The Morse spectrum of linear flows on vector bundles

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 Oseledec̆ 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 Oseledeč, 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.