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

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Abstract | References | Similar Articles | Additional Information

Abstract: For a linear flow on a vector bundle a spectrum can be defined in the following way: For a chain recurrent component on the projective bundle consider the exponential growth rates associated with (finite time) -chains in , and define the Morse spectrum over as the limits of these growth rates as and . The Morse spectrum of is then the union over all components . 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 contains all Lyapunov exponents of for arbitrary initial values, and the are closed intervals, whose boundary points are actually Lyapunov exponents. Using the fact that 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 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 ) 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 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.

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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