Stable manifolds of two-dimensional biholomorphisms asymptotic to formal curves

TL;DR

This paper studies the stable manifolds of two-dimensional holomorphic diffeomorphisms near formal invariant curves, proving finiteness and describing their structure under certain dynamical conditions.

Contribution

It establishes the existence and finiteness of stable manifolds for diffeomorphisms with invariant formal curves under hyperbolic or neutral conditions, generalizing to periodic curves.

Findings

Finitely many stable manifolds contain all orbits asymptotic to the formal curve.

Stable manifolds are either open domains or parabolic curves.

Results extend to formal periodic invariant curves.

Abstract

Let $F\in\mathrm{Diff}(\mathbb{C}^2,0)$ be a germ of a holomorphic diffeomorphism and let $\Gamma$ be an invariant formal curve of $F$. Assume that the restricted diffeomorphism $F|_{\Gamma}$ is either hyperbolic attracting or rationally neutral non-periodic (these are the conditions that the diffeomorphism $F|_{\Gamma}$ should satisfy, if $\Gamma$ were convergent, in order to have orbits converging to the origin). Then we prove that $F$ has finitely many stable manifolds, either open domains or parabolic curves, consisting of and containing all converging orbits asymptotic to $\Gamma$. Our results generalize to the case where $\Gamma$ is a formal periodic curve of $F$.