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
DOI:
https://doi.org/10.1090/S0002-9947-1973-0314084-0
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 $\left | t \right | \geq A$ 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.
- D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
- N. P. Bhatia and G. P. Szegő, Dynamical systems: Stability theory and applications, Lecture Notes in Mathematics, No. 35, Springer-Verlag, Berlin-New York, 1967. MR 0219843
- George D. Birkhoff, Dynamical systems, American Mathematical Society Colloquium Publications, Vol. IX, American Mathematical Society, Providence, R.I., 1966. With an addendum by Jurgen Moser. MR 0209095
- R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 251664, DOI https://doi.org/10.1090/S0002-9947-1969-0251664-4
- Herbert Busemann, The geometry of geodesics, Academic Press Inc., New York, N. Y., 1955. MR 0075623
- Patrick Eberlein, Geodesic flow in certain manifolds without conjugate points, Trans. Amer. Math. Soc. 167 (1972), 151–170. MR 295387, DOI https://doi.org/10.1090/S0002-9947-1972-0295387-4
- Patrick Eberlein, Geodesic flows on negatively curved manifolds. I, Ann. of Math. (2) 95 (1972), 492–510. MR 310926, DOI https://doi.org/10.2307/1970869
- Patrick Eberlein, Geodesic flows in manifolds of nonpositive curvature, Smooth ergodic theory and its applications (Seattle, WA, 1999) Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 525–571. MR 1858545, DOI https://doi.org/10.1090/pspum/069/1858545
- Anna Grant, Surfaces of negative curvature and permanent regional transitivity, Duke Math. J. 5 (1939), no. 2, 207–229. MR 1546119, DOI https://doi.org/10.1215/S0012-7094-39-00520-X
- Gustav A. Hedlund, The dynamics of geodesic flows, Bull. Amer. Math. Soc. 45 (1939), no. 4, 241–260. MR 1563961, DOI https://doi.org/10.1090/S0002-9904-1939-06945-0
- Gustav A. Hedlund, Fuchsian groups and mixtures, Ann. of Math. (2) 40 (1939), no. 2, 370–383. MR 1503464, DOI https://doi.org/10.2307/1968925
- Gustav A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J. 2 (1936), no. 3, 530–542. MR 1545946, DOI https://doi.org/10.1215/S0012-7094-36-00246-6
- Eberhard Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl. 91 (1939), 261–304 (German). MR 1464
- Eberhard Hopf, Statistik der Lösungen geodätischer Probleme vom unstabilen Typus. II, Math. Ann. 117 (1940), 590–608 (German). MR 2722, DOI https://doi.org/10.1007/BF01450032 B. O’Neill, (to appear).
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