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