A geometric criterion for positive toplogical entropy

Abstract:
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.

A closely related article is

• A surface with positive curvature and positive topological entropy by Gerhard Knieper and Howard Weiss. This paper is published in Journal of Differential Geometry 39(1994), 229-249.

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.