Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F10, 53C20, 53C70

Retrieve articles in all journals with MSC: 58F10, 53C20, 53C70

Additional Information

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