The morse spectrum of linear flows on vector bundles

Transactions of the American Mathematical Society

Volumn 348, Issue 11, 1996, Pages 4355-4388

  Colonius, Fritz ^a   Kliemann, Wolfgang ^b  

^a UNIVERSITY OF AUGSBURG   (Germany)

^b IOWA STATE UNIVERSITY   (United States)

Author keywords

Chain recurrence; Dichotomy spectrum; Ergodic theory; Floquet spectrum; Hyperbolic flows; Lyapunov exponents; Oselede spectrum; Shadowing lemma; Topological spectrum

Indexed keywords

EID: 21444456372     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-96-01524-3     Document Type: Article

References (37)

1. [Ak] Akin, Ethan, The General Topology of Dynamical Systems, American Mathematical Society, Graduate Studies in Mathematics 1 (1993).MR 94f:58041
2. [AN] Arnold, L., D.C. Nguyen, Generic properties of Lyapunov exponents, Random& Computational Dynamics 2 (1994), 335-345.MR 95m:28018
3. [Br1] Bronstein, I.U, Transversality implies structural stability, Soviet Math. Dokl. 23 (1981), 251-254,
4. (= Dokl. Akad. Nauk SSSR 257 (1981), 257-268).MR 82h:58038
5. [Br2] Bronstein, I.U., Nonautonomous Dynamical Systems (1984), Kishinev (in Russian).
6. [BC] Bronstein, I.U., V.F. Chernii, Linear extensions satisfying Perron's condition. I, Differential Equations 14 (1978), 1234-1243.MR 80c:54046 (Russian original)
7. [Ca] Carverhill, A., Flows of stochastic dynamical systems: ergodic theory, Stochastics 14 (1985), 273-317.MR 87c:58059
8. [CK1] Colonius, F., W. Kliemann, Lyapunov exponents of control flows, in Lyapunov Exponents, Arnold, L., H. Crauel, J.-P. Eckmann (eds.), Springer LN Mathematics 1486 (1991), 331-365.MR 93f:93118
9. [CK2] Colonius, F., W. Kliemann, Remarks on ergodic theory of stochastic flows and control flows, in Diffusion Processes and Related Problems in Analysis, Vol. II, M. Pinsky and V. Wihstutz (eds.), Birkhauser (1991), 203-240.MR 93i:60110
10. [CK3] Colonius, F., W. Kliemann, Some aspects of control systems as dynamical systems, J. Dynamics Diff. Equations 5 (1993), 469-494.MR 94g:93063
11. [CK4] Colonius, F., W. Kliemann, Linear control semigroups acting on projective space, J. Dynamics Diff. Equa. 5 (1993), 495-528.MR 94g:93064
12. [CK5] Colonius, F., W. Kliemann, Limit behavior and genericity for nonlinear control systems, J. Differential Equations 109 (1994), 8-41. CMP 94:11
13. [CK6] Colonius, F., W. Kliemann, The Lyapunov spectrum of families of time-varying matrices, Trans. Amer.Math. Soc. 348 (1996), 4389-4408.
14. [CK7] Colonius, F., W. Kliemann, Asymptotic null-controllability of bilinear systems, Geometry in Nonlinear Control and Differential Inclusions (Warsaw, 1993), Banach Center Publ., 32, Polish Acad. Sci., Warsaw, 1995, 139-148. CMP 96:05
15. [CKP] Coomes, B.A., H. Kocak, K.J. Palmer, A shadowing theorem for ordinary differential equations, Z. Angew.Math. Phys. 46 (1995), 85-106.MR 96b:58085
16. [Cn] Conley, C., Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series, no. 38, American Mathematical Society, Providence (1978).MR 80c:58009
17. [Cp] Coppel, W.A., Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, 1978.MR 58:1332
18. [DK] Daleckii, Ju.L., M.G. Krein, Stability of Solutions of Differential Equations in Banach Space, Translations of Math.Monographs, Vol. 43, Amer.Math. Soc., Providence, R.I. (1974).MR 50:5126
19. [EJ] Ellis, R., R. Johnson, Topological dynamics and linear differential systems, J.Diff. Equations 44 (1982), 21-39.MR 83c:54058
20. [FS] Franke, J.E., J.F. Belgrade, Hyperbolicity and chain recurrence, J.Diff. Equations 26 (1977), 27-36.MR 57:7685
21. [HPPS] Hirsch, M., J. Palis, C. Pugh, M. Shub, Neighborhoods of hyperbolic sets, Invent.Math. 9 (1970), 121-134.MR 41:7232
22. [JPS] Johnson, R.A., K.J. Palmer, G.R. Sell, Ergodic properties of linear dynamical systems, SIAM J.Math. Anal. 18 (1987), 1-33.MR 88a:58112
23. [Ka] Karoubi, M., K-Theory. An Introduction, Springer-Verlag, 1978.MR 58:7605
24. [La] Latushkin, Y., Exact Lyapunov exponents and exponentially separated subbundles, Partial Differential Equations, J. Wiener and J. Hale (eds.), Pitman, (1992), 91-95.
25. [LY] Ledrappier, F., L.-S. Young, Stability of Lyapunov exponents, Ergod. Th. Dynam. Sys. 11 (1991), 469-484.MR 921:58096
26. [Ma] Mane, R., Ergodic Theory and Differentiate Dynamics, Springer-Verlag, 1987.MR 88c:58040
27. [MS] Massera, J.L., J.J. Schaeffer, Linear differential equations and functional analysis, Ann. of Math 67 (1958), 517-573.MR 88c:58040
28. [NS] Nemytskii, V.V., V.V. Stepanov, Qualitative Theory of Dynamical Systems, Princeton University Press, (1960). (Russian edition 1949).MR 22:12258
29. [Os] Oseledec, V.I., A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans.Moscow Math. Soc. 19 (1968), 197-231.MR 39:1629
30. [Pa] Palmer, K.J., Exponential separation, exponential dichotomy and spectral theory for linear systems of ordinary differential equations, J. Diff. Equations 46 (1982), 324-345.MR 84e:34067
31. [Ro] Robinson, C., Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain Journal of Mathematics 7 (1977), 425-437.MR 58:13200
32. [SS1] Sacker, R.J., G.R. Sell, Existence of dichotomies and invariant splittings for linear differential systems 1, 3. Diff. Equations 15 (1974), 429-458.MR 49:6209
33. [SS2] Sacker, R.J., G.R. Sell, A spectral theory for linear differential systems, J. Diff. Equations 37 (1978), 320-358.MR 58:18604
34. [SZ] Salamon, D., E. Zehnder, Flows on vector bundles and hyperbolic sets, Trans. Amer.Math. Soc. 306 (1988), 623-649.MR 89f:58112
35. [SI] Sell, G.R., Lectures on Linear Differential Systems, School of Mathematics, University of Minnesota, Minneapolis, Minnesota, (1975).
36. [Sg] Belgrade, J., Isolated invariant sets for flows on vector bundles, Trans. Amer.Math. Soc. 203 (1975), 359-390.MR 51:4322
37. [Si] Sinai, Ya.G. (ed.), Dynamical Systems II, Encyclopedia of Mathematical Sciences, Springer-Verlag, 1989.MR 91i:58079