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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Invariant foliations near normally hyperbolic invariant manifolds for semiflows


Authors: Peter W. Bates, Kening Lu and Chongchun Zeng
Journal: Trans. Amer. Math. Soc. 352 (2000), 4641-4676
MSC (2000): Primary 37D30, 37L45; Secondary 53C12, 37D10, 37K55
Published electronically: June 14, 2000
MathSciNet review: 1675237
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Abstract:

Let $M$ be a compact $C^1$ manifold which is invariant and normally hyperbolic with respect to a $C^1$ semiflow in a Banach space. Then in an $\epsilon$-neighborhood of $M$ there exist local $C^1$ center-stable and center-unstable manifolds $W^{cs}(\epsilon)$ and $W^{cu}(\epsilon)$, respectively. Here we show that $W^{cs}(\epsilon)$ and $W^{cu}(\epsilon)$ may each be decomposed into the disjoint union of $C^1$ submanifolds (leaves) in such a way that the semiflow takes leaves into leaves of the same collection. Furthermore, each leaf intersects $M$ in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time.


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Additional Information

Peter W. Bates
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: peter@math.byu.edu

Kening Lu
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: klu@math.byu.edu

Chongchun Zeng
Affiliation: Courant Institute of Mathematical Sciences, New York University, New York, New York 10012
Email: zengch@math1.cims.nyu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02503-4
PII: S 0002-9947(00)02503-4
Received by editor(s): December 18, 1996
Received by editor(s) in revised form: June 5, 1998
Published electronically: June 14, 2000
Additional Notes: The first author was partially supported by NSF grant DMS-9622785 and the Isaac Newton Institute
The second author was partially supported by NSF grant DMS-9622853
The third author was partially supported by the Isaac Newton Institute
Article copyright: © Copyright 2000 American Mathematical Society