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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Geodesic flows on negatively curved manifolds. II


Author: Patrick Eberlein
Journal: Trans. Amer. Math. Soc. 178 (1973), 57-82
MSC: Primary 58F10; Secondary 53C20, 53C70
MathSciNet review: 0314084
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Abstract: Let M be a complete Riemannian manifold with sectional curvature $ K \leq 0$, SM the unit tangent bundle of M, $ {T_t}$ the geodesic flow on SM and $ \Omega \subseteq SM$ the set of nonwandering points relative to $ {T_t}$. $ {T_t}$ is topologically mixing (respectively topologically transitive) on SM if for any open sets 0, U of SM there exists $ A > 0$ such that $ \left\vert t \right\vert \geq A$ implies $ {T_t}(O) \cap U \ne \emptyset $ (respectively there exists $ t\;\varepsilon \;R$ such that $ {T_t}(O) \cap U \ne \emptyset $). For each vector $ v\;\varepsilon \;SM$ we define stable and unstable sets $ {W^s}(v),{W^{ss}}(v),{W^u}(v)$ and $ {W^{uu}}(v)$, and we relate topological mixing (respectively topological transitivity) of $ {T_t}$ to the existence of a vector $ v\; \in \;SM$ such that $ {W^{ss}}(v)$ (respectively $ {W^s}(v)$) is dense in SM. If M is a Visibility manifold (implied by $ K \leq c < 0$) and if $ \Omega = SM$ then $ {T_t}$ is topologically mixing on SM. Let $ {S_n} = $ {Visibility manifolds M of dimension n such that $ {T_t}$ is topologically mixing on SM}. For each $ n \geq 2$, $ {S_n}$ is closed under normal (Galois) Riemannian coverings. If $ M\; \in \;{S_n}$ we classify { $ v\; \in \;SM:\;{W^{ss}}(v)$ is dense in SM}, and M is compact if and only if this set = SM. We also consider the case where $ \Omega $ is a proper subset of SM.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0314084-0
PII: S 0002-9947(1973)0314084-0
Keywords: Geodesic flow, prolongational limit sets, nonwandering points, stable and unstable sets, topological transitivity, topological mixing, Axiom 1, Visibility manifold
Article copyright: © Copyright 1973 American Mathematical Society