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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The spectrum of vector bundle flows with invariant subbundles

Author: R. C. Swanson
Journal: Proc. Amer. Math. Soc. 83 (1981), 141-145
MSC: Primary 58F25; Secondary 58F19
MathSciNet review: 620000
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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^\char93 \} $ 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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Keywords: Vector bundle flows, infinitesimal generator of a semigroup, operator theory
Article copyright: © Copyright 1981 American Mathematical Society