C∞ Genericity of Positive Topological Entropy for Geodesic Flows on S2

Abstract

We show that there is a C∞ open and dense set of positively curved metrics on S2 whose geodesic flow has positive topological entropy, and thus exhibits chaotic behavior. The geodesic flow for each of these metrics possesses a horseshoe and it follows that these metrics have an exponential growth rate of hyperbolic closed geodesics. The positive curvature hypothesis is required to ensure the existence of a global surface of section for the geodesic flow. Our proof uses a new and general topological criterion for a surface diffeomorphism to exhibit chaotic behavior. Very shortly after this manuscript was completed, the authors learned about remarkable recent work by Hofer, Wysocki, and Zehnder [14, 15] on three dimensional Reeb flows. In the special case of geodesic flows on S2, they show that if the geodesic flow has no parabolic closed geodesics (this holds for an open and C∞ dense set of Riemannian metrics on S2), then it possesses either a global surface of section or a heteroclinic orbit. It then immediately follows from the proof of our main theorem that there is a C∞ open and dense set of Riemannian metrics on S2 whose geodesic flow has positive topological entropy. This concludes a program to show that every orientable compact surface has a C∞ open and dense set of Riemannian metrics whose geodesic flow has positive topological entropy.