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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The spectral characterization of normal hyperbolicity


Author: Richard Swanson
Journal: Proc. Amer. Math. Soc. 89 (1983), 503-509
MSC: Primary 58F15
MathSciNet review: 715875
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Abstract: In many cases hyperbolicity in dynamical systems can be expressed in terms of the spectrum of some canonically associated linear operator; e.g., the linearization at a fixed point. Such a characterization is known for Anosov diffeomorphisms and flows. We construct a vector bundle map, based on the tensor product, whose spectrum is decisive for detecting the normal hyperbolicity of a flow or diffeomorphism at an invariant manifold. This resolves a problem raised by Hirsch, Pugh and Shub. In the case of flows, our operator admits an infinitesimal formulation, which allows us to prove that normally hyperbolic systems are stable under reparameterization in many cases.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0715875-6
PII: S 0002-9939(1983)0715875-6
Keywords: Normally hyperbolic, invariant manifolds, vector bundle flows
Article copyright: © Copyright 1983 American Mathematical Society