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

Abstract

Given a germ of biholomorphism 𝐹 ∈ Dif f (ℂ𝑛, 0) with a formal invariant curve Γ such that the multiplier of the restricted formal diffeomorphism 𝐹|Γ is a root of unity or satisfies |(𝐹|Γ)′(0)| < 1, we prove that either Γ 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 toΓ andwhose union eventually contains every orbit asymptotic to Γ. This result generalizes to the case where Γ is a formal periodic curve.