The spectrum of vector bundle flows with invariant subbundles
Author:
R. C. Swanson
Journal:
Proc. Amer. Math. Soc. 83 (1981), 141145
MSC:
Primary 58F25; Secondary 58F19
MathSciNet review:
620000
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Abstract: A vector bundle flow on the vector bundle over a compact metric space induces a oneparameter group of bounded operators acting on the continuous sections of , with infinitesimal generator . An example is given by the tangent flow , if is a flow on a smooth manifold. In this article, the spectrum of the generator 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 . Auxiliary normal and tangential spectra are introduced, and their relationship and fine structure are explored.
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 C. Chicone and R. Swanson, The spectrum of the adjoint representation and the hyperbolicity of dynamical systems, J. Differential Equations 36 (1980), 2840. MR 571125 (81h:58047)
 [2]
 , Spectral theory for linearizations of dynamical systems, J. Differential Equations (to appear). MR 619131 (82h:58039)
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 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., SpringerVerlag, Berlin, 1977. MR 0501173 (58:18595)
 [6]
 R. Mañé, QuasiAnosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc. 229 (1977), 351370. MR 0482849 (58:2894)
 [7]
 F. Riesz and B. Sz.Nagy, Functional analysis, Ungar, New York, 1955. MR 0071727 (17:175i)
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 R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Differential Equations 27 (1978), 320358. MR 0501182 (58:18604)
 [9]
 , The spectrum of an invariant submanifold, J. Differential Equations 38 (1980), 135160. MR 597797 (82h:58040)
 [10]
 , A note on Anosov diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 278280. MR 0331432 (48:9765)
 [11]
 J. Selgrade, Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc 203 (1975), 359390. MR 0368080 (51:4322)
 [12]
 R. Swan, Vector bundles and projective modules, Trans. Amer. Math. Sooc. 105 (1962), 264277. MR 0143225 (26:785)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198106200004
PII:
S 00029939(1981)06200004
Keywords:
Vector bundle flows,
infinitesimal generator of a semigroup,
operator theory
Article copyright:
© Copyright 1981
American Mathematical Society
