A geometric criterion for hyperbolicity of flows
Authors:
R. C. Churchill, John Franke and James Selgrade
Journal:
Proc. Amer. Math. Soc. 62 (1977), 137143
MSC:
Primary 58F15; Secondary 34C35
MathSciNet review:
0428358
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Abstract: A chain recurrent set for a flow on a compact manifold is hyperbolic if and only if it is quasihyperbolic. This result gives an easy proof that the geodesic flow on a compact manifold of negative curvature is hyperbolic.
 [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, Ktheory, 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), 437463. 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 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), 113. MR 0377980 (51:14149)
 [10]
R. Mañé, Persistent manifolds are normally hyperbolic, Bull. Amer. Math. Soc. 80 (1974), 9091. 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), 429458. 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), 359390. MR 0368080 (51:4322)
 [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, Ktheory, 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), 437463. 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 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), 113. MR 0377980 (51:14149)
 [10]
 R. Mañé, Persistent manifolds are normally hyperbolic, Bull. Amer. Math. Soc. 80 (1974), 9091. 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), 429458. 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), 359390. MR 0368080 (51:4322)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197704283585
PII:
S 00029939(1977)04283585
Article copyright:
© Copyright 1977
American Mathematical Society
