The Morse spectrum of linear flows on vector bundles

For a linear flow Phi on a vector bundle pi : E --> S a spectrum can be defined in the following way: For a chain recurrent component M on the projective bundle PE consider the exponential growth rates associated with (finite time) (epsilon, T)-chains in M, and define the Morse spectrum Sigma(Mo)(M, Phi) over M as the limits of these growth rates as 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 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)(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 ail 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 PE) 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-infinity 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.