Flows on vector bundles and hyperbolic sets
Authors:
Dietmar Salamon and Eduard Zehnder
Journal:
Trans. Amer. Math. Soc. 306 (1988), 623649
MSC:
Primary 58F15; Secondary 34C35
MathSciNet review:
933310
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: This note deals with C. Conley's topological approach to hyperbolic invariant sets for continuous flows. It is based on the notions of isolated invariant sets and Morse decompositions and it leads to the concept of weak hyperbolicity.
 [1]
D.
V. Anosov, Geodesic flows on closed Riemannian manifolds of
negative curvature, Trudy Mat. Inst. Steklov. 90
(1967), 209 (Russian). MR 0224110
(36 #7157)
 [2]
D.
V. Anosov and Ja.
G. Sinaĭ, Certain smooth ergodic systems, Uspehi Mat.
Nauk 22 (1967), no. 5 (137), 107–172 (Russian).
MR
0224771 (37 #370)
 [3]
Charles
C. Conley, Hyperbolic sets and shift automorphisms, Dynamical
systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle,
Wash., 1974) Springer, Berlin, 1975, pp. 539–549. Lecture
Notes in Phys., Vol. 38. MR 0455043
(56 #13284)
 [4]
, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R.I., 1976.
 [5]
Neil
Fenichel, Persistence and smoothness of invariant manifolds for
flows, Indiana Univ. Math. J. 21 (1971/1972),
193–226. MR 0287106
(44 #4313)
 [6]
A. Floer, A topological persistence theorem for normally hyperbolic manifolds via the Conley index, preprint, RuhrUniversität Bochum, 1985.
 [7]
, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Bericht Nr. 42, RuhrUniversität Bochum, 1985.
 [8]
R.
Johnson and J.
Moser, The rotation number for almost periodic potentials,
Comm. Math. Phys. 84 (1982), no. 3, 403–438. MR 667409
(83h:34018)
 [9]
J.
Moser, On a theorem of Anosov, J. Differential Equations
5 (1969), 411–440 (German). MR 0238357
(38 #6633)
 [10]
Dietmar
Salamon, Connected simple systems and the
Conley index of isolated invariant sets, Trans.
Amer. Math. Soc. 291 (1985), no. 1, 1–41. MR 797044
(87e:58182), http://dx.doi.org/10.1090/S00029947198507970443
 [11]
James
F. Selgrade, Isolated invariant sets for flows on
vector bundles, Trans. Amer. Math. Soc. 203 (1975), 359–390.
MR
0368080 (51 #4322), http://dx.doi.org/10.1090/S0002994719750368080X
 [12]
R.
C. Churchill, John
Franke, and James
Selgrade, A geometric criterion for
hyperbolicity of flows, Proc. Amer. Math.
Soc. 62 (1976), no. 1, 137–143 (1977). MR 0428358
(55 #1382), http://dx.doi.org/10.1090/S00029939197704283585
 [13]
Robert
J. Sacker and George
R. Sell, A spectral theory for linear differential systems, J.
Differential Equations 27 (1978), no. 3,
320–358. MR 0501182
(58 #18604)
 [1]
 D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. 90 (1967). MR 0224110 (36:7157)
 [2]
 D. V. Anosov and J. G. Sinai, Some smooth ergodic systems, Russian Math. Surveys 22 (1967), 107172. MR 0224771 (37:370)
 [3]
 C. Conley, Hyperbolic sets and shift automorphisms, Dynamical Systems, Theory and Applications (J. Moser, ed.), Lecture Notes in Physics, vol. 38, Springer, New York, 1975, pp. 539549. MR 0455043 (56:13284)
 [4]
 , Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R.I., 1976.
 [5]
 N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971), 193226. MR 0287106 (44:4313)
 [6]
 A. Floer, A topological persistence theorem for normally hyperbolic manifolds via the Conley index, preprint, RuhrUniversität Bochum, 1985.
 [7]
 , A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Bericht Nr. 42, RuhrUniversität Bochum, 1985.
 [8]
 R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982), 403438. MR 667409 (83h:34018)
 [9]
 J. Moser, On a theorem of Anosov, J. Differential Equations 5 (1969), 411440. MR 0238357 (38:6633)
 [10]
 D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), 141. MR 797044 (87e:58182)
 [11]
 J. F. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc. 203 (1975), 359390. MR 0368080 (51:4322)
 [12]
 R. C. Churchill, J. Franke and J. F. Selgrade, A geometric criterion for hyperbolicity of flows, Proc. Amer. Math. Soc. 62 (1977), 137143. MR 0428358 (55:1382)
 [13]
 R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations 27 (1978), 320358. MR 0501182 (58:18604)
Similar Articles
Retrieve articles in Transactions 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/S00029947198809333109
PII:
S 00029947(1988)09333109
Article copyright:
© Copyright 1988
American Mathematical Society
