A geometric criterion for hyperbolicity of flows
HTML articles powered by AMS MathViewer
- by R. C. Churchill, John Franke and James Selgrade PDF
- Proc. Amer. Math. Soc. 62 (1977), 137-143 Request permission
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.
- D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
- M. F. Atiyah, $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1967. Lecture notes by D. W. Anderson. MR 0224083
- D. L. Rod, G. Pecelli, and R. C. Churchill, Hyperbolic periodic orbits, J. Differential Equations 24 (1977), no. 3, 329–348. MR 445545, DOI 10.1016/0022-0396(77)90003-1 C. C. Conley, The gradient structure of a flow: I, IBM Research, RC 3932 (#17806), Yorktown Heights, New York, July 17, 1972.
- J. J. Duistermaat, Fourier integral operators, Courant Institute of Mathematical Sciences, New York University, New York, 1973. Translated from Dutch notes of a course given at Nijmegen University, February 1970 to December 1971. MR 0451313
- Patrick Eberlein, When is a geodesic flow of Anosov type? I,II, J. Differential Geometry 8 (1973), 437–463; ibid. 8 (1973), 565–577. MR 380891
- John E. Franke and James F. Selgrade, Hyperbolicity and chain recurrence, J. Differential Equations 26 (1977), no. 1, 27–36. MR 467834, DOI 10.1016/0022-0396(77)90096-1 —, Abstract $\omega$-limit sets, chain recurrent sets, and basic sets for flows, Proc. Amer. Math. Soc. (to appear).
- Wilhelm Klingenberg, Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2) 99 (1974), 1–13. MR 377980, DOI 10.2307/1971011
- Ricardo Mañé, Persitent manifolds are normally hyperbolic, Bull. Amer. Math. Soc. 80 (1974), 90–91. MR 339283, DOI 10.1090/S0002-9904-1974-13366-5
- J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331
- Robert J. Sacker and George R. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations 15 (1974), 429–458. MR 341458, DOI 10.1016/0022-0396(74)90067-9
- James F. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc. 203 (1975), 359–390. MR 368080, DOI 10.1090/S0002-9947-1975-0368080-X
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 62 (1977), 137-143
- MSC: Primary 58F15; Secondary 34C35
- DOI: https://doi.org/10.1090/S0002-9939-1977-0428358-5
- MathSciNet review: 0428358