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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Isolated invariant sets for flows on vector bundles


Author: James F. Selgrade
Journal: Trans. Amer. Math. Soc. 203 (1975), 359-390
MSC: Primary 58F20
DOI: https://doi.org/10.1090/S0002-9947-1975-0368080-X
Erratum: Trans. Amer. Math. Soc. 221 (1976), 249.
MathSciNet review: 0368080
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Abstract: This paper studies isolated invariant sets for linear flows on the projective bundle associated to a vector bundle, e.g., the projective tangent flow to a smooth flow on a manifold. It is shown that such invariant sets meet each fiber, roughly in a disjoint union of linear subspaces. Isolated invariant sets which are intersections of attractors and repellers (Morse sets) are discussed. We show that, over a connected chain recurrent set in the base space, a Morse filtration gives a splitting of the projective bundle into a direct sum of invariant subbundles. To each factor in this splitting there corresponds an interval of real numbers (disjoint from those for other factors) which measures the exponential rate of growth of the orbits in that factor. We use these results to see that, over a connected chain recurrent set, the zero section of the vector bundle is isolated if and only if the flow is hyperbolic. From this, it follows that if no equation in the hull of a linear, almost periodic differential equation has a nontrivial bounded solution then the solution space of each equation has a hyperbolic splitting.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0368080-X
Keywords: Isolated invariant set, vector bundle, associated projective bundle, linear flow, cross ratio, expansion point, isolating block, attractor, repeller, chain recurrence, Morse set, Morse decomposition, hyperbolic flow, almost periodic system
Article copyright: © Copyright 1975 American Mathematical Society