The spectrum of vector bundle flows with invariant subbundles

Swanson, R. C.

doi:10.1090/S0002-9939-1981-0620000-4

The spectrum of vector bundle flows with invariant subbundles
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Proc. Amer. Math. Soc. 83 (1981), 141-145 Request permission

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.

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