 98208 Victoria Rayskin
 Degenerate Homoclinic Crossings
(35K, LATeX 2e)
Mar 18, 98

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Abstract. Let $F: M \rightarrow M$ denote a diffeomorphism of a $C^{\infty}$manifold $M$.
Let $p \in M$ be a hyperbolic fixed point with stable and unstable manifolds $W_S$ and $W_U$ respectively. Let $q$ be a point other than $p$ at which these manifolds meet. Suppose $W_U$ and $W_S$ meet nontransversally, i.e., they have a {\it degenerate homoclinic contact}.
It is shown that, subject to $C^1$linearizability and diagonalizability of the linear part of a map, a transverse crossing will arise and imply a horseshoe structure, if the dimension of one of the invariant manifolds is $1$. A more general manifolds are considered. Also, it is shown that in the planar case the Hirsch's linearizability assumption is unnecessary.
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