Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Isolated invariant sets for flows on vector bundles
HTML articles powered by AMS MathViewer

by James F. Selgrade PDF
Trans. Amer. Math. Soc. 203 (1975), 359-390 Request permission

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.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F20
  • Retrieve articles in all journals with MSC: 58F20
Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 203 (1975), 359-390
  • MSC: Primary 58F20
  • DOI: https://doi.org/10.1090/S0002-9947-1975-0368080-X
  • MathSciNet review: 0368080