Parabolic curves of diffeomorphisms asymptotic to formal invariant curves

TL;DR

This paper proves the existence of parabolic curves asymptotic to formal invariant curves for tangent to the identity diffeomorphisms in complex dimensions, extending to higher dimensions under certain conditions.

Contribution

It establishes the existence of parabolic curves asymptotic to formal invariant curves for tangent to the identity diffeomorphisms in any dimension, generalizing previous results.

Findings

Existence of parabolic curves asymptotic to formal invariant curves.

Extension of results to higher dimensions with additional conditions.

Use of blow-ups and ramifications for normal form reduction.

Abstract

We prove that if $F$ is a tangent to the identity diffeomorphism at $0\in\mathbb{C}^2$ and $\Gamma$ is a formal invariant curve of $F$ then there exists a parabolic curve (attracting or repelling) of $F$ asymptotic to $\Gamma$. The result is a consequence of a more general one in arbitrary dimension, where we prove the existence of parabolic curves of a tangent to the identity diffeomorphism $F$ at $0\in\mathbb{C}^n$ asymptotic to a given formal invariant curve under some additional conditions, expressed in terms of a reduction of $F$ to a special normal form by means of blow-ups and ramifications along the formal curve.