Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Morse spectrum of linear
flows on vector bundles


Authors: Fritz Colonius and Wolfgang Kliemann
Journal: Trans. Amer. Math. Soc. 348 (1996), 4355-4388
MSC (1991): Primary 58F25, 34C35, 34D08
DOI: https://doi.org/10.1090/S0002-9947-96-01524-3
MathSciNet review: 1329532
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a linear flow $\Phi $ on a vector bundle $\pi : E \rightarrow S$ a spectrum can be defined in the following way: For a chain recurrent component $\mathcal {M}$ on the projective bundle $\mathbb {P} E$ consider the exponential growth rates associated with (finite time) $(\varepsilon ,T)$-chains in $\mathcal {M}$, and define the Morse spectrum $\Sigma _{Mo}(\mathcal {M},\Phi )$ over $\mathcal {M}$ as the limits of these growth rates as $ \varepsilon \rightarrow 0$ and $T \rightarrow \infty $. The Morse spectrum $\Sigma _{Mo}(\Phi )$ of $\Phi $ is then the union over all components $\mathcal {M}\subset \mathbb {P}E$. This spectrum is a synthesis of the topological approach of Selgrade and Salamon/Zehnder with the spectral concepts based on exponential growth rates, such as the Oseledec spectrum or the dichotomy spectrum of Sacker/Sell. It turns out that $\Sigma _{Mo}(\Phi )$ contains all Lyapunov exponents of $\Phi $ for arbitrary initial values, and the $\Sigma _{Mo}(\mathcal {M},\Phi )$ are closed intervals, whose boundary points are actually Lyapunov exponents. Using the fact that $\Phi $ is cohomologous to a subflow of a smooth linear flow on a trivial bundle, one can prove integral representations of all Morse and all Lyapunov exponents via smooth ergodic theory. A comparison with other spectral concepts shows that, in general, the Morse spectrum is contained in the topological spectrum and the dichotomy spectrum, but the spectral sets agree if the induced flow on the base space is chain recurrent. However, even in this case, the associated subbundle decompositions of $E$ may be finer for the Morse spectrum than for the dynamical spectrum. If one can show that the (closure of the) Floquet spectrum (i.e. the Lyapunov spectrum based on periodic trajectories in $\mathbb {P} E$) agrees with the Morse spectrum, then one obtains equality for the Floquet, the entire Oseledec, the Lyapunov, and the Morse spectrum. We present an example (flows induced by $C^{\infty }$ vector fields with hyperbolic chain recurrent components on the projective bundle) where this fact can be shown using a version of Bowen's Shadowing Lemma.


References [Enhancements On Off] (What's this?)

