Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 38, pp. 1-9.
Title: Multidimensional singular $\lambda$-lemma
Author: Victoria Rayskin (Univ. of California at Los Angeles, USA)
Abstract:
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 $p$-disks 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.
Submitted November 4, 2002. Published April 11, 2003.
Math Subject Classifications: 37B10, 37C05, 37C15, 37D10.
Key Words: Homoclinic tangency; invariant manifolds; $\lambda$-Lemma;
order of contact; resonance