Flows on vector bundles and hyperbolic sets

Salamon, Dietmar; Zehnder, Eduard

doi:10.1090/S0002-9947-1988-0933310-9

Flows on vector bundles and hyperbolic sets
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by Dietmar Salamon and Eduard Zehnder

Trans. Amer. Math. Soc. 306 (1988), 623-649

DOI: https://doi.org/10.1090/S0002-9947-1988-0933310-9

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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.

References

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
D. V. Anosov and Ja. G. Sinaĭ, Certain smooth ergodic systems, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 107–172 (Russian). MR 0224771
Charles C. Conley, Hyperbolic sets and shift automorphisms, Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974) Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975, pp. 539–549. MR 0455043

Isolated invariant sets and the Morse index

Neil Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971/72), 193–226. MR 287106, DOI 10.1512/iumj.1971.21.21017

A topological persistence theorem for normally hyperbolic manifolds via the Conley index

A refinement of the Conley index and an application to the stability of hyperbolic invariant sets

R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982), no. 3, 403–438. MR 667409
J. Moser, On a theorem of Anosov, J. Differential Equations 5 (1969), 411–440. MR 238357, DOI 10.1016/0022-0396(69)90083-7
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, DOI 10.1090/S0002-9947-1985-0797044-3
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
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 428358, DOI 10.1090/S0002-9939-1977-0428358-5
Robert J. Sacker and George R. Sell, A spectral theory for linear differential systems, J. Differential Equations 27 (1978), no. 3, 320–358. MR 501182, DOI 10.1016/0022-0396(78)90057-8