ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

9. Ordinary Differential Equations and Dynamical Systems

1991 MS Classification: 34 Rayskin, Victoria Degenerate homoclinic crossings Let F : M → M denote a diffeomorphism of a C^{∞}-manifold M. Let p ∈ 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 cross. Suppose W_{U} and W_{S} cross non-transversally, i. e., they have a degenerate homoclinic crossing. 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 codimension of one of the invariant manifolds is 1. Also, it is shown that in the planar case the Hirsch’s linearizability assumption is unnecessary.