Stable manifolds of two-dimensional biholomorphisms asymptotic to formal 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 $F$.

References in corpus (1)

• Parabolic curves of diffeomorphisms asymptotic to formal invariant curves

Cited by in corpus (1)

• Characteristic directions of two-dimensional biholomorphisms