06-34 Victoria Rayskin

Tangential homoclinic crossings of invariant manifolds, having co-dimensions higher than one. (507K, Postscript) Feb 20, 06

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Abstract. Let $f: M \rightarrow M$ denote a diffeomorphism of a smooth manifold $M$. Let $p \in M$ be its hyperbolic fixed point with stable and unstable manifolds $W_S$ and $W_U$ respectively. Assume that $W_S$ is a curve. Suppose that $W_U$ and $W_S$ have a {\it degenerate homoclinic crossing} at a point $B\neq p$, i.e., they cross at $B$ tangentially with a finite order of contact. It is shown that, subject to certain conditions on the invariant manifolds and on the eigenvalues of the linear part of $f$, a transverse homoclinic crossing will arise arbitrarily close to $B$. This proves the existence of a horseshoe structure arbitrarily close to $B$, and extends our earlier result, restricted to the case when one of the invariant manifolds has codimension one. We also improve our other result, which is the key part of the proof: the Tangential Inclination lemma.

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