Parabolic curves of diffeomorphisms asymptotic to formal invariant curves

Abstract

Abstract. We prove that if F is a tangent to the identity diffeomorphism at 0 ∈ C 2 and Γ is a formal invariant curve of F then there exists a parabolic curve (attracting or repelling) of F asymptotic to Γ. The result is a consequence of a more general one in arbitrary dimension, where we prove the existence of parabolic curves of a tangent to the identity diffeomorphism F at 0 ∈ C n asymptotic to a given formal invariant curve under some additional conditions, expressed in terms of a reduction of F to a special normal form by means of blow-ups and ramifications along the formal curve.

Stable Manifolds of Two-dimensional Biholomorphisms Asymptotic to Formal Curves

“…As a consequence of our main result, in Section 7 we prove the following theorem, which generalizes results in [5] and [12].…”

“…Proof. We argue as in [18,Lemma 3.7]. Observe that, since D 2 (x) is diagonal and commutes with C 2 , E(x) is a fundamental solution of the linear system…”

“…In particular, they are obtained as fixed points of an adequate continuous map. Instead of using Banach fixed point theorem as in [15], [16], [18] or [17], we use Schauder fixed point theorem, which simplifies significantly the technical computations of the proof. The lack of uniqueness in Schauder's theorem will be compensated by an easy alternative argument.…”

“…In the two-dimensional case, the problem of the existence of stable manifolds of F has been addressed by several authors. The existence of one-dimensional stable manifolds, usually called parabolic curves (when they do not contain the origin), has been studied for example by Ueda [19] when F is semihyperbolic; by Écalle [7], Hakim [9], Abate [1], Abate, Bracci and Tovena [2], Molino [13], Brochero, Cano and López [6] and López and Sanz [12] when F is tangent to the identity; by Bracci and Molino [5] when F is quasi-parabolic. The existence of open stable manifolds has been treated for example by Ueda [18] in the semihyperbolic case; by Weickert [21], Hakim [9], Vivas [20], Rong [16] in the tangent to the identity case.…”

Characteristic directions of two-dimensional biholomorphisms

“…This question has been addressed by several authors.Écalle [Eca85] and Hakim [Hak98] gave a positive answer in the case where [v] is non-degenerate, showing that there exist at least k parabolic curves tangent to [v]. In the case where [v] is degenerate, there are partial answers by Molino [Mol09], Vivas [Viv12] and López et al [LS18,LRRS19], among others, which guarantee, under some additional hypotheses, the existence of parabolic curves tangent to [v] or of parabolic domains along [v]. In the particular case where F has an isolated fixed point at the origin, even if all of its characteristic directions are degenerate, Abate proved L. López-Hernanz and R. Rosas in [Aba01] (see also [BCL08]) that one of the characteristic directions of F supports at least k parabolic curves.…”