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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 $\left | t \right | \geq A$ 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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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