Electron. J. Diff. Eqns., Vol. 2003(2003), No. 38, pp. 1-9.

Multidimensional singular $\lambda$-lemma

Victoria Rayskin

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 $p$-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.

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

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Victoria Rayskin
Department of Mathematics
MS Bldg, 6363
University of California at Los Angeles, 155505
Los Angeles, CA 90095, USA
email: vrayskin@math.ucla.edu
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