THE MORSE SPECTRUM OF LINEAR FLOWS ON VECTOR-BUNDLES

For a linear flow Phi on a vector bundle pi : E --> S a spectrum can b e defined in the following way: For a chain recurrent component M on t he projective bundle PE consider the exponential growth rates associat ed with (finite time) (epsilon, T)-chains in M, and define the Morse s pectrum Sigma(Mo)(M, Phi) over M as the limits of these growth rates a s epsilon --> O and T --> infinity. The Morse spectrum Sigma(Mo)(Phi) of Phi is then the union over all components M subset of PE. This spec trum is a synthesis of the topological approach of Selgrade and Salamo n/Zehnder with the spectral concepts based on exponential growth rates , such as the Oseledec spectrum or the dichotomy spectrum of Sacker/Se ll. It turns out that Sigma(Mo)(Phi) contains all Lyapunov exponents o f Phi, for arbitrary initial values, and the Sigma(Mo)(M Phi) are clos ed intervals, whose boundary points are actually Lyapunov exponents. U sing the fact that Phi is cohomologous to a subflow of a smooth linear flow on a trivial bundle, one can prove integral representations of a ll Morse and ail Lyapunov exponents via smooth ergodic theory. A compa rison with other spectral concepts shows that, in general, the Morse s pectrum is contained in the topological spectrum and the dichotomy spe ctrum, but the spectral sets agree if the induced flow on the base spa ce is chain recurrent. However, even in this case, the associated subb undle decompositions of E may be finer for the Morse spectrum than for the dynamical spectrum. If one can show that the (closure of the) Flo quet spectrum (i.e. the Lyapunov spectrum based on periodic trajectori es in PE) agrees with the Morse spectrum, then one obtains equality fo r the Floquet, the entire Oseledec, the Lyapunov, and the Morse spectr um. We present an example (flows induced by C-infinity vector fields w ith hyperbolic chain recurrent components on the projective bundle) wh ere this fact can be shown using a version of Bowen's Shadowing Lemma.