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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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  • [1] C. Chicone and R. Swanson, The spectrum of the adjoint representation and the hyperbolicity of dynamical systems, J. Differential Equations 36 (1980), 28-40. MR 571125 (81h:58047)
  • [2] -, Spectral theory for linearizations of dynamical systems, J. Differential Equations (to appear). MR 619131 (82h:58039)
  • [3] H. R. Dowson, Spectral theory of linear operators, Academic Press, London, 1978. MR 511427 (80c:47022)
  • [4] E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc, Providence, R. I., 1957; reprint 1974.
  • [5] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Math., Springer-Verlag, Berlin, 1977. MR 0501173 (58:18595)
  • [6] R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc. 229 (1977), 351-370. MR 0482849 (58:2894)
  • [7] F. Riesz and B. Sz.-Nagy, Functional analysis, Ungar, New York, 1955. MR 0071727 (17:175i)
  • [8] R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Differential Equations 27 (1978), 320-358. MR 0501182 (58:18604)
  • [9] -, The spectrum of an invariant submanifold, J. Differential Equations 38 (1980), 135-160. MR 597797 (82h:58040)
  • [10] -, A note on Anosov diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 278-280. MR 0331432 (48:9765)
  • [11] J. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc 203 (1975), 359-390. MR 0368080 (51:4322)
  • [12] R. Swan, Vector bundles and projective modules, Trans. Amer. Math. Sooc. 105 (1962), 264-277. MR 0143225 (26:785)

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Keywords: Vector bundle flows, infinitesimal generator of a semigroup, operator theory
Article copyright: © Copyright 1981 American Mathematical Society

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