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

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 $|(F{|}_{\Gamma }{)}^{\prime }(0)|<1$, 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.

Part of section 5 in this article is an upgrade of some results in arXiv:1411.2945 that were not included in the final published version

References in corpus (3)

• The embedding flows of $C^{\infty }$ hyperbolic diffeomorphisms
• Finite dimensional groups of local diffeomorphisms
• Solution to Briot and Bouquet problem on singularities of differential equations