Multidimensional singular $\lambda$-lemma

Rayskin, Victoria

Electronic Journal of Differential Equations, Tome 2003 (2003).

Résumé

Summary: The well known $\lambda$-lemma [3] states the following: Let $f$ be a $C^1$-diffeomorphism of $\mathbb{R}^n$ with a hyperbolic fixed point at 0 and $m$- and $p$-dimensional stable and unstable manifolds $W^S$ and $W^U$, respectively ($m+p=n$). Let $D$ be a $p$-disk in $W^U$ and $w$ be another p-disk in $W^U$ meeting $W^S$ at some point $A$ transversely. Then $\bigcup_{n\geq 0} f^n(w)$ contains $p$-disk arbitrarily $C^1$-close to $D$. In this paper we will show that the same assertion still holds outside of an arbitrarily small neighborhood of 0, even in the case of non-transverse homoclinic intersections with finite order of contact, if we assume that 0 is a low order non-resonant point.

Détail

Classification : 37B10, 37C05, 37C15, 37D10
Keywords: homoclinic tangency, invariant manifolds, lambda-Lemma, order of contact