Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A geometric criterion for hyperbolicity of flows


Authors: R. C. Churchill, John Franke and James Selgrade
Journal: Proc. Amer. Math. Soc. 62 (1977), 137-143
MSC: Primary 58F15; Secondary 34C35
MathSciNet review: 0428358
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A chain recurrent set for a flow on a compact manifold is hyperbolic if and only if it is quasi-hyperbolic. This result gives an easy proof that the geodesic flow on a compact manifold of negative curvature is hyperbolic.


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

  • [1] D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, Trudy Mat. Inst. Steklov 90 (1967); English transl., Amer. Math. Soc., Providence, R. I., 1969. MR 36 #7157; 39 #3527. MR 0224110 (36:7157)
  • [2] M. Atiyah, K-theory, 2nd ed., Benjamin, New York, 1967. MR 36 #7130. MR 0224083 (36:7130)
  • [3] R. C. Churchill, G. Pecelli and D. L. Rod, Hyperbolic periodic orbits, J. Differential Equations (to appear). MR 0445545 (56:3885)
  • [4] C. C. Conley, The gradient structure of a flow: I, IBM Research, RC 3932 (#17806), Yorktown Heights, New York, July 17, 1972.
  • [5] J. J. Duistermaat, Fourier integral operators, Courant Inst, of Mathematical Sciences, New York Univ., New York, 1973. MR 0451313 (56:9600)
  • [6] P. Eberlein, When is a geodesic flow of Anosov type?. I, J. Differential Geometry 8 (1973), 437-463. MR 0380891 (52:1788)
  • [7] J. Franke and J. Selgrade, Hyperbolicity and chain recurrence, J. Differential Equations (to appear). MR 0467834 (57:7685)
  • [8] -, Abstract $ \omega $-limit sets, chain recurrent sets, and basic sets for flows, Proc. Amer. Math. Soc. (to appear).
  • [9] W. Klingenberg, Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2) 99 (1974), 1-13. MR 0377980 (51:14149)
  • [10] R. Mañé, Persistent manifolds are normally hyperbolic, Bull. Amer. Math. Soc. 80 (1974), 90-91. MR 49 #4043. MR 0339283 (49:4043)
  • [11] J. W. Milnor, Morse theory, Princeton Univ. Press, Princeton, N. J., 1963. MR 29 #634. MR 0163331 (29:634)
  • [12] R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations 15 (1974), 429-458. MR 49 #6209. MR 0341458 (49:6209)
  • [13] J. F. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc. 203 (1975), 359-390. MR 0368080 (51:4322)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F15, 34C35

Retrieve articles in all journals with MSC: 58F15, 34C35


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0428358-5
PII: S 0002-9939(1977)0428358-5
Article copyright: © Copyright 1977 American Mathematical Society