps, extending existing theories and applications.

Findings

Existence of a natural parametrization for unstable manifolds.

Parametrization validity for certain skew-product maps.

Application to constructing Fatou disks in parabolic maps.

Abstract

Given a parabolic map in one dimension $f(z) = z+O(z^2)$, $f \neq Id$, it is known that there exists the analogous of stable and unstable domains. That is, domains in which every point is attracted by $f$ (and 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 that are parabolic in each direction.