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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)

- Alexander Altland, Petr Braun, Fritz Haake, Stefan Heusler, Gerhard Knieper, Sebastian Müller
- 2009

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)

- JENS HEBER, GERHARD KNIEPER, HEMANGI M. SHAH
- 2005

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.

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