We prove that a diffeomorphism possessing a homoclinic point with a topological crossing (possibly with infinite order contact) has positive topological entropy, along with an analogous statement for heteroclinic points. We apply these results to study area-preserving perturbations of area-preserving surface diffeomorphisms possessing homoclinic and double heteroclinic connections. In the heteroclinic case, the perturbed map can fail to have positive topological entropy only if the perturbation preserves the double heteroclinic connection or if it creates a homoclinic connection. In the homoclinic case, the perturbed map can fail to have positive topological entropy only if the perturbation preserves the connection. These results significantly simplify the application of the Poincare-Arnold-Melnikov-Sotomayor method. The results apply even when the contraction and expansion at the fixed point are subexponential.
This article has appeared in Communications in Mathematical Physics 172(1995), 95-118.
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Authors' addresses: Keith Burns Department of Mathematics Northwestern University Evanston, IL 60208-2730 burns followed by math.northwestern.edu Howard Weiss Mathematics Department Pennsylvania State University University Park, PA 16802 weiss followed by math.psu.edu.