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… (More)
Long periodic orbits of hyperbolic dynamics do not exist as independent individuals but rather come in closely packed bunches. Under weak resolution a bunch looks like a single orbit in configuration space, but close inspection reveals topological orbit-to-orbit differences. The construction principle of bunches involves close self-" encounters " of an… (More)
Let M be a Hadamard manifold of dimension 3 whose sectional curvature satisfies −b 2 ≤ K ≤ −a 2 < 0 and whose curvature tensor satisfies ∇R ≤ C for suitable constants 0 < a ≤ b and C ≥ 0. We show that M is of constant sectional curvature, provided M is asymptotically harmonic. This was previously only known, if M admits a compact quotient.