98-208 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 non-transversally, 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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