  • [Ak] Akin, Ethan, The General Topology of Dynamical Systems, American Mathematical Society, Graduate Studies in Mathematics 1 (1993). MR 94f:58041
  • [AN] Arnold, L., D.C. Nguyen, Generic properties of Lyapunov exponents, Random& Computational Dynamics 2 (1994), 335--345. MR 95m:28018
  • [Br1] Bronstein, I.U, Transversality implies structural stability, Soviet Math. Dokl. 23 (1981), 251--254, (= Dokl. Akad. Nauk SSSR 257 (1981), 257--268). MR 82h:58038
  • [Br2] Bronstein, I.U., Nonautonomous Dynamical Systems (1984), Kishinev (in Russian).
  • [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)
  • [Ca] Carverhill, A., Flows of stochastic dynamical systems: ergodic theory, Stochastics 14 (1985), 273--317. MR 87c:58059
  • [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
  • [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.), Birkhäuser (1991), 203--240. MR 93i:60110
  • [CK3] Colonius, F., W. Kliemann, Some aspects of control systems as dynamical systems, J. Dynamics Diff. Equations 5 (1993), 469--494. MR 94g:93063
  • [CK4] Colonius, F., W. Kliemann, Linear control semigroups acting on projective space, J. Dynamics Diff. Equa. 5 (1993), 495--528. MR 94g:93064
  • [CK5] Colonius, F., W. Kliemann, Limit behavior and genericity for nonlinear control systems, J. Differential Equations 109 (1994), 8--41. CMP 94:11
  • [CK6] Colonius, F., W. Kliemann, The Lyapunov spectrum of families of time-varying matrices, Trans. Amer. Math. Soc. 348 (1996), 4389--4408.
  • [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
  • [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
  • [Cn] Conley, C., Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series, no. 38, American Mathematical Society, Providence (1978). MR 80c:58009
  • [Cp] Coppel, W.A., Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer--Verlag, 1978. MR 58:1332
  • [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
  • [EJ] Ellis, R., R. Johnson, Topological dynamics and linear differential systems, J.Diff. Equations 44 (1982), 21--39. MR 83c:54058
  • [FS] Franke, J.E., J.F. Selgrade, Hyperbolicity and chain recurrence, J.Diff. Equations 26 (1977), 27--36. MR 57:7685
  • [HPPS] Hirsch, M., J. Palis, C. Pugh, M. Shub, Neighborhoods of hyperbolic sets, Invent. Math. 9 (1970), 121--134. MR 41:7232
  • [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
  • [Ka] Karoubi, M., K-Theory. An Introduction, Springer-Verlag, 1978. MR 58:7605
  • [La] Latushkin, Y., Exact Lyapunov exponents and exponentially separated subbundles, Partial Differential Equations, J. Wiener and J. Hale (eds.), Pitman, (1992), 91--95.
  • [LY] Ledrappier, F., L.-S. Young, Stability of Lyapunov exponents, Ergod. Th. Dynam. Sys. 11 (1991), 469--484. MR 92i:58096
  • [Ma] Mañé, R., Ergodic Theory and Differentiable Dynamics, Springer-Verlag, 1987. MR 88c:58040
  • [MS] Massera, J.L., J.J. Schaeffer, Linear differential equations and functional analysis, Ann. of Math 67 (1958), 517--573. MR 88c:58040
  • [NS] Nemytskii, V.V., V.V. Stepanov, Qualitative Theory of Dynamical Systems, Princeton University Press, (1960). (Russian edition 1949). MR 22:12258
  • [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
  • [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
  • [Ro] Robinson, C., Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain Journal of Mathematics 7 (1977), 425--437. MR 58:13200
  • [SS1] Sacker, R.J., G.R. Sell, Existence of dichotomies and invariant splittings for linear differential systems 1, J. Diff. Equations 15 (1974), 429--458. MR 49:6209
  • [SS2] Sacker, R.J., G.R. Sell, A spectral theory for linear differential systems, J. Diff. Equations 37 (1978), 320--358. MR 58:18604
  • [SZ] Salamon, D., E. Zehnder, Flows on vector bundles and hyperbolic sets, Trans. Amer. Math. Soc. 306 (1988), 623--649. MR 89f:58112
  • [Sl] Sell, G.R., Lectures on Linear Differential Systems, School of Mathematics, University of Minnesota, Minneapolis, Minnesota, (1975).
  • [Sg] Selgrade, J., Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc. 203 (1975), 359--390. MR 51:4322
  • [Si] Sinai, Ya.G. (ed.), Dynamical Systems II, Encyclopedia of Mathematical Sciences, Springer--Verlag, 1989. MR 91i:58079

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58F25, 34C35, 34D08

Retrieve articles in all journals with MSC (1991): 58F25, 34C35, 34D08


Additional Information

Fritz Colonius
Affiliation: Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany
Email: colonius@uni-augsburg.de

Wolfgang Kliemann
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: kliemann@iastate.edu

DOI: https://doi.org/10.1090/S0002-9947-96-01524-3
Keywords: Chain recurrence, ergodic theory, Lyapunov exponents, dichotomy spectrum, topological spectrum, Oselede\u{c} spectrum, Floquet spectrum, hyperbolic flows, shadowing lemma
Received by editor(s): January 25, 1994
Received by editor(s) in revised form: March 31, 1995
Additional Notes: This research was performed during a stimulating visit at the Institute for Mathematics and Its Applications, Minneapolis. It was partially supported by DFG under grant no. Co 124/8-2 and by ONR grant no. N00014-93-1-0868.
Dedicated: Dedicated to J. L. Massera
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society