The spectrum of vector bundle flows with invariant subbundles

Abstract:

A vector bundle flow $({\Phi ^t},{\phi ^t})$ on the vector bundle $E$ over a compact metric space $M$ induces a one-parameter group $\{ \Phi _t^\# \}$ of bounded operators acting on the continuous sections of $E$, with infinitesimal generator $L$. An example is given by the tangent flow $(T{\phi ^t},{\phi ^t})$, if ${\phi ^t}$ is a flow on a smooth manifold. In this article, the spectrum of the generator $L$ is used to study the exponential growth rates of bundle trajectories in the neighborhood of a fixed invariant subbundle, e.g. the tangent bundle of a submanifold of $M$. Auxiliary normal and tangential spectra are introduced, and their relationship and fine structure are explored.