Gerhard Knieper and Howard Weiss
Genericity of Positive Topological Entropy for Geodesic Flows on $S^2$
(309K, pdf)

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