Stable manifolds of biholomorphisms in $\mathbb{C}^n$ asymptotic to formal curves

TL;DR

This paper investigates the structure of stable manifolds associated with biholomorphic germs in complex space, showing how orbits asymptotic to formal invariant curves are contained in finite stable manifolds, generalizing previous results.

Contribution

It establishes the existence and structure of stable manifolds asymptotic to formal invariant curves for biholomorphisms with specific multipliers, extending known results to formal periodic curves.

Findings

Either the formal invariant curve is contained in the set of periodic points.

Existence of finite stable manifolds with orbits asymptotic to the curve.

Union of stable manifolds covers all orbits asymptotic to the curve.

Abstract

Given a germ of biholomorphism $F\in\mathrm{Diff}(\mathbb{C}^n,0)$ with a formal invariant curve $\Gamma$ such that the multiplier of the restricted formal diffeomorphism $F|_\Gamma$ is a root of unity or satisfies $|(F|_\Gamma)'(0)|<1$, we prove that either $\Gamma$ is contained in the set of periodic points of $F$ or there exists a finite family of stable manifolds of $F$ where all the orbits are asymptotic to $\Gamma$ and whose union eventually contains every orbit asymptotic to $\Gamma$. This result generalizes to the case where $\Gamma$ is a formal periodic curve.