## 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.

Original language | English (US) |
---|---|

Pages (from-to) | 127-141 |

Number of pages | 15 |

Journal | Journal of Differential Geometry |

Volume | 62 |

Issue number | 1 |

DOIs | |

State | Published - 2002 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Geometry and Topology