| C ∞ genericity of positive topological entropy for geodesic flows (2008) | |||||||||||||||
Abstract | |||||||||||||||
| We show that there is a C ∞ open and dense set of positively curved metrics on S 2 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 S 2, 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 S 2), 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 S 2 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. | |||||||||||||||
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