02-480 Gerhard Knieper and Howard Weiss

Genericity of Positive Topological Entropy for Geodesic Flows on $S^2$ (309K, pdf) Nov 22, 02

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Abstract. We show that there is a $C^\infty$ open and dense set of positively curvedmetrics on $S^2$ whose geodesic flow has positive topological entropy, andthus 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 generaltopological 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, Wysochi, and Zehnder \cite{HWZ1, HWZ2} on three dimensional Reeb flows. In the special case of geodesic flows on $S^2$, they show that the geodesic flow for a $C^\infty$ dense set of Riemannian metrics on $S^2$ possesses either a global surface of section or a heteroclinic connection. It then immediately follows from theproof of our main theorem that there is a $C^\infty$ open and dense set ofRiemannian metrics on $S^2$ whose geodesic flow has positive topological entropy. This concludes a program to show that every orientable compact surface hasa $C^\infty$ open and dense set of Riemannian metrics whose geodesic flow haspositive topological entropy

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